EJGE Paper 2005-0514

 

 

Variational Slope Stability Analysis of Materials with Nonlinear Failure Criterion

 

Rafael Baker

Associate Professor, Faculty of Civil and Environmental Engineering,<br> Technion - Israel Institute of Technology, Technion City, Haifa. 32 000, Israel<br> E-mail: baker@techunix.technion.ac.il

Abstract

A rigorous variational slope stability analysis for materials with a nonlinear failure criterion is presented. The main advantage of the variational framework over classical slope stability procedures is that it does not introduce any geometrical or static assumptions. The present paper introduces general approuch, results, and the solution procedure associated with the variational methodology. Specific results (stability charts and failure modes) based on this methodology, and discussion of their engineering significance, are presented in a separate publication. Various properties of critical slip surfaces and normal strength functions are established. In particular it is shown that: a) Critical slip surfaces possess the characteristic property of log spirals, namely the angle between the normal and the radius vector is equal to the local value of the mobilized friction angle (which is not constant when nonlinear strength functions are considered). b) Critical normal stress distributions are convex functions possessing a single maximum in their range of definition. c) Critical slip surfaces are singular at points where normal stresses approach the tensile strength of the material, and this singularity has a major effect on the solution of the problem. Inspection of particular solutions illustrates that common static assumptions employed in classical limiting equilibrium procedures are not justified in the sense that they are not associated with the critical conditions. More important however, the present work shows that such static assumptions are not only unjustified; being in fact unnecessary.

Keywords: Non-linear failure criteria, Slope stability computations, Variational analysis.

Introduction

A substantial amount of experimental evidence (e.g. Penman (1953), Bishop et. al. (1965), Ponce and Bell (1971), Agar et. al. (1985), Atkinson and Farrar (1985), Day and Axten (1989), Maksimovic (1989)) suggests that failure criteria of many soils are not linear when large range of normal stresses is considered. Using approximate limiting equilibrium analysis Jiang et. al. (2003) and Baker (2003a) showed that nonlinearity of the failure criterion may have a very significant effect on slope design. Previous studies concerning the effect of nonlinear failure criteria on slope stability calculations (e.g. Maksimovic (1979), Lefebvre (1981), Charles and Soares (1984), Collins et. al. (1988), Perry (1994), Jiang et. al. (2002), Baker (2003a)) utilized approximate limiting equilibrium (L-E) procedures which are based on various assumptions. The present work investigates the implications of strength functions nonlinearity based on an extension of the rigorous variational approach to slope stability analysis presented by Baker and Garber (1977) and (1978).

Non-linear Mohr envelopes

A substantial amount of experimental evidence suggests that failure criteria of many soils are not linear, particularly in the range of small normal stresses. Considering a wide variety of geological materials; ranging from clays and silt to sand, gravel and rocks; Baker (2004a) verified that existing experimental information is consistent with the following nonlinear (N-L) strength function (Mohr envelope).

(1)

where are normal and shear stresses acting at failure on the failure plane, pa stands for atmospheric pressure and {A, n, T} are non dimensional strength parameters. Equation (1) is the Mohr form of the empirical strength criterion for rock masses introduced by Hoek and Brown (1980); expressed in terms of different strength parameters. The following comments are relevant with respect to Eq. (1).

(1) The class of strength functions defined by Eq. (1) includes the linear Mohr-Coulomb (M-C) criterion as a limiting case associated with  where are the conventional strength parameters cohesion and angle of internal friction respectively.

(2) Legitimate Mohr envelopes are real, non-negative, monotonically increasing and convex functions. Those requirements introduce the restrictions A > 0, and n£1. A legitimate Mohr envelope should not cross twice the same Mohr circle; Jiang et. al. (2003) showed that this requirement introduces the restriction .

(3) The parameter n controls the curvature of Str(sf). The parameter T is a non dimensional tensile strength and tNL = paT plays a similar role to  in the linear M-C framework. Most soils have very small (effective), tensile strength (Baker (2004a)) and setting T=0 in Eq. (1) defines a limiting class of zero tensile strength materials with nonlinear failure criterion. Numerical results in the present paper are restricted to this case. However most of the theoretical results derived in the present paper are valid for the more general case of , and it is convenient to not to introduce the restriction T=0 at the present stage.

(4) The derivative of Str(sf) with respect to sf can be interpreted as a stress dependent, tangential friction coefficient. The explicit form of this function is

(2)

where  is the corresponding tangential friction angle.

Equation (2) shows that . This result is a consequence of symmetry of Mohr circles and envelopes with respect to the  axis, and it is not related to physical friction (at the tensile strength point  the strength function represents tensile rather then frictional shear strength). It is noted that the M-C criterion satisfies the same relation; however this criterion has a slope discontinuity at the tensile strength point.

The slope stability problem

Classical formulation

As their name implies limiting-equilibrium (L-E) procedures are based on only two elements:

Equilibrium conditions for a test body bounded by a potential slip surface y(x) and the surface of the slope ys(x).

A limiting relation between normal and shear stress acting along a given slip surface. This relation introduces also the notion of safety factor with respect to shear strength F. The general form of the limiting relation used in the L-E framework is:

(3)

The function  is frequently called “mobilized strength envelope.” Equation (3) is fundamental to all L-E procedures and it constitutes the basic assumption of all such procedures. Baker (2003b) discussed various implications of this assumption.

