A new method to calculate the equivalent Mohr–Coulomb friction angle for cohesive and frictional materials

INTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICSInt. J. Numer. Anal. Meth. Geomech. 2011; 35:630–638Published online 20 April 2010 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/nag.922

A new method to calculate the equivalent Mohr–Coulomb frictionangle for cohesive and frictional materials

Hua Jiang1,∗,† and Xiaowo Wang2

1Key Laboratory for Bridge and Tunnel of Shaanxi Province, Chang’an University, Xi’an 710064, China2School of Civil Engineering, Purdue University, West Lafayette, IN 47907-1284, U.S.A.

SUMMARY

In this note, a new method to calculate the equivalent Mohr–Coulomb friction angle �′mc for cohesive and

frictional materials is presented. This method makes a transformation from the failure surface for cohesivematerials to the failure surface for cohesionless materials and obtains �′

mc as well as the principal stressratio �′

1/�′3 for cohesionless materials in the transformed space first, then obtains �′

mc for cohesivematerials by linking �′

1/�′3 in the transformed space and in the original space. In the application example,

an analytical solution of the invariant stress ratio L is derived from the failure function in the transformedspace. The influence of the intermediate effective principal stress �′

2 is also demonstrated using the alreadycalculated �′

mc. Copyright � 2010 John Wiley & Sons, Ltd.

Received 1 November 2009; Revised 9 February 2010; Accepted 10 February 2010

KEY WORDS: Mohr-Coulomb; Matsuoka–Nakai; failure surface; friction angle

1. INTRODUCTION

The concept of the equivalent Mohr–Coulomb (M–C) friction angle �′mc defined as the friction

angle of the M–C surface that would pass through the particular point on the periphery of thestrength criteria, was raised by Griffiths [1]. Two expressions of �′

mc for cohesionless materialsare given: one is expressed in terms of the invariants stress ratio L and the Lode angle � withexamples of three Drucker–Prager cones; the other is formulated in terms of the principal stressratio �′

1/�′3 and � with applications of the Matsuoka–Nakai (M–N) and the Lade–Duncan (L–D)

criteria.Recently, by adding the influence of cohesion c′, �′

mc of the extended Matsuoka–Nakai(E-M–N) criterion for cohesive materials was proposed by Griffiths and Huang [2], which canbe regarded as an extension of the second expression. In the first expression, L is obtained fromfailure functions of the strength criteria in Haigh–Westergaard space, whereas in the second andthe extended ones �′

1/�′3 is solved from its failure functions in the three-dimensional principal

stress space. In some cases, the solving process is rather cumbersome, for instance, the cubicequitation about �′

1/�′3 at each value of � needs to be solved for the M–N, L–D and the E-M–N

criteria [1, 2].In this note, a new expression of �′

mc for cohesive materials is derived by combing the firstexpression with the extended second one and transforming the failure surface for cohesive materialsto the failure surface for cohesionless materials. Finally, an example is given to demonstrate the

∗Correspondence to: Hua Jiang, School of Highway, Chang’an University, Middle Section of South Er Huan Road,Yanta District, Xi’an 710064, China.

†E-mail: [email protected]

Copyright � 2010 John Wiley & Sons, Ltd.

A NEW METHOD TO CALCULATE THE EQUIVALENT M–C FRICTION ANGLE 631

comparatively simple process of this new method, and a new use of �′mc is presented in addition

to being used to make direct comparisons between different failure criteria.

2. NEW METHOD TO CALCULATE EQUIVALENT M–C FRICTION ANGLEFOR COHESIVE AND FRICTIONAL MATERIALS

In the following discussions, compressive stress is considered positive and stress with a primeis always considered to be the effective value. For cohesionless materials, the principal stressand the equivalent M–C friction angle are denoted by �′

i (i =1,2,3) and �′mc. respectively; for

cohesive materials, with superposed carets �′i and �′

mc denoting, respectively, the correspondingparameters.

In order to include the cohesion in the frictional criteria, a translation of the principal stressspace along the hydrostatic axis is performed as shown in Figure 1, where o′ and o′ represent thetwo origins of �′

1, �′2,�′

3 and �′1, �′

2, �′3 coordinates’ system, respectively. The principal stress

on the new (translated) failure surface is expressed by

�′i = �i

′+c′ cot �′c (1)

where i =1,2,3, c′ and �′c are the cohesion and friction angle.

