Engineering Implications of the Relation between Static and Pseudo-Static Slope Stability Analysis

Robert Shukha

Graduate student at Faculty of Civil and Environmental Engineering,
Technion - I.I.T. Haifa, Israel

Rafael Baker 1

Associate Professor, Faculty of Civil and Environmental Engineering,
Technion - I.I.T., Technion City, Haifa, Israel.
Fax 972-4-8292322, E-mail baker@techunix.technion.ac.il

Dov Leshchinsky

Professor. Department of Civil and Environmental Engineering,
University of Delaware, Newark, Delaware, USA

1Corresponding Author

ABSTRACT

The results discussed in the present work are based on the observation that every pseudo-static slope stability problem can be transformed into an equivalent static problem with modified geometry and unit weight. This basic equivalency relation is not new, and it is mentioned, in passing, in several text books. However, the implications of this equivalency have not been previously investigated. The equivalency between pseudo static and static problems allows for a straightforward transfer of experience and results from the relatively mature static framework into the less developed pseudo static context. The main new results implied by this transformation include a general analytical solution for cohesionless materials; a criterion for prevention of tension crack during an earthquake event; and a limiting solution for cases resulting in deep critical slip surfaces. It is shown that the conventional pseudo-static format yields extremely conservative results for conditions resulting with deep critical slip surfaces. A conceptual framework for reducing this excessive conservatism is presented and discussed.

The analysis brings to light previously unnoticed discrepancy between the states of practice in the analysis of waterfront structures and submerged slopes. It is verified that under certain set of reasonable assumptions the equivalency between static and pseudo-static problems implies that earthquakes induce outward water flow in submerged slopes, and this effect is associated with very significant reduction in minimal safety factors.

Keywords: Slope stability, Limiting equilibrium, Pseudo static problems.

Introduction – The pseudo-static slope stability problem

Most design guidelines and codes (e.g. BSI 1995, FHWA 1997, Eurocode 8 1993) recommend a simple pseudo-static (PS) analysis as the standard procedure for evaluating stability of slopes under seismic conditions. This recommendation is restricted to situations for which liquefaction is not an issue, and the present work is limited to such situation.


Figure 1. The Pseudo-static slope stability problem.

The PS approach is a generalization of conventional limiting-equilibrium (LE) slope stability analysis in which seismic effects are represented by an additional body force dE as shown in Fig. 1. In this figure b, g, and H are the slope angle, unit weight of soil, and height of the slope, respectively; the functions y(x) and ys(x) represent a potential slip surface and the slope surface, respectively; is a differential body force due to gravity where da is a differential area of the potential test body ObDO; is a differential PS body force representing seismic effects, k is a non-dimensional PS coefficient which is usually specified by codes based on seismic information. For simplicity, the PS force dE in Fig. 1 is assumed to act in the horizontal direction (the usual assumption), but it is possible to perform PS analysis when this force acts in any specified direction.

Let t(x) and s(x) be the distributions of shear and normal stress over y(x), respectively; The basic assumption of the LE framework (Baker 2003) can be written as where F is the safety factor with respect to shear strength, and c, f are the shear strength parameters cohesion and angle of internal friction respectively. The safety factor F is associated with the particular slip surface y(x) shown in Fig. 1, i.e. F = F[y(x)]. Consistent with the general LE approach, the safety factor of the slope, Fs, is obtained by minimizing F[y(x)] with respect to all admissible slip surfaces y(x), i.e.  where is the critical slip surface.

The safety factor with respect to shear strength, F, depends on the given value of the PS coefficient k, i.e. Fs = Fs(k), and Fig. 1 represents the "basic format" of PS slope stability problems.  For certain purposes it is convenient to define a critical PS coefficient kc as the k value associated with a state of failure, at which Fs = 1 (Sarma and Bhave 1974), and to introduce an alternative safety factor Fk as Fk = kc/k. Such a representation is an alternative formulation (alternative format) of the same basic problem. The critical PS coefficient kc does not depend on information with respect to earthquakes. It is noted that Fs is a safety factor with respect to shear strength while Fk is essentially a safety factor with respect to loads. In general  and these two factors become equal only at failure when.