For the purpose of slope stability applications it is convenient to rewrite Eq. (3) in the following alternative form:

(4.1)

where:

(4.2)

and (4.3)

 is the total unit weight,  stands for the slope’s height,  is a non dimensional normal stress acting on the slip surface, STR( S ) is a non dimensional strength function. The parameter  relates the experimental normalization of stresses (with respect to pa) to the normalization with respect to  which is convenient in slope stability applications.

Basic elements of the slope stability analysis are introduced in Figure 1 which shows a straight homogeneous slope without pore pressure or external loads. The geometry of the slope is deified by the slope inclination  and slope’s height .


Figure 1. Basic conventions and definitions.

In this figure LTFCHDBL is a potential test body, with the distance HD representing an unloaded tension crack. Considering Fig. 1 and using Eq. (4) it is possible to derive the following non dimensional representation of the 3 equilibrium conditions for a general test body:

(5.1)

(5.2)

(5.3)

where

(5.4)

(5.5)

and  are non dimensional coordinates centered at the toe point of the slope (Fig.1).  is a potential slip surface.  and  are the “low” and “high” end points of  (slip surfaces with  or , define a slope with “reduced height”, and in homogeneous problems such surfaces can not be critical).  is a function representing the surface of the slope, and  is the depth of a tension crack at H.  is the derivative of .  represents the distribution of non dimensional normal stresses along . The dot in the notation  represents an arbitrary pair of functions.  are the resultant horizontal and vertical forces acting on the test body. Equations (5.1), and (5.2) are the conditions of horizontal and vertical equilibrium respectively.  is the resultant moment about the toe point T, and Eq. (5.3) is the condition of moment equilibrium.  depend on the functions , i.e. they are functionals. We use the convention that names of functionals are written in heavy bold letters.

Equations (5) combine the two basic elements utilized in L-E slope stability analysis (equilibrium of a test body, and definition of the safety factor), and they will be referred to as the basic equations of limiting equilibrium.

Experience with conventional limit equilibrium analysis shows that there exist many triplets ; satisfying Eq. (5). A pair  satisfying these equations constitutes a legitimate failure mechanism. Each legitimate failure mechanism is associated with some value of . Consequently the equations of limiting equilibrium define an implicit functional relation of the type. This functional is called here the safety functional. It is not possible to establish an explicit expression for this functional, however, for the present purpose, the important point is only that such a functional exists.

The basic slope stability problem is to find a pair of critical functions  which satisfy Eq. (5) and are associated with the minimum value  of the safety functional , i.e. the formal statement of the slope stability problem is:

(1)

(6)

Equation (6) establishes the variational character of slope stability problems. The following comments are relevant with respect to this problem:

The minimization in Eq. (6) is with respect to all pairs  satisfying the equations of L-E (Eq. (5)), and the above formulation does not include any arbitrary assumptions restricting the geometry of potential slip surfaces , or static assumptions restricting the form of .

(2) A characteristic feature of classical (rigorous) slope stability procedures (which satisfy relevant equilibrium conditions without arbitrary restrictions on the form of potential slip surfaces) is that they are all associated with two dimensional safety functionals of the form , where  is some “static function.” In the present formulation, (Eqs. (5) and (6)),  is identified with the normal stress function. In Morgenstern - Price (M-P) procedure (Morgenstern and Price (1965))  is identified with the inclination of the resultant force P acting on vertical lines inside the test body (Fig. 1). In Janbu’s procedure (Janbu (1973)),  is taken as the location (line of action),, of the same force (Fig. 1). All rigorous limiting equilibrium procedures address essentially the same problem, using different sets of "independent variables.” The transformation between these variables is trivial (at least in principle), and all rigorous formulations of the slope stability problem are in principle equivalent. The significant difference between classical procedures and the variational approach is related to the solution of the slope stability problem rather than its formulation. The minimization problem implied by classical procedures can be written in the general form where quantities written on the right of the vertical line are considered as given. A given function  is a “static assumption”, and results of classical procedures depend on the static assumptions implied by considering a given class of functions. The variational formulation avoids the need for static assumptions by minimizing the safety functional with respect to both  and. The static assumptions associated with commonly used functions  are not well motivated physically. The common justification for use of such assumptions is that they do not have significant effect on calculated safety factors. However Krahn (2003) demonstrated clearly that this is not necessarily always the case; thus casting a serious doubt on significance of results obtained using the M-P or Junbu procedures. It is noted that, in general, introducing constraints into a minimization problem increases the minimum value of the minimization criterion. As a result, using formal static assumptions (constraints), that are not well motivated physically, may result in un-conservative estimate of minimal safety factors. It can be stated that minimizing the safety functional with respect to  results with the most conservative set of “static assumptions” which is consistent with the general principles of L-E analysis as summarized in Eq. (5).

Variational Formulation

The safety functional  is an abstract conceptual relation without explicit representation, and Eq. (6) does not lead to a practical solution process. Considering the linear case of an M-C failure criterion, Baker and Garber (1978) constructed a solution process overcoming this difficulty. The present work generalizes the formulation of Baker and Garber to the case of the nonlinear strength function defined in Eq. (4.2). Considering this strength function it is convenient to introduce the following normalized safety factor :

(7.1)

Let Fn be the minimum value of Fm. Applying Eq. (7.1) at this particular  value results in:

(7.2)

and inverting Eq. (7.2) yields:

(7.3)

 Similar definition was used by Charles and Soares (1984). Equation (7.1) implies the following identity:

(8)

The utility of the definition (7.1) is due to the symmetrical role played by the pairs and  in Eq. (8).

