Kinematic Limit Analysis Approach for Seismic Active Earth Thrust Coefficients of Cohesive-Frictional Backfill Downloaded from ascelibrary.org by University Of Florida on 10/25/17. Copyright ASCE. For personal use only; all rights reserved. Jagdish Prasad Sahoo1 and R. Ganesh2 Abstract: A theoretical study has been performed for evaluating maximum resultant active force exerted by a cohesive-frictional backﬁll against a rigid wall with inclined back face undergoing outward horizontal translational movement in the presence of pseudostatic seismic loadings. With the application of a new kinematically admissible translational mechanism in the context of the upper bound limit theorem of plasticity, seismic active thrust has been derived as a function of nondimensional seismic active earth thrust coefﬁcients because of the contributions of soil unit weight, surcharge pressure, and cohesion of soil. The present analysis was performed by postulating a composite collapse mechanism comprised of a central radial shearing zone enclosed between triangular blocks at either side. The kinematically admissible velocity ﬁeld of the radial shear zone freely varies in between the velocity ﬁeld used for deﬁning the conventional circular to log spiral shear zones available in the literature. The inﬂuence of seismic acceleration coefﬁcients, location of surcharge pressure from the wall crest, backﬁll slope angles, characteristics of interface between soil and wall, orientations of wall, and properties of backﬁll on the magnitude of active earth force has thoroughly been examined. DOI: 10.1061/(ASCE)GM.1943-5622.0001030. © 2017 American Society of Civil Engineers. Author keywords: Active earth thrust; Kinematically admissible; Limit analysis; Retaining wall; Seismic loadings. Introduction The determination of active earth pressure induced by backﬁll soil mass against a rigid retaining wall is a major problem in the ﬁeld of geotechnical engineering. A vast number of research attempts have been made to estimate the static and seismic active earth pressure exerted by soil as backﬁll on a rigid retaining wall by using different approaches. The theories developed by Coulomb (1776) and Rankine (1857) are generally used to determine the active earth pressure against a rigid retaining wall. For computing the seismicinduced active earth force against rigid retaining walls with frictional backﬁll, Okabe (1924) and Mononobe and Matsuo (1929) have extended the static earth pressure theory of Coulomb (1776) by taking into account the pseudostatic seismic forces in the analysis; this modiﬁed Coulomb’s theory is commonly referred to as the Mononobe-Okabe theory. By perceiving the development of nonplanar failure surfaces in the backﬁll at their critical active state, thrust coefﬁcients were presented in the tabular form (Terzaghi 1943; Jumikis 1962; Kerisel and Absi 1990) for determining the static lateral earth thrust based on the limit equilibrium analysis of curved failure surfaces. Saran and Prakash (1968) have extended the Mononobe-Okabe theory for retaining walls with general cohesivefrictional soil as backﬁll and reported that the presence of cohesion in the backﬁll has a signiﬁcant inﬂuence on the magnitude of active earth thrust. Therefore, the contribution of soil cohesion cannot be omitted simply in the analysis and design of retaining structures. 1 Assistant Professor, Dept. of Civil Engineering, Indian Institute of Technology, Roorkee 247667, India (corresponding author). E-mail: jpscivil@gmail.com 2 Research Fellow, Dept. of Civil Engineering, Indian Institute of Technology, Roorkee 247667, India. E-mail: ravishivaganesh@gmail .com Note. This manuscript was submitted on January 30, 2017; approved on July 6, 2017; published online on October 25, 2017. Discussion period open until March 25, 2018; separate discussions must be submitted for individual papers. This paper is part of the International Journal of Geomechanics, © ASCE, ISSN 1532-3641. © ASCE Based on the upper bound theorem of limit analysis and by analyzing six different translational mechanisms, Chen and Rosenfarb (1973) have produced rigorous upper bound solutions for evaluating maximum static active earth thrust on a retaining wall with a general cohesive-frictional backﬁll. With the inclusion of pseudostatic seismic body forces and using the upper bound mechanism presented by Chen and Rosenfarb (1973), Chen and Liu (1990) computed seismic active earth thrust coefﬁcients on retaining walls with inclined back face due to sloping cohesive-frictional backﬁll. Das and Puri (1996) have addressed the limitations of the modiﬁed Mononobe-Okabe theory proposed by Saran and Prakash (1968) in which the backﬁll surface was considered horizontal and the effect of vertical seismic acceleration was neglected. Soubra and Macuh (2002) have computed the static active earth thrust coefﬁcients corresponding to components of surcharge and soil cohesion based on a rotational log spiral mechanism. The studies performed by Chen and Rosenfarb (1973), Chen and Liu (1990), Saran and Prakash (1968), and Das and Puri (1996) were based on the independent maximization scheme in which the seismic active earth thrust coefﬁcients, because of components such as soil weight, surcharge pressure, and cohesion of soil, were discovered with the help of the superposition rule. Based on the limit equilibrium approach, the total active earth force computed by Shukla et al. (2009) due to the contribution of soil weight and cohesion of soil without using the superposition rule was reported to differ from that determined by Saran and Prakash (1968) with the principle of superposition. By applying Coulomb’s rigid wedge planar mechanism the inﬂuence of line surcharge, uniformly distributed surcharge of inﬁnite and ﬁnite extends applied at a certain distance away from the wall crest on the magnitude of active earth forces, has been addressed (Motta 1994; Greco 2005, 2006). Nevertheless, the applications of these studies are limited only to purely frictional backﬁll. No further information seems to have been presented describing the inﬂuence of proximity of surcharge pressure on the maximum active force regarding varying wall geometry, backﬁll slope angle, properties of soil, and so forth, which are quite useful to better understand the design philosophy associated with a retaining wall under 04017123-1 Int. J. Geomech., 2018, 18(1): 04017123 Int. J. Geomech. Downloaded from ascelibrary.org by University Of Florida on 10/25/17. Copyright ASCE. For personal use only; all rights reserved. different backﬁll surcharges. Jarquio (1981) and Misra (1980) developed solutions for ﬁnding lateral earth pressure on walls due to different surcharges on the backﬁll based on the theory of elasticity in which the effect of the strength parameters of backﬁll were not taken into account. For estimating the total active thrust on retaining walls in the presence of live load surcharge, Kim and Barker (2002) used a mixed approach in which the active force was calculated by considering backﬁll to be simultaneously in an active state of failure and in the elastic state due to soil weight and surcharges, respectively. The applicability of solutions computed using the elastic theory and the mixed approach are found to be suitable only for unyielding walls, whereas its application to yielding walls, that is, for the case of active failure, is really questionable (Steenfelt and Hansen 1983, 1984; Motta 1994; Georgiadis and Anagnostopoulos 1998; Greco 2005, 2006). Mylonakis et al. (2007) have computed seismic earth thrust coefﬁcients in sand with the help of the lower bound theorem of plasticity. Nian and Han (2013) examined the effects of backﬁll cohesion, backﬁll slope angle, depth of tension crack, and pseudostatic seismic forces by extending Rankine’s (1857) earth pressure theory. Based on the limit-equilibrium concept considering the arching effect in backﬁll soil, theoretical solutions were developed for calculating the active pressure on rigid retaining walls (Li and Wang 2014; Rao et al. 2016; Chen et al. 2017). It is known that the solutions obtained from these methods, such as limit equilibrium, stress characteristics, and the lower bound theorem of limit analysis, may ensure the conditions required for static admissibility; however, the requirement of kinematic admissibility conditions may not always be satisﬁed (Huntington 1957; Kumar and Subba Rao 1997). Such kinematical admissibility conditions are essential to ensure the kinematics of the problem, or the deformations produced by the mechanism are practically feasible in accordance with ﬂow rule of the material. From the literature there are number of factors, in particular, backﬁll slope angle, continuous surcharge pressure on the backﬁll, soil cohesion, wall inclination, and seismic accelerations, which inﬂuence the magnitude of the active earth pressure of a retaining wall. In most of the previous studies, the effect of a continuous surcharge acting on the backﬁll has been considered; in practice, however, the uniform surcharge may act at a particular distance away from the wall on the backﬁll. There seems to be very little research done to tackle these kinds of problems, which are quite relevant to practical situations. All the reported studies are based on the assumption of a simple rigid wedge planar mechanism behind the wall, and its applications are limited to frictional backﬁll. A rigorous solution to address the inﬂuence of a uniform surcharge applied at some predetermined distance away from the wall on the cohesivefrictional soil is still lacking. In the present research, a new composite collapse mechanism has been developed, which consists of a radial shearing zone sandwiched between two triangular blocks. In this collapse mechanism, the kinematically admissible velocity ﬁeld of the radial shear zone converges on the velocity ﬁeld of conventional circular and log spiral shear zones when the inclination of velocities are equal to the internal friction angle of soil and zero, respectively, with the normal and all radial lines emerging from a singular point. By applying the upper bound theorem of limit analysis on this new collapse mechanism to the framework of the pseudostatic approach, the analysis was performed, and rigorous solutions were developed for determining the magnitude of total seismic active earth force against a rigid wall translating horizontally away from the cohesive-frictional soil backﬁll. The effect of the inclination of the wall back face and backﬁll surface, soil-wall interface friction angle and soil-wall interface adhesion force, and the location © ASCE of the ground surcharge from the wall have been examined. The inﬂuence of the superposition rule on the