Equivalency between static and pseudo-static slope stability problems



Figure 2. The basic transformation.
(a) Combination of body forces
(b) Rotation of coordinate system.

The basic PS problem shown in Fig. 1 can be redrawn as shown in Fig. 2a.  In this figure, the resultant vector of the differential forces dE and dW is denoted as dP. The inclination of this force is to the vertical, and its magnitude is where . In addition, introduce a new coordinate system, which is rotated at angle  to the original system as shown in Fig. 2a.  Finally, to simplify the presentation, Fig. 2a is rotated until the coordinate is horizontal; the results of this process are shown in Fig. 2b.

Figures 1 and 2b are merely different representations of the same slope stability problem (combining forces and changing coordinate systems does not affect the physical problem). Figure 1 show a PS problem; while Fig. 2b is a conventional static problem with modified g and slope geometry. Comparison of Figs. 1 and 2b establishes the basic Static-PS equivalency of slope stability problems. This equivalency means that solving the static problem in Fig. 2b will render the same results (same minimal safety factor and critical slip surface) as the original PS problem in Fig. 1. The following comments are made with regards to the established equivalency:

1. The equivalency is valid for all non-homogeneous problems with arbitrary geometry, external loads, and pore water pressure distribution.

2. The equivalency is valid for all LE slope stability methods (the transformation is applied to the problem, not to the solution).

3. For clarity of presentation Figs. 1 and 2a show slip surfaces passing through the toe, however, the equivalency is valid for any slip surface geometry.

4. The equivalency can be easily extended to situations in which the PS force dE is inclined rather than horizontal. In this slightly more general case  and  where  are horizontal and vertical PS coefficients respectively, and positivevalues means that the vertical component of the PS force is acting in the direction of gravity.

 

Combining results, a static problem which is equivalent to a given PS problem is characterized by:

(1a)

(1b)

(1c)

(1d)

Sloping base and crest lines inclined at

 and (1e)

respectively.

These relations show that the slope in the equivalent static problem is steeper, higher, and has larger unit weight than the original PS problem. Consequently, the above transformation renders a static problem in which all the aspects responsible for reduced stability are magnified. It is noted that using the positive sign in Eqs. (1a) and (1b) (i.e. considering a PS force acting in the downward direction), reduces  but increases the magnitude of. These two effects have opposite influence on safety factors and it is impossible to establish a priori which of the two possible direction of the vertical PS force is critical (i.e. when dealing with a specific problem both alternatives have to be checked).

An approach similar to the rotation transformation described above was previously used in the context of dynamic earth pressure calculations. Seed and Whitman (1970) mentioned that such a transformation can be used as an alternative way to derive the Mononobe-Okabe equations. They attributed this idea to a personal communication with Dr. Arango. Greco (2003) extended Mononobe-Okabe equations to more general geometries using a similar approach. In a slope stability context this transformation is mentioned in passing in number of textbooks (e.g.  Bromhead (1992),  Muir Wood (2004), Duncan and Wright (2005), but these authors did no investigated its engineering implications. It is important to realize that such a transformation is expected to be useful to all LE "extreme value problems" (Garber and Baker 1979) (e.g., active/passive lateral earth pressures, bearing capacity, slope stability etc.).

Figures 3a to 3c provide an "experimental verification" of the theoretical equivalency between static and PS problems.  This verification was carried out for the following simple case
.
The solutions for the static and PS problems were obtained using Bishop’s simplified method. It can be verified (Shukha 2004, Shukha and Baker 2003), that for homogeneous conditions Bishop's procedure provides a very good approximation for results obtained by more advance LE methods and strength reduction techniques (Zienkiewicz et. al. 1975). Solving the PS problem in the original coordinate system (i.e. without the transformation), PS forces are assumed to act at mid-height of each slice (implying that the resultant PS force acts at the center of gravity of the sliding mass).




Figure 3. Verification of the equivalency between static and pseudo-static problems.
(a) Solution in physical coordinates 
(b) Solution of the transformed problem.
(c) Comparison of solutions.