Derived from the failure function of the M–C criterion expressed in terms of L and � with c′ =0assumed, �′

mcfor the new failure surface can be written as

�′mc =arcsin

√3sin(60◦+�)L√

2+cos(60◦+�)L(2)

where invariant stress ratio L is defined as L = t/s; t =√SijSij; s = 1√3�ii; Sij is the deviator stress

tensor and is linked to the stress tensor �ij by Sij =�ij −(�ii/3)�ij. The similarity angle �(0��60◦)is given by

�= 1

3arccos

(3√

6J3

t3

)with J3 = 1

3sijsjkski (3)

which takes value of 0◦ for traxial compression (TC) and 60◦ for traxial extension (TE), is relatedto the Lode angle � by �=�+30◦.

Figure 1. Failure criterion in the three-dimensional stress space.

Copyright � 2010 John Wiley & Sons, Ltd. Int. J. Numer. Anal. Meth. Geomech. 2011; 35:630–638DOI: 10.1002/nag

632 H. JIANG AND X. WANG

Besides, �′mc can be equivalently formulated in terms of L and b (0�b�1) as

�′mc =arcsin

3L

2√

2√

b2 −b+1+(1−2b)L(4)

where b is a measurement of the intermediate effective principal stress and is defined byb= (�′

2 −�′3)/(�1

′−�′3). Equations (2) and (4) are interchangeable by using Equations (5)

and (6).

t =√

2√3

√b2 −b+1(�′

1 −�′3) (5)

�′1 +�′

3 = 2√3

s− 2b−1

3(�′

1 −�′3) (6)

If �′mc �=90◦, based on the failure function of the M–C criterion in the three-dimensional stress

space with c′ =0 assumed again, the principal stress ratio �′1/�′

3 denoted by R f in the new spaceis gained by

R f = tan2(45◦+�′mc/2) (7)

where �′mc, a computed value, is obtained from Equation (2) or (4).

When �′3 =0, the equivalent M–C friction angle �′

mc of the original failure surface can bewritten as

�′mc =2arctan

�′1

2c′ −90◦ (8)

According to Equation (1), �′1/c′ can be related to R f by

�′1 =c′ cot�′

c(R f −1) (9)

Equation (9) plays a transformation role between R f in the original space and �′1/c′ in the new

space. By substituting it into Equation (8), we can obtain

�′mc =2arctan

[(R f −1) cot�′

c

2

]−90◦ (10)

When �′3 �=0, the principal stress ratio �′

1/�′3 denoted by R f in the original space is determined by

R f = R f +(R f −1)c′

�′3

cot�′c (11)

Analogous to Equation (9), Equation (11) serves a transform role between the principal stress ratioin the original space and the new space.

Then �′mc of the original failure surface is given by

�′mc =2arctan

⎡⎣− c′

�′3+√(

c′

�′3

)2

+ R f

⎤⎦ −90◦ (12)



If setting x =c′/�′3, then x →∞ is derived using �′

3 →0, then substituting Equation (11) intosquare brackets of the above equitation, its approaching value is given as

lim�′

3→0

⎡⎣− c′

�′3+√(

c′

�′3

)2

+ R f

⎤⎦= lim

x→∞

[−x +

√x2 +(R f −1)cot�′

cx + R f

]

= limx→∞

(R f −1)cot �′c + R f

x√1+ (R f −1)cot �′

c

x+ R f

x2+1

= (R f −1)cot�′c

2(13)

Therefore, the approaching �′mc obtained from Equation (12) equals its exact value obtained from

Equation (10), and Equation (12) describes a continuous function about �′mc in the domain of

0��′3<∞.

If c′ =0, then Equation (12) is simplified to

�′mc =2arctan R f −90◦ (14)

which is the second expression of �′mc for cohensionless materials [1].

If �′mc =90◦, this case corresponds to the well-known singularity in triaxial extension for the

extended Von Mises criterion given by Bishop [3], where �′c =arcsin( 3

5 ) and �′3 =0. The equivalent

M–C friction angle �′mc can be obtained from Equation (12) by substituting Equations (15) and (16)

into it.

R f = −3

4

�′1

c′ (15)

c′

�′3

= −3

4(16)

In addition, when �′1/�′

3 is known, �′2/�′

3 can be written as

�′2

�′3=b

(�′

1

�′3−1

)+1 (17)

Then the influence of the intermediate principal stress can be clearly shown in the �′1/�′

3 − �′2/�′

3space, which can be taken as a new application of the equivalent M–C friction angle.

The process to calculate �′mc is summarized as follows:

1. Give �′c, c′ and �′

3.2. Using �′

c and �, get invariant stress ratio L according to failure function of strength criteria.3. Calculate the equivalent M–C friction angle �′

mc for cohesionless materials from Equation (2)or (4).

4. If �′mc �=90◦, �′

3 =0, calculate R f according to Equation (7) first, then get �′mcfrom Equa-

tion (10) or let �′3 →0 in the condition �′

mc �=900, �′3 �=0.