solutions has also been discussed. The effectiveness of the present approach was examined by comparing the solutions obtained in this paper with the experimental and theoretical solutions reported in the previous studies. For a larger negative inclination of wall back face, the selection of a simple planar mechanism without inclusion of any shear zone has been found to be more effective compared with the composite mechanism introduced in the present work or in the earlier studies in obtaining the maximum magnitude of the seismic active earth force. This may be due to the imposition of more constraints in the case of the composite mechanism than that of the planar mechanism for satisfying the kinematic admissibility conditions. Statement of Problem A rigid retaining wall with an inclined back face supporting cohesive-frictional soil as backﬁll is illustrated in Fig. 1. The height of the wall is h, and l refers to the length of the back face of the wall, which is sloped at an angle a with the vertical plane. The surface of the backﬁll makes an angle of b with the horizontal plane and loaded with a uniform surcharge pressure of magnitude q at a distance l l from the wall top, where l is a multiplier constant whose value varies from 0 to 1. The unit weight of backﬁll soil mass is denoted as g . The interface strength t int between the wall and backﬁll is assumed equal to cw þ s w tand , where cw = adhesion per unit length of wall whose magnitude is c(tan d /tanf ); s w = normal stress on the wall; d = soil-wall interface friction angle; and the parameters c and f = cohesion and internal friction angle of soil, respectively. In the case of rough walls, the resultant active force Pa is directed at an angle d from the normal to the wall surface, and a tangential adhesive force Cw is assumed to be acting along the surface of wall provided the soil is not fully cohesive ( f = 0). To perform the pseudostatic analysis, the earthquake acceleration in the horizontal direction is assumed to be distributed uniformly in the soil domain with a magnitude equal to khg, where kh and g = peak coefﬁcient of horizontal earthquake acceleration and the q λl α khq O β Cw E khγ l h δ γ Pa 04017123-2 Int. J. Geomech., 2018, 18(1): 04017123 A Fig. 1. Problem deﬁnition Int. J. Geomech. Downloaded from ascelibrary.org by University Of Florida on 10/25/17. Copyright ASCE. For personal use only; all rights reserved. acceleration due to gravity, respectively. The aim is to determine the magnitude of total active earth force on a wall with inclined back surface retaining a general cohesive-frictional sloping backﬁll with a surcharge placed at different positions. The soil mass is idealized as a rigid perfectly plastic material and obeys the MohrCoulomb failure criterion with an associated ﬂow rule. It is presumed that the properties of the backﬁll mass remain the same at all the points within the soil domain during the occurrence of an earthquake. The magnitude of d is assumed to be constant in the present analysis; however, the distribution of the magnitude of d along the length of the wall is not constant, which is largely dependent on the movement of the wall; addressing this issue is beyond the scope of this article. Theoretical Analysis The theoretical analysis was performed by using the upper bound theorem of limit analysis based on an assumed collapse mechanism satisfying kinematical admissibility conditions. The upper bound analysis enables the determination of the magnitude of collapse load in any kinematically admissible velocity ﬁelds by equating the rate of total work done by the body and external applied forces to the rate of total internal energy dissipation. Any velocity ﬁeld can be said to be kinematically admissible as long as it satisﬁes the ﬂow rule, velocity compatibility, and boundary conditions. The magnitude of the collapse load computed on the basis of an upper bound limit analysis will always remain either greater or equal to the magnitude of true collapse load, and unrestricted plastic ﬂow must impend when a kinematically admissible velocity ﬁeld exists. The virtual energy-work balance expression in the upper bound theorem of limit analysis can be written mathematically as follows: ð ð ð (1) Ti Vi dS þ Xi Vi dX ¼ D_ ɛ_ ij dX S X X where Ti = traction on the loaded boundary surface S; Xi = vector of distributed forces within the region X including horizontal and vertical body forces; Vi = kinematically admissible velocity ﬁeld along _ ɛ_ ij ¼ s_ ij ɛ_ ij = rate of incremental the boundary surface S; and D internal energy dissipation per unit volume corresponding to a strain rate ɛ_ ij in the kinematically admissible velocity ﬁeld, where s_ ij = stress tensor associated with ɛ_ ij . Thus, the right side of Eq. (1) implies the rate of total incremental internal energy dissipation during an incipient failure of soil medium. Collapse Mechanism and Proposed Radial Shear Zone With the assumption of six different kinds of collapse mechanisms, Chen and Rosenfarb (1973) have determined the active earth force on retaining walls subjected to translational displacement, in which the composite collapse mechanism consisting of either circular shearing zone or logarithmic spiral zone sandwiched between triangular rigid blocks, yielded better results when compared with others. The circular shear zone was formed by assuming the velocities or displacement rates inclined at an angle equal to the internal friction angle of soil ( f ) with the normal to the radial lines emerging from a singular point; whereas the log spiral shear zone in which the inclination of velocities are normal to the all radial lines emerging from a singular point. In a similar fashion, it is attempted in this paper to introduce a radial shear zone by taking the inclination of velocities with the normal to the radial lines emerging from the singular point as variable, which has been found to yield better upper bound solutions. Although the computed maximum magnitude of active earth force from different collapse geometry does not differ © ASCE much from those obtained from a simple planar mechanism, in the present study an improved composite collapse mechanism, as illustrated in Fig. 2, has been used for performing the analysis, in which the velocity ﬁeld is proposed using a radial shearing zone OBC sandwiched between the two rigid triangular blocks OAB and OCD that have planar discontinuities. Note that zone OBC is a continuous deforming shear zone in which the energy dissipation occurs within the region OBC and along the discontinuous shearing boundary surface BC, whereas blocks OAB and OCD are rigid where uniform velocity ﬁeld is implied because the energy would not be dissipated within the region of these blocks except along the planar rigid boundary surfaces AB and CD. In Fig. 2(a) the wall OA is prescribed to translate outward horizontally with a velocity V0, which will cause the triangular blocks OAB and OCD to slide downward with velocities V1 and V2, respectively. The relative velocity V01 of block OAB with respect to the wall is kept parallel to the back face of the wall, which is similar to that of Chen and Rosenfarb (1973), for accounting interface characteristics of the soil and wall. For the presented mechanism to be kinematically admissible, the direction of tangential velocity jump vector V01 between the wall and the adjoining soil block OAB must be directed downward for the problem under consideration. Fig. 2(b) gives the complete velocity hodograph of the proposed composite mechanism, and using this velocity hodograph the relationship among the velocities V1, V2, and V01, respectively, can easily be interpreted in terms of any reference velocity of the system, which in this paper is taken to be the velocity of wall V0. For a material obeying an associated ﬂow rule, the displacement rates of each block are an inclined angle f with the discontinuous rigid boundary ABCD. Unlike in the earlier studies in which velocities are assumed to be directed at an angle either equal to zero or f from the normal to the radii of the continuous shearing zone, it is considered in this paper that this makes an angle u , which is a variable whose value ranges from 0 to f . Velocity and Radius Increments in the Radial Shear Zone The radial shearing zone with an included angle « is constructed in this paper by assuming that it is comprised of n number of qL′ λl D khqL′ α E β O ε μ khW3 W3 dχ χ C rχ khWχ Cw V2 r0 V01 δ Pa φ V0 Wχ α +μ -θ khW2 khW1 W2 V1 Vχ φ W1 V0 φ B V01 α V1 ε V2 A (a) (b) Fig. 2. (a) Composite collapse mechanism and (b) velocity hodograph associated with composite collapse mechanism; the directions marked for angles and forces are considered as positive in the analysis (Note: The weights of various blocks denoted as W with subscripts 1, 2, and 3 indicate, respectively, the blocks OAB, OBC, and OCD) 04017123-3 Int. J. Geomech., 2018, 18(1): 04017123 Int. J. Geomech. Downloaded from ascelibrary.org by University Of Florida on 10/25/17. Copyright ASCE. For personal use only; all rights reserved. and inﬁnitesimal triangular rigid blocks and is characterized by a constant interior angle equal to D« (« /n), as illustrated in Fig. 3(a). These blocks share a common apex point (singular point), and their bases and sides constitute the discontinuous rigid shearing boundary surfaces separating the stationary and moving soil masses and the discontinuous velocity surfaces. Let the blocks Oa0a1, Oa1a2…, Oan-2an-1, and Oan-1an be prescribed to move with velocities equal to V0, V1…, Vn-1, and Vn, respectively. All these velocities are uniform, constant, and make an angle of u normal to the respective radial lines emerging from the singular point. To satisfy an associated ﬂow rule, the relative velocity of block n – 1 with respect to block n should be directed at an angle equal to f , as depicted in Fig. 3(b), which shows the velocity hodograph of a particular elemental block interface. Similarly, the inclination of the rigid shearing boundary surfaces with the block velocities should also be equal to f . Following the approach used by Chen (1975) and based on the compatible velocity diagram given in Fig. 3(b) for an inﬁnitesimal elemental block interface, the velocity of the nth block can be given by the following relation: 2 3 Dɛ 6 cos f þ u þ 7 2 7 6 Vn ¼ Vn1 6 (2) 7 4 Dɛ 5 cos f þ u 2 ε 2 Vn1 3 Dɛ 6 cos f þ u þ 7 2 7 6 ¼ Vn2 6 7 4 Dɛ 5 cos f þ u 2 (3) By substituting Eq. (3) into Eq. (2), Vn can be expressed in terms of Vn-2 as 2 32 Dɛ 6 cos f þ u þ 7 2 7 6 Vn ¼ Vn2 6 7 4 Dɛ 5 cos f þ u 2 (4) If the previous process is repeated n times, the velocity of the nth block could be written in terms of the initial velocity V0 as follows: 2 3n Dɛ 6 cos f þ u þ 7 2 7 6 Vn ¼ V0 6 