The Open Civil Engineering Journal, 2011, 5, 9-16 9 Open Access Estimation of the Active Earth Pressure with Inclined Cohesive Backfills: the Effect of Intermediate Principal is Considered

W. L. Yu1,2,*, J. Zhang1, R. L. Hu1, Z. Q. Li1, X. H. Sun2 and T. L. Li3

1Key Laboratory of Engineering Geomechanics, Institute of Geology and Geophysics, Chinese Academy of Sciences, Beijing 100029, China 2Exploration and Development Research Institute, Daqing Oil Field Company Ltd., Daqing, Heilongjiang 163712, China 3China Building Technique Group Co., Ltd, Beijing 100029, China

Abstract: Estimating active earth pressure accurately is very important when designing retaining wall. Based on the unified strength theory and plane strain assumption, an analytical solution has been developed to determine the active distribution on a retaining structure with the inclined cohesive backfill considering the effect of the intermediate principal stress. The solution derived encompasses both Bell’s equation (for cohesive or cohesionless backfill with a horizontal ground surface) and Rankine’s solution (for cohesionless backfill with an inclined ground surface). Keywords: Active earth pressure, inclined cohesive backfill, intermediate principal stress, Rankine’s theory, Bell’s equation.

1. INTRODUCTION When the backfill is cohesive ( > 0, c > 0) and Active earth pressure plays an important role in soil- horizontal, the active earth pressure, a, is calculated using structure interaction in many structures in civil engineering Bell’s eq. (3): such as retaining walls, retaining piles around a foundation  =  K  2cK (3) ditch, and so on. Therefore, estimating active earth pressure z a a accurately is very useful in geotechnical engineering, Where especially in the design of simpler retaining structures such K = tan2 (45o   /2) (4) as small gravity retaining walls. A theoretical framework for a earth pressure theory has been firmly established over the past couple of decades. Classical Rankine earth pressure Equations (1)–(4) are valid for smooth vertical walls. theory, one of the most important earth pressure theories, is Although the back of every real retaining wall is rough, still used because of its rigorous theory, clear concept and approximate values of the earth pressure can be obtained simple calculation. on the assumption that it is smooth [1]. Therefore, in this study, all the retaining walls are considered as smooth and Active earth pressure, , acting on a retaining structure vertical. with inclined cohesionless backfill with an angle  with the Beside the Rankine’s theory, there are some other horizontal (with , c=0, and angle, >0) is theoretical methods that have been developed to determine expressed by Rankine: the lateral earth pressures. Based on the assumption of a  = K  (1) logarithmic spiral failure surface, Caquot and Kerisel [2] a z developed tables of earth pressure coefficients. Sokolovski [3] presented a method based on finite-difference solution. Here, z is the vertical effective stress, and Ka is the Habibagahi and Ghahramani [4] developed a solution for active earth pressure coefficient, where lateral earth pressure coefficients based on zero extension line theory. cos  cos2   cos2  K (2) a = All the aforementioned methods can not be used in the cos cos2 cos2  +    case where the soil behind the wall is sloping and cohesive. With graphic method about Mohr’s circle of Stresses Nirmala Gnanapragasam [5] developed an analytical solution *Address correspondence to this author at the Key Laboratory of Engineering to determine the active lateral earth pressure distribution on a Geomechanics, Institute of Geology and Geophysics, Chinese Academy of Sciences, Beijing 100029, China; Tel: 86-010-82998610; retaining structure when a cohesive backfill is inclined. Its Fax: 86-010-82998610; E-mail: [email protected] can also be used to check the sustainability of a slope. With

