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Estimation of the Active Earth Pressure with Inclined Cohesive Backfills: the Effect of Intermediate Principal Stress Is Considered

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Estimation of the Active Earth Pressure with Inclined Cohesive Backfills: the Effect of Intermediate Principal Stress Is Considered 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 Stress 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 lateral earth pressure 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 cohesion, c=0, and friction 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 shear strength 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： N 2 2 (17) 1 + 3 4 1 3 1+ tan = 2z 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.
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