7 4 Dɛ 5 cos f þ u 2 (5) Initial radial line O Δε Δε Δε Δε rn rn-1 an Vn Θ Θ φ Vn-1 an-1 φ r1 Final radial line Note: Θ = Θ r0 π 2 φ Θ −θ V1 V0 φ a1 a0 (a) 1 (π − 2θ − Δε ) 2 Δε an-2 Surfaces a0O a1O . an-1O anO Description Discontinuous velocity surfaces that exist between the block boundaries a0a1 a1a2 . an-2an-1 an-1an Rigid shearing boundary surfaces between stationary and moving soil V0 V1 . Vn-1 Vn Respective block velocities which is constant within the block Velocity discontinuity surface Vn-1 Vn Vn-1, n δu δv φ δu: Tangential velocity jump vector δv: Normal velocity jump vector (b) Fig. 3. (a) Proposed velocity ﬁeld; (b) velocity hodograph for an inﬁnitesimal element in the radial shearing zone © ASCE 04017123-4 Int. J. Geomech., 2018, 18(1): 04017123 Int. J. Geomech. After some algebraic manipulations and substituting the limits for n tending to inﬁnity, the following expression for the velocity of the nth block can be obtained: Vn ¼ V0 eɛ tan ð f þu Þ (6) In a similar way, the following relation between the radius of initial and ﬁnal radial lines can also be presented Downloaded from ascelibrary.org by University Of Florida on 10/25/17. Copyright ASCE. For personal use only; all rights reserved. rn ¼ r0 eɛ tan ðf u Þ (7) Similar to Eqs. (6) and (7), the length of the radial line and velocity at an angle x from the initial radial line OB, as shown in Fig. 2, will be presented as follows in Eqs. (8a) and (8b): r x ¼ r0 e x V x ¼ V1 e x tanð f u Þ tanð f þu Þ (8a) (8b) Thus, the complete geometry of the collapse mechanism or the velocity ﬁeld can be constructed by using three independent variables, such as u , m , and « , where u = angle made by velocities with normal to the radial lines whose value ranges from 0 to f ; m = b b and « = BOC. AOB; Incremental Energy Dissipation in the Radial Shearing Zone During plastic shearing, the dissipation of energy is obvious along the discontinuous rigid boundary surfaces that exist between the moving and stationary masses, and along the block interfaces, which combined produce the radial shearing zone. Following Chen (1975) and referring to Fig. 3(b), the incremental energy dissipation rate per unit length along the block interface, which represents the velocity discontinuity surface, will be computed by the following expression: D_ ¼ crn1 d u (9) From Fig. 3(b), d u can be written as d u ¼ Vn1 sin Dɛ cos f = cos ð f þ u Dɛ=2Þ, and the Taylor series expansion of the resulting function from Eq. (9) for Dɛ ! 0 with the omission of higher order terms involving Dɛ gives D_ ¼ c cos f rn1 Vn1 Dɛ cosð f þ u Þ For an inﬁnitesimal rigid triangular block with an included angle of D« , substituting the length of the boundary surface, Ln1 ¼ rn1 = cosð f u ÞDɛ from Fig. 2(a), the following expression will be deduced from Eq. (12): cr0 V0 cos f eɛ x 0 1 D_ base ¼ (13) cosð f u Þ x0 Thus, the total incremental energy dissipation rate during plastic ﬂow in the radial shear zone becomes ɛx e 0 1 1 1 D_ radial zone ¼ cr0 V0 cos f þ cosð f u Þ cosð f þ u Þ x0 (14) It can easily be shown that Eqs. (6), (7), (11), and (13) obtained in this analysis are equivalent to the solutions of Chen (1975) presented for circular and log spiral shear zones by substituting u = f and u = 0, respectively. Thus, the velocity ﬁeld deﬁned using this new radial shear zone can have an obvious advantage of varying freely in between the velocity ﬁeld used for presenting the conventional circular to the log spiral shear zone; that is, when u varies from f to zero, the shear zone and its velocity ﬁeld changes corresponding to that of the circular to log spiral shear zone. The proposed shear zone is the generalized one in which the value of u varies from 0 to f ; thus, it has an advantage of analyzing the range of possible critical rupture surfaces for ﬁnding the critical collapse load. Seismic Active Earth Force Maximum Seismic-Induced Active Force After application of the upper bound theorem of limit analysis based on the virtual energy-work balance expression presented in Eq. (1), the resultant seismic active force Pa for an inclined rough wall in the proposed rigid translational mechanism, satisfying the kinematical admissibility conditions, can be presented in a simpliﬁed form as Pa ðu ; m ; ɛÞ ¼ (10) v_ 1 þ v_ 2 þ v_ 3 þ v_ q D_ 1 D_ 2 D_ 3 chV0 U1 ðu ; m Þ V0 U2 ðu ; m Þ (15) in which If the interior angles of blocks Dɛ are taken as this small, the integration of Eq. (10) within the limits 0 to « will provide the dissi_ sides along the interfaces of the blocks that form vepation energy D locity discontinuity surfaces within the radial shearing zone as follows: ɛx 0 _ sides ¼ cr0 V0 cos f e 1 D (11) cosð f þ u Þ x0 where x 0 ¼ tan ð f u Þ tan ð f þ u Þ. Again, the energy dissipation that takes place along the bases of triangular blocks, that is, along the discontinuous rigid boundary surface of the radial shear zone, can be expressed as ð _ base ¼ c Ln1 Vn1 cos f D K © ASCE (12) tan d U1 ð u ; m Þ ¼ tan f sinða þ m u Þ tan a cos a cosð m u Þ and U2 ð u ; m Þ ¼ sinða þ m u Þ sind þ cosða þ d Þ cosð m u Þ The terms v_ 1 , v_ 2 , and v_ 3 refer to the rate of work done by the horizontal and vertical body forces in the regions OAB, OBC, and OCD. v_ q is the work rate of the horizontal and vertical tractions on _ 2 , and D_ 3 represent the the loaded boundary surface ED, and D_ 1 , D rate of energy dissipation along the discontinuity surfaces AB, DC, and in the continually shearing zone OBC. Referring to Fig. 2, the total work done v_ 2 by the horizontal and vertical body forces in the radial shearing OBC can be given by the following expression: 04017123-5 Int. J. Geomech., 2018, 18(1): 04017123 Int. J. Geomech. v_ 2 ¼ 8 >ðɛ g< 2> : Kaq g s kh r2x V x cosða þ m þ x u Þd x þ r2x V x sinða þ m þ x u Þd x 9 = ; (16) Kaq g cs ¼ 0 Downloaded from ascelibrary.org by University Of Florida on 10/25/17. Copyright ASCE. For personal use only; all rights reserved. The total work done v_ q by the horizontal and vertical tractions on the loaded boundary surface ED can be presented as v_ q ¼ qL0 V2 ½kh cos ða þ m þ ɛ u Þ þ sin ða þ m þ ɛ u Þ (17) where L0 is the effective length of the surcharge in the critical collapse mechanism, which can be expressed as Eqs. (18a) and (18b) h cosð m þ f u Þ λ 0 ɛ tanð w u Þ e L ¼ cos a sinða þ m þ ɛ þ f b u Þ cos b (18a) cos b cosð m þ f u Þ eɛ tanð f u Þ λ sinða þ m þ ɛ þ f b u Þ (18b) 1 U2 ð u ; m Þ ( ) 3 8 X X Cn ðu ; m ;ɛÞ þ Nq C4 ðu ; m ;ɛÞ Nc Cn ðu ; m ;ɛÞ n¼1 Pa ðu ; m ; ɛÞ ¼ g h2 Ka g s þ qhKaqs chKacs 2 (19) g h2 Kaq g s chKacs 2 (20) g h2 Kaq g cs 2 (21) Pa ðu ; m ; ɛÞ ¼ Pa ðu ; m ; ɛÞ ¼ where Ka g s , Kaqs , and Kacs = individual seismic active earth thrust coefﬁcients due to the components of soil unit weight, surcharge, and soil cohesion, respectively; Kaqg s = combined seismic active earth thrust coefﬁcient, which describes the effect of both the soil unit weight and surcharge components in terms of a nondimensional factor Nq (2q/ g h); and Kaqg cs = combined seismic active earth thrust coefﬁcient, which describes the effect of all components in terms of nondimensional factors Nq (2q/ g h) and Nc (2c/ g h). The expressions for ﬁnding coefﬁcients Ka g s , Kaqs , and Kacs will be written as follows in Eqs. (22a)–(22c): Ka g s ¼ 3 X 1 Cn ðu ; m ; ɛÞ U2 ðu ; m Þ n¼1 (22a) C4 ðu ; m ; ɛÞ U2 ð u ; m Þ (22b) 8 X 1 Cn ðu ; m ; ɛÞ U2 ðu ; m Þ n¼5 (22c) Kaqs ¼ Kacs ¼ Similarly, the following expressions can be deduced for ﬁnding the combined seismic active earth force coefﬁcients n¼5 (24) The earth thrust coefﬁcients presented in Eqs. (19)–(24) correspond to the seismic condition. By simply putting kh = 0 into Eqs. (15)–(17), the expressions given in Eqs. (15)–(24) will be applicable for static condition. The notations used for seismic earth thrust coefﬁcients Ka g s , Kaqs , Kacs , and Kaq g cs will be replaced with Ka g , Kaq , Kac , Kaq g , and Kaq g c , respectively, for denoting earth thrust coefﬁcients under static conditions. The value of Cn ðu ; m ; ɛÞ for n = 1–8 has been presented in Appendix I. The critical resultant active force can be found by maximizing Pa given in Eqs. (19)–(21) with respect to the variables u , m , and « , which are used to deﬁne the shape of the chosen composite collapse mechanism. Thus, the maximum magnitude of the resultant active force can be determined with the help of the following equality conditions in Eqs. (25a)–(25c): The resultant seismic active force Pa given in Eq. (15) can be expressed in the following forms: © ASCE (23) 0 ðɛ when ( ) 3 X 1 ¼ Cn ðu ; m ; ɛÞ þ Nq C4 ðu ; m ; ɛÞ U2 ðu ; m Þ n¼1 ∂Pa ðu ; m ; ɛÞ ¼0 ∂u (25a) ∂Pa ðu ; m ; ɛÞ ¼0 ∂m (25b) ∂Pa ðu ; m ; ɛÞ ¼0 ∂ɛ (25c) Application of the superposition principle enables the computation of the thrust coefﬁcients deﬁned in Eqs. (19)–(20), in which the thrust coefﬁcient corresponding to one component can be determined to be independent of other component(s). Using the principle of superposition, solutions have been obtained for earth pressure problems in static conditions (Chen and Rosenfarb 1973; Soubra and Macuh 2002) and in seismic conditions (Saran and Prakash 1968; Chen and Liu 1990; Das and Puri 1996). However, it was found that for the same properties of backﬁll and the same magnitudes of seismic acceleration coefﬁcients, the total active force obtained by Shukla et al. (2009) due to cohesion and self-weight of backﬁll differs largely from that computed by Saran and Prakash (1968), although the method of analysis is the same in both studies. In the analysis by Saran and Prakash (1968), the failure surfaces from the effect of soil weight and cohesion were optimized separately, which is of course too far from reality; whereas the total active force was obtained due to the combined contribution of soil weight and cohesion by optimizing a single failure plane (Shukla et al. (2009). In the present analysis, the difference in the solutions obtained with and without the superposition principle depend on the magnitudes of kh, Nq, Nc, l , and f , which is discussed later. Theoretically, the deﬁnition of the thrust coefﬁcient given in Eq. (21) is exact when the factors g , q, and c are all greater than zero. This is because of the realization of the unique failure surface from a mathematical standpoint. However, such a deﬁnition requires two additional unbounded factors, such as Nq and Nc, for presenting the numerical solutions. Thus, in the present study, the computations were performed with and without the superposition principle, 04017123-6 Int. J. Geomech., 2018, 18(1): 04017123 Int. J. Geomech. Downloaded from ascelibrary.org by University Of Florida on 10/25/17. Copyright ASCE. For personal use only; all rights reserved. depending on the dependence of the thrust coefﬁcients on factors