1874-1495/11 2011 Bentham Open 10 The Open Civil Engineering Journal, 2011, Volume 5 Yu et al. the trial wedge (graphical) method NAVFAC [6] determined strength theory may be regarded as a theoretical system to the active lateral force for each case using a force polygon. cover a series of regular strength criteria. When b=0; 0.5 and 1, the Tresca yield criterion, linear approximation of the von In general, the most favorable backfill materials are Mises yield criterion and the twin-shear stress yield criterion permeable coarse-grained soils with well settlement, are obtained, respectively [8]. Fig. (1) shows the loci of UST preferably with little silt or no clay content, but such in the deviatoric plane. materials may be unavailable or too expensive. Therefore, the cohesive or poor quality granular soils are used as backfill in some areas where the well-drained granular soils are in shortage [7]. Based on multi-slip mechanism and the model of multi- shear element, M.H.Yu established the unified strength theory (UST) which takes into consideration the different contribution of all stress components on the yield of failure of materials [8, 9]. The UST encompasses the twin theory [10-12] and single strength theory (Morh- Coulomb 1900). The excellent agreement between the predicted results by the UST and the experiment results indicates that the UST is applicable for a wide range of stress states in many materials, including metal, rock, soil, concrete, and others [13, 14].

At present, few investigations have been conducted about the effect of the intermediate principal stress on the active Fig. (1). The loci of the unified strength theory in the deviatoric earth pressure in soil. It is meaningful to develop a method to plane (by M.H. Yu 2001). determine the active earth pressure considering the all three Substituting eqs. (7) and (8) into eq. (5) and eq. (6), principal stresses. The purpose of this paper is to determine we get: the active lateral earth pressure distribution on a retaining structure against an inclined cohesive backfill considering  +   If 1 3 1 3 sin , the intermediate principal stress.  2  +  2 2 2. REVIEW OF THE UNIFIED STRENGTH THEORY 2       Then  = tan 45°  1+ b   b  21+ b tan 45°  c (9) 3 2 ()1 2 () 2 The UST has a unified model and simple unified     mathematical expression that is suitable for various materials. The mathematical expression can be introduced as 2    b +    Else  = tan 45°  2 1  2tan 45°  c (10) follows [12]: 3 2 1 b 2   +   If  1 +  3 ,  2  3. THEORETICAL ANALYSIS MODEL 1+  The following conditions are assumed in deriving the (b + ) Then 2 3 ; (5) analytical solution: (1) the soil is isotropic and homogeneous  1  =  t 1+ b and has both internal angle of friction and cohesion; the friction and cohesion are constant and remain independent of 1 Else (6) depth; (2) the force on the wall is acting parallel to the slope. ( 1 + b 2 )   3 =  t 1+ b Consider a retaining wall with a cohesive backfill of Where b is coefficient of intermediate principal stress, slope angle  ( is positive when the surface of the backfill the parameter, , is the ratio of tensile strength, t, and slopes upwards from the top of the wall), as shown in Fig. compressive strength, c. In geotechnical,  and t can be (2). The upright surface MN represents the retaining wall. If expressed by shear strength parameters: the backfill of the retaining wall moves to the wall, the force  1 sin on the wall decreases gradually. When the soil is in the active  = t = (7) earth pressure state, then the pressure acting on the wall is the  1+ sin c active earth pressure. On the contrary, if the retaining wall moves to the backfill behind the wall, the force on the wall 2ccos (8)  t = increases gradually. For an element of soil at the dept z at the 1+ sin back of the wall, the construction of Mohr’s circle of stresses According to unified strength theory, the parameter b in active stress state is shown in Fig. (3). The vertical stress plays a significant role. With different values of b, the on the soil element is denoted as OB and is expressed as unified strength theory represents or approximates all the OB = z cos traditionally strength or yield criteria. Hence, the unified Estimation of the Active Earth Pressure with Inclined Cohesive Backfills The Open Civil Engineering Journal, 2011, Volume 5 11

Where  is the unit weight of the soil, and z is the depth 3.1. Active Earth Pressure Strength pa below ground surface. The lateral stress on the soil element When the soil is in active limit equilibrium state, the force is denoted by OA' in Fig. (3). OB and OA' will hereafter be is denoted as the active earth pressure pa. As seen from Fig. referred to as z and , respectively. However, it should be (3), lines OH and OH' are symmetrical, the lengths of lines noted that in this paper z and  are not stresses acting OA and OA' are the same; the lengths of lines OB and line normal to their respective planes. OB' are also the same. When the force on the wall is the active earth pressure, then