Fig. 3a shows the results of the stability analysis for static and PS conditions. Fig. 3b shows the equivalent static problem associated with the PS problem shown in Fig. 3a, and the solution of this equivalent problem. Using Eq. (1), the parameters defining the equivalent static problem are
,
and base/crest lines sloping at 8.5°.  It is noted that solving the problem in the rotated coordinate system results in a critical slip surface in that system. The corresponding solution in the physical coordinates is obtained by rotating Fig. 3b backwards by the angle. Figure 3c compares the results obtained with and without the rotational transformation.  The following comments are made with regard to the results shown in Figs. 3:

1. Figure 3c shows that the solutions of the original PS problem and the equivalent static problem are identical for both minimal safely factor and critical slip surface. This observation constitutes an "experimental" verification of the general equivalency  between static and PS problems.

2. Figures 3 show that critical slip surfaces for static and PS problems are not identical.  It can be verified that generally the PS critical surface defines a larger and deeper sliding body than the static one, i.e. where  are critical slip surfaces for the PS and static cases respectively. The difference between static and PS critical slip surfaces is large for situations resulting in deep seated critical slip surfaces, and it is relatively small in cases for which critical slip surfaces pass through the toe of the slope. 

3.  The numerical effort required for solving the problems in Figs. 3a and 3b is similar. Consequently, the rotational transformation is not expected to provide a useful numerical tool for solution of practical (non homogeneous) PS problems. However, since the transformed problem represents static conditions, the transformation allows one to apply the vast knowledge, experience intuition and judgment, accumulated using static problems, to PS situations for which no similar knowledge-base exists.  Stated differently, the main utility of the established equivalency between static and PS slope stability problems is that it allows transfer of known results from the mature static framework into a PS context. In addition; using this transformation it is possible to utilize computer programs that do not include a PS option for solution of PS problems.  Furthermore; using the equivalency between static and PS problems, benchmark cases can be established to assess correctness of computer codes (i.e. a situation in which a given computer program does not yield the same solution for a pair of corresponding static and PS problems is an indication of a programming "bug", and this feature provides a very convenient mean of "debugging" existing programs).  

Implication of the equivalency between static and PS problems

The present section establishes various implications resulting from the equivalency between static and PS slope stability problems. For simplicity, all cases discussed in the present section are based on the assumption  and  (i.e. the usual practice of assuming that PS forces act in the horizontal direction), however the obtained results are valid also in the general case of arbitrary given values of .

a) Cohesionless soils

Baker and Garber (1978) introduced a general variational LE approach to static slope stability problems. The main advantage of the variational approach compared with conventional LE formulations is that it does not introduce kinematic or static assumptions. Using this approach Baker (1981) verified that for materials the critical slip surface coincides with the steepest section of the slope surface, and the factor of safety is given by where is the inclination of the steepest section of the slope surface. This solution is a generalizations of classical results which are usually derived in the infinite slope context (where there is no difference between  and), and it is valid for finite height slopes with general surface geometries.


Figure 4. Pseudo-static solution for cohesionless materials.

Consider a PS analysis of c = 0 materials. Based on the general relation  (Fig. 3a) it is not obvious that the limiting relation is valid for PS conditions. However, considering the rotated geometry in Fig. 4 as a conventional static problem, it can be seen that the classical relations are valid for PS problems in materials. The equivalency between static and PS problems implies therefore:

(2)

where quantities written to the right of the vertical line are considered as given. Furthermore, setting  in Eq. (2), and solving for the critical PS coefficient results in:

(3)

Eqs (2) and (3) are, apparently, new general analytical solutions of PS problems for c = 0 materials. These solutions are direct consequences of the equivalency between static and PS problems.


Figure 5. Numerical results for cohesionless materials.

Leshchinsky and San (1994) applied the variational procedure of Baker and Garber in order to solve  PS problems without application of the rotational transformation. They present a complete stability chart for the case of materials. Using this chart it is possible to establish the values shown as point in Fig. 5, and compare them with the analytical solution given by Eq. (3). The following comments are made with respect to Fig. 5:

1) The points in Fig. 5 were obtained by digitizing graphical results presented by Leshchinsky and San.  The scatter of these points is a consequence of both numerical inaccuracies in Leshchinsky and San's presentation (occurring while solving the set of several highly nonlinear equations resulting from the variational framework), and digitization of their results. Ignoring numerical errors, the results in Fig. 5 provide a mutual and independent substantiation of Eq. (3) and the numerical results derived by Leshchinsky and San.