Nq and Nc. Consideration of the Influence of the Preexisting Crack in the Backfill The previously developed formulation has been extended to consider that the possible effect of tension crack in the cohesivefrictional backﬁll on the magnitude of active earth force is essential for both static and seismic conditions. It is also known that the region up to the crack depth in the backﬁll does not permit any development of soil resistance against the failure of soil mass under the active state behind a wall. Therefore, the soil mass in the cracked region should be treated as an additional surcharge that acts on the remaining part of backﬁll lying below the cracked zone while evaluating the total active force. The maximum expected depth of crack (zc) to be developed vertically under soil tension may be readily taken into account if available or computed alternatively using the classical Rankine earth pressure theory 2c p f zc ¼ tan þ (26) g 4 2 It can be noted that the analytical expression provided by Mazindrani and Ganjali (1997) for the depth of crack is the same as that given in Eq. (26), which is independent of backﬁll slope angle. This is contradictory to the ﬁnding of Nian and Han (2013), in which the depth of crack is a function of backﬁll slope angle. Nevertheless, the present formulation addresses only the effect of preexisting dry cracks on the seismic active earth thrust; therefore, the depth of crack given in Eq. (26) is expressed independent of the seismic earth thrust coefﬁcient, backﬁll slope angle, and orientation of wall. The use of this approximation for accounting the effects of a preexisting crack in the formulation of seismic active earth thrust has also been reported by Das and Puri (1996). Further, the present formulation also can be used explicitly if the expected depth of crack is known. Accordingly, the seismic active earth thrust coefﬁcient presented in Eq. (24) will be rewritten considering the inﬂuence of preexisting crack as 9 8 3 > > X > > > > 2 0 > > ð j Þ C u ; m ; ɛ þ N C u ; m ; ɛ 1 ð Þ ð Þ > > n q 4 = < 1 0 n¼1 Kaq g cs ¼ 8 X > U2 ð u ; m Þ > > > > > ð1 j ÞNc Cn ðu ; m ; ɛÞ > > > > ; : n¼5 (27) and from Eq. (21) Pa ð u ; m ; ɛ Þ ¼ g h2 0 K 2 aq g cs (28) where j (zc/h) = normalized depth of tension crack whose value ranges from 0 to 1. The value of C04 ðu ; m ; ɛÞ is given in Appendix. I. Eq. (28) is therefore a generalized expression that can be used to ﬁnd the resultant static and seismic active earth force exerted by cohesive-frictional or purely frictional soil as backﬁll supported by a wall while considering the inﬂuence of surcharge loadings and tension crack in the backﬁll. Solutions and Discussions The active earth thrust coefﬁcients deﬁned in Eqs. (22)–(27), which are used for determining the resultant active earth thrust, have been computed in this paper separately by using computer code in © ASCE MATLAB. The numerical solutions for active earth thrust coefﬁcients obtained from the present analysis under both static and seismic conditions have been presented for two situations of surcharge on the backﬁll: (1) uniform surcharge applied throughout the backﬁll (l = 0) and (2) uniform surcharge applied at a certain distance away from the backﬁll (l > 0). Uniform Surcharge Applied throughout the Backfill (k = 0) Static Condition. The variation of static active earth thrust coefﬁcients Ka g and Kaq , and Kac with b / f for different combination of parameters, such as d /f , a, and f , are shown in Figs. 4 and 5. Fig. 4 shows that for any given value of f , the magnitude of both coefﬁcients Ka g and Kaq increases continuously with an increase in the values of b / f from negative to positive, and attains its maximum values corresponding to the limiting value of b / f equal to 1 beyond which the sloping backﬁll of purely frictional material can no longer be in stable condition. This is one of the consequences of the application of the superposition rule in obtaining solutions that are applicable for cohesive-frictional backﬁll in which the sloping backﬁll can be stable by itself, even for b / f > 1. The computed magnitude of coefﬁcients Ka g and Kaq shows an increasing trend when the value of a changes from negative to positive. This increase in the magnitude of thrust coefﬁcients due to the change in orientation of the wall from negative to positive is attributed to the involvement of a large plastic region of soil mass behind the wall at the time of collapse under the active state. There is a marginal decrease or increase in the magnitude of coefﬁcients Ka g and Kaq with respect to change in the values of the soil-wall interface friction depending on the parameters f , b / f , and a. Further, a backﬁll with higher internal friction angle reduces signiﬁcantly the magnitude of coefﬁcients Ka g and Kaq . It is conﬁrmed that the magnitude of coefﬁcient Kac can be related to Kaq for the case in which b = a = 0 by using the theorem of Caquot and Kerisel (1948), and it is expressed in the following: 1 Kaq Kac ¼ cos d tan f (29) The parameters u , m , and « , which deﬁne the critical collapse geometry for ﬁnding coefﬁcients Kac and Kaq , remain exactly the same, implying that the solutions are free from errors resulting from the superposition rule. This is realized mainly because of the adoption of the magnitude c(tan d /tan f ) as the soil-wall interface adhesion per unit length of the wall. For given values of d / f , a, and f , the magnitude of coefﬁcient Kac is found to increase when the value of b / f changes from its negative to positive values. Signiﬁcant reduction in the values of Kac occurs when there is an increase in the values of f and a decrease in the value of d . The changes in wall orientation from positive to negative values have been found to reduce the magnitudes of Kac in almost all cases. Seismic Condition. Computations revealed that the magnitude of Kacs is independent of earthquake acceleration coefﬁcient kh. The seismic independency of Kacs can also be easily conﬁrmed by looking into Eq. (22c); hence, it can be said that Kacs ¼ Kac . This seismic independency of coefﬁcient Kacs arises mainly due to the application of the superposition rule in obtaining solutions for soils with g = 0. Also, the use of the coefﬁcients Kacs for designing retaining structures with cohesive-frictional backﬁll in seismic conditions may become uneconomical unless the error between the values computed and the true values obtained including the weight of backﬁll mass becomes marginal. Despite this fact, only the variation of Ka g s and Kaqs with kh for different combinations of parameters, 04017123-7 Int. J. Geomech., 2018, 18(1): 04017123 Int. J. Geomech. 1.4 1.4 α = +20° = +10° = 0° = -10° = -20° = +10° = 0° = -10° = -20° 1 = +10° = 0° = -10° = -20° Kaγ , Kaq Kaγ , Kaq Kaγ , Kaq 0.6 Kaγ Kaq Ka γ Kaq 0.2 -0.5 0 0.5 1 -0.5 β/φ 0 0.5 0.2 1 -0.5 = +10° = 0° = -10° = -20° α = +20° φ = 30°, δ/φ = 0.5 = +10° = 0° = -10° = -20° 0.9 Kaγ , Kaq Ka γ Kaq 0.5 0.5 -0.5 0 0.5 0.1 -0.5 1 0 0.5 1 -0.5 1.2 1.2 α = +20° φ = 40°, δ/φ = 0 = +10° = 0° = -10° = -20° 0.8 α = +20° φ = 40°, δ/φ = 0.5 = +10° = 0° = -10° = -20° Kaγ , Kaq Kaγ , Kaq Kaγ , Kaq 0.4 Ka γ Kaq 0 0 -0.5 0 0.5 1 -0.5 β/φ Ka γ Kaq 0.4 0.4 0 φ = 40°, δ/φ = 1 0.8 0.8 Ka γ Kaq 1 (f) 1.2 = +10° = 0° = -10° = -20° 0.5 β/φ (e) α = +20° 0 β/φ β/φ (d) Ka γ Kaq 0.5 0.1 0.1 φ = 30°, δ/φ = 1 0.9 Kaγ , Kaq Kaγ , Kaq Ka γ Kaq 1 1.3 α = +20° φ = 30°, δ/φ = 0 0.9 0.5 (c) 1.3 1.3 = +10° = 0° = -10° = -20° 0 β/φ (b) α = +20° Kaγ Kaq β /φ (a) φ = 20°, δ/φ = 1 1 0.6 0.2 (g) α = +20° φ = 20°, δ/φ = 0.5 1 0.6 Downloaded from ascelibrary.org by University Of Florida on 10/25/17. Copyright ASCE. For personal use only; all rights reserved. 1.4 α = +20° φ = 20°, δ/φ = 0 0 0.5 -0.5 1 (h) 0 0.5 1 β/φ β/φ (i) Fig. 4. Variation of Kag and Kaq with b / f and a for (a) f = 20° and d / f = 0, (b) f = 20° and d / f = 0.5, (c) f = 20° and d / f = 1, (d) f = 30° and d / f = 0, (e) f = 30° and d / f = 0.5, (f) f = 30° and d / f = 1, (g) f = 40° and d / f = 0, (h) f = 40° and d / f = 0.5, and (i) f = 40° and d / f = 1 © ASCE 04017123-8 Int. J. Geomech., 2018, 18(1): 04017123 Int. J. Geomech. 2.2 φ = 20°, δ/φ = 0 1.7 Kac Kac 1.4 α = +20° 0.5 1 β /φ (a) 1.2 -0.5 0 1.7 φ = 30°, δ/φ = 0 0.5 Kac Kac Kac 1.3 α = +20° 0.5 1 β /φ (d) 0.9 -0.5 0 1.5 1.2 0.5 -0.5 1 β /φ (e) φ = 40°, δ/φ = 0 Kac Kac Kac 1.1 α = +20° 0.5 α = +20° = +10° = 0° = -10° = -20° 0.7 0.6 0 φ = 40°, δ/φ = 1 = +10° = 0° = -10° = -20° 0.6 1 -0.5 β /φ 1 1.5 = +10° = 0° = -10° = -20° -0.5 0.5 β /φ 1.9 φ = 40°, δ/φ = 0.5 0.9 α = +20° 0 (f) 1.2 0.9 α = +20° = +10° = 0° = -10° = -20° 0.8 0 φ = 30°, δ/φ = 1 = +10° = 0° = -10° = -20° 0.8 1 1.7 = +10° = 0° = -10° = -20° -0.5 0.5 β /φ 2.1 φ = 30°, δ/φ = 0.5 1.1 α = +20° 0 (c) 1.4 1.1 1.5 -0.5 1 β /φ (b) 1.4 (g) = +10° = 0° = -10° = -20° 1 0 α = +20° = +10° = 0° = -10° = -20° 0.9 1.7 1.6 α = +20° = +10° = 0° = -10° = -20° -0.5 φ = 20°, δ/φ = 1 2 1.8 1.3 Downloaded from ascelibrary.org by University Of Florida on 10/25/17. Copyright ASCE. For personal use only; all rights reserved. 2.4 φ = 20°, δ/φ = 0.5 Kac 2.1 0 0.5 -0.5 1 (h) 0 0.5 1 β /φ β /φ (i) Fig. 5. Variation of Kac with b / f and a for (a) f = 20° and d / f = 0, (b) f = 20° and d / f = 0.5, (c) f = 20° and d / f = 1, (d) f = 30° and d / f = 0, (e) f = 30° and d / f = 0.5, (f) f = 30° and d / f = 1, (g) f = 40° and d / f = 0, (h) f = 40° and d / f = 0.5, and (i) f = 40° and d / f = 1 © ASCE 04017123-9 Int. J. Geomech., 2018, 18(1): 04017123 Int. J. Geomech. such as d / f , b / f , a, and f , are shown in Figs. 6–8. According to the concept of shear ﬂuidization (Richards et al. 1990) and stability criteria of slopes (Sarma 1999), the maximum value of b / f is restricted in this paper to obtaining the results satisfying the following expression: Downloaded from ascelibrary.org by University Of Florida on 10/25/17. Copyright ASCE. For personal use only; all rights reserved. jb j tan1 ðkh Þ 1 f f (30) Figs. 6–8 show that the magnitude of both the coefﬁcients, Ka g s and Kaqs , has been found to increase always with an increase in the magnitude of kh irrespective of the magnitudes of d / f , b / f , a, and f . This shows that the assumed direction for earthquake acceleration is critical; thus, the larger magnitudes of Kaqs are obtained when the critical direction of kh is considered to act along the direction of the wall movement. Any increase in the values of f reduces the magnitudes of coefﬁcients Ka g s and Kaqs . On the other hand, the changes in the values of d / f may either increase or decrease the magnitudes of Ka g s and Kaqs depending on the parameters b / f , a, and f . Nevertheless, this increase or decrease in the magnitudes of Ka g s and Kaqs with respect to changes in d / f is only marginal for negatively