Solving the moment equation for, the slope stability problem may be written in the equivalent form:

(9.1)

where

(9.2)

Subject to satisfaction of the constraints:

(9.3)

The functionals  and  are obviously different, ( does not have an explicit representation while  is a ratio of integrals). The explicit expression for  (Eq. (9.2)) was obtained by solving the equation of moment equilibrium, and this guaranties satisfaction of moment equilibrium. In Eq. (9) the conditions of vertical and horizontal equilibrium are considered as integral constraints. The basic idea of considering equilibrium equations as constraints in L-E analysis is due to Kopacsy (Kopacsy (1955), (1957), (1961)). However, Kopacsy tried to formulate a slope stability problem without using the notion of safety factors, and for no obvious reason he chose to minimize the weight of the test body subject to the requirement that the three equilibrium equations are satisfied. Such a formulation has no obvious physical justification. The process of defining a minimization criterion by solving one equilibrium equation, treating the other two equilibrium conditions as constraints, was introduced by Baker and Garber (1977) and (1978). In the present work we use the equation of moment equilibrium in order to define; considering the remaining two equilibrium conditions as integral constraints. This choice is arbitrary, and the same final solution is obtained using any one of Eq. (5) for the definition of ; treating the remaining two equilibrium conditions as constraints. This is a consequence of the isoperimetric theorem of variational calculus (Petrov (1968)).

Variational analysis

Transformations of the Physical Slope Stability Problem

Consider first Eq. (9.1) ignoring the integral constraints in Eq. (9.3). Equation (9.2) defines  as ratio of integrals while standard variational techniques (e.g. Euler’s equations), are applicable only to functionals defined by a single integral. It can be verified however (e.g. Petrov (1968)), that stationary points of a ratio can be identified by considering the following auxiliary problem:

Extremize an auxiliary functional defined as:

(10.1)

Subject to the constraint:

(10.2)

The following comments are relevant with respect to Eq. (10).

(1) Fn is considered as an unknown constant during the extremization of . The result of the extremization process are two potentially critical functionsand depending on the unknown value of . Substituting these functions into the definition of , this functional is reduced to an ordinary function , and Eq. (10.2) is a nonlinear equation for the unknown value of . We adopt the convention that when a functional is reduced to a function the name of this function is written in a text format.

(2)  is an auxiliary functional and the nature (maximum, minimum, inflection), of its stationary points is not relevant. For this reason we use the non-committed term extremization in Eq. (10). The important point is that  and  have the same stationary points (pair of critical functions ). In essence Eq. (10) is simply an indirect numerical process for identification of stationary points of a ratio.

Equations (10) specify the solution process for Eq. (9.1) without the constraints in Eq. (9.3). These constraints are incorporated into the problem using the well known Lagrange undetermined multipliers method. Application of this method to the present problem requires definition and extremization of a second auxiliary functional  where  are Lagrange multipliers. Values of the multipliers are established by application of the constraints  to the results of the extremization process. Application of this program results with the following problem:

Extremize an auxiliary functional defined as:

(11.1)

where

(11.2)

Subject to satisfaction of the following system of constraints:

(11.3)

(11.4)

(11.5)

The following comments are appropriate with respect to Eq. (11):

(1) The first form of  in Eq. (11.2) is obtained by substituting the definitions of  and  into the basic definition  of the Lagrange functional. Comparing this form with the expression for moment equilibrium (Eq. (5.3)) it is possible to verify that, physically,  is the resultant moment about a point P with the coordinates. The second form of Eq. (11.2) is the consequence of this “physical interpretation” of the formal Lagrange’s multipliers. In the following we consider  rather then as the basic system of unknowns.

(2) The comments following Eq. (10) apply also with respect to Eq. (11), in particular:

a) The parameters  are considered as unknown constants during extremization of .  is defined by a single integral (Eq. (11.1)), and its extremization can be done using standard variational tools (Euler’s equations).

b) The result of the extremization process are two potentially critical functions , depending on .the unknown parameters . Inserting the critical functions into the definitions of , and  reduces these functionals to functions of . Equations (11.3) and (11.4) are direct consequences of the Lagrange procedure in which the magnitude of multipliers is established by imposing the constraints. Satisfaction of the constraint equations  results in, and Eq. (11.5) becomes identical to Eq. (10.2). Equations (11.3) to (11.5) are a system of 3 simultaneous nonlinear equations in 3 unknowns. Physically this system represents conditions of equilibrium for the critical test body in which  is the condition of moment equilibrium about point P rather then T.

(3)  is an auxiliary Lagrange functional, and the nature of its stationary points is not relevant. The important point is that the Lagrange multipliers technique and the definition of guaranty that stationary points of  occur at the same “place” (pairs of functions ) as the stationary points of the physical slope stability problem defined by Eq. (9). The nature of stationary points of the physical slope stability problem is obviously significant. However, Baker (2003b) verified that subject to some weak formal restrictions (which are satisfied automatically in all practical situations), this problem has a regular stationary minimum for all nonlinear failure criteria which are associated with finite tensile strength. Consequently one of the stationary points identified by the solution of Eq.(11) must coincide with the solution of the physical slope stability problem.

