Stability of geosynthetic reinforced soil structures

33 The Horseshoe Covered Bridge Farms Newark, DE. 19711, USA Copyright© 2002, ADAMA Engineering, Inc. All Rights Reserved (www.GeoPrograms.com) Written by Dov Leshchinsky, Ph.D.

1 INTRODUCTION

Soil is an abundant construction material that, simi-lar to concrete, has high compressive strength but virtually no tensile strength. To overcome this weakness, soils, like concrete, may be reinforced. The materials typically used to reinforce soil are relatively light and flexible, and though extensible, possess high tensile strength. Examples of such ma-terials include thin steel strips and polymeric materi-als commonly known as geosynthetics (i.e., geotex-tiles and geogrids). When soils and reinforcement are combined, a composite material, the so-called 'reinforced soil', possessing high compressive and tensile strength (similar, in principle, to reinforced concrete) is produced.

The increase in strength of the reinforced earth structure allows for the construction of steep slopes, embankment over soft foundation, or various types of retaining walls. Compared with all other alterna-tives, geosynthetic reinforced soil structures are cost-effective. As a result, earth structures reinforced with geosynthetics are being constructed worldwide with increased frequency, even in permanent and critical applications (e.g., Tatsuoka and Leshchin-sky, 1994).

This paper describes a design process for geosyn-thetic-reinforced slope. It includes details of stability analyses used to determine the required layout and strength of the reinforcing material. This process serves as the basis for the computer program ReSlope (Leshchinsky, 1997, 1999). Recognizing the limita-tions of ReSlope (e.g., available strength of the rein-

forcement at its front-end can be less than the required strength; analysis of complex geometries; stability of embankment reinforced at its base), the rational for a more complete stability analysis is presented. This has resulted in program ReSSA (Leshchinsky, 2002). Finally, the design of walls, which customarily adopt lateral earth pressure approach, is briefly discussed. This approach is used by national design procedures such as AASHTO or NCMA (Collin, 1997) methods, which serve as the basis for program MSEW (Lesh-chinsky, 1999, 2000). An appendix provides com-parative summary of programs ReSlope, MSEW and ReSSA. 2 DESIGN-ORIENTED ANALYSIS 2.1 General Limit equilibrium analysis has been used for dec-ades in the design of earth slopes and embankments. Attractive features of this analysis include experi-ence of practitioners with its application, simple in-put data, useful (though limited) output design in-formation, and results that can be checked for 'reasonableness' through a different limit equilibrium analysis method, charts, or even hand calculations. Consequently, extension of this analysis to the de-sign of geosynthetics reinforced slopes, embank-ments and retaining walls, where the reinforcement is tangibly modeled, is desirable. The main draw-backs of limit equilibrium analysis are its inability to deal with displacements and its limited representa-tion of the interaction between dissimilar or incom-

ABSTRACT: A framework for stability analysis of reinforced soil structures is presented. It produces eco-nomical design of stable reinforced walls, slopes and embankments. Elements such as local, compound, global and direct sliding stabilities are ensured. This framework was implemented in program ReSlope. More complex and versatile stability analysis methods can use the presented framework as a generic template (e.g., program ReSSA uses it in an analysis-oriented fashion). Following the conceptual analyses is an instructive parametric study. General guidelines about the selection of long-term geosynthetic and soil strengths and a comparison with a case history are discussed. The meaning of “factor of safety” in the context of reinforced soil structures is investigated showing it to be different for MSE walls and slopes. Some of the factors of safety used in programs ReSSA, ReSlope and MSEW are not defined in the same way thus their numerical value has to be examined independently; however, when the factor of safety is one, all definitions are equivalent. An appendix provides comparative summary of programs ReSlope, MSEW and ReSSA.

patible materials comprising the soil structure. Typically, adequate selection of materials properties and safety factors should ensure acceptable dis-placements, including safe level of reinforcement deformation.

In principle, inclusion of geosynthetic reinforcement in limit equilibrium analysis is a straightforward process in which the tensile force in the geosynthetic material is introduced directly in the equilibrium equations to assess its effects on stability. However, the inclination of this tensile force at the assumed slip surface must be assumed. Physically, its angle may vary between the as-installed (typically horizontal) and the tangent to the potential slip surface. By using a log spiral mechanism, Leshchinsky and Boedeker (1989) have demonstrated that for typical cohesionless backfill, this inclination has little effect on both the required strength and layout of reinforcement. Conversely, Leshchinsky (1992) pointed out that for problems such as reinforced embankments over soft soil, the inclination of the reinforcing geosynthetic, located at the foundation and backfill interface, plays a significant role. The long-term value of cohesion used in design of manmade reinforced steep slopes or walls is negligibly small and hence, inclination has little effects. Therefore, the force in such structures may be assumed horizontal without being overly conservative. In case of basal reinforcement of embankment over soft soil, the uncertainties associated with defining the foundation properties make it prudent to be conservative and assume the reinforcement force is horizontal. Consequently, based on a practical argument, the force inclination is assumed horizontal.

A potentially significant problem in limit equilibrium analysis of reinforced soil is the need to know the reactive force in each reinforcement layer at the limit state. Physically, this force may vary between zero and the ultimate strength when the slope is at a global state of limit equilibrium. Assuming the actual force is known in advance, as is commonly done in analysis-oriented approach, implies the reinforcement force is actually active, regardless of the problem. The designer then assumes the available ‘active’ force of each reinforcement layer to ensure that overall satisfactory state of limit equilibrium is obtained. The end result of such assumption may yield an actual slope in which some layers actually provide more force than their long-term available strength while other layers are hardly stressed. To overcome the potential problem of local instability (reinforcement overstressing), a rational methodology to estimate the required (i.e., reactive) reinforcement tensile resistance of each layer is introduced via a 'tieback analysis' or internal stability analysis.

Consequently, the designer can verify whether an individual layer is overstressed or understressed, regardless of the overall stability of the slope. Once this problem of 'local stability' is resolved, overall stability of the slope is assessed through rotational and translational mechanisms. The rotational mechanism (termed 'compound stability' or ‘pullout analysis’) examines slip surfaces extending between the slope face and the retained soil. The force in the geosynthetic layers in this limit-state slope stability analysis is taken directly as the maximum available long-term value for each layer. The translational analysis ('direct sliding') is based on the two-part wedge method in which the passive wedge is sliding either over or below the bottom reinforcement layer, or along the interface with the foundation soil.

The common factor of safety in stability analysis of reinforced soil is equally applied to all failure-resisting components (i.e., soils and reinforcement). This im-plies that all resisting elements are equally mobilized. Practice proves that such an approach combined with the ability of geosynthetic to greatly deform produce structures in which all reinforcement layers are typi-cally mobilized uniformly (i.e., efficient use of rein-forcement). This definition of safety factor is used in ReSSA thus making it applicable to marginally stable slopes where the overall factor of safety needs to be increased via reinforcement.

A modified concept included in this paper relates to a versatile definition of factor of safety suitable for in-herently unstable unreinforced structures. It suggests a rational and physically meaningful alternative to the conventional factor of safety used in slope stability. In fact, this factor of safety can be measured in an actual structure. This factor of safety is used in ReSlope.

2.2 On the factor of safety in reinforced soil structures

Limit equilibrium analysis deals with systems that are on the verge of failure. However, existing slopes are stable. To analyze such slopes, the concept of factor of safety, Fs, has been introduced. In unrein-forced slopes, Fs is used to replace the existing soil with an artificial one, in which the shear strength is φm = tan-1(tanφ/Fs) and cm = c/Fs where φm and cm are the ‘design’ shear strength parameters of the arti-ficial soil. Alternatively, these values represent the average mobilized shear strength of the actual soil. Employing the notion of Fs in limit equilibrium re-duces the statical indeterminacy of a stable slope formulation via use of Mohr-Coulomb failure crite-rion. It also provides an object for minimization in which the lowest value of Fs, considering all poten-

tial failure surfaces and mechanism, is sought. The physical significance of the conventional factor of safety can be accepted in an average sense only; i.e., the average reduction of shear strength so that the sliding mass will globally be at the verge of failure. Extensive experience with limit equilibrium analysis has produced engineering database providing ac-ceptable values of Fs.