sloped backﬁll surfaces compared with that of the horizontal and positively sloped backﬁll surfaces. Further, some peculiar observations between the magnitudes of Ka g s and Kaqs with changes in the values of wall orientation and backﬁll slope angles are noticed. The magnitude of Kaqs in the case of negatively sloped backﬁll surface is found to be (1) always greater than or equal to Ka g s for positively inclined walls and (2) lower or higher than the magnitude of Ka g s for negatively inclined walls depending on the values of kh, d / f , and f .On the other hand, in the case of positively sloped backﬁll surface, the magnitude of Kaqs is found to be (1) always less than or equal to Ka g s for positively inclined walls and (2) always slightly higher than the value of Ka g s for negatively inclined walls irrespective of the value of kh. A similar kind of trend has also been observed for all the values of d / f and f that have been taken into consideration in this analysis. Uniform Surcharge Applied at a Certain Distance Away from the Backfill (k > 0) The usage of the superposition rule leads to the conclusion that the coefﬁcient Kac remains unaffected with changes in the magnitude of l . This can also be conﬁrmed from the expressions presented for Kac , as given in Eq. (22c). In fact, this observation makes no sense for retaining walls with a general cohesive-frictional backﬁll because this type of presentation inherently neglects the effects of the combined interaction that exists among the various factors. In such situations, the accuracy of the solution, therefore, largely depends on the absolute magnitude of factors, such as Nq and Nc. Nevertheless, in this section with the assumption of Nc = 0, the solutions have been presented. Unlike in the case l = 0, the total active earth resistance cannot be derived from the independent contribution of the surcharge and soil weight components based on the principle of superposition. Such an application of this principle in obtaining solutions for l > 0 does not guarantee accuracy unless the difference between the solutions obtained with and without the superposition rule are negligible. Hence, the following sections present the numerical solutions for static and seismic active earth thrust coefﬁcients based on the deﬁnition introduced in Eq. (23). The necessity of this kind of deﬁnition for thrust coefﬁcients is explained brieﬂy later. Static Condition. The variations of Kaq g with l for different combinations of Nq (2q/ g h), b / f , d / f , a, and f are presented in Figs. 9–11. The magnitudes of Kaq g decrease continuously with an © ASCE increase in the value of l up to a certain critical value, which is referred to as the critical surcharge distance ratio l cr, beyond which the magnitudes of Kaqg become exactly equal to that of Ka g , indicating that the resultant active force is independent of the magnitudes of Nq. However, the value of critical surcharge distance ratio l cr is found to vary largely depending on the magnitude of Nq for any given combination of parameters, such as b / f , d / f , a, and f . For a given orientation of wall, higher magnitudes of Kaq g are always realized with a positively sloped backﬁll surface compared with that of the horizontal backﬁll. A considerable decrease or increase in the magnitudes of Kaq g is found depending on the values of Nq and f . Generally, the value of l cr has been found to increase with an increase in Nq, whereas the same decreases with an increase in f . Further, it has been found that the value of l cr for retaining walls (1) with negative inclination is always lower and (2) with positive inclination is higher compared with that of vertical walls. For any given orientation of wall and backﬁll friction angle, the changes in the magnitude of l cr may also become signiﬁcant when the backﬁll slope angles change from positive to negative values. Generally, the magnitude of l cr is found to reduce when the backﬁll slope angles change from positive to negative values. Seismic Condition. In Figs. 12–14, the variations of Kaq g s with kh and l for d / f = 2/3 with different combinations of Nq, f , b / f , and a have been presented. Similar to the static case, a signiﬁcant reduction in the magnitudes of Kaq g s has been observed with an increase in the value of l up to a certain critical surcharge distance ratio l cr for any values of kh. Under any given set of parameters, the minimum values of Kaqg s become exactly equal to that of Ka g s . This shows that for values of l equal to or greater than l cr, the magnitudes of Kaq g s are independent of surcharge pressure. Any increase in the values of kh always results in (1) an increase in the magnitude of Kaq g s and (2) a reduction in the magnitude of l cr. The values of critical surcharge distance ratio l cr are found to be highly dependent on the magnitudes of kh, Nq, f , b / f , and a. Generally, the magnitude of l cr decreases when the backﬁll sloping surface and orientation of the wall changes from its positive to negative values. A similar trend can also be noticed with an increase in the values of f . Influence of Tension Cracks in the Backfill For b / f = 0.5, d / f = 0.5, a = 0°, and Nc = 0.1, the variations of 0 Kaq g cs with l for different combinations of Nq, f , and kh have been shown in Fig. 15. Two cases have been considered in this paper: (1) backﬁll free from any crack, i.e., j = 0 and (2) backﬁll possessing a crack, i.e., j > 0 in which the value of the crack depth is calculated based on Eq. (26). The consideration of the preexisting crack in the formulation may not always be a critical scenario, 0 that is, the magnitudes of Kaq g cs computed for j > 0 may be either less than or greater than that computed without considering crack in the backﬁll depending on the magnitudes of Nc and Nq. However, a clear-cut distinction between the values of critical surcharge distance ratio l cr computed for the two cases has been noted. The deﬁnition of l cr carries the same meaning that was 0 explained in previous sections, that is, the magnitudes of Kaq g cs corresponding to l cr will be free from the inﬂuence of the magnitude of backﬁll surcharge. For given values of kh, Nq, f , b / f , and a, the values of l cr always have been found to reduce in the presence of crack in the backﬁll. This reduction in the values of l cr may be due to the fact that the presence of vertical cracks decreases the effective span of surcharge (L0 ) toward estimating 0 the magnitudes of Kaq g cs from the critical active failure mechanisms formed in the backﬁll. For the same reason, depending on the magnitudes of Nc and Nq, the potential driving forces due to external surcharge pressure acting on the cracked zone and its 04017123-10 Int. J. Geomech., 2018, 18(1): 04017123 Int. J. Geomech. 1 1 φ = 20°, δ/φ = 0, β/φ = -0.15 α = +20° α = +20° = +10° = 0° = -10° = -20° = +10° = 0° = -10° = -20° = +10° = 0° = -10° = -20° Kaγs, Kaqs 0.7 0.6 Ka γ s Kaqs 0.4 0.4 0.4 0.1 kh 0.2 0.3 (a) 0 0.1 kh 0.2 1.4 φ = 20°, δ/φ = 0.5, β/φ = 0 α = +20° α = +20° = +10° = 0° = -10° = -20° = +10° = 0° = -10° = -20° = +10° = 0° = -10° = -20° 0.9 1 0.6 Ka γ s Kaqs 0.2 Ka γ s Kaqs 0.3 0.1 kh 0.2 0.3 (d) 0.2 0 0.1 kh 0.2 0.3 (e) 0.9 1.2 φ = 20°, δ/φ = 1, β/φ = 0 α = +20° α = +20° = +10° = 0° = -10° = -20° = +10° = 0° = -10° = -20° = +10° = 0° = -10° = -20° 0.9 Kaγs, Kaqs 0.9 0.6 Ka γ s Kaqs 0.3 0.2 0.3 Ka γ s Kaqs 0.3 kh 0.2 0.6 Ka γ s Kaqs 0.1 kh φ = 20°, δ/φ = 1, β/φ = +0.15 α = +20° 0.6 0 0.1 (f) 1.2 φ = 20°, δ/φ = 1, β/φ = -0.15 0 Kaγs, Kaqs 1.2 0.3 0.6 Ka γ s Kaqs 0 0.2 Kaγs, Kaqs Kaγs, Kaqs 0.5 kh φ = 20°, δ/φ = 0.5, β/φ = +0.15 α = +20° Kaγs, Kaqs 0.8 0.1 (c) 1.2 φ = 20°, δ/φ = 0.5, β/φ = -0.15 0 0.3 (b) 1.1 (g) Kaγs Kaqs Kaγs Kaqs Kaγs, Kaqs Downloaded from ascelibrary.org by University Of Florida on 10/25/17. Copyright ASCE. For personal use only; all rights reserved. 1 Kaγs, Kaqs 0.8 0.6 0 φ = 20°, δ/φ = 0, β/φ = +0.15 α = +20° Kaγs, Kaqs 0.8 1.3 φ = 20°, δ/φ = 0, β/φ = 0 0.3 0.3 0 0.1 kh 0.2 (h) 0.3 0 0.1 kh 0.2 0.3 (i) Fig. 6. Variation of Kag s and Kaqs with kh and a for f = 20° with (a) d / f = 0 and b / f = –0.15, (b) d / f = 0 and b / f = 0, (c) d / f = 0 and b / f = þ0.15, (d) d / f = 0.5 and b / f = –0.15, (e) d / f = 0.5 and b / f = 0, (f) d / f = 0.5 and b / f = þ0.15, (g) d / f = 1 and b / f = –0.15, (h) d / f = 1 and b / f = 0, and (i) d / f = 1 and b / f = þ0.15 © ASCE 04017123-11 Int. J. Geomech., 2018, 18(1): 04017123 Int. J. Geomech. 0.8 0.8 φ = 30°, δ/φ = 0, β/φ = -0.4 = +10° = 0° = -10° = -20° Kaγs, Kaqs Kaγs, Kaqs Kaγs, Kaqs 0.5 Ka γ s Kaqs Ka γ s Kaqs Ka γ s Kaqs 0.2 0.2 0.1 kh 0.2 0.2 0 0.3 (a) 0.1 kh 0.2 0.3 (b) 0.7 = +10° = 0° = -10° = -20° 0.8 0.4 1.4 φ = 30°, δ/φ = 0.5, β/φ = 0 Ka γ s Kaqs 0.6 0.2 0.3 α = +20° = +10° = 0° = -10° = -20° 1 Kaγs, Kaqs Kaγs, Kaqs kh φ = 30°, δ/φ = 0.5, β/φ = +0.4 = +10° = 0° = -10° = -20° 0.4 0.3 0.1 α = +20° Kaγs, Kaqs 0.5 0 (c) 0.8 φ = 30°, δ/φ = 0.5, β/φ = -0.4 0.6 α = +20° = +10° = 0° = -10° = -20° 0.1 0 0.1 kh 0.2 Kaγs Kaqs (d) 0.2 0 0.1 kh 0.2 0.3 (e) 1 1.7 φ = 30°, δ/φ = 1, β/φ = 0 α = +20° α = +20° = +10° = 0° = -10° = -20° = +10° = 0° = -10° = -20° = +10° = 0° = -10° = -20° 1.2 Kaγs, Kaqs 0.7 0.4 Ka γ s Kaqs 0.1 0.1 kh 0.2 0.2 0.3 0.3 0.7 Ka γ s Kaqs 0.1 kh φ = 30°, δ/φ = 1, β/φ = +0.4 α = +20° 0.4 0 0.1 (f) φ = 30°, δ/φ = 1, β/φ = -0.4 0.7 0 Kaγs, Kaqs 1 (g) Ka γ s Kaqs 0.2 0.3 Kaγs, Kaqs Downloaded from ascelibrary.org by University Of Florida on 10/25/17. Copyright ASCE. For personal use only; all rights reserved. α = +20° = +10° = 0° = -10° = -20° 0.6 0.4 0 φ = 30°, δ/φ = 0, β/φ = +0.4 α = +20° α = +20° 0.6 1.1 φ = 30°, δ/φ = 0, β/φ = 0 Kaγs Kaqs 0.2 0 0.1 kh 0.2 (h) 0.3 0 0.1 kh 0.2 0.3 (i) Fig. 7. Variation of Kag s and Kaqs with kh and a for f = 30° with (a) d / f = 0 and b / f = –0.4, (b) d / f = 0 and b / f = 0, (c) d / f = 0 and b / f = þ0.4, (d) d / f = 0.5 and b / f = –0.4, (e) d / f = 0.5 and b / f = 0, (f) d / f = 0.5 and b / f = þ0.4, (g) d / f = 1 and b / f = –0.4, (h) d / f = 1 and b / f = 0, and (i) d / f = 1 and b / f = þ0.4 © ASCE 04017123-12 Int. J. Geomech., 2018, 18(1): 04017123 Int. J. Geomech. 