(4) In principle, may have more then one stationary point. Each one of these stationary points is associated with a different, potentially critical, “mode of failure” of the slope stability problem. Considering a slope with horizontal force acting on its surface Baker (2003b) showed that such a problem has both a minimum and a maximum. The minimum being associated with the conventional “active” slope stability problem in which the critical test body moves down, while the solution associated with the maximum represents a “passive” failure mode in which horizontal force is large enough to cause the test body to fail by moving up. In such a problem  has at least two stationary points, and. the system of nonlinear equation ((11.3) to (11.5)) is satisfied by two different sets of. The solution process implied by Eq. (11) is valid for both of these failure modes. Considering a particular stability problem it is necessary to select the relevant solution of Eq. (11).

(5) Equations (11) constitute the numerical framework by which the solution (minimum), of the physical slope stability problem (Eq. (9)), can actually be established. This process includes two distinct stages:

Extremization of the auxiliary functional - This step results with two families of functions (potential extremals),  and  depending on the three unknown parameters.

 Solution of the system of simultaneous nonlinear equations (11.3) - (11.5) - Solving this system imposes the conditions of horizontal, vertical and moment equilibrium for the critical test body, and establishes critical values of , resulting with the final solution triplet  of the physical slope stability problem.

The solution process formalized by Eq. (11) is valid for the nonlinear strength function defined in Eq. (1), and it constitutes a generalization of corresponding results derived and applied by Baker and Garber (1977), (1978) and Baker(1981), (that are valid for the particular case of a linear M-C failure criterion).

Eulers Equations

In order to derive the potentially critical functions  and  it is necessary to apply Euler’s equations to the term  defined in Eq. (11.2). The general form of Euler’s equations for the 2D functional  is:

(12.1)

(12.2)

 does not depend on , and the first Euler equation is reduced to . Applying this equation to the expression for  (Eq. 11.2), and using the identity (8.2) results in:

(13)

This differential equation is simplified considerably with the following co-ordinate transformation:

(14.1)

(14.2)

where are polar coordinates centered at the point , with the angle  measured counter clockwise from the vertical as shown in Fig. (2). In terms of these coordinates Eq. (13) simplify to:

(15.1)


Figure 2. The generalized log-spiral property of critical slip surfaces.

Following similar process with the second Euler’s equation yields:

(15.2)

where .

Equations (15) are pair of non linear, first order, differential equations for. These equations are coupled, and they must be solved simultaneously. The following features of these equations are significant:

1) Consider first the limiting linear case associated with . In that case Eq. (7.1) is reduced to  and the identity in Eq. (8) results with. Substituting this result into the polar form (Eq. 15.1) of Euler’s first equation reduces this equation to  resulting therefore with decupling of Eq. (15). The general solution of the differential equation is the family of log-spirals, and utilizing the characteristic property of log-spirals (resultant of differential normal and frictional forces passing through the pole) it is possible to evaluate safety factors without specification of normal stress distributions. Baker and Garber (1978) termed this result “The basic theorem of limiting equilibrium.” The present, more general, perspective shows however that this result is a consequence of decupling of Euler’s equations resulting from the assumed linearity of the failure criterion, and it is not generally valid.

2) The utility of the polar form of Euler’s equations (Eq. (15)) is due to the fact that this form does not depend on the unknown constants. Therefore the solution of Eq. (15) is also independent of these constants. Introducing the solution forinto the transformation equations (14), results with the functionsand, and the Cartesian form  of potentially critical slip surfaces and normal stress functions is obtained by the parametric representations  and.

3) Consider the solution of Eq. (15) as an initial value problem. In order to solve this system it is necessary to specify 4 parameters (at this stage  are arbitrary given numbers). For each given list  it is possible to numerically integrate Eq. (15) and obtain the functions. However, the numerical solution process brakes down at an  value for which the normal stress is equal to the negative of the tensile strength and. The following comments are relevant with respect to this observation:

a) Eq. (15.1) is singular at points where, and the numerical solution can not be extended past the singularity.

b) The result that at the tensile strength limit  is valid for all legitimate strength functions (even M-C) so such a singularity always exists.

c) Starting with the same list of initial values and integrating Eq. (15) "backwards" (towards smaller values of q) the process will again brake down at a  value where q. Consequently, the system of Eq. (15) has a solution only between two limiting values at which  is equal to the tensile strength, and the structure of Eq. (15) results with a definition of a finite range  in which these equations have a solution. Equation (14.1) implies that the solution of Euler’s equations exists in a finite  range. In the following we will refer to this range as the “solution range” of Euler’s equations. The “physical range” of the slope stability problem extends from  to  and it is part of formal solution range, i.e. . The singular points are associated with the tensile strength limit .

d) Baker (1981) and (2003b) introduced a "cracking hypothesis" which states that a point  at which the normal stress is equal to the negative of the tensile strength (i.e. when, is associated with a vertical tension crack extending from  to the surface of the slope. Baker (2003b) showed that in order to guaranty that the slope stability problem is well defined (having a stationary minimum), it is necessary to exclude from considerations test bodies with internal tension cracks. The above formal considerations show that the variational formulation automatically satisfy this restriction (it is impossible to extend the solution past the singular points which are associated with a tension crack, so internal tension cracks are automatically excluded).

e) It is noted that the stress state in the vicinity of the lower end point L is essentially "passive" and there can not be a tension crack at this point. Consequently, the physical range can not include , i.e. in general . It will be verified shortly that for certain conditions can be equal to.

4) At singular points  and . Inserting the relation  into the second Euler equation (Eq. (15.2)) results with where  stands for  or. Consider first the conditions in the vicinity of;  values in the vicinity of this point are positive (Fig. 2), and the above relation implies, i.e.  is a decreasing function of  in the vicinity of. Figure 2 shows that  values in the vicinity of  are negative, and similar considerations imply that at that point  is an increasing function of. Consequently  must have at least one maximum in the range. A more detailed investigation shows that  can have only a single maximum in this range; resulting with the general conclusion that normal stress functions  which are solutions of Euler's equations are convex (satisfying ) in their entire range of definition. The physical range of the problem is part of the solution range, and the above conclusion is valid also in that range.