Leshchinsky and Reinschmidt (1985) applied Fs equally to all shear-resisting components; i.e., soils or reinforcement. This renders a factor of safety that is equivalent to the one used in unreinforced slopes (e.g., symbolizing the same average reduction of strength of dissimilar materials that are attaining a limit equilibrium state simultaneously). In fact, this definition is used also in most slope stability analy-ses of reinforced slopes (e.g., program ReSSA). Such definition produces a single number that signi-fies the state of global stability of a reinforced sys-tem, similar to unreinforced slopes, homogeneous or stratified.

Another definition of Fs that also ‘globalizes’ the reinforced system is presented in the federal design guidelines in the US (Elias and Christopher, 1997):

where Fsu is the factor of safety for the unreinforced slope; Mr and Md are the resisting moment due to reinforcement layers and the total driving moment, respectively. Mr and Md are calculated for the same slip surface as Fsu. It should be noted that the sur-face (typically circle) yielding the minimum Fsu is not necessarily the one yielding the minimum Fs; the critical surface in reinforced problems is deeper than the unreinforced one. Such an approach yields an overall factor of safety whose physical meaning is only valid in a global sense. However, it treats the reinforcement as pure moment (i.e., only Mr result-ing from reinforcement force is considered; actual force is not included in the equilibrium equations). Programs ReSSA and MSEW can use this definition of Fs as an option.

Extension of limit equilibrium stability analysis to reinforced steep slopes provides an opportunity to introduce a modified definition for Fs. Rather than extending the conventional definition of Fs, one can use the fact that unstable soil structures are suffi-ciently stable solely due to the reinforcement tensile resistance. Hence, ‘Fs’ for the soil alone in this case is unity everywhere along a slip surface (i.e., a plas-tic ‘hinge’ develops mobilizing the full available strength of the soil). For this state, the required rein-forcement force needed to restore a state of limit equilibrium can be calculated. As an example, see

Figure 2 where a log spiral mechanism is used. The stability of the slope now hinges on the reinforce-ment strength. Hence, the actual factor of safety can be defines as: Fs

)2(

t required

t available=

where tavailable is the long-term available strength and trequired is the strength required for stability (i.e., for a limit equilibrium state of the composite reinforced system). This definition signifies a factor of safety with respect to the available strength of the rein-forcement. Such Fs can actually be measured.

This modified definition of Fs is based on the premise that the soil will attain its full strength be-fore the reinforcement ruptures; i.e., the soil will at-tain an ‘active’ state exactly as assumed in design of retaining walls including those reinforced with geo-synthetics. Geosynthetic materials are ductile, typi-cally rupturing at strains greater than 10% thus may allow sufficient deformations to develop within the soil to reach active state. In reality, most of the de-formation for the active state will occur during con-struction as the geosynthetic mobilizes its strength. In fact, this definition is similar to the one used in MSE walls (e.g., Elias and Christopher, 1997; Collin, 1997); the design (available) shear strength parameters of the soil are fully used and then a fac-tor of safety is applied on the long-term strength of the reinforcement only. Details of the consequences of this definition are given elsewhere (Leshchinsky, 2000). Programs ReSlope and MSEW allow the user to use this definition of Fs while program ReSSA can reproduce it upon some manipulation (i.e., analyze a reinforced system repetitively while reducing the strength of the reinforcement until the resulted overall Fs is 1.0; the soil now is in an active state; increase the reinforcement strength to obtain safe long-term value).

)1(/ MdMrFsuFs +=

Figure 1. Notation and convention

Figure 2. Log spiral slip surface and its statical

implications

2.3 Internal stability analysis Internal stability analysis is used to determine the required tensile resistance of the each layer needed to ensure that the reinforced mass is safe against in-ternal collapse due to its own weight and surcharge loading. In the context of retaining walls, this analysis identifies the tensile force needed to resist the active lateral earth pressure at the face of a steep slope. That is, the tensile force needed to restrain the unstable slope from sliding. The reinforcement tensile force capacity is made possible through suf-ficient anchorage of each layer into the stable soil zone located behind the active zone. It is assumed that at the face of the slope, some type of facing re-stricts soil movement relative to the reinforcement; hence, the full long-term strength of the geosynthetic is available at the face of the slope. This assump-tion is utilized in ReSlope; however, the actual strength available at the face (connection strength) is used in ReSSA or MSEW. While MSEW considers internal stability explicitly (as does ReSlope), ReSSA looks for the most critical situation regard-less whether it is surficial, deep, compound or direct sliding.

Figure 1 shows notation and convention. Rein-forcement is comprised of primary and secondary layers. Only primary layers are considered in ReSlope; in ReSSA or MSEW the effects of inter-

mediate reinforcement are considered. Furthermore, ReSSA is applicable also to base-reinforced em-bankments over soft soil. In practice, secondary layers allow for better compaction near the face of the steep slope and thus reduce the potential for sloughing. In walls it may alleviate connection loads (Leshchinsky, 2000). The secondary layers are narrow (typically 1 m wide), installed only if the primary layers are spaced far apart (e.g., more than about 0.6 m apart). At the slope face, the geosyn-thetic layers may be wrapped around the exposed portion of the soil mass or, if some cohesion exists, the layers may simply terminate at the face as shown in Figure 1.

In general, the following rational could be used with any type of stability analysis. It is most conven-ient to use it in conjunction with log spiral stability analysis since the problem then is statically determi-nate. This analysis produces the location of the criti-cal slip surface and subsequently, the necessary reac-tive force in the reinforcement. While ReSlope utilizes the log spiral, ReSSA is using for rotational failure circular arcs combined with Bishop stability analysis. MSEW uses planar slip surfaces for internal stability following Rankin or Coulomb lateral earth pressure theories (MSEW is restricted to very steep slopes having an angle larger than 70°, i.e., walls).

The log spiral mechanism makes the problem stati-cally determinate. For an assumed log spiral failure surface, fully defined by the parameters xc, yc and A, the moment equilibrium equation about the pole can be written explicitly without resorting to statical as-sumptions (Figure 2). Consequently, by comparing the driving and resisting moments, one can check whether the mass defined by an assumed log spiral is stable for the design values of the shear strength parameters: φd and cd and the distribution of reinforcement force tj. This check is repeated for other potential slip surfaces until the least stable system is identified. That is, until the maximum required restoring reinforcement force is found. The terms Kh and Kv (Figure 2) represent the seismic coefficients introducing pseudo-static force components. It is assumed to act at the center of grav-ity of the critical mass. To simplify the presentation, no surcharge is shown in Figure 2; however, including it in the moment equation is straightforward.

Figure 3 illustrates the computation scheme for es-timating the tensile reaction in each reinforcement layer. In STEP 1, the soil mass acting against Dn is considered. Note that layer n is wrapped around the slope face to form ‘facing’ Dn (Figure 3) thus making it physically feasible for a mass of soil to be laterally supported rendering local stability. That is, a 'facing unit' Dn (i.e., an imaginary facing element in the front edge of the reinforced soil mass) prevents slide of un-

stable soil above it. This facing is capable of provid-ing lateral support through the development of the necessary tensile force in the geosynthetic (reaching, at most, its long-term strength). While this assump-tion exists in ReSlope, MSEW and ReSSA allow for reduced strength at the front-end signifying possible low-strength connection to a facing element (MSEW) or simply front-end pullout (ReSSA). Note that mas-sive stabilization of slope requires reinforcement away from the face thus making the front-end strength less significant unless surficial stability is of concern.

