Passive earth pressures in the presence of hydraulic gradients

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Passive earth pressures in the presence of hydraulic gradients A. H. SOUBRA, Ã R. KA STNER { and A. BENMANSOUR Ã The paper describes a variational approach applied to the limit equilibrium method for calculating the effective passive pressures of a cohesionless soil, taking into consideration the seepage ¯ow.It is shown that in the general case of non-homogeneous and non-isotropic hydraulic properties of the soil medium, the shape of the slip surface which veri®es the three limiting equilibrium equations of the soil mass at failure is a log-spiral.It is also shown that the passive earth pressure calculation is independent of the normal stress distribution along this surface.The variational limit equilibrium method is equivalent to the upper bound method in limit analysis for a rotational log-spiral mechanism.Numerical results of the coef®cients of passive earth pressures in the presence of seepage ¯ow are presented and discussed.KEYWORDS: earth pressure; ®lters; limit state design/analysis; pore pressures; seepage; sheet piles and cofferdams.
Cet article pre Âsente une approche variationnelle applique Âe a Á la me Âthode du prisme de rupture permettant le calcul de la pression passive effective des terres en pre Âsence d'e Âcoulement dans le cas d'un sol purement pulve Ârulent.On montre que dans le cas ge Âne Âral d'un sol aux proprie Âte Âs hydrauliques non homoge Ánes et non isotropes, la forme de la surface de rupture qui ve Âri®e les trois e Âquations d'e Âquilibre est une spirale logarithmique.Nous montrons aussi que le calcul de la pression passive ne de Âpend pas de la distribution des contraintes normales agissant le long de cette surface.La me Âthode du prisme de rupture variationnelle est e Âquivalente a Á la me Âthode de la borne supe Ârieure en analyse limite pour un me Âcanisme rotationnel en spirale logarithmique.Des valeurs nume Âriques du coef®cient de bute Âe en pre Âsence d'e Âcoulement sont pre Âsente Âes et discute Âes.

INTRODUCTION
The design of deep sheeted excavations is often dominated by the ¯ow of water around the sheet piles.The seepage ¯ow in¯uences the stability of the excavation where bulk heave or piping may occur.While the piping takes place at the excava tion level, the heaving is more catastrophic and its risk is usually evaluated by considering a rectangu lar failure mechanism adjacent to the wall (Terzaghi, 1943).The vertical force equilibrium of this soil mass is then considered by neglecting the vertical frictional forces along the vertical faces of this mechanism.
Based on laboratory model tests, Kastner (1982) has shown that the failure of the sheet piling structures in the presence of seepage ¯ow is not only due to the heaving phenomenon but may also occur due to the reduction of the passive earth pressures in front of the wall.Our aim in this paper is to propose an outline for the calculation of the effective passive pressures, taking into ac count the seepage forces.
Looking for a simple model capable of correctly describing the behaviour of soil in the passive state in the presence of seepage ¯ow, we opted for the limit equilibrium method.This approach is based on a priori hypotheses concerning the shape of the slip surface (kinematic function) and the normal stress distribution (static function) along this sur face.The variational approach applied to the limit equilibrium method has been employed in order to avoid the restrictions of such a priori hypotheses.

OVERVIEW OF PREVIOUS VARIATIONAL ANALYSIS
The variational limit equilibrium method has been used by several authors in geotechnical en gineering.Kopacsy (1957) applied this approach to the three dimensional slope stability problem; however, no explicit solution is offered.Several investigators, for example Dorfman (1965), Garber (1973), Revilla & Castillo (1977) and Ly (1979), have used the calculus of variations to avoid introducing an assumption concerning the shape of the slip surface but they made an assump tion concerning the normal stress distribution along this surface.A more interesting analysis consists of ®nding the two unknown functions without any a priori assumptions.Thus, Baker & Garber (1977, 1978) applied the variational approach to the two dimensional slope stability problem, and Garber & Baker (1977) and Castillo & Luceno (1978) ap plied this approach to the problem of the bearing capacity of a strip footing.Then, Garber & Baker (1979) treated the problems of slope stability, bear ing capacity and earth pressure distribution in a uni®ed manner as a single problem.Later, this method was used by Leshchinsky et al. (1985), Ugai (1985) and Leshchinsky & Baker (1986) to study the three dimensional slope stability problem, and by Leshchinsky & Reinschmidt (1985), who applied it to the reinforced slope stability problem.Finally, Leshchinsky & San (1994) applied the variational limit equilibrium method to the seismic stability of slopes, and presented interesting results in the form of design charts.It is to be noted here that the variational approach has been the subject of some controversy.Particularly, the work by Castillo & Luceno (1982) shows that the functional has no minimum.Notice, however, that experience indicates that the slip surface determined by the variational analysis reasonably duplicates reality (Leshchinsky & San, 1994).Consequently, the solutions given by the variational limit equilibrium method are very interesting.We present in the following the application of this method to the passive earth pressure problem, taking into account the seepage forces.