M pa =  =OA'=OA, z=OB'=OB,  2 A= pacos, B= zcos=z cos  (14)  Z  Where A and B are the two roots of eq. (13):

2   +   +  4  1+ tan2  1m ()1 3 ()1 3 1 3 () (15)   A = 2   21()+ tan   2  dz  + +  +  4  1+ tan2  90- ()1 3 ()1 3 1 3 () (16)  B = 21+ tan2   ()  From eqs. (14) and (16), the following equations are obtained：

2 N  +  4  1+ tan2  = 2z   + (17) ()1 3 1 3 ()()1 3 Fig. (2). General soil-structure system.   = z  +   2 z2 cos2  (18) 1 3  ()1 3   According to eqs. (14), (15) and (16), eq. (19) is obtained:  +  1 3 p cos + cos (19)  A + B = 2 = a   z  E' H 1+ tan  Equating eq. (19), we get: B 2  +  2z  + A p = ()1 3  = 1 3   (20) E a z z z  2z z O   A' 3.2. Definition of Intermediate Principal Stress

Pa Plane strain state is widely existent in geotechnical B' engineering such as slope, strip foundation, retaining wall H' etc. Intermediate principal stress can be determined by using the generalized Hook’s law when analyzing the strength and deformation condition of rock and soil body with the

nonlinear method. Suppose that the cross section of retaining Fig. (3). Mohr’s circle to derive the analytical active earth pressure wall as x-z plane, thus the direction vertical to the cross expression. section is y-direction. With the elastic solution of plane The equations of the Mohr’s circle and Line OH can be strain problem, written as, respectively y = 0 2 2   +  2    According to the generalized Hook’s law:   1 3 + ()  0 =  1 3 (11) 2 2 1       y =  y  ()z + x  E  =  tan  (12) In x-z plane: z+x=1+3. The intermediate principal So we get a unary quadratic equation of : stress is tentatively defined as: 1+ tan2     +  +  = 0 (13) (21) ()()1 3 1 3  2 = ()1 + 3 12 The Open Civil Engineering Journal, 2011, Volume 5 Yu et al.