ReSlope uses the moment equilibrium equation to find the critical log spiral producing max(tn), employ-ing the free-body diagram shown in Figure 3 while examining many potential surfaces. The resulted tn counterbalances the horizontal pressure against Dn and thus, signifies the reactive force in layer n. That is, the resulted tn represents the force needed to restore equilibrium and hence stability. Note that Dn was chosen to extend down to layer n. This tributary area implies a 'toe' failure activating the largest possible re-action force. In MSEW the reinforcement reaction is calculated based on lateral earth pressure satisfying horizontal equilibrium at each elevation. In ReSSA, the user can verify that any given layer supplies suffi-cient force to render satisfactory Fs.

In STEP 2, the force against Dn-1 is calculated. Dn-1 extends from layer n to layer (n-1). Using the moment equilibrium equation, max(tn-1), required to retain the force exerted by the unstable mass against Dn-1, is calculated. When calculating tn-1, the reac-tion tn, determined in STEP 1, is known in magni-tude and point of action. Hence, the reactive force in layer (n-1) is the only unknown to be determined from the moment equilibrium equation.

Figure 3 shows that by repeating this process in ReSlope, the distribution of reactive forces for all re-inforcing layers, down to t1, are calculated while supplying the demand for a limit equilibrium state at each reinforcement level. Application of appropri-ate factor of safety to the required reinforcement strength should ensure selection of geosynthetic pos-sessing adequate long-term strength. In MSEW, the reaction is determined by using the lateral earth pressure and the tributary area of each layer. Con-versely, in ReSSA, the available Fs at each elevation are checked while considering rotational and transla-tional failure and the existing long-term strength of the reinforcement. In ReSSA the approach is analy-sis-oriented (i.e., given the layout and strength of re-inforcement, find the minimum Fs for the structure) whereas in ReSlope it is design-oriented (i.e., given the desired Fs, find the layout and strength of rein-forcement).

Note that cohesive steep slopes are stable up to a certain height. Consequently, the scheme in Figure 3 may produce zero reactive force in top layers. Though these layers may not be needed for local sta-bility, they may be needed to resist compound failure as discussed in the next section.

The outermost critical log spiral in ReSlope defines the extreme surface as dictated by Layer 1. In conventional internal stability analysis (e.g., MSEW) it signifies the extent of the 'active zone'; i.e., it is the boundary between the sliding soil mass and the stable soil. Consequently, reinforcement layers are anchored into the stable soil to ensure their capacity to develop the calculated tensile reaction tj (Figure 4). The 'sta-ble' soil, however, may not be immediately adjacent to this outermost log spiral and therefore, some layers should be extended further to ensure satisfactory sta-bility (see next section).

Note in Figures 3 and 4 that the reinforcement lay-ers are wrapped around the overlying layer of soil to form the slope face. However, in slopes that are not as steep (say, i<50°), typically there is no wrap around the face nor is there any other type of facing. In this case, load transfer from each unstable soil mass to the respective reinforcement layer is feasible due to a 'co-herent' mass formed at the face. This mass may be formed by soil arching, by a trace of cohesion and by closely spaced reinforcement layers. The end result is a soil 'plug' that acts, de facto, as a facing unit thus enabling the load transfer into the primary reinforce-ment layer. It should be pointed out that 'closely spaced reinforcement' does not necessarily mean closely spaced primary reinforcement layers; simply, this 'plug' can be formed by the combination of secon-dary and primary layers acting together to create a co-herent mass. Since reinforcement layers, including primary and secondary layers, are spaced approxi-mately 30 cm apart in practice, and since the secon-dary layers extend at least about 1 m into the slope, the contribution of secondary layers to the formation of a 'facing' needs not be ignored. With time, surface vegetation and its root mat enhances this 'facing.' The end result of forming a coherent face is not just an ef-ficient load transfer from the deeply unstable soil mass to the reinforcement, but also improved surficial stability and erosion resistance. While such transfer is needed to ensure that the front-end available strength assumption in ReSlope is valid, ReSSA assess the sta-bility based on actual front-end strength. MSEW is limited to walls thus uses facia in its analysis. Note that when planar reinforcement is closely spaced, the load carried by each layer can be small. Consequently, even if the full geosynthetic strength cannot develop at the face, its overall effect on stability may not be critical (parametric studies of practical cases show it).

Figure 3. Scheme for calculating tensile reaction in reinforcement layers

2.4 Compound and pullout stability analysis For a given geometry, pore-water pressure distribution and (φd and cd), the internal stability analysis provide the required tensile resistance at the level of each rein-forcement layer. It also yields the trace of the outer-most log spiral defining the 'active' soil zone, a notion commonly used in conjunction with analysis of retain-ing walls. In reinforced soil structures, the capacity of the reinforcement to develop the required tensile resis-tance depends also on its pullout resistance; i.e., the length anchored into the stable soil zone. If the boundary of this stable zone is indeed defined by the 'active' one, then potential slip surfaces that extend into the soil mass further than the outermost log spiral in Figure 4, outside or within the effective anchorage length, will never be critical. However, such potential surfaces may render reduced pullout resistance since the effective anchorage length is shortened. That is, the reduced tensile resistance capacity along these sur-faces could potentially produce a globally unstable system. Consequently, a conventional slope stability approach is used to determine the required reinforce-

ment length so that compound failures (i.e., surfaces extending into the unreinforced soil zone) will not be likely to occur. The term ‘conventional’ refers to the nature of the analysis in which global stability is sought (recall that internal stability looks at local sta-bility at the elevation of each reinforcing layer). The objective of the compound analysis is to find the minimum length of each reinforcement layer needed to ensure adequate stability against rotational failures.

Internal stability analysis yields the required rein-forcement strength at each level (in ReSlope and MSEW). In actual practice, however, specified rein-forcement layers will have allowable strengths in ex-cess of that required (i.e., tj ≤ t(allowable)j whereas tallowable ≤ tavailable and tavailable is the long-term strength). The end result of specification of reinforcement stronger than needed is that actually only m reinforcement lay-ers, extending outside the ‘active’ zone and into the stable soil, are globally needed. That is, the m layers are sufficient to maintain stability of the active mass. Internally, however, layers (m+1) through n are also needed to ensure local stability as implied in the scheme presented in Figure 3. The minimum number of layers, m, is calculated using the following equa-tion:

)3(

11)( ∑

=≥∑

=

n

jt j

m

jt jallowable

Note that m is the number of layers, counting from the bottom, capable of developing a total tensile resistance equal to (or slightly greater than) the net total rein-forcement force obtained from the internal stability analysis. When m = n, the compound stability degen-erates to that introduced by Leshchinsky (1992). The m layers are assumed to contribute their full allowable strength simultaneously to global stability when com-pound stability of the reinforced system is examined. The assumption of simultaneous availability of rein-forcement strength is commonly used in limit equilib-rium stability analysis of reinforced slopes and is sup-ported by (scattered) field data.

Embedding the layers immediately to the right of the outermost log spiral obtained in the internal sta-bility analysis, so that tallowable for layers 1 through m and tj for layers (m+1) through n can develop through pullout resistance, ensures that, in an aver-age sense, the mobilized friction angle, φmob, along this log spiral is equal to, or slightly less than, φd. The upper layers (m+1) through n (see points A, B and C in Figure 5) are not needed for the global sta-bility of the ‘active’ mass and therefore, from a theo-retical view point could be ignored at points A, B and C. Note that the mobilized friction angle, φmob,

represents the required friction angle to produce a limit equilibrium state while using the allowable re-inforcement strength. Hence, when φmob < φd, a fic-titious situation is analyzed; i.e., the system is actu-ally stable since the available soil strength, as expressed by φd, is larger than needed, φmob, for a limit equilibrium state. Only when φmob = φd limit equilibrium state achieved.

