AP04-Bittnar1.vp 1 Introduction The Coulomb failure condition is defined by the equation f c� � � �� � �tan 0 (1) where � and � are the shear and normal traction components respectively on the critical plane in the material, c is the ap- parent cohesion and � is the angle of shearing resistance (internal friction). The usual sign convention is used for the normal stress �, compression is negative. In the classical Mohr-Coulomb formulation, the critical plane normal is in- clined by the angle � � �� �4 2 from the �1 direction to the �3 direction. Ordered principal stresses � � �1 2 3� � are assumed. This orientation of the plane follows from the postulated condition that the Mohr circle in the � �1 3� plane touches the envelope (1) as shown in Fig. 1. Stresses �cx, �cz and �c are implied in the coordinate frame associated with the critical plane. The Mohr-Coulomb condition is natural but the assumed orientation of the critical plane in fact lacks a rigorous substantiation. Other orientations could be assumed. A rational modification of the Mohr-Coulomb con- dition can be obtained when the critical plane orientation is not a priori restrained. Instead, it can be determined so that f attains its maximum on the critical plane. The resulting criterion should be more severe than the classical one. 2 Mohr-Coulomb criterion based on an extreme property Direct notation is used in the development, and a general triaxial stress is assumed for full generality. Stress tensor � is assumed to have principal stresses �i with direction vectors ni. The unknown critical plane normal is denoted n. The nor- mal and tangential traction components on the plane are � �� � � � �n n n n� �� 2 . (2) The extreme of f is sought when n is subject to variation with subsidiary condition n n� �1. Lagrange multiplier � is in- troduced and the extended criterion � � � � �f f �( )n n 1 is differentiated with respect to n to yield 1 2 4 2 2 2 � � �( ( ) ) tan� � � � � �n n n n n n� � �� � 0 . (3) The equation is contractively multiplied by n and the re- sulting scalar equation is used to eliminate � from Eq. 3. As- suming that � � 0, equation { } ( tan ) [ (( ) tan )] �� � � �� � � � � � � � � � � � � 2 2 2 n n n n n n n n n � � � � � 0 (4) is obtained for the unknown n. The equation can be rewritten in a comprehensive form when tensor � is introduced � �� � �� � � �2( tan )n n � � , (5) ( )� �� � �n n n 0 . (6) The eigenvectors of � deliver extremes of f. Eigenvectors of � and � are the same, however, so these extremes are minima (� � 0) of f. In order to find the other extremes, all variables are decomposed in terms of the eigenvalues �i and principal vectors ni of �: � �� � � � � � i i i i i i i i i i i n n n n n n n 2 (7) s n n n n i i i i i i i i i i i i � � � � � � � � � n n n n � �� � � � � � 2 2 2 2 2 2 � � � � 2 (8) � � � � � r r s i i i i i i i n n � � ��( ( tan )).2 (9) © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ 93 Acta Polytechnica Vol. 44 No. 5 – 6/2004 Mohr-Coulomb Failure Condition and the Direct Shear Test Revisited P. Řeřicha An alternative critical plane orientation is proposed in the Mohr-Coulomb failure criterion for soils with an extreme property. Parameter identification from the direct shear test is extended to incude the lateral normal stress. Keywords: soil strength, shear failure, direct shear test. Fig. 1: Inclination � of the critical plane in the classical Mohr- -Coulomb yield condition Equation (6) becomes ( ) ,r r n n ii j j j i� � � 2 0 1 3. (10) Six relevant solutions can be best presented in terms of the cyclic permutations of indices i, j, k: n n ni j k, tan tan ,� � � � � � � � � � 1 2 1 4 0 2 � � . (11) It is apparent that the critical plane normal lies always in the plane of two principal stresses directions, in the same plane as the classical Mohr-Coulomb normal. Back substitu- tions yield then � � � � � � � � �� � � � � � i j i j 4 1 22tan , ( tan ) (12) and the modified Coulomb condition on the critical plane of maximum f: f ci j i j� � � � � � � � � � � � � � � � 1 2 2 4 0 2 ( ) tan tan ( ) tan� � � � � � � (13) It is interesting to compare the equation with the original Mohr-Coulomb condition 1 2 1 02( ) tan ( ) tan� � � � � �i j i j c� � � � � �� � �� � � , (14) and with the Coulomb condition (1) applied on the plane of the maximum shear stress 1 2 0[( ) ( ) tan� � � � ��i j i j c� � � � � . (15) The latter condition represents the third option for the critical plane orienation. For plane stress conditions �2 0� the graphic representation of all three yield locuses is in Fig. 2. The modified yield locus is the most severe, as expected. Intersections of the modified (A) and classical (M) yield locuses with the rendulic plane � �1 2� are shown in Fig. 3. Functions fA and fM are important for parameter calibration of the model by the triaxial test. The modified Coulomb (A) condition intersection with the rendulic plane is: f x c x A( ) ( tan )( tan ) ( tan ) tan tan � � � � � � � � 2 2 3 4 3 3 6 3 4 2 � � � � 2 � (16) whereas for the original Mohr-Coulomb f x c x M( ) ( tan ) tan tan � � � � � � 2 2 3 3 3 3 1 2 � � � . (17) Positive signs pertain to the lower branches of the yield lo- cus intersections with the rendulic plane. The three options for the critical plane orientation distin- guish three slightly different material models of the Mohr- -Coulomb type. The practical value of these modifications can be assessed in connection with the solutions of actual prob- lems. The problem tackled below is the parameter identifica- tion in the direct shear test. 3 Evaluation of the direct shear test Most applications of constitutive equations include a) the parameter calibration and b) solution of the actual task ana- lytically or numerically. Let us assume first that the triaxial test is used in the first step. Tests provide points in the rendulic plane and parameters c and tan � are selected to best fit the points. Other