Substituting eq. (21) into eqs. (9) and (10), then eqs. (22) root. Obviously, the inclined angle should be smaller than and (23) can be obtained: the internal friction angle. Substituting eq. (28) into eq. (24), we can obtain: If  1 + 3  1  3 ,  2  + sin 2 2 2 ()G ± G  4Q ()1+ b  sin  bsin  b  b sin  2(1 sin)(1+ b)  2b(1+ sin)  3 = (29) Then 1   (22) 2(1+ sin)(1+ b) (1+ sin)(2 + 2b) = 4c(1+ b)cos 2c(1+ b)cos 3  (1+ sin)(1+ b) (2+ 2b)(1 sin) +2b(1 sin)  2(1+ sin)(1+ b) Else 1   (23) In eq. (28), the negative root 1 becomes the minimum  = 4c(1+ b)cos 3 principal stress in active earth pressure circle (while the positive root  becomes the maximum principal stress in Based on the UST and plane strain assumption, we derive 1 passive earth pressure circle); Likewise, in eq. (29), the the active earth pressure in two different cases: smaller 3 is the minimum principal stress in active earth 4. FORMULA DERIVATION OF P IF pressure circle, and the other is the maximum principal stress A in passive earth pressure circle.  +   1 3 1 3 sin  2  +  In the active earth pressure circle: 2 2 G  G2  4Q 2 + b  bsin 4.1. Formula Derivation of pa ()2c(1+ b)cos  + = ()  (30) 1 3 2(1+ sin)(1+ b) (1+ sin)(1+ b) If  1 + 3  1  3 ,  2  + sin 2 2 Substituting eqs. (28) and (29) into eq. (20), we can Minimum principal stress 3 can be obtained according to obtain: eq. (22):  2 G  G  4Q 2 + b  bsin 2c(1+ b)cos p = cos  () (31)  1+ b  sin  bsin  b  b sin  2c(1+ b)cos a 2 (1+ sin)(1+ b) (1+ sin)(1+ b)  = 1 () (24)  3 (1 sin )(1 b) +  +  z cos Uniting eqs. (18) and (24), the following equation is Eq. (31) can be used to calculate the active earth pressure obtained: strength with an inclined cohesive backfill considering the effect of the intermediate principal stress. 2c(1+ b)cos + z cos2  2 + b  bsin  2  () 1 1 1+ b  sin  bsin  b  b sin  +   (25) 4.2. Cases of pa when 1 3 1 3  2  + sin 2z cos2 c(1+ b)cos +  2 z2 cos2 (1+ sin)(1+ b) 2 2 + = 0 The validity of eq. (31) can be verified for simplified 1+ b  sin  bsin  b  b sin soil–structure scenarios. Eq. (25) is a unary quadratic equation of 1, we here introduce parameters of G and Q, Case A: cohesive soil with horizontal backfill surface behind the wall not considering the intermediate principal 2c(1+ b)cos + z cos2  2 + b  bsin stress G = () (26) 1+ b  sin  bsin  b  b sin Substituting =0, b=0 into eqs. (26) and (27), we get 2 2z cos2 c(1+ b)cos +  2 z2 cos2 (1+ sin)(1+ b) 2c(1+ b)cos + z cos  2 + b  bsin 2ccos + 2z Q = (27) G = ()= (32) 1+ b  sin  bsin  b  b sin 1+ b  sin  bsin  b  b sin 1 sin So eq. (25) can also be expressed as: 2z cos2 c(1+ b)cos +  2 z2 cos2 (1+ sin)(1+ b) Q =  2  G  + Q = 0 1+ b  sin  bsin  b  b sin (33) 1 1 2zccos +  2 z2 (1+ sin) And its solution is = 1 sin 2 G ± G  4Q (28)  1 = Then 2 G  G2  4Q Here, we have to make sure that discriminant of the = z (34) square root is positive (G2-4Q > 0) in order to get the real 2 Estimation of the Active Earth Pressure with Inclined Cohesive Backfills The Open Civil Engineering Journal, 2011, Volume 5 13

2  1+ b+ sin + bsin  b + b sin + 2c(1+ b)cos G  G  4Q 2 + b  bsin 3 () (40) ()2c(1+ b)cos  1 = () (1 sin )(1 b)  1 + 3 =    +  2(1+ sin)(1+ b) (1+ sin)(1+ b) Based on eqs. (18) and (40), the following expression is Substitute eq. (34) into eq.(31): obtained:

2 2 b bsin cos z cos2 2 b bsin 2c(1 b)cos G  G  4Q +    2ccos 2   + +   +  p = ()   () a 2 (1+ sin)(1+ b) (1+ sin) (35) 3 3 1+ b+ sin + bsin  b + b sin (41) 2 2 2 2 2    z cos (1 sin)(1+ b)  2zc(1+ b)cos cos  z = z tan (45°  )  2c tan(45°  ) + = 0 2 2 1+ b+ sin + bsin  b + b sin

Which is identical to Bell’s expressions for cohesive Eq. (41) is a unary quadratic equation about 3, we soils, eqs. (3) and (4). introduce another two parameters G' and Q': Case B: granular soil with an inclined backfill surface not z cos2  2 + b+ bsin  2c(1+ b)cos considering the intermediate principal stress (when b=0) G ' = () (42) 1+ b+ sin + bsin  b + b sin Substituting c=0, b=0 in eqs. (26) and (27), we get 2 2 2 2  z cos (1 sin)(1+ b) 2c(1+ b)cos + z cos  ()2 + b  bsin 2ccos + 2z cos2  Q' = G = = (36) 1+ b+ sin + bsin  b + b sin (43) 1+ b  sin  bsin  b  b sin 1 sin 2zc(1+ b)cos cos2   2z cos2 c(1+ b)cos +  2 z2 cos2 (1+ sin)(1+ b) 1+ b+ sin + bsin  b + b sin Q = 1+ b  sin  bsin  b  b sin (37) So the solution to eq. (40) about 3 is obtained:  2 z2 cos2 (1+ sin) = (G ' G '2 4Q') 1 sin ±  (44)    3 = 2 And From eqs. (40) and (44),