0.7 φ = 40°, δ/φ = 0, β/φ = -0.55 = +10° = 0° = -10° = -20° = +10° = 0° = -10° = -20° 0.8 Kaγs, Kaqs 0.3 0.4 0.3 Ka γ s Kaqs Kaγs Kaqs 0.1 0 0.1 0 0.1 kh 0.2 0.3 (a) 0 0.1 kh 0.2 α = +20° α = +20° = +10° = 0° = -10° = -20° = +10° = 0° = -10° = -20° 0.5 1 Kaγs, Kaqs Ka γ s Kaqs Ka γ s Kaqs 0.1 0 0.1 kh 0.2 0.3 (d) 0 0.1 kh 0.2 2.1 φ = 40°, δ/φ = 1, β/φ = 0 α = +20° α = +20° = +10° = 0° = -10° = -20° = +10° = 0° = -10° = -20° 0.6 1.4 Kaγs Kaqs 0 0 kh Kaγs Kaqs 0 0 0.3 0.3 0.7 Kaγs Kaqs 0.2 0.2 Kaγs, Kaqs 0.3 0.1 kh φ = 40°, δ/φ = 1, β/φ = +0.55 = +10° = 0° = -10° = -20° 0.3 0 0.1 α = +20° Kaγs, Kaqs 0.6 0 (f) 0.9 φ = 40°, δ/φ = 1, β/φ = -0.55 Ka γ s Kaqs 0 0.3 (e) 0.9 0.3 0.5 0.3 0 0.2 φ = 40°, δ/φ = 0.5, β/φ = +0.55 = +10° = 0° = -10° = -20° 0.23 (g) 1.5 φ = 40°, δ/φ = 0.5, β/φ = 0 α = +20° Kaγs, Kaqs 0.46 kh (c) 0.7 φ = 40°, δ/φ = 0.5, β/φ = -0.55 0.1 Kaγs, Kaqs 0.69 0 0.3 (b) Kaγs, Kaqs Downloaded from ascelibrary.org by University Of Florida on 10/25/17. Copyright ASCE. For personal use only; all rights reserved. α = +20° = +10° = 0° = -10° = -20° 0.5 Kaγs, Kaqs 0.5 φ = 40°, δ/φ = 0, β/φ = +0.55 α = +20° Ka γ s Kaqs α = +20° 1.2 φ = 40°, δ/φ = 0, β/φ = 0 Kaγs, Kaqs 0.7 0.1 kh 0.2 (h) 0.3 0 0.1 kh 0.2 0.3 (i) Fig. 8. Variation of Kag s and Kaqs with kh and a for f = 40° with (a) d / f = 0 and b / f = –0.55, (b) d / f = 0 and b / f = 0, (c) d / f = 0 and b / f = þ0.55, (d) d / f = 0.5 and b / f = –0.55, (e) d / f = 0.5 and b / f = 0, (f) d / f = 0.5 and b / f = þ0.55, (g) d / f = 1 and b / f = –0.55, (h) d / f = 1 and b / f = 0, and (i) d / f = 1 and b / f = þ0.55 © ASCE 04017123-13 Int. J. Geomech., 2018, 18(1): 04017123 Int. J. Geomech. 1.2 β/φ = +0.5 β /φ = 0 β/φ = -0.5 Kaqγ Kaqγ φ = 20°, δ/φ = 0, α = -20° 0.4 0.8 (a) λ 1.2 0.8 φ = 20°, δ/φ = 0, α = 0° 0.3 1.6 2 0 0.4 0.8 (b) λ 1.2 0.4 0.8 λ 1.2 1.6 0.9 1.6 2 β/φ = +0.5 β /φ = 0 β/φ = -0.5 Nq = 1.0 = 0.8 = 0.6 = 0.4 = 0.2 = 0.0 Kaqγ Kaqγ 0 β /φ = 0 β/φ = -0.5 Nq = 1.0 = 0.8 = 0.6 = 0.4 = 0.2 = 0.0 0.6 2 β/φ = +0.5 β /φ = 0 β/φ = -0.5 0.8 1.6 (c) 1.2 β/φ = +0.5 φ = 20°, δ/φ = 0, α = +20° 0.4 Nq = 1.0 = 0.8 = 0.6 = 0.4 = 0.2 = 0.0 1.2 Kaqγ 0 Nq = 1.0 = 0.8 = 0.6 = 0.4 = 0.2 = 0.0 1.2 0.6 0.8 0.6 0.4 φ = 20°, δ/φ = 0.5, α = -20° 0 0.4 0.8 (d) λ 1.2 φ = 20°, δ/φ = 0.5, α = 0° 0.3 1.6 2 0 0.4 0.8 (e) λ 1.2 0.4 0.8 λ 1.2 1.6 Kaqγ 2 β /φ = 0 β/φ = -0.5 Nq = 1.0 = 0.8 = 0.6 = 0.4 = 0.2 = 0.0 0.9 1.6 β/φ = +0.5 β /φ = 0 β/φ = -0.5 Nq = 1.0 = 0.8 = 0.6 = 0.4 = 0.2 = 0.0 0.6 0 2 β/φ = +0.5 β /φ = 0 β/φ = -0.5 0.8 1.6 (f) 1.2 β/φ = +0.5 φ = 20°, δ/φ = 0.5, α = +20° 0.4 Nq = 1.0 = 0.8 = 0.6 = 0.4 = 0.2 = 0.0 1.2 Kaqγ 0.2 Kaqγ Downloaded from ascelibrary.org by University Of Florida on 10/25/17. Copyright ASCE. For personal use only; all rights reserved. Nq = 1.0 = 0.8 = 0.6 = 0.4 = 0.2 = 0.0 0.9 0.5 0.3 β/φ = +0.5 β /φ = 0 β/φ = -0.5 β/φ = 0 β/φ = -0.5 Nq = 1.0 = 0.8 = 0.6 = 0.4 = 0.2 = 0.0 0.7 1.6 β/φ = +0.5 Kaqγ 0.9 0.6 0.8 0.4 φ = 20°, δ/φ = 1, α = -20° 0.2 0 (g) 0.4 0.8 λ 1.2 φ = 20°, δ/φ = 1, α = 0° 0.3 1.6 0 2 (h) 0.4 0.8 λ 1.2 φ = 20°, δ/φ = 1, α = +20° 0.4 1.6 2 0 (i) 0.5 1 λ 1.5 2 2.5 Fig. 9. Variation of Kaqg with l , Nq, and b / f for f = 20°: (a) d / f = 0 and a = –20°; (b) d / f = 0 and a = 0°; (c) d / f = 0 and a = þ20°; (d) d / f = 0.5 and a = –20°; (e) d / f = 0.5 and a = 0°; (f) d / f = 0.5 and a = þ20°; (g) d / f = 1 and a = –20°; (h) d / f = 1 and a = 0°; (i) d / f = 1 and a = þ20° © ASCE 04017123-14 Int. J. Geomech., 2018, 18(1): 04017123 Int. J. Geomech. β /φ = 0 β/φ = -0.5 Nq = 1.0 = 0.8 = 0.6 = 0.4 = 0.2 = 0.0 0.6 Kaqγ Kaqγ β/φ = +0.5 β/φ = 0 β/φ = -0.5 β /φ = 0 β/φ = -0.5 0.8 Nq = 1.0 = 0.8 = 0.6 = 0.4 = 0.2 = 0.0 0.5 1.4 β/φ = +0.5 β/φ = +0.5 Nq = 1.0 = 0.8 = 0.6 = 0.4 = 0.2 = 0.0 1 Kaqγ 0.7 0.3 0.6 φ = 30°, δ/φ = 0, α = -20° 0 0.2 0.4 (a) λ 0.7 φ = 30°, δ/φ = 0, α = 0° 0.2 0.6 0.8 0 1 0.5 (b) 1 λ Nq = 1.0 = 0.8 = 0.6 = 0.4 = 0.2 = 0.0 Kaqγ Kaqγ Nq = 1.0 = 0.8 = 0.6 = 0.4 = 0.2 = 0.0 0.6 φ = 30°, δ/φ = 0.5, α = -20° 0.4 (d) λ 0.6 0.8 β /φ = 0 β/φ = -0.5 0.5 (e) 1 λ 1 0 1.5 Nq = 1.0 = 0.8 = 0.6 = 0.4 = 0.2 = 0.0 Kaqγ φ = 30°, δ/φ = 1, α = -20° λ 0.8 1.8 β/φ = +0.5 Nq = 1.0 = 0.8 = 0.6 = 0.4 = 0.2 = 0.0 1.1 0.7 φ = 30°, δ/φ = 1, α = 0° 0.1 1.2 1.2 β/φ = 0 β/φ = -0.5 0.4 0.4 λ 1.5 β/φ = +0.5 0.7 0.2 0 0.6 β /φ = 0 β/φ = -0.5 Nq = 1.0 = 0.8 = 0.6 = 0.4 = 0.2 = 0.0 0 φ = 30°, δ/φ = 0.5, α = +20° 0.2 (f) β /φ = 0 β/φ = -0.5 0.4 Nq = 1.0 = 0.8 = 0.6 = 0.4 = 0.2 = 0.0 1 φ = 30°, δ/φ = 0.5, α = 0° 0 β/φ = +0.5 1.5 0.6 0.2 1.2 1 β/φ = +0.5 Kaqγ 0 λ 1.4 0.4 0.1 0.5 β /φ = 0 β/φ = -0.5 0.3 (g) 0 β/φ = +0.5 β /φ = 0 β/φ = -0.5 0.5 1.5 (c) 0.8 β/φ = +0.5 φ = 30°, δ/φ = 0, α = +20° 0.2 Kaqγ 0.1 Kaqγ Downloaded from ascelibrary.org by University Of Florida on 10/25/17. Copyright ASCE. For personal use only; all rights reserved. 0.4 0 0.5 λ φ = 30°, δ/φ = 1, α = +20° 0.3 1 (h) 1.5 0 0.6 λ 1.2 1.8 (i) Fig. 10. Variation of Kaqg with l , Nq, and b / f for f = 30°: (a) d / f = 0 and a = –20°; (b) d / f = 0 and a = 0°; (c) d / f = 0 and a = þ20°; (d) d / f = 0.5 and a = –20°; (e) d / f = 0.5 and a = 0°; (f) d / f = 0.5 and a = þ20°; (g) d / f = 1 and a = –20°; (h) d / f = 1 and a = 0°; (i) d / f = 1 and a = þ20° © ASCE 04017123-15 Int. J. Geomech., 2018, 18(1): 04017123 Int. J. Geomech. 0.7 β/φ = +0.5 β /φ = 0 β/φ = -0.5 Nq = 1.0 = 0.8 = 0.6 = 0.4 = 0.2 = 0.0 Kaqγ 0.5 Kaqγ 0.2 β/φ = +0.5 β /φ = 0 β/φ = -0.5 β /φ = 0 β/φ = -0.5 Nq = 1.0 = 0.8 = 0.6 = 0.4 = 0.2 = 0.0 0.3 1.1 β/φ = +0.5 Nq = 1.0 = 0.8 = 0.6 = 0.4 = 0.2 = 0.0 0.8 Kaqγ 0.4 0.5 0.3 φ = 40°, δ/φ = 0, α = -20° 0 0.3 (a) λ 0.3 0.6 φ = 40°, δ/φ = 0, α = 0° 0.1 0.9 0 0.4 (b) Kaqγ Kaqγ Nq = 1.0 = 0.8 = 0.6 = 0.4 = 0.2 = 0.0 φ = 40°, δ/φ = 0.5, α = -20° 0.3 0.6 0.4 (e) 0 Nq = 1.0 = 0.8 = 0.6 = 0.4 = 0.2 = 0.0 Kaqγ 0.5 φ = 40°, δ/φ = 1, α = -20° λ 0.6 1 1.4 β/φ = +0.5 β/φ = +0.5 1 φ = 40°, δ/φ = 1, α = 0° 0 (h) Nq = 1.0 = 0.8 = 0.6 = 0.4 = 0.2 = 0.0 0.6 0.1 0.9 1.5 β /φ = 0 β/φ = -0.5 0.3 0.3 λ β /φ = 0 β/φ = -0.5 0.1 0 0.5 (f) 0.7 Nq = 1.0 = 0.8 = 0.6 = 0.4 = 0.2 = 0.0 0 φ = 40°, δ/φ = 0.5, α = +20° 0.2 0.8 λ β /φ = 0 β/φ = -0.5 (g) 0.8 φ = 40°, δ/φ = 0.5, α = 0° 0 0.9 Nq = 1.0 = 0.8 = 0.6 = 0.4 = 0.2 = 0.0 0.5 0.1 β/φ = +0.5 0.2 β/φ = +0.5 Kaqγ (d) λ 1.2 β /φ = 0 β/φ = -0.5 0.3 0.3 0.8 1.1 β/φ = +0.5 0.5 0.1 0 λ β /φ = 0 β/φ = -0.5 Nq = 1.0 = 0.8 = 0.6 = 0.4 = 0.2 = 0.0 0 0.4 (c) β /φ = 0 β/φ = -0.5 0.2 0 0.8 λ 0.7 β/φ = +0.5 φ = 40°, δ/φ = 0, α = +20° 0.2 Kaqγ 0 Kaqγ Downloaded from ascelibrary.org by University Of Florida on 10/25/17. Copyright ASCE. For personal use only; all rights reserved. 0.1 0.4 λ φ = 40°, δ/φ = 1, α = +20° 0.2 0 0.8 (i) 0.5 λ 1 1.5 Fig. 11. Variation of Kaqg with l , Nq, and b / f for f = 40°: (a) d / f = 0 and a = –20°; (b) d / f = 0 and a = 0°; (c) d / f = 0 and a = þ20°; (d) d / f = 0.5 and a = –20°; (e) d / f = 0.5 and a = 0°; (f) d / f = 0.5 and a = þ20°; (g) d / f = 1 and a = –20°; (h) d / f = 1 and a = 0°; (i) d / f = 1 and a = þ20° © ASCE 04017123-16 Int. J. Geomech., 2018, 18(1): 04017123 Int. J. Geomech. 1.6 kh = 0.3 kh = 0.2 kh = 0.1 Nq = 1.0 = 0.8 = 0.6 = 0.4 = 0.2 = 0.0 1.2 φ = 20°, β/φ = -0.15, α = -20° 0.3 0 0.7 λ 1.4 φ = 20°, β/φ = -0.15, α = 0° 0 0.8 λ 1.6 Nq = 1.0 = 0.8 = 0.6 = 0.4 = 0.2 = 0.0 Nq = 1.0 = 0.8 = 0.6 = 0.4 = 0.2 = 0.0 1.4 0.7 φ = 20°, β/φ = 0, α = -20° 0 0.8 λ 1.6 φ = 20°, β/φ = 0, α = 0° 2.4 0 0.9 λ 1.8 2.7 Nq = 1.0 = 0.8 = 0.6 = 0.4 = 0.2 = 0.0 1.6 φ = 20°, β/φ = +0.15, α = -20° 0.2 0 0.9 λ 1.8 2 3 kh = 0.3 kh = 0.2 kh = 0.1 Nq = 1.0 = 0.8 = 0.6 = 0.4 = 0.2 = 0.0 1.4 φ = 20°, β/φ = +0.15, α = 0° 0.4 0 2.7 λ 2.3 1 0.8 1 3.2 kh = 0.3 kh = 0.2 kh = 0.1 Kaqγs 1.4 0 Kaqγs Nq = 1.0 = 0.8 = 0.6 = 0.4 = 0.2 = 0.0 φ = 20°, β/φ = 0, α = +20° (f) 2.2 kh = 0.3 kh = 0.2 kh = 0.1 Nq = 1.0 = 0.8 = 0.6 = 0.4 = 0.2 = 0.0 0.5 (e) 2 2.7 1.2 0.4 (d) 1.8 kh = 0.3 kh = 0.2 kh = 0.1 1.9 0.9 0.3 λ 2.6 kh = 0.3 kh = 0.2 kh = 0.1 Kaqγs Kaqγs 1.1 0.9 (c) 1.9 kh = 0.3 kh = 0.2 kh = 0.1 0 2.4 Kaqγs 1.5 φ = 20°, β/φ = -0.15, α = +20° 0.4 (b) (a) Nq = 1.0 = 0.8 = 0.6 = 0.4 = 0.2 = 0.0 1 0.4 2.1 kh = 0.3 kh = 0.2 kh = 0.1 1.6 0.8 Kaqγs Downloaded from ascelibrary.org by University Of Florida on 10/25/17. Copyright ASCE. For personal use only; all rights reserved. 0.7 (g) Nq = 1.0 = 0.8 = 0.6 = 0.4 = 0.2 = 0.0 Kaqγs Kaqγs 1.1 2.2 kh = 0.3 kh = 0.2 kh = 0.1 Kaqγs 1.5 1.1 λ 2.2 (h) φ = 20°, β/φ = +0.15, α = +20° 0.5 3.3 0 1.1 λ 2.2 3.3 (i) Fig. 12. Variation of Kaqg s with l , Nq, and kh for f = 20° and d / f = 2/3: (a) b / f = –0.15 and a = –20°; (b) b / f = –0.15 and a = 0°; (c) b / f = – 0.15 and a = þ20°; (d) b / f = 0 and a = –20°; (e) b / f = 0 and a = 0°; (f) b / f = 0 and a = þ20°; (g) b / f = þ0.15 and a = –20°; (h) b / f = þ0.15 and a = 0°; (i) b / f = þ0.15 and a = þ20° © ASCE 04017123-17 Int. J. Geomech., 2018, 18(1): 04017123 Int. J. Geomech. 1.1 kh = 0.3 kh = 0.2 kh = 0.1 Nq = 1.0 = 0.8 = 0.6 = 0.4 = 0.2 = 0.0 0.8 φ = 30°, β/φ = -0.40, α = -20° 0.1 0 0.4 λ 0.8 φ = 30°, β/φ = -0.40, α = 0° 0 0.4 λ 0.8 1.2 (b) 1 1.1 Nq = 1.0 = 0.8 = 0.6 = 0.4 = 0.2 = 0.0 0.8 0.4 φ = 30°, β/φ = 0, α = -20° 0 0.5 λ 1 φ = 30°, β/φ = 0, α = 0° 0 1.5 0.6 λ 1.2 1.8 0 Nq = 1.0 = 0.8 = 0.6 = 0.4 = 0.2 = 0.0 1.4 φ = 30°, β/φ = +0.40, α = -20° 0.2 0 0.7 λ 1.4 1.4 2.1 kh = 0.3 kh = 0.2 kh = 0.1 Nq = 1.0 = 0.8 = 0.6 = 0.4 = 0.2 = 0.0 1.4 φ = 30°, β/φ = +0.40, α = 0° 0.2 0 2.1 λ 2.3 0.8 0.6 0.7 3.2 kh = 0.3 kh = 0.2 kh = 0.1 Kaqγs 1 φ = 30°, β/φ = 0, α = +20° Kaqγs Nq = 1.0 = 0.8 = 0.6 = 0.4 = 0.2 = 0.0 Nq = 1.0 = 0.8 = 0.6 = 0.4 = 0.2 = 0.0 (f) 2 kh = 0.3 kh = 0.2 kh = 0.1 1.5 kh = 0.3 kh = 0.2 kh = 0.1 0.3 (e) 1.4 1 0.8 0.2 (d) λ 1.3 0.5 0.1 0.5 1.8 kh = 0.3 kh = 0.2 kh = 0.1 Kaqγs Kaqγs 0.7 0 (c) kh = 0.3 kh = 0.2 kh = 0.1 Nq = 1.0 = 0.8 = 0.6 = 0.4 = 0.2 = 0.0 φ = 30°, β/φ = -0.40, α = +20° 0.3 Kaqγs (a) Nq = 1.0 = 0.8 = 0.6 = 0.4 = 0.2 = 0.0 0.7 0.2 1.2 kh = 0.3 kh = 0.2 kh = 0.1 1.1 0.5 Kaqγs Downloaded from ascelibrary.org by University Of Florida on 10/25/17. Copyright ASCE. For personal use only; all rights reserved. 0.3 (g) Nq = 1.0 = 0.8 = 0.6 = 0.4 = 0.2 = 0.0 Kaqγs Kaqγs 0.5 1.5 kh = 0.3 kh = 0.2 kh = 0.1 Kaqγs 0.7 1 λ 2 (h) φ = 30°, β/φ = +0.40, α = +20° 0.5 3 0 1.1 λ 2.2 3.3 (i) Fig. 13. Variation of Kaqg s with l , Nq, and kh for f = 30° and d / f = 2/3: (a) b / f = –0.40 and a = –20°; (b) b / f = –0.40 and a = 0°; (c) b / f = – 0.40 and a = þ20°; (d) b / f = 0 and a = –20°; (e) b / f = 0 and a = 0°; (f) b / f = 0 and a = þ20°; (g) b / f = þ0.40 and a = –20°; (h) b / f = þ0.40 and a = 0°; (i) b / f = þ0.40 and a = þ20° © ASCE 04017123-18 Int. J. Geomech., 2018, 18(1): 04017123 Int. J. Geomech. 0.7 kh = 0.3 kh = 0.2 kh = 0.1 Nq = 1.0 = 0.8 = 0.6 = 0.4 = 0.2 = 0.0 0.5 φ = 40°, β/φ = -0.55, α = -20° 0 0 0.2 λ 0.4 φ = 40°, β/φ = -0.55, α = 0° 0 0.3 λ 0.6 0.9 (b) 0.6 Nq = 1.0 = 0.8 = 0.6 = 0.4 = 0.2 = 0.0 Nq = 1.0 = 0.8 = 0.6 = 0.4 = 0.2 = 0.0 0.7 0.2 φ = 40°, β/φ = 0, α = -20° 0 0.3 λ 0.6 φ = 40°, β/φ = 0, α = 0° 0.9 0 0.4 λ 0.8 1.2 φ = 40°, β/φ = +0.55, α = -20° 0 0.6 λ 1.2 λ 1.2 1.8 Nq = 1.0 = 0.8 = 0.6 = 0.4 = 0.2 = 0.0 1.4 φ = 40°, β/φ = +0.55, α = 0° 0.2 1.8 kh = 0.3 kh = 0.2 kh = 0.1 2.3 0.7 0.1 0.6 3.2 Nq = 1.0 = 0.8 = 0.6 = 0.4 = 0.2 = 0.0 1.2 0.4 0 kh = 0.3 kh = 0.2 kh = 0.1 Kaqγs 0.7 φ = 40°, β/φ = 0, α = +20° Kaqγs Nq = 1.0 = 0.8 = 0.6 = 0.4 = 0.2 = 0.0 Nq = 1.0 = 0.8 = 0.6 = 0.4 = 0.2 = 0.0 (f) 1.7 kh = 0.3 kh = 0.2 kh = 0.1 0.9 kh = 0.3 kh = 0.2 kh = 0.1 0.3 (e) 1 0.6 0.7 0.1 (d) λ 1.1 0.4 0 0.3 1.5 kh = 0.3 kh = 0.2 kh = 0.1 Kaqγs Kaqγs 0.4 0 (c) 1 kh = 0.3 kh = 0.2 kh = 0.1 φ = 40°, β/φ = -0.55, α = +20° 0.2 Kaqγs (a) Nq = 1.0 = 0.8 = 0.6 = 0.4 = 0.2 = 0.0 0.5 0.1 0.6 kh = 0.3 kh = 0.2 kh = 0.1 0.8 0.3 Kaqγs Downloaded from ascelibrary.org by University Of Florida on 10/25/17. Copyright ASCE. For personal use only; all rights reserved. 