VARIATIONAL APPROACH OF THE PASSIVE EARTH PRESSURE PROBLEM
Figure 1 shows a double walled cofferdam sub jected to a seepage ¯ow where H is the total head loss and u(x, y) represents the distribution of the pore water pressures in the soil medium.
The assumptions made in the analysis can be summarized as follows: (a) The soil is cohesionless.It is homogeneous and isotropic with respect to its angle of internal friction ö.(b) The soil medium is non homogeneous and non isotropic with respect to the hydraulic properties.It is composed of n permeable layers overlying impermeable rock.Each layer is characterized by its coef®cients of per meability K hi and K vi .(c) The breadth B 0 (Fig. 1) is large enough so that there is no interaction of the two failure mechanisms which develop in front of the two walls of the cofferdam.
(d) The resultant P P of the effective passive pressures is assumed to act at the bottom third of the penetration depth (Fig. 2(a)).This force can be expressed as follows: where K p is the coef®cient of passive earth pres sure in the presence of seepage ¯ow, ã9 is the submerged unit weight of the soil, and f is the penetration depth.The variational approach is brie¯y presented in this paper.For more details, refer to Soubra (1989).Fig. 2(a) illustrates the formulated problem and shows the notation used.A potential slip sur face y(x) is subjected to a total normal stress ó (x).Both functions y(x) and ó (x) are assumed to be continuous.Using Coulomb's failure criterion ô(x) [ó (x) u(x)]tan ö ó 9(x)tan ö, the global limiting equilibrium equations for the soil mass (Fig. 2(b)) can be written as Notice that one of the endpoints x 0 is null  2b) and (2c), one rea lizes that the effective passive force P p is a func tional of two functions, y(x) and ó 9(x).The mathematical problem of the passive earth pressure is to ®nd those functions which give the minimum value of the passive force functional and simulta neously satisfy all three equations of limiting equi librium (equations (2a), ( 2b) and ( 2c)).
Using equation (2b) to de®ne P p , while consider ing the other two equilibrium equations (equations (2a) and (2c)) as constraints (i.e.equations that must be satis®ed), the passive earth pressure problem is a variational isoparametric one with a variable end point.This variational problem is equivalent to the minimization of an auxiliary functional G: where L 0 , L 1 and L 2 are given as follows: ë 1 and ë 2 are the Lagrange undetermined multi pliers.Finally, the two extremal functions y(x) and ó 9(x) must satisfy the following conditions: (a) The system of Euler's differential equations for the functional G: A, we have x A y A 0. At the variable end point B, a variational condition must be satis ®ed.This condition is called the `transversality condition' and it can be written as follows: where ä is a variational operator.