2) It is interesting to note that the limit expressed in Eq. (3) (i.e. ) is exactly the limit at which the well-known Mononobe-Okabe equation breaks down as the argument in its square root becomes negative for.  Similar situation occurs in the static Coulomb’s lateral earth pressure equation which breaks down when the inclination of the crest line, exceeds.  Eq. (3) provides a physical interpretation to the mathematical breakdown in both cases. This breakdown is associated with surficial instability occurring when considering impossible (unstable), inclinations of natural (unsupported) part of the slope.


Figure 6. Pseudo-static conditions associated with tension cracks.

b)  Tension cracks

Consider a case in which both  and  are large enough so that in the equivalent static problem .  In that case the face of the slope in the rotated representation (Fig. 2b) will form an overhanging cliff as shown in Fig. 6.  Baker and Leshchinsky (2003) studied failure mechanisms involving overhanging cliffs and showed that such slopes are stable only in materials possessing significant tensile strength.  However, tensile strength of most soils is negligibly small and a soil slope with an overhanging cliff will crack, losing its stability.  Hence, in order to prevent tension cracks during an earthquake event, slope angles should be limited to:

This very simple design recommendation follows directly from the equivalency between PS and static problems. k values recommended by most codes rarely exceeds 0.35, resulting in and Eq. (4) implies the sensible recommendation . It should be noted that this recommendation is necessary, but probably not sufficient in order to eliminate formation of tension cracks during an earthquake. Physically, a PS force pointing away from the main body of the slope decreases normal stress over the slip surface, possibly generating tensile stresses near the crest where overburden stresses are small.  The geometry of the overhanging cliff in the rotated representation generates an equivalent state of tensile stresses. 

 

c) Deep slip surfaces

Baker and Leshchinsky (2001) generalized conventional slope stability calculations by introducing the notion of "safety maps". A safety map is a 2D functionspecifying the minimal safety factor at each interior point of the slope. This function is obtained by minimizing with respect to all potential slip surfaces passing through a given spatial point in the interior of the slope. The notion of safety maps provides a powerful diagnostic and interpretive tool which has significant advantages over the common practice of examining only minimal safety factors and critical slip surfaces. For example, inspecting such a map, it is possible to identify zones within a slope in which safety factors are lower than required design value.  Hence, zones where retrofit (e.g., reinforcement) is needed can easily be identified.



Figure 7. Deep slip surfaces.
(a) Solution in physical coordinates.
(b) Solution of transformed problem.

Consider a homogeneous problem defined by the input information  . Figures 7a and 7b show the results (in the original and rotated representations respectively), obtained while attempting to solve this problem using the simplified Bishop method. The lines ab and cd in Fig. 7a represent the distribution of local safety factors along the lower and upper boundaries of the slope (i.e. ab represents the function and cd is the function ). These functions should be interpreted in the safety maps context.

In the example shown in Fig. 7a the functionsand decrease monotonically and they do not indicate presence of a minimum. The search process for the critical slip circle of this problem was terminated when the "critical circle" emerged  before the toe point, and it had a radius of. The safety factor along this limiting slip circle was found to be . Exactly the same safety factor was obtained for the transformed problem shown in        Fig. 7b. The "solutions" in Figs. 7 are obviously not legitimate (critical conditions occurring on slip surfaces passing through the boundary of the search region). The above results indicate that the "true" critical conditions for this case would be realized on a slip circle having an infinite radius, and such problems cannot be analyzed numerically.

Examination of the results in Fig. 7a (physical coordinates) is not particularly enlightening. However inspection of Fig. 7b shows clearly that this figure represents essentially an infinite slope problem in which the slope is inclined at an angle  ( is the inclination of the short section between the toe (T) and crest (C) points in Fig 7b, and for deep enough slip surfaces this section can be ignored). Stated differently, for deep enough slip surfaces the limiting form of the pseudo-static problem corresponds to the simple case of infinite slope, and it is possible to utilize the well known static solution of this problem in order to derive results relevant to the PS case. This observation illustrates again the significant intuitive and suggestive power of the rotational transformation.