At this stage of ReSlope’s analysis, which uses de-sign-oriented approach, layers 1 through m are length-ened to a test body defined by an arbitrary log spiral extending between the toe and the crest, to the right of the outermost log spiral (Figure 5). Each layer be-yond the slip surface is embedded so that the calcu-lated t(allowable)j can be developed; φmob for this surface will be smaller than φd used in design (i.e., for this layout, the internal stability outermost surface is most critical). The upper layer is truncated in a numerical sense (i.e., tm = 0), and the moment equilibrium equa-tion for the arbitrary log spiral is used to check whether φmob = φd. If φmob = φd than layer m is suffi-ciently long (see point D in Figure 5); otherwise, lengthen this layer and repeat calculations until satis-factory length is found. A satisfactory length implies that the critical log spiral passing through point D yields a stable system for the design friction angle, φd; all feasible log spirals between this one and the out-ermost log spiral from the internal stability have φmob < φd indicating they represent less critical mechanisms (note that the strength of layers 1 through m is avail-able between these two log spirals).

The process is repeated to find the required length of layer (m-1) (Figure 5). Since layers above were al-ready ‘truncated’, they no longer contribute tensile re-sistance to deeper slip surfaces. Once the process has been repeated for all layers down to layer 1, the length of all layers (curve DEFGH in Figure 5), required to ensure that φmob does not exceed φd for all possible log spiral failure surfaces, has been determined. The proc-ess in ReSlope is slightly conservative since the full anchorage lengths to resist pullout are specified be-yond points D, E, F and G. This simplification is con-servative since, contrary to the compound analysis procedure, it ensures the following: t(allowable)m at point D (not zero resistance at D); t(allowable)m-1 at point E (not zero resistance at E); and so on. However, since the anchorage length of planar geosynthetic sheet is typi-cally small relative to its total required length in prac-tical problems, this simplification is reasonably con-servative. Programs ReSSA and MSEW do not use this simplification; the actual available strength of re-inforcement at its intersection with the slip surface is calculated and used in the stability analysis.

Compound critical surfaces emerging above the toe are also possible and consequently, the procedure in

Figure 5 is repeated for slip surfaces emerging through the face of the slope. Subsequently, layers previously truncated are lengthened, if necessary, to ensure that φmob ≤ φd. While other surfaces can pass through the reinforcement and the foundation, ReSlope ignores those (it assumes competent founda-tion). However, ReSSA fully accounts for such sur-faces.

A layout similar to the envelope ABCDEFG will contain, at least, m potential slip surfaces, all having the same minimal safety factor against rotational fail-ure (Figure 5). However, because of practical consid-erations, a uniform or linearly varying length of layers is specified in practice. As a result, the number of such equally critical slip surfaces is reduced in actual structure since most layers are longer and typically, some are stronger than optimally needed. ReSlope ignored the extra stability attained by longer than needed reinforcement (recall that its objective is to find the minimum length of reinforcement that pro-duces a target value of Fs against rotational failure). ReSSA considers the actual layout by accounting for the actual specified length and strength of reinforce-ment (its objective is to calculate Fs for a given layout and strengths).

Finally, anchorage lengths are calculated to resist pullout forces that are equal to the required allowable strength of each layer multiplied by a factor of safety Fs-po. In these calculations the overburden pressure along the anchored length and the parameter defining the shear strength of the interface between soil and re-inforcement are used. In ReSlope, this parameter, Ci, termed the interaction coefficient. It relates the inter-face strength to the reinforced soil design strength pa-rameters: tan(φd) and cd. In ReSSA and MSEW it re-lates to the full strength of the soil but a factor of safety ensures that the actual capacity would be at least 1.5 times greater than that needed.

The interaction coefficient is typically determined from a pullout test. The required anchorage length of layer j must equal tj / {σj⋅Ci⋅[tan(φd)+cd]} where σj signifies the average overburden pressure above the anchored length. Adding the anchorage length to the length needed to resist compound failure produces the total length required to resist internal and compound failures.

Figure 4. Tensile reaction transferred into soil next

to active zone

2.5 Direct sliding analysis Specifying reinforcement layout that satisfies a pre-scribed φd against rotational failure does not ensure sufficient resistance against direct sliding of the re-inforced mass along its interface with the foundation soil, or along any reinforcement layer. The rein-forcement length required to ensure stability against failure due to direct sliding, Lds, can be determined from a limit equilibrium analysis that satisfies force equilibrium; i.e., the two-part wedge method. Such a conventional approach is used in ReSlope and MSEW. However, ReSSA is consistent with LE analysis and therefore, it uses Fs against direct slid-ing (Spencer method) that accounts for the strength of the reinforcement should failure propagate through the geosynthetic layers.

Figure 6 shows the notation used in defining the geometry and forces in the two-part wedge analysis. First, an initial value of Lds is assumed. Then, for an assumed interwedge force inclination, δ, the maxi-mum value of the interwedge force, Pmax, is found by varying θ while solving the two force equilibrium equations for the active Wedge A. This interwedge force signifies the resultant of the lateral earth pres-sure exerted by the backfill soil on the reinforced soil. Next, the vertical force equilibrium equation for Wedge B is solved considering the vertical component of the lateral thrust of the active wedge (i.e., Pmaxsinδ). The reaction NB is obtained and the base sliding resist-ing force of Wedge B, TB, is calculated. While this procedure is used in ReSlope and, with some limita-

tions in MSEW, in ReSSA the method used is Spencer and all equations of equilibrium are satisfied.

When calculating TB, the coefficient Cds is used (Cds = the interaction coefficient between the rein-forcement and the soil as determined from a direct shear test). If the bottom layer is placed directly over the foundation soil, two values of Cds are needed: one for the interface with the reinforced soil and the other for the interface with the foundation soil.

In ReSlope and MSEW, the actual factor of safety against direct sliding, Fs-ds, is calculated by comparing the resisting force with the driving force:

)4(

cos δPT BF dss =−

This factor of safety corresponds to the assumed value of Lds. In case it is unsatisfactory, the value of Lds is changed and the process is repeated for Wedge A and Wedge B until the computed factor of safety against direct sliding equals to the prescribed value. In ReSSA the definition of the factor of safety against direct sliding is equally applied to the soils shear strength and reinforcement layers intersection the slip surface. That is, in ReSSA the rotational and translational Fs have the sane physical signifi-cance; in ReSlope the significance is different and thus comparing values rendered by these two pro-grams could be misleading.

The assumed value of δ may have significant in-fluence on the outcome of the analysis. Selecting δ>0 implies the retained soil will either settle rela-tive to the reinforced soil and/or the reinforced soil will slide slightly as a monolithic block thus allow-ing interwedge friction to develop. Some rein-forcement layers will typically intersect the inter-wedge interface (especially if i < 70° ). However, unlike program ReSSA, ReSlope ignores the tensile resistance of these reinforcement layers. Conse-quently, selecting a value of δ in between (2/3)φd and φd could be viewed as a conservative choice.

The technique for incorporating seismicity into the force equilibrium analysis is shown in Figure 6. In a pseudo-static approach, however, large seismic coeffi-cients may produce unrealistically large reinforced soil block, Wedge B. In this case, a permanent dis-placement type of analysis is recommended (i.e., Newmark's stick-slip model; e.g., Ling, Leshchinsky and Perry, 1997). Alternatively, one may eliminate inertia from Wedge B, analogous, in a sense, to Mononobe-Okabe model used in analysis of gravity walls. Only the 'dynamic' effects on P are superim-posed then on the statical problem. ReSlope allows for the elimination of wedge B; ReSSA and MSEW do not allow for such elimination.