The ®rst Euler equation
Combining equations ( 3) and (5a) yields a dif ferential equation.The solution of this equation in a polar coordinate system is a log spiral (Fig. 3) whose equation is given as follows: Note that the log spiral function has a particular property, that the resultant of the forces (ó 9 dl) and (tan öó 9 dl) passes through the pole of the spiral.Hence, the moment equation about the pole is independent of the stress distribution ó 9(x), and may be used for the determination of the effective passive force.The two remaining equilibrium equa tions may be satis®ed by every ó 9(x) distribution that has two degrees of freedom.Thus, one has to ®nd the critical è 0 and è 1 angles which satisfy the moment equilibrium equation and give the mini mum value of the effective passive force P p .This is done by a two dimensional minimization proce dure of P p with respect to è 0 and è 1 .
The independence of the effective passive force from the normal stress distribution can also be shown, due to the special property of the present functional.This functional can be written as fol lows: G is linear in ó 9 and is independent of ó 9.The ®rst Euler equation implies that f (x, y, y) 0. Substituting this equation into equation ( 8), one can see that this functional becomes independent of ó 9 as follows: G g(x, y, y).This result is a direct consequence of the shape of the slip surface.Thus, the ®rst Euler equation transforms the pas sive functional from a functional of two unknown functions to a functional of a single function.Therefore, it is possible to solve the passive earth pressure problem by simply minimizing the new functional G without specifying the normal stress distribution.Finally, it is easy to see that the moment equation of the rotational log spiral me chanism around the centre is identical to the work equation for the same mechanism in the upper bound method in limit analysis.Thus, solving the passive earth pressure problem by writing the moment equation around the centre of the log spiral will give an upper bound solution of the exact solution for an associated ¯ow rule Coulomb material.

The second Euler equation and the transversality condition
Since the aim of this study is the determination of the critical effective passive force, the results obtained so far are enough to solve the problem.Indeed, it has been shown by Baker & Garber (1977) that the second Euler equation and the transversality condition give the normal stress dis tribution.In this work, this is also the case, but since the ó 9(x) distribution is not necessary to assess the effective passive force, we will not express these equations.

Existence of a minimum of the functional
The second variation of the passive earth func tional shows that this functional is degenerated.Thus, we cannot say mathematically that there is a minimum (Castillo & Luceno, 1982).This dif® culty concerning the existence of a minimum can be overcome due to the equivalence between the variational limit equilibrium method and the upper bound method in limit analysis for a rotational mechanism as has been shown before.This equiva lence has been established in a more general way by Castillo & Luceno (1983) and Leshchinsky et al. (1985).

CALCULATION SCHEME OF THE PASSIVE EARTH PRESSURES
As mentioned before, the solution of the passive earth pressure problem by a variational limit equi librium method is equivalent to saying that the equation of moment equilibrium must be satis®ed for the soil mass bounded by the log spiral and the ground surface.As shown in Fig. (2b), the forces acting on the collapse mechanism are (a) the saturated weight of the soil mass between the log spiral surface and the ground surface, (b) the effective passive force P p which is inclined at ä to the normal of the sheet pile, (c) the pore water pressures along the penetration depth and the log spiral surface, and (d) the effective normal and tangential stress distributions along y(x).The moment equilibrium equation can be written as follows: W sat (r 0 cos è 0 X ) P p sin ä(r 0 cos è 0 ) where M 1 and M 2 represent the moments of the force U 1 and of the pore water pressures along y(x) respectively.They are given as follows: From equation ( 9), one can easily see that P p is a function of the two parameters è 0 and è 1 which completely describe the failure mechanism.The most critical K p value is obtained by a minimiza tion procedure of the K p coef®cient given by equa tion (1), that is [K P (2P P )a(ã9 f 2 )] with respect to the two parameters mentioned above.A compu ter program has been developed for assessing the minimal K p values and the corresponding critical slip surfaces.

NUMERICAL RESULTS OF THE PASSIVE EARTH PRESSURES
Case of no seepage ¯ow Table 1 compares the passive earth pressure coef®cient K p for ö 408 and äaö 1a2 ob tained from the present analysis with that of other authors in the case of no seepage ¯ow.The com parison of the present upper bound solution with the upper and lower bound solutions given respec tively by Chen & Rosenfarb (1973) and Lysmer (1970) shows that the difference with the lower bound solution is smaller than 3%, which means that the present solution is very close to the exact solution for an associated ¯ow rule Coulomb material.On the other hand, the currently accepted values given by Sokolovski (1960) and Caquot & Ke Ârisel (1948) lie in the range between the best upper and lower bound solutions given by the limit analysis method.Finally, the comparison between the present result and the ones given by the limit equilibrium methods (Coulomb, 1776;Shields & Tolunay, 1973) shows that the traditional limit equilibrium method may greatly overestimate or underestimate the passive earth pressure coef® cients due to the a priori assumptions concerning the shape of the slip surface and the normal stress distribution along this surface.