5). Introducing the definition  into the Cartesian form of Euler’s first equation (Eq. 13), and using the coordinate transformation (14), it is possible to show that , where  is the inclination of the slip surface at . This relation implies the following “algebraic form” of Euler’s first equation:

(16)

Let be the angle between the normal and the polar radius vector at some point a on  (Fig.2). Considering the definitions of  and  shows that. Combining this result with Eq. (16) yields therefore. Consequently, slip surfaces that are solutions of Euler’s first equation posses the characteristic property of log-spirals, namely the angle between the normal and the radius vector is equal to the local value of mobilized friction angle. It appears appropriate therefore to call these solutions “Generalized log-spirals”, and identify the point  as the pole of the generalized spiral.

6) In the linear M-C framework is a constant and the log-spiral property makes it possible to evaluate safety factors without specifying the normal stress distribution (considering only the condition of moment equilibrium). In the present nonlinear setting the log spiral property is much less consequential since is stress dependent, and it is necessary to solve both of Euler’s equations in order to establish the distribution of along the slip surface. The unusual conditions associated with the linear M-C criterion (decupling of Euler's equations, and the possibility of establishing safety factors without specification of the normal stress distribution) are not typical of the general nonlinear case. These special conditions "distort" the essentially simple and consistent structure of L-E analysis; resulting with various apparent inconsistencies. In particular; the basic framework of the variational approach incorporates the 3 equilibrium conditions for a test body. However the result that critical slip surfaces are log-spirals shows that solution of this problem may be obtained based on moment equilibrium alone without specification of the normal stress function or application of horizontal or vertical equilibrium conditions. (i.e. the second Euler equation (15.2) is derived but not applied) . Considering the linear case as a limit of the nonlinear criterion associated with  removes these inconsistencies, and shows that the normal stress distribution associated with solution of Eq. (15.2) is relevant even in the linear case. Stated differently; the linear case merely allows derivation of a partial solution of the variational problem, (which includes the critical slip surface and minimal safety factor), and this partial solution is consistent with the general variational framework. In the general case all three equilibrium equations have the same stature, and the functions  are of equal significance. The preoccupation of the profession with the linear M-C criterion results with over emphasizing the significance of slip surfaces, and downgrading the role played by normal stress functions. This attitude is reflected in the classical M-P and Janbu procedures where  (which corresponds to ), is considered as "given."

Tranversality equations and boundary conditions

Euler’s equations deliver potential extremals when these extremals are restricted to pass through fixed end points. When the end points are not fixed but can move along Ys(X) it is necessary to supplement Euler’s equations by a system of variational boundary conditions called the Transelsality equations which are applied at the end points (Elsgolc (1962)). The general form of the transversality equations for the present problem is:

(17)

where  may be either or .

 and , in these ranges  is equal to zero and Eq. (17) is reduced to. Using the definition of  (Eq. (11.2)) and the coordinate transformation (Eq. (14)), this expression becomes:

(18)

In the following sections we study the implications of this relation separately at each end point.

Boundary conditions at H

Applying Eq. (18) at the high point H the term  is the depth of tension crack at this point, and Eq. (18) becomes:

(19)

Equation (19) admits two different types of solutions corresponding to situations with and without tension cracks.

a) If the tensile strength is high enough its magnitude does not affect the stability of the slope, and there is no tension crack, i.e. . In that case  and Eq. (19) can be written in the form:

(20.1)

Following the solution of Euler's second equation, the functional form of  is known and Eq. (20.1) is a nonlinear equation for determination of  and .

b) The cracking hypothesis implies that when a tension crack exists; the normal stress  at the tip of the crack is equal to . , and taking into account that  can not be equal to zero, Eq. (19) is reduced to:      

(20.2)

In this case the transversality equation at H controls the depth of tension cracks rather than  .

Equation (20.2) is associated with. The last of these relations implies that, i.e. when a tension crack exists it is located at a singularity of Euler's equations.

The two solutions given by Eq. (20) represent two possible solutions of the transversality equation at H. These two solutions define alternative failure modes (one with, and one without, tension crack). In general both of these failure modes have to be considered separately, and the failure mode associated with a smaller value of  is the critical one.

Consider however the limiting case of zero tensile strength materials . Equation (20.2) shows that in this case, i.e. as expected, zero tensile strength materials can not support vertical tension cracks. This result is associated with  and  which satisfy also Eq. (20.1). Consequently, the particular case of zero tensile strength materials has the convenient property that the two alternative failure modes specified by Eqs. (20.1) and (20.2) coincide, and there is no need to consider them separately. It is convenient to interpret this result as meaning that the variational solution for zero tensile strength materials has a zero length tension crack.