When  the static solution of the infinite slope problem implies that the critical slip surface is realized at the boundary  between the soil and underlying bedrock, and the safety factor is given by the well known relation . In order to apply this solution to the rotated geometry shown in Fig. 7b, replace in the above equation  by,  by (Eq. 1b), and d by  where is the height of the slope and - a non-dimensional depth factor.  Performing these substitutions results in:

(4)

Setting in Eq. (4) and solving for  yields:

(5)

The following comments are relevant with respect to Eqs (4) and (5):

1) Inspection of Figs. 7 makes it clear that for the input data considered in that figure the critical conditions are realized on a slip circle passing through minus infinity. In this case the term  in Eq. (4) approaches zero, and this equation yields . The numerical "solutions" in Figs 7 yielded  because using a numerical process circles with infinite radii can not be considered. Combination of the rotational transformation with the infinite slope solutions makes it possible to reach the limiting result "directly", illustrating again the utility of this transformation. 

2) Loukidis et.al. (2003). derived Eq. (5) based on limit analysis. In their paper this equation is presented as being valid for all values. However, the present derivation shows clearly that       Eq. (5) is valid only if (without this restriction the rotated geometry in Fig. 7b is not equivalent to an infinite slope). Baker et. al. (2005) verified this limitation of Eq. (5) using numerical results based on FLAC strength reduction method (ITASCA 2002).

3)   Baker (2004) showed that under static conditions occurs only if  is identically equal to zero and. The results in Fig. 7 suggest that in PS problems this singular situation occurs also for finite  values.  This is a consequence of the general relation   (Fig. 3a) which shows that the critical slip surface in PS problem is deeper than the corresponding static one. As a result, a hard bottom has a more significant effect on results of PS problems than in the static case (conditions resulting with the critical slip surface being constrained by the hard bottom can be realized for a larger ranges ofand  values).

d)  Coupled PS analysis

Consider applying Eq. (5) to the particular case of and. Under such conditions              for static conditions; the relation  implies that also in the PS case. As a result Eq. (5) yields  for all values of . This result implies that the slope will not be stable under any earthquake, regardless of the magnitude of its cohesive strength. This unreasonable conclusion is the result of the fact that test bodies having infinite volume are acted upon by an infinitely large resultant PD force. In order to eliminate this inconsistency it is convenient to reconsider some basic elements forming the conceptual basis of the PS approach.

Seed and Martin (1966) applied a one dimensional shear beam model in order to evaluate PS coefficients for earth dams. Their approach was based on the assumption that the PS coefficient is proportional to the average acceleration that the earthquake induces in the dam, and they averaged the local spatial acceleration function over the fixed cross section of the dam. In principle  depends also on time, but the time variable is usually eliminated by focusing attention on the moment at which the input bedrock acceleration attains it maximum value. This basic assumption can be formalized by a relation of the form  where  is a constant of proportionality and aavr is the average of a(x, y). Ashford and Sitar (2002) generalized this approach by averaging over the cross sections of potential test bodies used in LE analysis and this practice will be followed here.

The local spatial acceleration is a pseudo-periodic spatial function (Kramer1996). The average of such functions depends on the size of the averaging zone, and it approaches zero when the size of the averaging zone approaches infinity.  These observations imply that where  is a characteristic size of the critical test body resulting from LE calculations and  is a characteristic spatial wave length of the earthquake. The function k(B/l) introduces a coupling between seismic and LE aspects of PS stability analysis. A simplified version of such an approach (based on infinite slope analysis) was discussed by Bray et. al. (1998). Some of their results can be interpreted in terms of the function, and these results verify the limiting relation.

The common practice of specifying PS coefficients based on seismological considerations alone ignores the coupling between seismic and LE aspects of PS stability analysis. Such a simplification is probably reasonable for conditions resulting with relatively shallow critical slip surfaces in which variations of B values are not large; however, it leads to unacceptable results for conditions resulting with very deep critical slip surfaces. Considering the coupling between seismic and LE aspects of PS stability analysis the safety factor can be written as. Eq. (5) implies that in purely cohesive slopes  for conditions resulting with infinitely deep slip surfaces. However, under these conditions also approaches zero, so  may remain finite, thus eliminating the inconsistency that flat non frictional slopes cannot withstand any earthquake. The above arguments provide a conceptual framework for coupled PS analysis. At this stage the main practical conclusion based on the above discussion is that application of conventional (un-coupled) PS analysis to situations resulting in deep critical slip surfaces is very conservative.

e)  Pseudo-static analysis of submerged slopes

Application of PS analysis to submerged slopes is questionable for a number of reasons, the most important one being a strength loss due to liquefaction. Nevertheless it is instructive to consider the implication of the rotational transformation with respect to such problems, and the following discussion is limited to situations in which liquefaction is not an issue (e.g. dense sand or heavily over consolidated clays).