Case of seepage ¯ow
From equation ( 9), the determination of the effective passive force P P requires the determina tion of the terms M 1 and M 2 .The hydraulic head distribution j(x, y, z) in the soil medium is gov erned by the following equation: In some simple cases, such as the case of the single sheet pile driven into a semi in®nite homo geneous soil medium, the pore water pressure distribution is given analytically (Soubra & Kast ner, 1992).For more complex geometry or for a multi layered soil medium with different coef® cients of permeability, the pore water pressure dis tribution cannot be known analytically and it requires a numerical resolution of equation ( 12).
For the double walled cofferdam considered in this paper, the numerical method used for the determination of the potential ®eld in the soil medium is the well known ®nite difference meth od where the differential equation (equation 12) is approximated by a ®nite difference equation.The boundary conditions used are shown in Fig. 4. The medium is discretized by a rectangular mesh (Fig. 5).The ®nite difference equations written at the different nodes form a system of linear equations whose unknowns are the values of the hydraulic head at the nodes.This system is solved by the Gauss Seidel method, using over relaxation in order to accelerate the rate of convergence.
The determination of the moments M 1 and M 2 has been made by numerical integration using the Gaussian quadrature method where the pore water pressures along both the sheet pile and the log spiral surface are determined by numerical interpolation.Let us pass now to the presentation of some numerical results obtained from the com puter program.Notice that all subsequent results concern the case when ã sat aã w 2 and B 0 a2 10 m (Fig. 1).