procedures are available for identifying of the parameters using, for instance, the modified and alternate Mohr-Coulomb diagrams as recommended in [1] and [4]. Different parameter values are obtained for the three versions of the yield locus. The calibrated locus remains nearly the same for all versions, however. Application of the three ver- sions in any actual problem solution does not thus make any difference in the results, in spite of the difference in the pa- rameter values. Differences might occur when direct shear apparatus is used in the first step, see Fig. 4. The failure plane orientation is imposed by the test arrangement in this case. Strictly speak- 94 © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ Acta Polytechnica Vol. 44 No. 5 – 6/2004 Fig. 2: The modified Mohr-Coulomb (A), the original Mohr-Cou- lomb (M) and the maximum shear plane condition (S) for tan . , .� �� �0 5 26 6° Fig. 3: The modified Coulomb (A) and the original Mohr-Cou- lomb (M) for tan . , ( . )� �� � �0 5 26 6 , c � 1 in the rendulic plane � �1 2� . Axis �1 and projection of axes �1 and �2 are also shown. ing, there is no homogeneous stress in the specimen and from this point of view the test is not suitable for direct parameter calibration. Nevertheless, in a layer adjacent to the failure plane approximately homogeneous stress conditions can be assumed. It is assumed forthwith that normal stress �z and the corresponding limit shear stress � are determined in the di- rect shear test in the failure plane, see Fig. 4. Corresponding techniques are specified, e.g., in [2] or [3]. The point is that the failure plane in this test does not coincide with the critical plane in the Mohr-Coulomb failure condition for any of the three versions considered here. Consequently, the line obtained by fitting the � �z : points from the test is not the Mohr-Coulomb failure condition. The confining normal stress �x in the slip direction is unknown. The Coulomb condition (1) does not depend on the latter stress. The Mohr-Coulomb condition with both its modifications (14) – (16) however, depends on both principal stresses in the problem plane and therefore depends on �z and �x. Consequently, the direct shear test cannot directly determine c and � since �x is not known. The arrangement of the shear test admits the approxi- mate assumption of proportionality between the confining and active stresses � �x z� with constant parameter . The direct shear test can now be simulated with the three versions of the failure criterion. Assuming stress components � � �z x z, � , and � in the layer adjacent to the slip plane at failure, standard expres- sions for the principal stresses are substituted in the respective failure criterion and explicit formulas for � are derived. These formulas represent the correct failure limits. The respective limits read, for the modified Mohr-Coulomb: � �� � � � � � � � � � � � � � � � � � � 1 2 1 4 4 1 1 2 2 2 4 tan ( ) ( tan ( ) tan ( z z� � �) tan ( tan ) ,� �� �c c4 1 2 1 2 (18) for the classical Mohr-Coulomb: � � � � � � �� � � � � � � � � � � ! " # 1 2 4 1 1 1 4 1 2 2 2 2 1 z z zc tan ( ) tan ( ) 2 , (19) and for the Coulomb on the maximum shear plane: � �� �� � �� � � � �1 2 1 1 42 2 2 2 2 1 2z c( ) ( ) tan . (20) Each criterion can be perceived as a batch of curves � �( )z with parameter . Standard evaluation of this fictitious test would deliver a straight line from the point � z � 2 on the hori- zontal axis to the point � �1 on the vertical axis – the conven- tional Mohr-Coulomb envelope. It is apparently wrong to use in the Mohr-Coulomb material models the parameter values obtained in the direct shear test by the standard evaluation. Instead, the three parameters c, � and should be deter- mined to best fit the measured data. Low values < 0.3 obviously are not realistic. The other extreme, �1, is closest to the conventional Mohr-Coulomb envelope that would be obtained by the standard test evalua- tion with the same material. However, not even this extreme curve coincides with the conventional envelope except for the maximum shear orientation of the critical plane in Fig. 7. It is worth noting that the introduction of parameter allows for curved locuses, which are often observed in practice [1], and that parameter , a side product of the parameter fitting, can be used to determine the elastic properties of the soil. © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ 95 Acta Polytechnica Vol. 44 No. 5 – 6/2004 Fig. 4: Stress components at yield in the direct shear test Fig. 5: Mohr-Coulomb for tan . , ( . ),� �� � � �0 5 26 6 1c and sev- eral values of parameter Fig. 6: Modified Mohr-Coulomb for tan . , ( . ),� �� � � �0 5 26 6 1c and several values of parameter Fig. 7: Coulomb for the maximum shear plane tan . ,� � 0 5 ( . ),� � � �26 6 1c and several values of parameter References [1] US Army Corps of Engineers (2003). “Slope stability” US Army Corps of Engineers Documents, pub. no: em1110-2-1902, appendix d. [2] US Army Corps of Engineers (1970) “Slope stability”, Appendix IX, Drained (S) direct shear test US Army Corps of Engineers Documents, pub. no: em1110-2-1906. [3] American society for testing and materials (1998) “Stan- dard test method for direct shear test of soils under consolidated drained conditions.” American society for testing and materials, ASTM D3080-98, Annual book of ASTM Standards, STM, West Conshohocken, PA, USA, (1998). [4] Fredlund D. G., Vanapalli S. K.: “Shear stress in unsatu- rated soils.” Chapter 2.7 in Handbook of Agronomy Soil Science Society of America, 2002. Prof. Ing. Petr Řeřicha, DrSc. phone: +420 224 354 478 e-mail: petr.rericha@fsv.cvut.cz / rer@cml.fsv.cvut.cz Department of Structural Mechanics Czech Technical University in Prague Faculty of Civil Engineering Thákurova 7 166 29, Praha 6, Czech Republic 96 © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ Acta Polytechnica Vol. 44 No. 5 – 6/2004