0.2 (g) Nq = 1.0 = 0.8 = 0.6 = 0.4 = 0.2 = 0.0 Kaqγs Kaqγs 0.4 1.1 kh = 0.3 kh = 0.2 kh = 0.1 Kaqγs 0.6 0 0.9 λ 1.8 (h) φ = 40°, β/φ = +0.55, α = +20° 0.5 2.7 0 1 λ 2 3 (i) Fig. 14. Variation of Kaqg s with l , Nq, and kh for f = 40° and d / f = 2/3: (a) b / f = –0.55 and a = –20°; (b) b / f = –0.55 and a = 0°; (c) b / f = −0.55 and a = þ20°; (d) b / f = 0 and a = –20°; (e) b / f = 0 and a = 0°; (f) b / f = 0 and a = þ20°; (g) b / f = þ0.55 and a = –20°; (h) b / f = þ0.55 and a = 0°; (i) b / f = þ0.55 and a = þ20° © ASCE 04017123-19 Int. J. Geomech., 2018, 18(1): 04017123 Int. J. Geomech. Downloaded from ascelibrary.org by University Of Florida on 10/25/17. Copyright ASCE. For personal use only; all rights reserved. Fig. 15. Variation of K0 aqg cs obtained with and without considering tensile crack in the backﬁll corresponding to l and f for Nc = 0.1, b / f = 0.5, d / f = 0.5, and a = 0° with (a) kh = 0.0, (b) kh = 0.1, (c) kh = 0.2, and (d) kh = 0.3 self-weight can be either smaller or larger than the resisting forces arising from internal stresses. Thus, the smaller and larger magni0 tudes of Kaq g cs are realized for the case in which the effect of crack in the backﬁll is included. Furthermore, and not surpris0 ingly, it is seen that the magnitudes of Kaq g cs and l cr always increase considerably with an increase in the values of kh for both j = 0 and j > 0. The present study follows the assumption that the cracks existing in the backﬁll are perfectly vertical. However, the possibility of the existence of nonvertical cracks in the backﬁll can be expected, which will be a subject for future research. Comparison of Results The present theoretical analysis has been validated by comparing solutions produced in this paper with the experimental results © ASCE reported by Fang et al. (1997) and theoretical solutions of Coulomb (1776), Chen and Rosenfarb (1973), Kerisel and Absi (1990), Motta (1994), Soubra and Macuh (2002), and Shukla et al., (2009). In Fig. 16, the present values of Ka g are compared with the laboratory experimental results of Fang et al. (1997) for different backﬁll slope angles ( b ). The results of Fang et al. (1997) corresponded to a vertical retaining wall (a = 0) with a height of 0.3 m. A dry sand backﬁll was used with the unit weight ( g ) and internal friction angle ( f ) of 15.5 kN/m3 and 30.9°, respectively. The friction angle of the soil-wall interface (d ) has been reported to be 19.2°. In Fig. 16, the solutions obtained from Coulomb’s theory for different values of d are also presented. For d = 19.2°, both solutions from the Coulomb theory and the present analysis are lower than that of experimental results. However, present theoretical formulation shows improvement greater than that of Coulomb’s theory for different values of b and d . Table 1 shows a comparison of earth thrust coefﬁcients Ka g 04017123-20 Int. J. Geomech., 2018, 18(1): 04017123 Int. J. Geomech. 0.5 obtained in this study with those reported by Chen and Rosenfarb (1973) for an inclined wall with horizontal backﬁll. The values of Ka g obtained from three different collapse mechanisms by Chen and Rosenfarb (1973) do not differ much from that of the present solutions. Table 1 shows that for a = –20°, the planar failure mechanism of Chen and Rosenfarb (1973) provided better solutions compared with their composite collapse mechanism and the mechanism used in the present analysis. This is because of the requirement of additional constraints for the satisfaction of kinematic admissible conditions in the composite collapse mechanism due to which the composite mechanisms of Chen and Rosenfarb (1973) and the present study cannot degenerate effectively to those of planar mechanisms. However, a very close agreement among the tabulated solutions can be noted. Fig. 17 provides a comparison of the present values of static earth thrust coefﬁcients Ka g and Kaq with the theoretical results of Kerisel and Absi (1990) for f = 45° and d / f = 1 in the case of (1) a vertical wall retaining an inclined backﬁll and (2) an inclined wall retaining a horizontal backﬁll. The present upper bound solutions are slightly lower than the limit equilibrium solutions of Kerisel and Absi (1990) for negatively sloped surfaces of backﬁll and wall back; however, a close agreement for the coefﬁcients Ka g and Kaq computed between the two solutions can be observed. The solutions from the limit equilibrium approaches may φ = 30.9° Present analysis Coulomb (1776) Fang et al. (1997) Kaγ 0.4 δ 0° 10° 19.2° 0.2 -30 -10 10 30 β Fig. 16. Comparison of results from the present analysis with the solutions obtained from Coulomb’s theory and the experimental results reported by Fang et al. (1997) for f = 30.9° and a = 0° Table 1. Comparison of Ka g Obtained for Horizontal Backﬁll in the Present Study with That Reported by Chen and Rosenfarb (1973) Chen and Rosenfarb (1973) LSM f (degrees) 30 40 CSM PLM Present study a (degrees) d / f = 1/2 d /f = 1 d / f = 1/2 d /f = 1 d / f = 1/2 d /f = 1 d / f = 1/2 d /f = 1 þ20 0 –20 þ20 0 –20 0.475 0.302 0.188 0.368 0.200 0.095 0.501 0.302 0.178 0.428 0.214 0.095 0.475 0.301 0.181 0.365 0.197 0.090 0.501 0.297 0.168 0.418 0.210 0.088 0.476 0.301 0.215 0.370 0.199 0.115 0.501 0.297 0.237 0.428 0.210 0.146 0.476 0.303 0.189 0.369 0.200 0.096 0.502 0.302 0.179 0.429 0.214 0.096 Note: LSM = log sandwich; CSM = circular sandwich; PLM = planar mechanism. 0.4 0.5 Present analysis Kerisel and Absi (1990) Present analysis Kerisel and Absi (1990) 0.4 0.3 Kaγ , Kaq 0.3 Kaγ , Kaq Downloaded from ascelibrary.org by University Of Florida on 10/25/17. Copyright ASCE. For personal use only; all rights reserved. 0.3 Kaq 0.2 Kaq 0.2 Kaγ Ka γ 0.1 0 β = 0°, φ = 45°, δ/φ = 1, λ = 0 -0.5 (a) -0.3 -0.1 0.1 α/φ 0.3 α = 0°, φ = 45°, δ/φ = 1, λ = 0 0.1 -1 0.5 -0.7 (b) -0.4 -0.1 β /φ 0.2 0.5 0.8 Fig. 17. Comparison of the present analysis with those reported by Kerisel and Absi (1990) for f = 45°, d / f = 1, and l = 0 for (a) vertical wall with different backﬁll slope surfaces and (b) inclined wall with horizontal backﬁll © ASCE 04017123-21 Int. J. Geomech., 2018, 18(1): 04017123 Int. J. Geomech. Remarks Applicability of the Principle of Superposition In most of the earlier studies, the solutions were obtained based on an independent maximization scheme in which the total active force has been expressed as the summation of the maximum force induced due to the individual contribution of unit weight, cohesion, and surcharge; thus, this involves the inherent assumption that the principle of superposition is valid. Generally, the use of the superposition principle in the computation of the total active earth force for a general cohesive-frictional backﬁll with surcharge loadings requires three distinct critical slip surfaces with respect to soil weight, cohesion, and surcharge components rather than predicting a unique slip surface that has been observed in the ﬁeld or laboratory trials. The use of the principle of superposition in computing total active force will be on the safer side; however, depending on the wall geometry, properties and conﬁguration of backﬁll, surcharge loadings, and so forth, the magnitude of total active force is sometimes overestimated highly compared with the actual values. Thus, for examining the applicability of the superposition principle, a comparison of the normalized values of resultant seismic active force (2Pa/ g h2) computed with and without the superposition principle is presented Tables 3 and 4 for l = 0 and l > 0, respectively, with d / f = 2/3, b / f = 1/3, and a = 0. Table 3 shows that the computed values of 2Pa/ g h2 using the superposition principle always remains higher than the corresponding solutions obtained without the superposition rule. For any given value of Nq, the error involved in the computation of the resultant active force using the superposition principle increases when (1) the magnitudes of kh and Nc are higher and (2) the magnitudes of f are lower. This is a consequence mainly due to the omission of interaction effects associated with the previously mentioned factors. However, when the effect of the tension crack is included, the error in the computation due to the superposition principle has been reduced, which is caused by the imposition of an additional surcharge in the formulation for the cracked region. For the computation of active earth thrust with purely frictional backﬁll, individual coefﬁcients obtained with the help of the superposition principle can be applied for lower magnitudes of kh because the error involved in the computations is practically 0.7 Present analysis Motta (1994) 0.6 Nq = 1.0 = 0.5 = 0.25 = 0.0 0.5 Kaqγ Downloaded from ascelibrary.org by University Of Florida on 10/25/17. Copyright ASCE. For personal use only; all rights reserved. present theory compared with that of Shukla et al. (2009) for lower values of kh. For higher values of kh, there is no signiﬁcant difference between the two solutions. be more conservative than the limit analysis solutions; however, the kinematic admissibility of the collapse mechanism cannot always be ensured. In Fig. 18, the present static earth thrust coefﬁcient Kaqg has been compared with the numerical results of Motta (1994) for f = 30° and d / f = 1 in the case of a vertical wall supporting a horizontal frictional backﬁll. The numerical results of Motta (1994) and the present theoretical results are found to be exactly the same. For different combinations of f , d / f , and b / f , Table 2 presents a comparison of static earth thrust coefﬁcients Ka g , Kaq , and Kac computed from the present theory with the upper bound solutions of Soubra and Macuh (2002) based on rotational log spiral collapse mechanisms. The present solution has been found to be practically the same when compared with the rigorous upper bound solutions obtained by Soubra and Macuh (2002). Nevertheless, the theory of Soubra and Macuh (2002) could not be extended simply to compute 0 the combined thrust coefﬁcients Kaqg s and Kaq g cs or to address the effect of the position of the surcharge placed on the backﬁll. Fig. 19 0 presents a comparison of the seismic earth thrust coefﬁcient Kaq g cs obtained using the present theory with the limit equilibrium solutions of Shukla et al. (2009) for different values of Nc in the case of a smooth vertical wall retaining horizontal backﬁll. There is mar0 ginal improvement in the coefﬁcients Kaq g cs computed from the 0.4 0.3 φ = 30°, β/φ = 0,δ/φ = 0.5, α = 0° 0.2 0 0.2 0.4 λ 0.6 0.8 1 Fig. 18. Comparison of the