Figure 6. Two-part wedge mechanism used in direct

sliding analysis

Figure 5. Length required to resist compound and pullout failures

2.6 Commentary 1. The factor of safety used in program ReSlope is

compatible with that used in reinforced walls in internal stability analysis (e.g., MSEW). Like walls, unreinforced unstable slopes thus enabling the soil to mobilize its full strength (i.e., attain an active state).

2. The presented approach assumes the foundation to be competent and therefore, deepseated failures were not considered. This approach was implemented in ReSlope. However, the computational procedure can be modified for slip surfaces that penetrate the foundation soil. Program ReSSA uses a generic approach that allows for soft foundations, complex geometry including reinforced embankments.

3. The approach can be modified to include any type of limit equilibrium analysis. In case of generalized approach, separation into direct sliding and compound stability is not needed (e.g., ReSSA). However, search routines in generalized methods must be capable of capturing critical surfaces of greatly different geometries (ReSSA allows for rotational failure using Bishop’s and 2- and 3-part wedges using Spencer’s).

4. Possibility of surficial failure is ignored in the presented procedure (i.e., ReSlope). It can be modified to deal with this issue by assigning low or zero reinforcement strength at the face provided the geosynthetic is not wrapped around. However, for steep slopes, strict limit equilibrium analysis will indicate insufficient stability at the surface. The empirical concept of soil ‘plug’ is assumed to be valid for closely spaced reinforcement layers. Programs ReSSA and MSEW directly address the potential issue of surficial stability.

5. Program MSEW follows accepted practice for the design and analysis of MSE walls. Hence, it includes checks for bearing capacity (considering the reinforced soil as a coherent mass) and eccentricity (or overturning). Both failure modes are adopted from conventional retaining wall design and may not be applicable for flexible MSE structures. Deepseated stability (used in ReSSA and ReSlope) serves as a much more rational approach than bearing capacity (using Meyerhof approach for eccentric load). Overturning failure is unrealistic mode of failure.

In the strict context of analysis, log spiral slip surface is valid for homogenous soil only. However, in the compound failure analyses (Figure 5), this surface passes through both reinforced and retained soil and possibly, even through the foundation soil. As an ap-proximation, one can use an averaging technique, considering the compound failure surface lengths in the reinforced soil and in the retained soil, to find equivalent values for φd and cd to be used in analysis. The value of the equivalent φd is used to define the trace of the log spiral passing through the reinforced and retained soils. This approximation approach is used by ReSlope. Program ReSSA considers the ac-tual soil properties in each zone through which the slip surface passes. The trade off is using a ‘less rigorous’ stability analysis (from statical standpoint): Bishop and Spencer. In practice, however, both methods typically yield quite accurate results.

3 DESIGN CONSIDERATIONS

3.1 General The presented approach is based on the state of lim-iting equilibrium. Such a state deals, by definition, with a slope that is at the onset of failure. Applica-tion of adequate safety factors should ensure accept-able margins of safety against the various failure

mechanisms analyzed. It is implicitly assumed that the different materials involved (i.e., the geosyn-thetic materials and soils) will all contribute their design strengths simultaneously to attain a state of limit equilibrium. For materials reaching a constant plastic shear strength after some deformation (e.g., soils), such an assumption is realistic. However, not all materials in the reinforced soil system possess this idealized plasticity. Consequently, the follow-ing guidance is provided for selecting material prop-erties.

3.2 Progressive failure and soil shear strength Slip surface development in soil is a progressive phenomenon, especially in reinforced soil where re-inforcement layers delay the formation of a surface in their vicinity (e.g., Huang et al., 1994), or it may be overstressed locally thus greatly deforming or creeping locally. Leshchinsky et al. (1995) recom-mended that the design values of φ and c (i.e., φd, cd) should not exceed the residual strength of the soil. This would ensure that at the state of a fully devel-oped slip surface, the shear strength used in the limit equilibrium analysis is indeed attainable all along the slip surface.

Use of residual strength has clear cost implica-tions in the design of reinforced slopes. The re-quired strength of the reinforcement increases somewhat; however, the required length of rein-forcement increases significantly since deeper slip surfaces are predicted. For compacted granular soil, an increase in length of 30 to 50% might typically be required. This additional length makes construction more difficult, especially if space constraint exists (e.g., widening existing embankment), thus render-ing construction more expensive than just the cost of extra reinforcing material. Hence, this combined with what currently appears as overly conservative designed reinforced slopes create a need to introduce a less conservative design approach.

Based on some experimental evidence, Leshchin-sky (2001) suggested the following hybrid procedure for design when granular compacted fill is used: a. Use φpeak and limit equilibrium analysis to locate

the critical slip surfaces. These surfaces will be utilized to determine the required layout of geo-synthetic layers (i.e., length and spacing).

b. Use φresidual along traces of the critical slip sur-faces determined in (a) to compute the required geosynthetic strength. That is, in internal stabil-ity use φpeak to locate the slip surface and the use φresidual in the limiting equilibrium equations to determine the geosynthetic reactive force. In compound analysis use φresidual in the limiting

equilibrium equations to assess the required rein-forcement strength along slip surfaces deter-mined using φpeak.

It is entirely possible that the backfill in flexibly re-inforced slopes will deform (during or after con-struction) mobilizing the soil beyond its peak strength. Therefore, the stability of such slopes may hinge then upon the strength of the reinforcement. Consequently, the reinforcement strength becomes critical to stability in case residual strength develops. Note that the hybrid approach recognizes that slip surfaces will form and have a trace based on the soil peak strength. However, possible development of progressive failure is also recognized and at this state, the ductile and potentially creeping reinforce-ment should be sufficiently strong to keep the sys-tem stable. It should be noted that in a sense, Ta-tsuoka et al. (1998) proposed a similar hybrid approach, however, it was limited to seismic design of reinforced walls.

The proposed procedure may result in signifi-cantly shorter reinforcement as compared to using φresidual. However, the required reinforcement strength will be somewhat larger than that computed when using φpeak. Leshchinsky (2001) proposed a simple procedure when using ReSlope. For ReSSA or MSEW the process is straightforward.

If cohesive fill is used, extreme care should be used when specifying the cohesion value. Cohesion has significant effects on stability and thus the required re-inforcement strength. In fact, a trace of cohesion may indicate that no reinforcement at all is needed at the upper portion of the slope. However, over the long run, cohesion of manmade embankments tends to drop and nearly diminish (normally consolidated clay). Since long-term stability of reinforced slopes is of major concern, it is perhaps wise to ignore the co-hesion altogether. It is therefore recommended to limit the design value of cohesion to a maximum of about 5 kPa. It should be pointed out, however, that end-of-construction analysis must also be conducted if a soft foundation is present. In this case stability against deepseated failure must be ensured.

3.3 Reduction factors related to geosynthetics Limit equilibrium analysis assumes that the geosyn-thetic will not mobilize its full strength before the design strength of the soil is attained. Formally, there is no consideration of deformations. One can envision a scenario in which very stiff reinforcement will have its strength mobilized, potentially reaching its design value before the soil mobilizes its strength. This may lead to overstressing and subse-

quently, premature rupture of the reinforcement, vio-lating the analysis premise that its tensile resistance will be available simultaneously with the soil strength. The result might be local, or even global, collapse. However, since geosynthetics are ductile (typically, rupture strain greater than 10%), large strains may develop locally in response to over-stressing thus allowing the soil to deform and mobi-lize its strength as assumed in the analysis and as needed for stability. Over twenty years of experi-ence indicate that lack of stiffness compatibility is not a problem when using limit equilibrium design.