2 2 2 z cos cos cos cos 2 G  G  4Q        (G '± G '  4Q')1+ b+ sin + bsin  b + b sin = () (38)  = () 1 (45) 2 1 sin 2(1 sin)(1+ b) 2c(1+ b)cos Substitute eq. (38) into eq. (31): + (1 sin)(1+ b) G  G2  4Q ()2 + b  bsin cos 2ccos pa =   z cos Similarly, in eq. (44), the negative root 3 becomes the 2 (1+ sin)(1+ b) (1+ sin) minimum principal stress in active earth pressure circle cos  cos2   cos2  (39) (while the positive root 3 becomes the minimum principal = z cos stress in passive earth pressure circle); Likewise, in eq. (45), 2 2 cos + cos   cos  the smaller 1 is the maximum principal stress in active earth 2 2 pressure circle, and the other is the maximum principal stress cos  cos   cos  in passive earth pressure circle. =  z cos cos2 cos2  +    In active earth pressure state: Which is identical to Rankine’s expressions with an G  G2  4Q inclined backfill surface for cohesionless soils, eqs. (1) and 2 + b+ bsin + 2c(1+ b)cos 2 () (46) (2).  1 + 3 = (1 sin)(1+ b) 5. FORMULA DERIVATION OF PA IF Finally, the active earth pressure is obtained:  1 + 3  1  3 sin  2  2    G ' G ' 4Q' 2 + b+ bsin 2c(1 b)cos 2 2   ()+  (47) pa = cos  +  2 (1 sin)(1+ b) (1 sin)(1+ b) 5.1. Formula Derivation of Active Earth Pressure pa z cos The derivation process, very similar to section 4, is as Here, we have to make sure that discriminant of the follow: square root is positive (G2-4Q > 0) in order to get the real According to eq. (10) maximum principal stress 1 can be root. Obviously, the inclined angle should be smaller than obtained, the internal friction angle. 14 The Open Civil Engineering Journal, 2011, Volume 5 Yu et al.

2  1 + 3  1  3 G ' G ' 4Q' 2 + b+ bsin cos 5.2. Cases of pa when sin   ()2ccos  2  +  p = + 2 2 a 2 (1 sin)(1+ b) (1 sin) (55) In order to verify the validity of eq. (47), there are cos  cos2   cos2  simplified soil–structure scenarios. z = z cos cos + cos2   cos2  Case A: cohesive soil with horizontal backfill surface behind the wall not considering the intermediate principal Which agrees with Rankine’s expressions for cohesive stress soils with an inclined backfill surface for cohesionless soils, eqs. (1) and (2). Substituting =0, b=0 into eqs. (42) and (43), we get z cos2  2 + b+ bsin  2c(1+ b)cos 6. COMPARISON BETWEEN THE NEW RESULTS G ' = () AND THE CLASSICAL ONES 1+ b+ sin + bsin  b + b sin (48) In order to show the differences between the classical 2z  2ccos = (b=0) and the present magnitudes of the earth pressure, 1+ sin we, firstly, should know whether 2 is bigger than  +    2 z2 cos2 (1 sin)(1+ b) 1 3 + 1 3 sin or not. The result counts on the values Q' = 2 2 1 b sin bsin b b sin + +  +    +   of Poisson’s Ratio () and internal friction angle (). The 2zc(1+ b)cos cos2  (49) value of  is, mostly, less than 0.5, so  is the smaller one.  2 1+ b+ sin + bsin  b + b sin Two cases are listed as follow:  2 z2 (1 sin)  2zccos Case A: comparison between the new results and = Rankin’s theory: 1+ sin There are four figures (Figs. 4-7) to compare the new Then results with Rankin’s theory by changing different para- meters. The results show that active earth pressures G ' G '2  4Q' z(1 sin)  2ccos = (50) computed by Rankin’s theory are obviously higher than the 2 1+ sin new results. The pressures become small when b increases. Substituting eq. (50) into eq. (47), we obtain: The pressures decrease to a great degree with Poisson’s ratio and internal friction angle increasing, and increase slightly G ' G '2  4Q' 2 + b+ bsin cos 2ccos with slope angle increasing. p = ()+ a 2 (1 sin)(1+ b) (1 sin) (51) 30 28 2   z = z tan (45°  )  2c tan(45°  ) 26 2 2 24 22 Which is identical to Bell’s expressions for cohesive 20 b=0 soils, eqs. (3) and (4). 18 b=0.25 Case B: cohesionless soil with an inclined backfill 16 b=0.5 14 b=0.75 surface not considering the intermediate principal stress active earth pressure/kPa 12 b=1 (when b=0) 10 Substituting c=0, b=0 in eqs. (42) and (43), we get 33.544.55 height /m 2 2c(1+ b)cos + z cos  2 + b  bsin 2ccos + 2z cos2  G ' = ()= (52) Fig. (4). Pressures at different heights changing with b. 1+ b  sin  bsin  b  b sin 1 sin 31 2z cos2 c(1+ b)cos +  2 z2 cos2 (1+ sin)(1+ b) 29 Q' = 1+ b  sin  bsin  b  b sin (53) 27 25  2 z2 (1+ sin) = 23 b=0 1 sin   21 b=0.25 b=0.5 Then 19 b=0.75