Case of a homogeneous and isotropic soil medium
Case of a single layer of in®nite depth.Soubra & Kastner (1992) published the results of the passive earth pressure coef®cients in the presence of seepage ¯ow in the case of a single sheet pile wall driven into a homogeneous and isotropic semi in®nite soil medium where the hydraulic head can be known analytically.Note that the same results have also been obtained by the present analysis using the ®nite difference method for the determi nation of the hydraulic heads.
Figure 6 shows the variation of the passive earth pressure coef®cient with Ha f for ö 308 and for four values of äaö (äaö 0, 1a3, 1a2, 2a3).For a zero K p value, the corresponding Ha f value is the same for different äaö values ( Ha f 2 X 78).This means that the angle of friction at the soil structure interface has no effect on the Ha f value causing failure by heaving.This fact can Fig. 5. Finite difference mesh 4. Boundary conditions for the seepage ¯ow of the double-walled cofferdam be explained as follows: when the effective passive force vanishes, there is no interaction at the soil structure interface and we have the tradi tional heaving phenomenon.For the same case, Terzaghi's approach gives a value of Ha f at fail ure equal to 2´82.Furthermore, the piping phe nomenon which appears for the critical hydraulic gradient at the point D (Fig. 4) occurs for a value of Ha f equal to 3´14.It should be mentioned that the numerical results have shown that in the case of a homogeneous and isotropic semi in®nite soil medium, the failure by heaving will occur before the piping phenomenon as long as ö is smaller than 458.
Figure 7 shows some charts of the variation of the passive earth pressure coef®cient as a function of Ha f for different values of ö (ö 208, 258, 308, 358, 408) and äaö (äaö 0, 1a3, 2a3).From these ®gures, the reduction of this coef®cient is quasi linear for the Ha f values varying from 0 to 2´5.
Finally, it should be mentioned that the K p value increases with ã sat .This increase is to be expected, since the soil weight has the favourable effect of increasing the stability of the soil mass in front of the sheet pile.For example, when ö 358, äaö 2a3 and Ha f 2, the increase of the passive earth pressure coef®cient attains 22% when ã sat aã w increases from 2 to 2´2.Case of a single layer of ®nite depth D. The numerical results obtained from the present pro gram have shown that the passive earth pressure coef®cient increases with the Da f decrease.The minimum relative depth Da f necessary to obtain the results of the semi in®nite case must be greater than or equal to 6.
Figure 8 shows some charts of the variation of the passive earth pressure coef®cient as a function of Ha f for different values of ö (ö 208, 258, 308, 358, 408) and äaö (äaö 0, 1a3, 2a3) and for Da f 2.
Case of a homogeneous and non isotropic soil medium.Figure 9 shows the variation of the passive earth pressure coef®cient with K h aK v when ö 358, Ha f 2 and äaö 0, 1a3, 2a3 and 1.There is a large decrease of the passive earth pressure coef®cient up to K h aK v 100.Beyond this limit, the passive earth pressure coef®cient tends to an asymptote.This can be explained by the fact that the equipotential lines in front of the sheet pile become quasi horizontal beyond a certain value of K h aK v and, thus, the potential ®eld does not change any more in the zone concerned with the failure mechanism.
Figure 10 shows some charts of the variation of the passive earth pressure coef®cient as a function of Ha f for different values of ö (ö 208, 258, Fig. 9. K p versus K h aK v for ö 358 and Haf 2 in the case of a homogeneous and non-isotropic semi-in®nite medium 308,358,408) and äaö (äaö 0, 1a3, 2a3) and for Da f 2 when the permeability ratio K h aK v 50.
Case of an isotropic two layered soil medium.In this section, we consider the frequent case of a cofferdam driven into a two layered soil medium.
Case where the bottom of the sheet pile lies in the upper layer (case A). Figure 11 shows the case of an isotropic two layered soil medium where the permeability coef®cients of the upper and lower layers are respectively K 1 and K 2 .
Figure 12 shows the variation of the passive earth pressure coef®cient with K 1 aK 2 when ö 358, äaö 2a3 and Ha f 2.
For the case of a single layer (K 1 aK 2 1), the passive earth pressure coef®cient is equal to 3´81.When K 1 aK 2 .10, one obtains the passive earth pressure coef®cient corresponding to the case of a single layer of limited depth (ö 358, äaö 2a3, Ha f 2 and Da f 2), since the lower layer can be considered as an impermeable substra tum.Notice, however, that for cases when the lower layer has a greater permeability coef®cient than the upper layer (K 1 aK 2 , 1), most of the head loss is concentrated in the upper layer, resulting in greater pore water pressures.Consequently, the passive earth pressure coef®cient decreases with the K 1 aK 2 decrease.
Case where the bottom of the sheet pile lies in the lower layer (case B). Figure 13 shows the case of an isotropic two layered soil medium.The unique difference from the previous case is that the bottom of the sheet pile wall lies in the lower layer.
Figure 14 shows the variation of the passive earth pressure coef®cient with K 1 aK 2 when ö 358, äaö 2a3 and Ha f 2. As in the previous section (case A), the passive earth pressure coef®cient in the case of an isotro pic single layer (K 1 aK 2 1) is equal to 3´81.For K 1 aK 2 .100, the head loss takes place solely in the lower layer, and the upper layer can be consid ered as a ®lter increasing the global stability of the soil in front of the sheet pile.Consequently, the increase of the passive earth pressure coef®cient with the K 1 aK 2 increase is a logical phenomenon.On the other hand, for cases when the lower layer has a greater permeability coef®cient than the upper layer (K 1 aK 2 , 1), most of the head loss takes place in the upper layer, resulting in a sig ni®cant reduction of its resultant body force (buoy ant weight seepage force).Consequently, the passive earth pressure coef®cient decreases with the K 1 aK 2 decrease.However, it should be noted that this calculation scheme is only valid as long as the vertical hydraulic gradient in the upper layer is smaller than the critical gradient i c ã9aã w , as otherwise failure by heaving of the upper layer will occur due to the fact that the saturated weight of this layer is equal to the resultant of the pore water pressures on the base of this layer.This limitation is shown in Fig. 14 by a vertical dotted line.