The result  implies also and. Consequently, considering zero tensile strength materials the high end point H of  is located at a singularity of Euler's equation. This result has number of significant implications:

a) In the general case integration of Euler's equations requires specification of the 4 parameters . The result makes it convenient to start numerical integration of Eq. (15) with some assumed values and, integrating “backwards” towards smaller values of. It is impossible to set  to zero since such a setting will result with  and it is impossible to start the numerical integration process. Usually we set  to some small number such as 10-5, and numerical experimentation showed that changing this setting by 2 orders of magnitude does not affect physically significant results.

b) Starting the integration of Euler’s equations at some, the integration process is carried as far as possible, i.e. it is carried down to  at which is again approaching zero. Thus; for each potential input triplet; numerical solution of Euler’s equations results with the functions,  and a value of .

c) Equation (20.2) shows that zero tensile strength materials can not support tension cracks, and this result implies. Introducing this requirement into Eq. (14.2) results in , and the coordinate transformation  ( Eq. 14.2) results in complete specification of .

Boundary conditions at L

The nature of the slope stability problem requires that. In order to satisfy this requirement it is necessary to impose the restriction:

 where the minimization is done in the range. Input triplets  failing to satisfy this requirement are not legitimate, and they must be excluded from considerations. The requirement  is a nonlinear equation for. It can be verified that this equation may have 1 or 2 solutions; defining alternative failure modes. A case in which has only one solution is associated with a shallow toe failure mechanism satisfying. Input lists resulting with the equation having two solutions are associated with two possible failure mechanisms; a) Deep toe solutions in which  but , and b) Base failure - which satisfies ,  and .

Following the determination of  the only remaining unknown is. This unknown is established by different considerations in the cases of toe and base failure mechanisms:

a) Toe failure – In that case  is known, and there no need for a transverality equation at L. Inserting the requirement  into Eq. (14.1) and solving for  results in .

b) Base failure – Investigating circular slip surfaces Taylor (1943) showed that the critical condition for the base failure mode occurs when the center  of the circle is located on a vertical line through the mid-point of the slope i.e. . Baker (1981) showed that similar result is valid for conventional log-spirals. It can be verified that the generalized log-spiral property (Eq. (16)) of  ensures that similar result is applicable also in the present non linear setting.

Following the determination of, the coordinate transformation  ( Eq. 14.1) results in complete specification of .

In principle the setting  replaces the formal transversality equation at L. Nevertheless it is instructive to consider also the transversality equation at L in order to derive a criterion for establishing which one of two deep failure modes (base or toe) is the critical one. Applying Eq. (18) at L using the fact that  results in:

(21)

Following the determination of  as a solution of the nonlinear equation, all the terms defining are known, and  may or may not be equal to zero. Transersality equations are optimality requirements and the base failure mode is critical if. For zero tensile strength materials  and Eq. (21) can be satisfied only if  (an obvious result for the base failure mode).

Numerical solution process for nonlinear zero tensile strength materials

Figure 3 summarizes the numerical procedure implied by the above considerations. The last element of this procedure involves evaluation of the unbalanced forces and moment , , . The Cartesian form of these quantities is given in Eq. (5). It is noted that application of Euler’s and Transversality equations resulted with a parametric representation of the extremals (i.e. the actually established functions are,  and  ). In order to evaluate the unbalanced forces it is necessary to express Eq. (5) in terms of integrals with respect to . The resulting expressions are rather bulky and they are not reproduced here.



Figure 3. Variational solution procedure for nonlinear zero tensile strength materials.

For each given pair the procedure in Fig. 3 is a numerical definition of 3 non linear functions in the 3 unknowns. Solving these equations; results in the following 3 2D functions  and. The significance of these results is established in the following discussion:

(1) A homogeneous slope stability problem in zero tensile strength material is defined by the list of input parameterswhere D stands for the term Data. The above solution process results in the 2D function. Inserting this function into Eq. (7.3) gives

(22)

Equation (22) shows that availability of the function  makes it possible to evaluate the minimum safety factor for all possible combinations of parameters in the input list. The function  has the same conceptual significance as Taylor's stability chart (Taylor's chart is relevant for the M-C failure criterion, while  is relevant to a zero tensile strength material obeying the H-B failure criterion). It is noted however that unlike Taylor's stability chart, the function  does not depend on, and there is no need for iterations in order to evaluate minimum safety factors. Graphical representation of  in the form of stability chart is given in Baker (2004b).

(2) The solution process resulting with the function  delivers also a parametric representation of  and describing critical slip surfaces and normal stress functions. Using the definitions of and  it is possible to write:

(23.1)

(23.2)

where  and  are dimensional representations of critical slip surfaces and normal stress functions. The most significant feature of Eq. (23) is that critical slip surfaces and normal stress functions are in fact independent of the strength parameter. It can be verified that this result is valid only in the particular limiting case of zero tensile strength materials. It is noted in passing that the above conclusion is consistent with well known results for the purely frictional strength model which is a limiting case corresponding to  of the present analysis. In this limiting case the critical slip surface coincides with the surface of the slope, and the critical normal stress function is identically equal to zero, i.e. and .

(3) Evaluation of each “point” of the 3 non linear functions defined by Fig. 3 involves numerical solution of a pair of coupled non linear differential equations and numerical evaluation of 3 integrals. Consequently establishing the stability chart  is numerically challenging. However this is not a serious limitation for a "one time" job such as establishing the stability function. All the numerical work was done using the commercial program Mathematica IV (Wolfram (2002)), which provides convenient tools for numerical solution of differential equations, numerical integration, and numerical solutions of nonlinear equations.

Example problems

Complete results (stability charts, and failure modes), based the above analysis, and discussion of their engineering significance, were presented in (Baker (2004b)). Here we consider only two example problems illustrating various features of the variational solution.