Figure 8. Pseudo-static analysis of submerged slopes
(a) The physical problem.
(b) The transformed problem.

There are two different "schools of thought" with respect to PS analysis of submerged slopes:

Approach I

- The first school considers the effect of the free water standing near the slope (Fig. 8a), as a "surcharge" which is not affected by the earthquake. It is noted that an extensive literature search failed to yield even a single reference explicitly justifying the assumption that earthquakes do not affect the distribution of water pressures (surcharge) acting on the submerged face OA of the slope. However such an assumption is implied by the structure of almost all common slope stability programs, which do not include a method for modifying the static surcharge as a function of the PS coefficient k.

Approach II

- In the second approach free water is considered essentially as a zero shear strength material, which is acted upon by PS forces. The PS forces acting in the free water reduces the supporting hydrostatic pressure which the free water applies to the slope, thus reducing its stability. This type of approach is common for stability analysis of waterfront structures (e.g. Ebling and Morison 1992). When used in the waterfront design framework, the effect of seismic loads acting in the free water is modeled by a hydrodynamic pressure (Westergaard 1933) which reduces the supporting static water pressure acting on the slope (dam). Westergaard's original solution was restricted to vertical dams; however Chwang and Housner (1978) extended this solution to slopes having an arbitrary inclination. Matsuzawa et. al. (1985) presented experimental results supporting the validity of the hydrodynamic pressure notion.

Consider now formal implications of the rotational transformation on PS analysis of submerged slopes. Figures 8a and 8b show a schematic submerged slope in the physical and transformed coordinates respectively. Two features of the problem defined in Fig. 8b are significant for the present discussion:

(1) The rotational transformation results with the inclined phreatic surface CD in Fig. 8b. Such a surface implies that the equivalent static problem represents a situation in which there is a uniform seepage in the direction  (parallel to the rotated phreatic surface). Stated differently, the rotational transformation implies that earthquakes induce flow in submerged slopes. This result illustrates again the very significant intuitive power of this transformation. The physical "source" of such a flow is related to the reduction of the hydrostatic pressure acting on the slope's face due to the effect of the earthquake on the free standing water. Consequently the rotational transformation is consistent with Approach II.

(2) For c = 0 materials the solution of the static problem shown in Fig. 8b is well known (Bolton 1979).  The critical slip surface in this solution coincides with the slope's face (the heavy dashed line OA in Fig. 8b), and the safety factor is given by  where a is the angle between the flow and surface directions, gT = Total unit weight, gw = Unit weight of water, and gsub = gT - gw is the submerged unit weight. This solution is valid for steady state flow rather than the transient conditions existing during an earthquake. However the basic premise of PS analysis is the assumption that dynamic problems can be approximated by equivalent static problems. Consequently the above solution is consistent with the PS framework.

In order to apply this static solution to PS problems replace in the above equation b by b + q and a by b (the rotational transformation does not change the angle between the slope and phreatic surfaces). It is noted that in the present case there is no need to modify unit weights according to Eq. (1b) since the effect of this modification cancels out. Performing the above transformations results in:

(7)

Equation (7) is the general PS solutions for submerged slopes, based on Approach II.

Consider the following specific input list           and assume that the static phreatic surface is located at the top of the slope. Under static conditions the existence of this phreatic surface is not consequential, and the safety factor is equal to, while application of Eq. (7) yields .