To ensure that indeed some overstressing of the re-inforcement without breakage is possible, an overall factor of safety is specified. This factor multiplies the calculated minimal required reinforcement strength at each level. Typical values for this factor range from Fs-u=1.3 to 1.5. The strength of the factored rein-forcement should be available throughout the design life of the structure. To achieve this, reduction factors for installation damage (RFid), durability (RFd), and creep (RFcr) should be applied so that geosynthetics possessing adequate ultimate strength, tult, could be se-lected. That is, the specified geosynthetic should have the following short-term ultimate strength:

t

)7()()()()( RFrRFdRFcrRFid

F ustrequiredult⋅⋅⋅

− )( ⋅⋅=

Table 1 shows typical range of values used for vari-ous polymeric materials. The values of RFid and RFd are site specific. For typical reinforced soil con-ditions (i.e., near neutral pH), degradation should not be a problem when using typical reinforcing polymeric materials. The creep reduction factor, RFcr, depends, to large extent, on the polymer type and the manufacturing process. The term ultimate strength, tult, should correspond to the result obtained from the short-term wide-width tensile test, following, for example, ASTM D4595-86 procedure. Typically, the strength at 5% elongation strain in the wide-width test is reported as well. Some designers concerned with performance prefer to use this value as 'tult.' It should be noted that performance (i.e., deformations) of slope and embankments is less critical than that of walls and therefore, the 5% 'limit' is unnecessary and is overly conservative for most practical purposes. In fact, it is conservative even for walls. Finally, if seismicity is considered in the design, the reduction factor against creep can be set to one. Simply, since the duration of the superimposed

pseudo-static seismic load is short, significant creep is not an issue. However, the designer should verify that the required seismic strength is no higher than the required long-term value for static stability where creep is feasible; the larger strength value from static and pseudo-seismic should prevail.

Table 1. Preliminary reduction factors

Polymer Type RFid RFd RFcr

Polyester 1.0 to 1.5 1.0 to 2.0 1.5 to 2.0 Polypropylene 1.0 to 1.5 1.0 to 2.0 3.0 to 5.0 Polyethylene 1.0 to 1.5 1.0 to 2.0 2.5 to 5.0

PVA 1.0 to 1.5 1.0 to 1.5 1.4 to 1.8

3.4 Other specified safety factors The factor of safety against direct sliding, Fs-ds, en-sures that the force causing direct sliding is ade-quately smaller than the force available to resist it. In ReSlope it is a straightforward adaptation of analysis from reinforced retaining walls (MSEW) or gravity walls. It is recommended to use Fs-ds=1.5 to 2.0 to avoid possible progressive failure associated with peak shear strength of soil. However, if the de-sign value of the soil shear strength used in analysis is lower than its residual strength, one can use Fs-ds = 1.0 since safety is already manifested in the reduced shear strength. In ReSSA the factor of safety for di-rect sliding is applied to soils shear strength as well as reinforcement strength (i.e., slope stability ap-proach); hence, a value of 1.3 to 1.5 should be satis-factory.

Note that the coefficient Cds is related to direct sliding mechanism. There are two such coefficients. The first signifies the ratio of shear strength of the interface between the reinforcement and reinforced soil to the shear strength of the reinforced soil alone. The second coefficient signifies a similar ratio but with respect to the strength of the foundation soil. This coefficient reflects a mechanism in which soil slides over the reinforcement sheet or vice versa. Its value can be determined from direct shear tests in which the shear strength of the interface between the relevant types of soil and the reinforcement is as-sessed under realistic normal loads. Typically, Cds will vary between 0.5 and 1.0, depending on the type of soil and reinforcement. For typical granular soils and geosynthetics, Cds is about 0.8. In many cases the required length of bottom layer (i.e., see LB in Figure 7) may increase significantly as Cds decreases below 0.8.

The factor of safety against pullout, Fs-po, multi-plies the calculated required allowable tensile force

of each reinforcement layer. Anchorage length then is calculated to provide pullout resistance for this factored tensile force. Typically, Fs-po value is speci-fied as 1.5. Ci signifies an interaction coefficient. It relates the strength of the interface between the rein-forcement and soil to the shear strength of the rein-forced soil or the foundation soil, reflecting geosyn-thetic movement relative to the confining soil. The required anchorage length is calculated based on Ci. The value Ci is normally determined from a pullout test. Typically, the value of Ci varies between 0.5 and 1.0, depending on the type of soil and rein-forcement. For granular soils, the typical value of Ci is about 0.7. It should be pointed out that for typical reinforced slope or wall, anchorage length is quite small relative to the total required length in the final layout (i.e., 30 to 60 cm vertical spacing). Conse-quently, the interaction coefficient may just be con-servatively assumed in design.

3.5 Practical layout of reinforcement Two practical options for specifying reinforcement length are common in practice (Figure 7). The first option simplifies construction by specifying all lay-ers to have uniform length. This length is selected as the longest value obtained from the internal stability analysis, the pullout and compound failure analysis, or the direct sliding analysis. In walls it also include overturning and bearing capacity modes of failure.

The second safe option is to specify LB and LT at the bottom and top, respectively, where LB is the longest length from all analyses and LT is the longest length obtained from internal stability and com-pound/pullout analyses. Length of layers in between is linearly interpolated. This specification is more economical; however, it may result in misplaced layers at the construction site.

Figure 7 shows primary and secondary reinforcing layers. In ReSlope stability analyses, only primary layers are considered. However, layers spaced too far apart may promote localized instability along the slope face. Therefore, secondary reinforcement layers should be used. Their width should extend at least 1m back into the fill and their strength, for practical pur-poses, may be the same as the adjacent primary rein-forcement. The vertical spacing of secondary rein-forcement layer is typically be limited to 30 cm. Secondary reinforcement creates a 'coherent' mass at the face, a factor important for local stability, espe-cially in ReSlope where surficial stability is not checked. Furthermore, it allows for better compaction of the soil at the face of the steep slope. This, in turn,

increases the sloughing resistance and prevents surfi-cial failures. If wrap-around is specified (necessary in slopes steeper than about 50°), secondary reinforce-ment can be used to wrap the slope face as well. It should be backfolded at least 1 m back into soil, same as the wrapping primary layers. Secondary rein-forcement can also be used with ‘wire cage’ facing, gabions, or even modular blocks in segmental retain-ing walls.

Figure 7. Practical layout of reinforcement

4 RESULTS AND CASE HISTORY

4.1 Typical results Figure 8, reproduced from Leshchinsky and Boedeker (1989), shows the required tensile force calculated using internal stability analysis versus φpeak and slope inclination. It should be noted that Leshchinsky and Boedeker (1989) used a variational calculus technique to facilitate the generation of re-sults, however, the results are identical to those pro-duced using the scheme presented in this paper. This figure is limited to cohesionless slopes. The ordinate K represents the non-dimensional value of the calculated tj and, in a sense, is equivalent to ½Ka in lateral earth pressures (Ka is equivalent to Ran-kin’s if the reinforcement force is horizontal and to Coulomb’s if this force is inclined). Notice that for reasonable range of φpeak, the difference in required K as a function of assumed reinforcement force ori-entation at the slip surface is small. This difference is the largest for vertical slopes. The value of each tj can be calculated from this chart following the ra-tional presented in Figure 3. That is, start with j = 1 and top layer to find tn for which H equals Dn, then go to j = 2 and layer n -1 to find tn-1 where H equals Dn+Dn-1 and tn is known, and so on. Alternatively,

one can use this chart as an approximation. That is, the overburden pressure at the middle of a tributary area of a reinforcing layer can be calculated and then be multiplied by the tributary area and by the coeffi-cient K obtained from the chart. Note that soil pos-sessing low φ such as 15° or 20° is not likely to ex-hibit peak shear characteristics; it is presented in this and following figures for instructive purposes unless one uses the chart for a case where φdesign = φresidual = φpeak.