Problem I. The N-L strength law for problem I is defined by the parameters , which are appropriate for London clay (Perry (1994), Baker (2004a)). Additional information defining this problem is  (1 to 4 slope),  and. This input information results with a base failure mode and the minimum value of the normalized safety factor is. Substituting this value into Eq. (22) yields , i.e. this slope satisfies normal design criteria.

Problem II. The N-L strength law for problem II is defined by the parameters  which are appropriate for compacted Israeli clays (Baker 2004a)). The slope for Problem II is defined by, and. This input information results with a shallow toe failure mode and  which implies, i.e. the slope is in a state of failure.

These two problems are very different; illustrating a wide range of situations ( values close to the two theoretical limits of 0.5 and 1, gentle and steep slope inclinations, stable and unstable conditions, deep and shallow critical slip surfaces etc.). Detailed results obtained for these two problems are presented in Figs. 4 to 6 and they are discussed below:

Figure 4 shows critical normal stress distributions obtained for Problems I and II. Figures 4a and 4c show that in both problems  has a single maximum in this range, i.e. it is strictly convex function. This result confirms the general conclusion obtained previously.



Figure 4. Critical normal stress functions for example problems.

The physical range of the solution is between the low to the high end points L and H. In zero tensile strength materials the high end point is located at and low end point is established by the requirement  (Fig. 3). Problem I is associated with a base failure mode. In the present case  is close to  resulting with  (Fig. 4b). The critical slip surface for Problem II is a shallow surface passing through the toe, and  (Fig. 4c). Taylor (1937) suggested that critical normal stress distributions can be approximated as clipped  functions. The result for Problem I supports this proposition (Fig. 4a), however the critical normal stress function for Problem II is highly asymmetrical and it clearly can not be approximated this way. Janbu (1957) suggested approximating normal stress distribution as and this function is shown as the dashed lines in Fig. (4a) and (4c) (it is noted that Janbu’s normal stress function is not related to his slope stability procedure). Inspection of Figs. (4a) and (4c) shows that Janbu’s proposition is reasonable for base failure, but it does not yield consistent results for the toe failure mode in which. The normal stress functions shown in Fig. 4 are derived results, and they replace the static assumptions introduced by Taylor and Janbu. These functions are associated with minimal value of the safety factor and they represent the “most conservative” static assumption that is consistent with general principles of L-E analysis.



Figure 5. Critical slip surfaces for example problems.

Figure (5a) is a general view of the critical slip surface for Problem I. Figure (5b) is a magnification of this surface in the vicinity of the low end point L (note that the coordinates in Fig. (5b) are drawn with different scales). Figure (5b) shows that  has a shallow local maximum at point b. By itself this maximum is not significant, being located outside the physical range  to. However existence of this maximum shows that  is not uniformly convex, including both convex and concave sections. The concave section for problem I extends from a to L. This feature of critical slip surfaces is associated with N-L strength functions and it disappears when a linear failure criterion is considered (conventional log-spirals are always convex). Inspection of Fig. (5a) shows that the concave section is so “flat” that practically it can be considered as straight. Similar results were obtained for all combinations of input data lists D (i.e. the concave section is always almost flat). Limiting equilibrium procedures are essentially global, and they are not sensitive to small local variations in the shape of slip surfaces (the minimum of these procedures is rather flat). Consequently, the fact that the critical slip surface shown in Figs. (5a) and (5b) has a slightly convex section is not practically significant, being essentially a mathematical curiosity. It is of interest to note however the result that in the linear case, the sufficient conditions ensuring that the general slope stability problem is well set, possessing a stationary minimum (Baker (2003b)), do not require critical slip surfaces to be convex.

Figure (5c) shows the critical slip surface for Problem II. In this problem the function  decrease monotonically in the solution range (D to H), resulting therefore with a shallow toe failure. Close inspection of this figure shows that the critical slip surface has a slightly concave section near the toe point T, but this section is not practically significant. Comparison of Figs. (5a) and (5c) shows that (as expected), critical slip surfaces become shallower as  and/or . In the limiting case of  the N-L law degenerates into the purely frictional model. The solution triplet for this limiting case is well known, consisting of (surface failure). From a L-E perspective this solution is a singularity since it is not associated with a finite test body (it is impossible to write equilibrium conditions for such a zero volume “test body” and one of the two basic elements of the L-E approach is satisfied only in a certain limiting sense). The N-L strength law eliminates this singularity for all.  is associated with  and ; in that case  (Eq. (2)), and the term  approaches infinity; i.e. in the present framework surface failure is never critical and the singularity associated with this failure mode is removed (surface failure is an erosion rather than stability problem).

Figures (6a) and (6b) show the functions  obtained (using Eq. (16)) for the two example problems. The most significant feature of these functions is the essentially vertical sections that they have in the vicinity of. Inspection of Fig. 6a shows that every value of  in the range 70o to 130o can be considered as valid. This observation shows that certain secondary quantities (among them the local inclinationof the slip surface) are not well defined in the vicinity of a singularity.



Figure 6. Distributions of secondary quantities in example problems (a)  for Problem I, (b)  for Problem II. (c)  for Problem I. (d)  for Problem II.

Figures (5a) and (5c) show that a “global” value of  near H is well defined, and only the local  value, in the immediate vicinity of the singularity is undefined. It is noted that the indeterminate situation with respect to  (and other secondary variables) does not affect physically significant aspects of the solution. The L-E equations (Eq. (5)), relate various integrals of. These integrals are well behaved, and the behavior of  at the singular point itself is not consequential. However the inability to establish a reliable value of  has an effect on the solution procedure shown in Fig. (3), making it impossible to use the parameter  in the numerical solution process.