5. If �′mc �=90◦, �′

3 �=0, calculate R f and R f according to Equations (7) and (11), then make

use of Equation (12) to get �′mc.

6. If �′mc =90◦, calculate �′

mc by substituting Equations (14) and (15) into Equation (12).

It is worth noting that Equation (2) or (4), resembling the first expression of �′mc for cohesionless

materials [1], plays a role in getting �′mc of the new failure surface; Equation (12), which has

a similar expression as the extended second expression of �′mc for cohesive materials [2], plays



a role in getting �′mc for the original failure surface; �′

mc �=90◦, �′3 =0 and �′

mc =90◦ can betaken as special cases of �′

mc �=90◦, �′3 �=0.

The new method demonstrated above can be seen as a general expression of �′mc for cohesive

and frictional materials, by which �′mc of the Drucker–Prager, M–N [4–8], L–D [9, 10] and Moji

criteria [11, 12] for cohesive materials can be easily obtained provided that their invariants stressratio L is known. In the subsequent sections, the E-M–N criterion for cohesive material is used asan example.

3. INVARIANTS STRESS RATIO L OF EXTENDED M–NFAILURE CRITERION

Matsuoka and Nakai [4–6] proposed a criterion for cohesionless soil based on the SpatiallyMobilized Plane theory, whose failure function can be expressed in terms of invariants of stresstensor as

I1 I2

I3=k with k =9+8tan2 �′

c (18)

Here k is a constant and is defined such that the M–N criterion coincides with the M–C criterion intriaxial compression and extension, the value of k is 9 at the hydrostatic axis in which �′

1 =�′2 =�′

3;I1, I2 and I3 are the first, second and third invariants of the stress tensor �ij, respectively, whichare given by

I2 = 12�ij�ij with I2 = I 2

1 /3− J2 (19)

I3 = 13�ij�jk�ki with I3 = J3 − I1 J2/3+ I 3

1 /27 (20)

For cohesive soils, cohesion is included in the failure function [7, 8], and the M–N criterion iscalled the E-M–N criterion, whose principal stresses can be shifted to that of the M–N criterionaccording to Equation (1).

Substituting Equations (19) and (20) into Equation (18) and expressing I1, J2 in terms of s, t ,then the M–N criterion can be expressed as

(s

t

)3 − 3

2

k−3

k−9

s

t+

√2

2

k

k−9cos3�=0 (21)

A general form of the above cubic equation is written as

z3 + pz+q =0 (22)

with p=− 32 (k−3)/(k−9), q =

√2

2 (k/(k−9))cos3� and z =s/t =1/L .Making Vieta’s substitution z = x − p/(3x), a converted form of Equation (22) is obtained by

(x3)2 +qx3 − p3

27=0 (23)

For k>9,

�= q2

4+ p3

27=−k2(k−9)sin2 3�+27(k−1)

8(k−9)3<0



Therefore, two conjugate complex roots about x3 are obtained from Equation (23).

x3 =− 1

2√

2

B

k−9

(k

Bcos3�± i

k

B

√sin2 3�+ 27(k−1)

k2(k−9)

)with B =

√k2 +27

k−1

k−9>0 (24)

Then two conjugate complex roots about x are obtained from Equation (24).

x1 =a+bi

x2 =a−bi(25)

with

a = − 1√2

3

√B

k−9cos

[1

3arccos

(k cos3�

B

)]

b = − 1√2

3

√B

k−9sin

[1

3arccos

(k cos3�

B

)]

Further, there are three roots for Equation (22) together

z1 = x1 +x2 =−√

2 3

√B

k−9cos

[1

3arccos

(k cos3�

B

)]

z2 = x1w+x2w2 =

√2 3

√B

k−9cos

[1

3arccos

(k cos3�

B

)−60◦

]

z3 = x1w2 +x2w=

√2 3

√B

k−9cos

[1

3arccos

(k cos3�

B

)+60◦

](26)

with w2 +w=−1.Since the cross-sectional shape of the failure surface must have a threefold symmetry charac-

teristic for an isotropic material, only the sector 0��60◦ needs to be considered and the intervalof three roots are

−√

2 3

√B

k−9� z1�− 1√

23

√B

k−9

1√2

3

√B

k−9� z2�

√2 3

√B

k−9

− 1√2

3

√B

k−9� z3�

1√2

3

√B

k−9

(27)

It is noted that 3√

B/(k−9)>0, z1<0 and z2>0 in the sector of 0��60◦, whereas the sign ofz3 changes from negative to positive as � varies from 0 to 60◦, which never appears in invariantsstress ratio L . If tension-positive of the stress is assumed, s<0 is required that leads to z1 as adesired answer. For compression-positive sign taken in the article, only z2 is the desired answerwith a compact expression as

z2 =k1 cos

[1

3arccos(k2 cos3�)−60◦

]with k1 =

√2 3

√B

k−9, k2 =k/B (28)

Considering that Equation (28) takes the form of the L–D [9, 10, 13] and Ottosen criterion [13, 14],Equation (28) can be seemed as an explicit expression of the E-M–N criterion in Haigh–Westergaardspace and the invariants stress ratio L is determined by the parameter �′

c at each value of �.