The K value in Figure 8 is the same as Rankin’s for horizontal reinforcement force; for vertical walls it would produce the same tensile force mobilized in each reinforcement layer as in MSEW (thus making ReSlope useful in terms of accepted design for both walls and slopes). ReSSA and the compound stabil-ity module of ReSlope will require half the maxi-mum strength rendered from Figure 8 since it as-sumes uniform mobilization of the reinforcement force at a LE state.

Figure 9 shows the outermost traces of critical log spirals obtained from internal stability analysis. It is for the horizontal inclination of geosynthetic force (for traces when reinforcement is tangential, see Leshchinsky and Boedeker, 1989). Notice that for vertical slopes, the surfaces are practically planar in-clined at angle of (45°+ φpeak/2). Also notice that as φpeak decreases, the slip surfaces become signifi-cantly deeper thus implying longer required length of reinforcement.

No charts are shown for required length based on compound stability analysis. The results in this case will depend on the selected reinforcement strength. The interested reader is referred to Leshchinsky et al. (1995) to view some typical surfaces. In general, compound failure will not control the length in near vertical slopes provided the reinforcement is closely spaced and uniform in strength. However, this would not necessarily be the case if geosynthetic layers with variable length and/or strength is speci-fied. Program ReSlope and ReSSA are ideally suited for this mode of failure.

Figure 10 shows the length of reinforcement re-quired to resist direct sliding. It is constructed for strength related to peak shear strength, direct sliding coefficient, Cds, equals one, and a factor of safety to resist direct sliding, Fs-ds, equals 1.5. Figure 10 (top) represents the case where full friction is developed along the interface between the two wedges (i.e., δ = φpeak) while Figure 10 (bottom) shows the conserva-tive case where δ = 0. Generally, it can be seen that as the slope flattens, the length of reinforcement in-creases. Also, the friction angle and the interwedge angle have significant effects on length. Notice that for 45° slopes combined with φ = 45°, no reinforce-

ment is needed, however, if one uses φdesign < φpeak the required length will increase. In this case, one could use lower Fs-ds in lieu of smaller φ. While these results correspond to ReSlope and MSEW, ReSSA uses slope stability analysis for direct sliding and therefore layers length may depends also on layers strength.

4.2 Case history

Fannin and Herman (1990) report the results of a field test of a well-instrumented full-scale slope. One tested slope in which no intermediate rein-forcement layers were used is adequate for compari-son with the discussed progressive failure approach and program ReSlope.

Figure 8. Calculated tensile reaction for cohesionless slopes

Figure 10. Required length to resist direct sliding as function of peak shear angle and slope inclination (assuming all soils possess same strength/density)

The slope height was 4.8 m and its inclination was 1H:2V (Figure 11). The backfill soil was a uni-formly graded medium to fine sand, compacted to a unit weight of 17 kN/m3. The plane strain residual internal angle of friction is reported to be 38°. Un-fortunately, the peak angle is not reported. The lay-out of the uniformly spaced geogrids is shown in Figure 11. The force distribution in each geogrid layer was measured using load cells. Only the fac-ing was constructed of a wire mesh, which is con-sidered equivalent to wrap-around face. Following construction, the wall was surcharged with soil placed to a depth of 3 m. Since no details are given, it is assumed that the slope of the this surcharge fill was 2H:1V.

T

apprwith38°.surflargevenreinstrenvalu

Figure 9. Outermost traces of internal slip surfaces

he outermost internal failure surfaces using the oach presented in this paper are contained in the reinforced zone (Figure 11) for φresidual = Since φpeak is unknown, the corresponding slip ace is not plotted, however, because φpeak is er than φresidual, the critical slip surfaces would be shallower (i.e., certainly contained within the

forced zone). The long-term allowable geogrid gth is not reported, but it can be verified that its e is much larger than the measured forces.

Hence, all compound slip surfaces are also con-

tained within the reinforced soil. Assessment of di-rect sliding reveals that Fs-ds for the layout used is between 1.5 and 2.0.

The actual layout is not the same as required in Figure 5 (i.e., not minimum lengths but rather uni-form lengths) and therefore, back-analysis using the presented design-oriented analysis (ReSlope) can only suggest a probable range of feasible values. The probable range for each layer is between the re-

quired forces needed for internal stability and for compound failure. The approach (ReSlope) speci-fies the maximum value of this probable range in design. Table 2 shows the comparison between measured values and those predicted using φresidual = 38°.

The agreement exhibited in Table 2 is considered good. Repeating calculations for the problem for φpeak = 43°, one gets Σtj = 11.1 kN/m; for φpeak = 41°, one gets Σtj = 13.3 kN/m. The measured (actual) value of Σtj was 15.3 kN/m. Fannin and Herman re-port only the total sum of forces for the surcharged case. The measured value is 22.2 kN/m whereas the calculated one (φresidual = 38°) is 21.1 kN/m.

Strain measurements by Fannin and Herman indi-cates the location of maximum force is shallower than that implied by φresidual (i.e., implied by the trace of slip surface shown in Figure 11). Use of φpeak as proposed by Leshchinsky (2001) will also produce shallower surfaces. Looking at the measured tension values (Table 2), one sees that the mobilized force in the reinforce-ment is approximately uniform among all layers. Such observation supports the approach used in ReSSA and in ReSlope if one examines the com-pound failure mode.

Figure 11. Configuration of Norwegian Wall

5 CONCLUSION

A framework for assessing the stability of slopes and embankments reinforced with geosynthetics has been presented. The analyses involved are based on limit equilibrium. These analyses ensure that the reinforced mass is internally and externally stable. A physically meaningful definition of factor of safety, which is relevant to an unstable soil structure unless reinforced, is introduced.

The presented stability analyses include recommendations regarding the selection of soil shear strength parameters and safety factors. Recognizing the limitations of limit equilibrium analysis, especially when applied to soil structures comprised of materials possessing different properties (i.e., such as soil and polymeric materials) and the potential for progressive failure, a hybrid approach for selecting soil shear strength is recommended. The peak shear strength parameters of the soil should be used to determine the critical slip surfaces (i.e., the reinforcement layout). Superimposing on these critical slip surfaces the residual strength of the soil and solving the limit equilibrium equations provide an estimate of the required reinforcement strength in case progressive failure is likely to develop.

The presented design procedure has been imple-mented in ReSlope (Leshchinsky, 1997, 1999). The mechanism and analysis used can be replaced with other stability methods such as program ReSSA (Leshchinsky, 2002). The approach is comprehensive and economical; experience proves it is safe. While ReSSA is based on ‘pure’ slope stability approach, ReSlope is based on a hybrid approach. That is, its rigorous Internal Stability mode yield results conven-tionally used in the design of MSE walls reinforced with geosynthetics whereas its Compound Stability mode corresponds to reinforced slope stability analy-sis. Consequently, its results are compatible with those of program MSEW (Leshchinsky, 1999, 2000); however, it does not deal with stability aspects that could be important for walls (e.g., connection strength). It provides a layout that ‘automatically’ can resist compound failure, an aspect that cannot be ad-dressed by lateral earth pressure methods used in de-sign of walls.

REFERENCES

Collin, J. (1997). Design Manual for Segmental Retaining Walls. 2nd Edition, National Concrete Masonry Association (NCMA). Elias, V. and Christopher, B.R. 1997. Mechanically Stabilized Earth Walls and Reinforced Steep Slopes, Design and Construction Guidelines. FHWA Demonstration Project 82. Report No. FHWA-SA-96-071. Fannin, J. and Herman, S. 1990. Performance data for sloped reinforced soil wall. Canadian Geotechnical Journal, 27(5), 676-686.