In order to verify that the very dramatic reduction in the minimal safety factor associated with a relatively moderate PS coefficient  is not an error; the same problem was solved numerically using Bishop's method. Figures 9a and 9b show complete safety maps (Baker and Leshchinsky 2001) obtained in the physical coordinates using the commercially available program ReSSA(2.0) (Leshchinsky and Han, 2003). The boundaries between the colored zones in Figs. 9 are safety factor contours along which  are constants, and the line AB in these figures is the safety contour associated with.



Figure 9. Numerical Pseudo-static solutions of submerged slopes.
(a) Approach II.
(b) Approach I.

Figure 9a shows results based on Approach II. Invoking effective stress analysis, ReSSA computes the porewater pressure along the slip surface using the phreatic surface.  Under static conditions the effect of the ponding water on the surface of the slope is computed based on the resultant water pressure acting on top of each slice.  In case of seismicity, this resultant includes an additional horizontal force component that is equal the weight of the water column above the slice multiplied by .  The following observations are relevant with respect to Fig. 9a:

a) The numerical solution process converged to a critical slip surface approaching the surface of the slope and a safety factor of (compared with the theoretical value of 0.31 based on         Eq. (7)). It is realized that the way seismicity is applied to pounding water in ReSSA is just one possible approximation, and there are alternative ways to formulate the same problem. However the correspondence between the numerical (Fig. 9a) and analytical (Eq. (7)) results provides a mutual substantiation for both the formal validity of the general solution given in       Eq. (7) and the numerical process incorporated in ReSSA.

b) The critical slip surface is not physically significant when the minimal safety factor is less then unity. Under such condition the contour line associated with defines the lower boundary AB of the unstable zone and only this line is expected to have physical manifestations in the field. It is noted that the numerical solution for the critical slip surface in Fig. 9a is realized near a singularity and this introduces significant difficulties in the derivation of this solution. However the magnitude of safety factors along the surface of the slope is not practically significant, and this singularity is not present at interior points of the safety map (i.e. the general structure of this map is not affected by the singularity). Both of these observations illustrate the significant utility of the safety map notion introduced by Baker and Leshchinsky.

Figure 9b shows similar results based on Approach I (i.e. neglecting earthquake effects on the water pressures acting on the face of the slope, which is the common practice in routine PS slope stability analysis). In this case too, the critical conditions occur for , however the most significant difference between the results shown in Figs. 9a and 9b is that the minimal safety factor associated with Approach I is , and this value is significantly larger than the corresponding value (0.30) based on Approach II.

The dramatic reduction of safety factors based on Approach II is the result of application of PS forces to the free water standing near the slope. It is not obvious that that PS coefficients for soil and free water are equal, but it is doubtful that setting the PS coefficient in the water to zero (as implied by Approach I) is the proper solution to this question.  At this stage it is not clear which one of the two approaches (I or II) is more appropriate. In view of the practical significance of PS analysis of submerged slopes (inner face of water reservoirs, stability of sea bottom slopes etc.), it is rather surprising that this issue has not received a more complete treatment in the professional literature. The attitude of the authors is that in order to ensure a safe design, PS static analysis of submerged slope should be based on Approach II, until it is definitely established that the more liberal Approach I is really justified.

Summary and conclusions

The results established in the present work are based on the observation that every PS slope stability problem can be transformed into an equivalent static problem with modified geometry and unit weight. The equivalency between PS and static problems results in a straightforward mechanism allowing transfer of known results from the relatively mature static framework into the less developed PS context.  The main new results implied by this transformation include:

- A general analytical solution for cohesionless materials.

- A criterion for prevention of tension crack during an earthquake event.

- A limiting solutions for cases resulting in deep critical slip surfaces. It is shown that the conventional (un-coupled) PS format yields extremely conservative results for condition resulting with deep critical slip surfaces. A conceptual framework aimed at reducing this excessive conservatism is presented and discussed.

- Under certain set of reasonable assumptions (PS forces acting on the free water), the equivalency between static and PS problems implies that earthquakes induce outward water flow in submerged  slopes, and such flow is associated with very significant reduction in minimal safety factors. Discussion of this case identified an unresolved difficulty associated with PS analysis of submerged slopes. This discussion shows also a previously unnoticed inconsistency between the state of practice used in PS analysis of slopes and waterfront structures.

References

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  9. Bolton, M. (1979) A Guide to Soil Mechanics.  John Wiley and Sons N.Y.

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