The present approach to L-E analysis is completely global, involving quantities defined only on the boundaries  and of the test body. Classical procedures (e.g. M-P, Janbu and many other) are formulated in terms of internal variables (inter-slice forces) inside the test body. It is of interest therefore to consider the implications of the present analysis with respect to such internal variables. Considering the conditions of horizontal and vertical equilibrium for the part BDHCFB of the critical test body (Fig. 1) it is possible to derive the following relations:

(24.1)

(24.2)

(24.3)

where t is a formal integration variable, and  are the horizontal and vertical components of the inter-slice force P (Fig. 1), and  is the inclination of this force (M-P inter-slice force function). Considering moment equilibrium of BDHCFB it is possible to derive also Janbu’s function  but we do not present results with respect to this function. Figure (6c) and (6d) show the functions  obtained for problems I and II respectively. The following comments are relevant with respect to these figures:

a) The M-P procedure is specified for a given function, i.e. the engineer is expected to have a priori information about the form of this function. The functions  shown in Figs. (6c and 6 e) are derived results which are associated with the minimal safety factor. These functions do not have a simple form, and it is clearly not practical to expect an engineer to specify such functions a priori. Stated differently, the M-P procedure is under-specified. Janbu’s procedure suffers from the same limitation.

b) Spencer (1967) removed the above difficulty by assuming that is a constant function (parallel inter-slice forces). This assumption has no rational basis and it is introduced only for reasons of convenience. The simplified Bishop method (Bishop (1955)) is based on the even stronger assumption that. Figures (6c) and (6d) show clearly that both of these static assumptions are not justified (they are not associated with a minimal safety factor). Krahn (2003) demonstrated that for certain situations different inter-slice force functions  are associated with significantly different estimated minimum safety factors. Consequently; applying the M-P procedure with essentially arbitrary, and definitely not critical, inter-slice force function  may result in significant over estimations of the minimal safety factor.

c) The present work shows that static assumptions are not only un-justified, they are in fact not necessary. Minimizing the safety functional with respect to both Y(X) and S(X) (Eq. (6)), automatically generates the most conservative static assumption which is consistent with general principles (Eq. (5)) of L-E analysis.

Summary and conclusions

Variatinal slope stability analysis for materials with a nonlinear failure criterion is presented. The present work is based on the Mohr form of Hoek and Brown empirical failure criterion. This form depends of 3 non-dimensional strength parameters . Zero tensile strength materials are characterized by T = 0.

The variational formulation has two basic advantages over conventional approaches to slope stability analysis:

a) The variational approach does not include geometrical or static assumptions. These assumptions are replaced by minimization of the safety functional with respect to the normal stress function. Results of the variational analysis correspond to the most conservative set of static assumptions which is consistent with general principles of limiting equilibrium analysis.

b) The variational formulation provides a practical solution process leading to identification of the critical slip surface and the critical normal stress function acting on it. The variational solution process is presented in Fig. 3. Complete results (stability chart and failure modes) based on this process are reported by Baker (2004b).

The following general results are implied by the variational analysis:

(1) In the variational framework critical slip surfaces and normal stress functions are solutions of a pair of coupled, first order, nonlinear, differential equations (Euler's equations). In the limiting case when the failure criterion is linear these equations decuple; resulting in various apparent inconsistencies which do not occur in the general non linear case.

(2) The tensile strength value t of a failure criterion is characterized by and  (symmetry of Mohr circles and envelopes). This limiting normal stress is a singularity of Euler's differential equations, and these equations have a solution only in a finite zone bounded by two points at which normal stresses are equal to the negative of the tensile strength. The physical range of the problem is part of solution range. These results show the fundamental role played by the tensile strength in slope stability analysis. This effect is not sufficiently emphasized in conventional presentations of this subject.

(3) In the variational framework depth (and existence) of tension cracks is a derived result rather then a priory assumption. In cases for which a tension crack exists; the upper end point H of potentially critical slip surfaces is located at a singularity. In zero tensile strength materials tension cracks exist, but their depth is equal to zero, and such materials can not support vertical slopes. This result implies that in zero tensile strength materials the upper end point of critical slip surfaces is located at a singularity; and various secondary quantities (among them the inclination of critical slip surface) are not well defined in the vicinity of this singularity.

(4) Potentially critical normal stress distributions are convex functions having a single maximum in the solution range.

(5) Potentially critical slip surfaces are generalized log-spirals possessing the property that the angle between the normal and the radius vector is equal to the local value of mobilized tangential friction angle. In non linear materials this angles depend on normal stresses and it varies along the slip surface. The resulting family of generalized log-spirals is much more general then conventional spirals, including functions having both concave and convex sections.

(6) The family of generalized log spirals has a characteristic point which can be identified as the pole of the spiral. In the base failure mode the pole of the critical slip surface is located on a vertical line through the mid point of the slope.

(7) The solution process implied by the variational analysis results in solution triplets

consisting of normalized forms of critical slip surfaces, critical normal stress functions, and the minimum safety factor. The function  has the same conceptual significance as Taylor's stability chart. This function is presented in Baker (2004b). Considering the definitions of  and , shows that in zero tensile strength materials critical slip surfaces and normal stress functions do not depend on the strength parameter A.

(8) Investigation of particular solutions showed that common static assumptions employed in classical limiting equilibrium procedures are not justified in the sense that they are not associated with the critical conditions. More important however, the present work shows that such assumptions are not only unjustified; being in fact unnecessary.

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