4. RESULTS BASED ON THE NEW METHOD

In Figure 2, �′c =30◦ is assumed for the E-M–N failure criterion, �′

mc =�′c occurs at �=0 and

�=60◦ because the E-M–N failure criterion coincides with the M–C criterion in TC and TE, and�′

mc increases with the increasing value of c′/�′3 with �′

3 =0(c′/�′3 =∞) and c′ =0(c′/�′

3 =0) asan upper bound and a low bound, respectively, which is the same as the findings in Reference [2].

Figure 3 presents �′2/�′

3 verses �′1/�′

3 when c′ =0, in which the E-M–N criterion for cohesivesoils reduces to the M–N criterion for cohesionless soils. In taking �′

3 =1, �′1 increases from the

initial value to a submit corresponding to the maximum �′mc, then decreases to the initial value

again as �′2 increases from �′

2 = �′3 (TC) to �′

2 = �′1 (TE), which demonstrates the influence of

the intermediate effective principal stress �′2.

When c′/�′3 =10, similar result as in the case of c′/�′

3 =0 is shown in Figure 4. �′1/�′

3increases significantly as c′/�′

3 varies from 0 to 10 by comparing Figure 3 with Figure 4.

Figure 2. �′mc versus � and b for different values of c′/�′

3 of the E-M–N criterion when �′c =30◦.

Figure 3. �′2/�′

3 versus �′1/�′

3 of the E-M–N criterion when c′/�′3 =0.



Figure 4. �′2/�′

3 versus �′1/�′

3 of the E-M–N criterion when c′/�′3 =10.

5. CONCLUSION

A new way to calculate the equivalent M–C friction angle �′mc for cohesive and frictional materials

is presented, which makes a translation from the failure surface for cohesive materials to the failuresurface for cohesionless materials.

In the application example, an analytical solution of the invariant stress ratio L is obtained inthe translated principal stress space, and the results based on the new methods agree well with theones in reference 2 based on the numerical solution.

The relationship between �′1/�′

3and �′2/�′

3 for cohesive and frictional materials is easilyderived using calculated �′

mc, and the influence of the intermediate principal stress is clearlyshown.

ACKNOWLEDGEMENTS

The author is grateful to Professor Griffths DV, who reviewed the draft of the note.

REFERENCES

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2. Griffiths DV, Huang JS. Observations on the extended Matsuoka–Nakai failure criterion. International Journalfor Numerical and Analytical Methods in Geomechanics 2009; 33(17):1889–1905.

3. Bishop AW. The strength of soils as engineering materials. Géotechnique 1966; 16(2):91–130.4. Matsuoka H, Nakai T. Stress-deformation and strength characteristics of soil under three different principal

stresses. Proceedings of JSCE 1974; 232:59–74.5. Matsuoka H. Stress–strain relationships of sands based on the mobilized plane. Soils and Foundations 1974;

14(2):47–61.6. Matsuoka H. Stress–strain relationships of clays based on the mobilized plane. Soils and Foundations 1974;

14(2):77–87.7. Ohmaki S. Strength sand deformation characteristics of overconsolidated cohesive soil. Proceedings of the Third

International Conference on Numerical Methods in Geomechanics, Aachen, vol. 1, 1979; 465–474.8. Houlsby GT. A general failure criterion for frictional and cohesive materials. Soils and Foundations 1986;

26(2):97–101.9. Lade P, Duncan J. Elasto-plastic stress–strain theory for cohesionless soil. Journal of Geotechnical Engineering

Division (ASCE) 1975; 101(10):37–53.10. Lade PV. Elasto-plastic stress–strain theory for cohesionless soil with curved yield surfaces. International Journal

of Solids and Structures 1977; 13(11):1019–1035.



11. Ewy R. Wellbore-stability predictions by use of a modified Lade criterion. SPE Drill Completion 1999; 14(2):85–91.

12. Mogi K. Fracture and flow of rocks under high triaxial compression. Journal of Geophysical Research 1971;76(5):1255–1269.

13. Podgórski J. General failure criterion for isotropic media. Journal of Engineering Mechanics 1985; 111(2):188–201.

14. Ottosen NS. A failure criterion for concrete. Journal of the Engineering Mechanics Division 1977; 103(4):527–535.


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A new method to calculate the equivalent Mohr–Coulomb friction angle for cohesive and frictional materials