present analysis with those reported by Motta (1994) for f = 30°, b / f = 0, d / f = 0.5, and a = 0° Table 2. Comparison of Active Thrust Coefﬁcients Obtained for Vertical Walls in the Present Study with That Reported by Soubra and Macuh (2002) Soubra and Macuh (2002) Ka g f (degrees) d / f 20 30 40 © ASCE 0 1/2 1 0 1/2 1 0 1/2 1 Present analysis Kaq Kac Ka g Kaq Kac b / f = 0 bf = 2/3 b / f = 0 b / f = 2/3 b / f = 0 bf = 2/3 b / f = 0 b / f = 2/3 b / f = 0 b / f = 2/3 b / f = 0 b / f = 2/3 0.490 0.449 0.436 0.333 0.303 0.304 0.217 0.200 0.215 0.611 0.577 0.570 0.441 0.415 0.425 0.296 0.282 0.310 0.490 0.450 0.441 0.333 0.304 0.309 0.217 0.201 0.219 0.628 0.593 0.586 0.469 0.442 0.453 0.332 0.316 0.347 1.400 1.553 1.712 1.155 1.267 1.466 0.933 1.029 1.295 1.616 1.755 1.917 1.353 1.450 1.656 1.079 1.164 1.445 0.49 0.448 0.434 0.333 0.303 0.302 0.217 0.200 0.214 04017123-22 Int. J. Geomech., 2018, 18(1): 04017123 0.611 0.577 0.571 0.441 0.415 0.426 0.296 0.282 0.311 0.490 0.451 0.449 0.333 0.305 0.315 0.217 0.202 0.224 0.628 0.593 0.591 0.469 0.442 0.458 0.332 0.316 0.352 1.401 1.549 1.691 1.155 1.265 1.454 0.933 1.028 1.289 1.610 1.743 1.883 1.346 1.440 1.635 1.073 1.158 1.433 Int. J. Geomech. Downloaded from ascelibrary.org by University Of Florida on 10/25/17. Copyright ASCE. For personal use only; all rights reserved. Fig. 19. Comparison of the present analysis with those reported by Shukla et al. (2009) for a smooth vertical wall retaining a horizontal c- f backﬁll with no load acting on it with (a) Nc = 0, (b) Nc = 0.05, (c) Nc = 0.1, and (d) Nc = 0.2 Table 3. Comparison of Normalized Resultant Seismic Active Thrust (2Pa/ g h2) Computed with and without Using the Principles of Superposition for a = 0°, d / f = 2/3, b / f = 1/3, l = 0, and j = 0 f = 20° f = 30° Nq = 0.5 kh 0 0.3 Nc 2Paw/ g h 0.025 0.05 0.1 0.2 0.025 0.05 0.1 0.2 0.695 0.653 0.568 0.399 1.834 1.791 1.706 1.537 2 Nq = 1 2Pawo/ g h 0.694 0.649 0.562 0.389 1.772 1.668 1.46 1.068 2 2Paw/ g h 2 Nq = 0.5 2 0.943 0.901 0.816 0.646 2.462 2.419 2.334 2.165 2 Nq = 1 2Pawo/ g h 2Paw/ g h 2Pawo/ g h 0.941 0.897 0.809 0.636 2.400 2.296 2.088 1.672 0.479 0.444 0.373 0.232 1.165 1.129 1.059 0.918 0.478 0.441 0.368 0.222 1.134 1.068 0.941 0.723 2 2Paw/ g h 0.653 0.618 0.547 0.406 1.569 1.534 1.463 1.322 2 2Pawo/ g h2 0.651 0.614 0.541 0.395 1.538 1.472 1.339 1.108 Note: 2Paw/ g h2 and 2Pawo/ g h2 refer to 2Pa/ g h2 computed with and without using the superposition principle, respectively. insigniﬁcant. Table 4 shows that the application of the superposition principle leads to poor solutions for estimating the values of critical surcharge distance ratio l cr. The error involved in the © ASCE computation of 2Pa/ g h2 and l cr has been found to be over 20 and 200%, respectively, depending on the magnitudes of kh and Nq. Therefore, it is recommended to take into account the combined 04017123-23 Int. J. Geomech., 2018, 18(1): 04017123 Int. J. Geomech. Table 4. Comparison of Normalized Resultant Seismic Active Thrust (2Pa/ g h2) Computed with and without Using the Principles of Superposition for a = 0°, d / f = 2/3, b / f = 1/3, and Nc = 0 kh = 0 kh = 0.1 Downloaded from ascelibrary.org by University Of Florida on 10/25/17. Copyright ASCE. For personal use only; all rights reserved. Nq = 0.5 f l 2Paw/ g h 20° 0.3 0.6 0.9 1.2 1.5 1.8 2.1 2.4 2.7 0.680 0.638 0.606 0.580 0.558 0.541 0.527 0.515 0.506 2 Nq = 1 2Pawo/ g h 0.671 0.614 0.561 0.514 0.490 0.490 0.490 0.49 0.49 2 2Paw/ g h 2 Nq = 0.5 2Pawo/ g h 0.870 0.786 0.722 0.669 0.627 0.591 0.563 0.539 0.522 2 0.857 0.750 0.658 0.578 0.508 0.490 0.490 0.490 0.490 2Paw/ g h 0.867 0.829 0.797 0.770 0.748 0.728 0.711 0.695 0.680 2 Nq = 1 2Pawo/ g h 2 2Paw/ g h 0.863 0.813 0.767 0.725 0.686 0.650 0.616 0.611 0.611 2 2Pawo/ g h2 1.124 1.047 0.983 0.930 0.885 0.846 0.812 0.781 0.749 1.118 1.023 0.941 0.868 0.802 0.740 0.687 0.635 0.611 Note: 2Paw/ g h2 and 2Pawo/ g h2 refer to 2Pa/ g h2 computed with and without using the superposition principle, respectively. marginal decrease or increase in the magnitude of active earth thrust has been observed with respect to the change in the values of soil-wall interface friction depending on the parameters f , b , and a. For given values of f , d , and b , the magnitude of earth thrust has been found to always reduce with the change of wall orientation from positive to negative values. The location of surcharge (l ) applied on the backﬁll has a larger impact on reducing the active earth thrust up to a critical surcharge distance ratio l cr beyond which the computed values of active earth force are independent of the magnitude of the surcharge pressure. For the case of purely ﬁctional backﬁll, this quantity becomes exactly equal to that resulting from the component of soil unit weight. However, the value of l cr also increases with an increase in the values of the factors Nq and kh. The magnitudes of earth thrust coefﬁcient and critical surcharge distance ratio always increase with an increase in the values of kh for the backﬁll with and without any tension crack. However, the consideration of the preexisting crack in the formulation may not always form a critical scenario for the case of seismic loadings and surcharge located at some distance from the wall. Proper attention, therefore, must be paid in evaluating total active earth force because it depends mainly on the magnitudes of Nc and Nq. The composite failure mechanism with shear zone in the form of a log spiral or circular sandwich between two triangular blocks considered in the previous studies and the composite mechanism introduced in the present analysis are not effective compared with the planar failure mechanism for higher values of negative wall inclination. effect of the ground surcharge and soil self-weight components as presented in Figs. (9)–(14) to determine the resultant active earth thrust. 3. Applicability of Present Solutions for Nonassociated Flow Rule Material and in the Presence of Vertical Earthquake Acceleration For computing the magnitude of the active earth thrust based on the nonassociated ﬂow rule, the solutions obtained in the present study can still be used by using the corrections suggested by Drescher and Detournay (1993) and Michalowski and Shi (1995). The total active earth thrust in the presence of vertical earthquake acceleration can be computed from the solutions produced in the present analysis by following the approach devised by Somers (2003). Thus, the earthquake acceleration in the vertical direction is not included in the present theoretical formulation. 4. 5. Conclusions In this paper, extensive useful design charts for determining the static and pseudostatic seismic resultant active earth force against an inclined wall undergoing a horizontal translational movement have been developed by performing a theoretical analysis. With the application of the upper bound theorem of limit analysis and the consideration of a composite collapse mechanism involving a new kinematically admissible velocity ﬁeld, the analysis was performed. The magnitudes of resultant active force under different wall orientations and backﬁll properties can be estimated by taking into account the inﬂuence of the location of continuous surcharge pressure and the tension crack in the backﬁll. The speciﬁc conclusions drawn from the present analysis are summarized as follows: 1. From using the superposition principle, it can be said that the active earth thrust coefﬁcient corresponding to the cohesion component remains independent of seismic loading. However, considerable increase in the active earth thrust coefﬁcients corresponding to soil self-weight and ground surcharge components has been noticed with an increase in the magnitudes of seismic acceleration coefﬁcients. 2. For a given wall orientation (a), the magnitude of active earth thrust increases continuously with a decrease in the values of f and with an increase in the values of b . However, a © ASCE 6. Appendix I. Analytical Functions for Finding the Seismic Active Earth Thrust Coefficients Various functions required to obtain the seismic active earth thrust coefﬁcients with the help of Eqs. (22)–(24) and (27) are presented as follows: C1 ðu ; m ; ɛÞ ¼ 04017123-24 Int. J. Geomech., 2018, 18(1): 04017123 cosð m þ f u Þcosða þ m u Þsin m cosa cosð f u Þcosð m u Þ ½kh þ tanða þ m u Þ (31) Int. J. Geomech. 8* +9 M0 ɛ > > ð Þ cos k a þ m u a þ m u þ ɛ 1 þ M cos ð Þ ð Þe > > 0 h = < M0 ɛ 2 ð Þ þ M sin k a þ m u a þ m u þ ɛ sin ð Þ ð Þe cos ð m þ f u Þ 0 h > C2 ðu ; m ; ɛÞ ¼ > > > ; cosa cos2 ð f u Þcosð m u Þ : 1 þ M20 C3 ðu ; m ; ɛÞ ¼ cos2 ð m þ f u Þcosða þ m þ ɛ b Þcosða þ m þ ɛ u ÞeM0 ɛ ½kh þ tanða þ m þ ɛ u Þ cosa cosð f u Þcosð m u Þsinða þ m þ ɛ þ f b u Þ Downloaded from ascelibrary.org by University Of Florida on 10/25/17. Copyright ASCE. For personal use only; all rights reserved. cosða þ m þ ɛ u Þ C4 ðu ; m ; ɛÞ ¼ cosð m u Þ ( cosð m þ f u ÞeM1 ɛ λeɛ tanð f þu Þ sinða þ m þ ɛ þ f b u Þ cos b (32) (33) ) ½kh þ tanða þ m þ ɛ u Þ (34) 8 9 > ð1 j Þcosð m þ f u ÞeM1 ɛ ðλ j sinaÞeɛ tanð f þu Þ > < = cosða þ m þ ɛ u Þ C04 ðu ; m ; ɛÞ ¼ ½kh þ tanða þ m þ ɛ u Þ sinða þ m þ ɛ þ f b u Þ cos b > > cosð m u Þ 00 : ; þC ðu ; m ; ɛÞ 4 (35) " # cosða b Þ 2 j ð1 j Þcos b cosð m þ f u ÞeM1 ɛ C 4 ðu ; m ; ɛ Þ ¼ þ j 2 sina eɛ tanð f þu Þ cosa cos b Nq sinða þ m þ ɛ þ f b u Þ 00 C5 ðu ; m ; ɛÞ ¼ sin m cos f cosð f u Þcosð m u Þ (37) cosð m þ f u Þ½1 eM1 ɛ cosð f þ u Þ C6 ðu ; m ; ɛÞ ¼ 1þ cosð f u Þ 2 sin f cosð m u Þ C7 ðu ; m ; ɛÞ ¼ cosða þ m þ ɛ b Þcosð m þ f u Þcos f eM1 ɛ cosð f u Þcosð m u Þsinða þ m þ ɛ þ f b u Þ C8 ðu ; m ; ɛÞ ¼ tan d tan f M0 ¼ 2 tanð f u Þ þ tanð f þ u Þ (41) M1 ¼ tanð f u Þ þ tanð f þ u Þ (42) References Caquot, A., and Kerisel, L. (1948). Traite de mecanique des sols, Gauthier Villars, Paris. Chen, J., Li, M., and Wang, J. (2017). “Active earth pressure against rigid retaining walls subjected to conﬁned cohesionless soil.” Int. J. Geomech., 10.1061/(ASCE)GM.1943-5622.0000855, 06016041. Chen, W. F. (1975). Limit analysis and soil plasticity, Elsevier, Amsterdam, Netherlands. Chen, W. F., and Liu, X. L. (1990). Limit analysis in soil mechanics, Elsevier, Amsterdam, Netherlands. © ASCE sinða þ m u Þ tan a cos a cosð m u Þ (36) (38) (39) (40) Chen, W. F., and Rosenfarb, J. L. (1973). “Limit analysis solutions of earth pressure problems.” Soils Found., 13(4), 45–60. Coulomb, C. A. (1776). “Essai sur une application des regies des maximis et minimis a quelques problemes de statique relatifs a l’arquiteture.” Memoirs Academie Royal Pres. Div. Sav. 7, Paris, 343–382. Das, B. M., and Puri, V. K. (1996). “Static and dynamic active earth pressure.” Geotech. Geolog. Eng., 14(4), 353–366. Drescher, A., and Detournay, E. 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