active earth pressure/kPa 17 2 2 b=1 2 z cos cos cos cos G ' G '  4Q'        15 = () (54) 0.3 0.35 0.4 2 1+ sin Poisson's ratio Substituting eq. (54) into eq. (47), we obtain: Fig. (5). Pressures of different Poisson’s Ratios changing with b. Estimation of the Active Earth Pressure with Inclined Cohesive Backfills The Open Civil Engineering Journal, 2011, Volume 5 15

50 30

45 25 40

35 b=0 20 b=0 b=0.25 30 b=0.25 b=0.5 25 b=0.75 15 b=0.5 active earth pressure /kPa b=0.75 b=1 active earth pressure /kPa 20 b=1 20 25 30 10 internal friction angle /° 0.3 0.35 0.4 poisson's ratio Fig. (6). Pressures of different internal friction angles changing with b. Fig. (9). Pressures of different Poisson’s Ratios changing with b.

30 33 31 25 29 27 20 25 b=0 15 23 b=0.25 b=0 21 b=0.5 10 b=0.25 19 b=0.75 b=0.5

active earth pressure /kPa 17 b=1 5 b=0.75 active earth pressure /kPa 15 b=1 0 12 15 18 20 25 30 slope angle of backfill /° internal friction angle /° Fig. (7). Pressures of different slope angles changing with b. Fig. (10). Pressures of different internal friction angles changing with b

30 30 b=0 25 b=0.25 25 b=0.5 20 20 b=0.75 15 b=1 15 b=0 10 b=0.25 10 b=0.5 5 active earth pressure /kPa 5 b=0.75 0 active earth pressure /kPa b=1 3 3.5 4 4.5 5 0 Height /m 81216 cohesion /kPa Fig. (8). Pressures at different heights changing with b. Fig. (11). Pressures of different cohesion changing with b.

Case B: comparison between the new results and Bell’s wall with a cohesive inclined backfill considering the effect Equation: of the intermediate principal stress based on UST and the Similarly, we make another four figures (Figs. 8-11) to plane strain assumption. The results show that the Poisson’s compare the new results with Bell’s equation by changing Ratio, , and coefficient of intermediate principal stress, b, have large effect on the earth pressure. The analytical different parameters. The results show that active earth solution is a generalized expression that can be used for pressures computed by Bell’s theory are obviously higher retaining structures against inclined various soil and rock than the new results. The pressures become small when b backfills. The results show that the analytical solution increases. The pressures decrease with Poisson’s ratio, encompasses Rankine’s solution. internal friction angle and cohesion increasing. However, it should be noted that the principal stress y 7. CONCLUSIONS AND DISCUSSIONS be not always the intermediate principal in the direction of An analytical solution has been developed to determine y=0. Only in the plane strain condition, the principal stress the active lateral earth pressure distribution on a retaining y is always the intermediate principal in the direction of 16 The Open Civil Engineering Journal, 2011, Volume 5 Yu et al. plane strain. It is a pity that, in this paper, there is no  = Lateral stress in soil comparison between the calculated earth pressure and the = Vertical stress in soil at depth of z actual earth pressure, mostly because the actual earth z pressure is hard to be measured. z = Depth below ground surface