Huang, C.-C., Tatsuoka, F., and Sato, Y. 1994. Failure mechanisms of reinforced sand slopes loaded with a footing. Soils and Foundations, Journal of the Japanese Society of Geotechnical Engineering, 34(2), 27-40. Leshchinsky, D. 1992. Keynote paper: Issues in geosynthetic-reinforced soil. Proceedings of the International Symposium on Earth Reinforcement Practice, held in Nov. 1992 in Kyushu, Japan. Editors: Ochiai, Hayashi and Otani. Published by Balkema, 871-897. Leshchinsky, D. 1997. Software to Facilitate Design of Geosynthetic-Reinforced Steep Slopes. Geotechnical Fabrics Report, Vol. 15, No. 1, 40-46. Leshchinsky, D. 1999. Putting Technology to Work: MSEW and ReSlope for Reinforced Soil-Structure Design. Geotechnical Fabrics Report, Vol. 17, No. 3, April, pp. 34-38. Leshchinsky, D. 2000. Alleviating Connection Load. Geotechnical Fabrics Report, October/November, Vol. 18, Number 8, 34-39. Leshchinsky, D. 2000. On the Factor of Safety in Reinforced Steep Slopes. ASCE, Geotechnical Special Publication, Ed.: Zornberg and Christopher, No. 103, 2000, pp. 337-345. Leshchinsky, D. 2001. Design Dilemma: Use Peak or Residual Strength of Soil. Geotextiles and Geomembranes, Vol. 19, No. 2, pp. 111-125. Leshchinsky, D. 2002. Design Software for Geosynthetic-Reinforced Soil Structures. Geotechnical Fabrics Report, Vol. 19, Marc/April, pp. 44-49. Leshchinsky, D. and Boedeker, R. H. 1989. Geosynthetic reinforced earth structures. Journal of Geotechnical Engineering, ASCE, 115(10), 1459-1478. Leshchinsky, D., Ling, H. I., and Hanks, G. 1995. Unified Design Approach to Geosynthetic-Reinforced Slopes and Segmental Walls. Geosynthetics International, Vol. 2, No. 5, 845-881. Leshchinsky, D. and Reinschmidt, A.J. 1985. Stability of membrane reinforced slopes. Journal of Geotechnical Engineering, ASCE 111(11), 1285-1300. Ling, H.I., Leshchinsky, D. and Perry, E.B. Seismic Design and Performance of Geosynthetic-Reinforced Soil Structures. Geotechnique, Vol. 47, No. 5, 1997, pp. 933-952. Tatsuoka, F. and Leshchinsky, D. 1994. Editors: Recent Case Histories of Permanent Geosynthetic-Reinforced Soil Retaining Walls,

Proceedings of SEIKEN Symposium, held in November, 1992 in Tokyo, Japan, published by Balkema, 349 pages. Tatsuoka,F., Koseki, J., Tateyama, M., Munaf, Y. and Hori, N. 1998. Seismic stability against high seismic loads of geosynthetic-reinforced soil retaining structures. Keynote lecture, Proceedings of the 6th International Conference on Geosynthetics, Atlanta, Georgia, Vol. 1, 103-142. Taylor, D.W. 1937. Stability of earth slopes. Journal of the Boston Society of Civil Engineering, 24(3), 197-246. Yoshida, T. and Tatsuoka, F. 1997. Deformation property of shear band in sand subjected to plane strain compression and its relation to particle characteristics. Proceedings of the 14th International Conference on Soil Mechanics and Foundation Engineering, Hamburg, September, 237-240, Balkema.

Layer No. j

/ Elevation

[m] 1 /0.0 2 / 0.6 3 / 1.2 4 / 1.8 5 / 2.4 6 / 3.0 7 / 3.6 8 / 4.2 Σtj

Table 2. Reinforcement forces under self-weight loading

Calculated Measured Internal

Failure, tj [KN/m]

Compound Failure, Σtj/n

[kN/m] Probable Range

[kN/m] Maximum

Force, [kN/m]

4.1 2.13 2.1 – 4.1 1.06 3.5 2.13 2.1 – 3.5 2.25 2.9 2.13 2.1 – 2.9 2.01 2.4 2.13 2.1 – 2.4 2.34 1.8 2.13 1.8 – 2.1 2.00 1.3 2.13 1.3 – 2.1 1.46 0.8 2.13 0.8 – 2.1 1.92 0.2 2.13 0.2 – 2.1 2.26 17.0 17.0 15.3

Appendix

ReSSA, ReSlope, MSEW: Comparative Summary For complete details, see features of each program posted at www.GeoProgram.com

Program Applicable to MSE Structure:

Slope Angle

Reinforce-ment Geometry Maximum No.

of Soils Water

MSEW Walls1 70°-90° Geosynthetic or Metallic

Simple, two-tiered,

bridge abut-ment, back to

back

Reinforced soil, retained soil, and 5 other soils

Phreatic surface used

only in global stability

ReSlope Slopes & Walls2 10°-90° Geosynthetic Simple

Reinforced soil, retained soil,

foundation soil

Phreatic surface

ReSSA Slopes, Walls & Embankment3 10°-90° Geosynthetic or

Metallic

Nearly any complex ge-

ometry

25 different soils; reinforce-ment can be em-

bedded in all soils

Phreatic surface or pie-zometric lines; effective, total or mixed stress

analysis 1 MSEW is strictly for MSE walls (following AASHTO or NCMA). A slope stability module (Bishop) is available to check global stability. Reinforcement must be embedded in a prismatic shape non-cohesive homogeneous soil; the retained soil is non-cohesive. Additional 5 layers of soil can be specified for global stability analysis. Water is invoked only in global stability.

2 ReSlope is a design-oriented program that conducts local and global stability checks. The local stabil-

ity check is analogous with the one used in MSEW. It does not deal with facia (it assumes 100% connec-tion strength). It also does not deal with eccentricity, overturning, and bearing capacity (though deepseated failure is assessed). It inherently assumes competent foundation.

3 ReSSA uses a global slope stability framework (i.e., it assumes all reinforcement layers are equally mobilized). This means that if used in walls, the reinforcement strength might be insufficient for local sta-bility (experience shows that this is not an issue with geosynthetics). It considers various failure mecha-nisms; however, no overturning and bearing capacity are explicitly checked.

Program Strength

of Connec-tion:

Surcharge Unreinforced Slopes/Walls

Types of Stability Analysis2

Mechanisms

MSEW 0 - 100%

Uniform and strip (live and

dead), horizon-tal, point, and

isolated

No

Internal, direct sliding, eccentricity

(overturning), connection,

pullout, bear-ing capacity, and global

Planar, 2-part wedge (simpli-

fied), Meyerhof, Circular (Bishop)

ReSlope 100%1 Uniform and strip No

Internal, compound, deepseated

Log spiral and circular for deep-seated (Bishop)

ReSSA 0 - 100% Uniform and strip

Yes (can run as a generic slope sta-

bility program)

Rotational and transla-

tional

Circular (Bishop), 2- and 3-

part wedge (Spencer); effects of reinforcement s included if inter-sects slip surface

1 ReSlope ignores surficial failure assuming 100% connection strength 2 Factor of safety in ReSSA is consistent regardless whether rotational or translational analysis is

used; this factor applies equally to all elements resisting failure (i.e., shear strength parameters of soils and reinforcement resistance, if available); the factor for pullout represents a ratio of resisting force and pullout force. In MSEW this factor is applied only on the reinforcement strength in Internal Stability; it represents ratio of resisting force and driving force in direct sliding and pullout; it represent ratio of mo-ments in overturning; it represents the ultimate foundation capacity over the actual load, considering ec-centric load and Meyerhof method. In ReSlope the user can specify different factor for the soil shear strength and for the reinforcement strength when dealing with Internal Stability or Compound (thus mak-ing this approach adaptable to conventional approach to walls or to slopes); ratio of forces in pullout re-sistance and in direct sliding; reduction of soils shear strength when deepseated failure (Bishop) is used.