CONCLUSION
The variational limit equilibrium method was applied to the passive earth pressure problem, tak ing into account the seepage ¯ow due to dewater ing.It showed that the failure mechanism, in the general case of non homogeneous and non isotro pic hydraulic properties of the soil medium, is a log spiral.This method is equivalent to the upper bound method in limit analysis for a rotational log spiral mechanism.
In the case of no seepage ¯ow, the present upper bound solution is the smallest one existing in the literature and is very close to the currently ac cepted results of Caquot & Ke Ârisel (1948).
In the case of seepage ¯ow, the present mechan ism allowed us to determine the reduction of the passive earth pressures.For the limiting case corre sponding to zero passive pressures, the present mechanism describes the traditional heaving phe nomenon in front of the sheet pile.The numerical results have shown that:  shown that: (i) for great values of K 1 aK 2 , when the bottom of the sheet pile lies in the upper layer, one obtains the case of a single layer of limited depth, and when the bottom of the sheet pile lies in the lower layer, the upper layer can be considered as a ®lter; and (ii) for small values of K 1 aK 2 , most of the head loss takes place in the upper layer, thus resulting in a signi®cant reduction of the effective passive pressures.
Finally, one can see that the effect of soil aniso tropy and non homogeneity is signi®cant for the reduction of the passive earth pressures.Thus, the assessment of the reduction of the effective passive pressures taking into account these parameters is of great interest in the practice of geotechnical engi neering.
NOTATION B 0 breadth of the double-walled cofferdam D layer depth dl elementary surface along the slip surface f penetration depth H total hydraulic head loss h, v coordinate system whose origin is at point F i c critical hydraulic gradient K 1 , K 2 isotropic permeability coef®cients of layers 1 and 2 K h , K v horizontal and vertical permeability coef®cients K p coef®cient of passive earth pressure K x , K y , K z permeability coef®cients in the principal directions x, y and z M 1 moment of the force U 1 M 2 moment of the pore water pressures acting on the slip surface P p effective passive force r 0 , r 1 initial and ®nal radius of the log-spiral slip surface U 1 resultant of pore water pressures at the soil structure interface u pore water pressure W sat saturated weight of the soil mass OAB X distance between the wall and the line of action of the force W X 1 distance between the bottom of the wall and the normal component of the effective passive force X 2 distance between the bottom of the wall and the line of action of force U 1 y(x) equation of the slip surface in the (x, y) coordinate system y d y dx ä friction angle at the soil structure interface ö angle of internal friction of the soil j hydraulic head ã9 submerged unit weight of the soil ã sat saturated unit weight of the soil ã w unit weight of water ë relaxation factor ë 1 , ë 2 Lagrange's undetermined multipliers è 0 , è 1 angles de®ning the log-spiral slip surface ó (x) total normal stress distribution along the slip surface ó 9(x) effective normal stress distribution along the slip surface ó 9 dó 9 dx ô(x) tangential stress distribution along the slip surface

Fig. 1 .
Fig. 1.Double-walled cofferdam in a multi-layered soil medium The constraint equations.(c) The boundary conditions: at the ®xed endpoint

Fig. 2 .
Fig. 2. (a) Slip surface for passive earth pressure analysis.(b) Free body diagram Fig. 3. Log-spiral slip surface for passive earth pressure analysis

Fig. 8 .
Fig. 8. K p versus Haf for different values of ö and ä in the case of a homogeneous isotropic single layer of ®nite depth (Daf 2)

Fig. 12 .
Fig. 10.K p versus Haf for different values of ö and ä in the case of a homogeneous and non-isotropic single layer (Daf 2) (a) The heaving of a soil mass in front of the sheet pile occurs before the piping phenomenon in the case of a homogeneous and isotropic semi in®nite soil medium as long as ö is smaller than 458.(b) The effective passive pressures increase with the decrease of the layer depth in the case of a single layer problem.

Fig. 14 .
Fig. 13.Case of an isotropic two-layered soil medium: case B

Ã
Ecole Nationale Supérieure des Arts et Industries de Strasbourg { Institut National des Sciences Applique Âes de Lyon.

Table 1 .
Passive earth pressure coef®cient K p as given by different authors for ö 408, äaö 1a2