ACKNOWLEDGEMENTS REFERENCES The authors acknowledge financial support from Chinese [1] K. Terzaghi, R. B. Peck, and G. Mesri, in National Science and Technology Support Program (No. Engineering Practice, New York: John Wiley and Sons, Inc., 1996. [2] A. Caquot , and J. Kerisel, Tables for the calculation of the passive 2008BAK50B04-3), key project (No. KZCX2-YW-Q03-2), pressure, active pressure and bearing capacity of foundations, ([2009]03-02-03) and National Natural Science Foundation Paris: Gauthier-Villars, 1948. of China (No. 41072226). [3] V. V. Sokolovski, Statics of soil media, 2nd ed. Butterworth & Co. Ltd., London, 1960. LIST OF SYMBOLS [4] K. Habibagahi and A. Ghahramani, “Zero extension line theory of earth pressure”, Journal of the Geotechnical Engineering Division,  = Maximum principal stress ASCE, vol. 105, no. GT7, pp. 881-896, 1979. 1 [5] N. Gnanapragasam, “Active earth pressure in cohesive soils with an inclined ground surface”, Canadian Geotechnical Journal, vol. 2 = Intermediate principal stress 37, pp. 171-177, 2000. 3 = Minimum principal stress [6] NAVFAC, DM7.2: Foundations and earth structures. Naval Facilities Engineering Command, United States Department of the b = Coefficient of intermediate principal shear Navy, Alexandria, Va., 1982. stress [7] C. R. I. Clayton, I. F. Symons, and J. C. Hiedra-Cobo, "The pressure of clay backfill against retaining structures", Canadian t = Tensile strength Geotechnical Journal, vol. 28, pp. 282-297, 1991. [8] M. H. Yu, New system of strength theory, Xi’an: Xi’an Jiaotong c = Compressive strength University Press: 1992 (in Chinese). [9] M. H. Yu, “Unified strength theory for geomaterials and its c = Soil cohesion application”, Chinese Journal of Geotechnical Engineering, vol. 16, no. 2, pp. 1-9, 1994.  = Internal friction angle of soil [10] M. H. Yu, “Twin shear stress yield criterion”, International  = The ratio of tensile strength  and compressive Journal of Mechanical Sciences, vol. 25, no. 1, pp. 71-74, 1983. t [11] M. H. Yu, and L. N. He, “A new model and theory on yield and strength c failure of materials under complex stress state”, Mechanical Behavior of Materials-6, Oxford: Pergamon Press, 1991, pp. 841- E = Modulus of elasticity 846.  = Unit weight of soil [12] M. H. Yu, L. N. He, and L.Y. Song, “Twin shear stress theory and its generalization”, Scientia Sinica (Series A), vol. 28, no. 12, pp.  = Poisson’s Ratio of filling, 0<<0.5 1174-1183, 1985 (in Chinese). [13] G. W. Ma, and I. Shouju, “plastic limit analysis of circular plates  = Angle that backfill makes with the horizontal with respect to unified yield criterion”, International Journal of Mechanical Sciences, vol. 10, no. 10, pp. 963-976, 1998. Pa = Active lateral earth pressure [14] M. H. Yu, J.C. Li and Y.Q. Zhang, “Unified characteristics line theory of spacial axisymmetric plastic problem”, Sciences in China  = Shear strength of soil (Series E), vol. 44, no. 2, pp. 207-215, 2001 (in Chinese).

Received: October 05, 2010 Revised: December 15, 2010 Accepted: December 20, 2010

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