An integrated limiting equilibrium approach for design of reinforced soil retaining structures: Part I—formulation

An integrated limiting equilibrium approach for design of reinforced soil retaining structures: Part I—formulation

ARTICLE IN PRESS Geotextiles and Geomembranes 22 (2004) 119–150 An integrated limiting equilibrium approach for design of reinforced soil retaining ...

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ARTICLE IN PRESS

Geotextiles and Geomembranes 22 (2004) 119–150

An integrated limiting equilibrium approach for design of reinforced soil retaining structures: Part I—formulation R. Baker1,*, Y. Klein2 Faculty of Civil and Environmental Engineering, Technion—Israel Institute of Technology, Technion City, Haifa 32 000, Israel Received 22 June 2003; received in revised form 21 September 2003; accepted 17 October 2003

Abstract The present work presents a fully integrated limiting equilibrium process for design of reinforced soil retaining structures. Design examples based on this procedure are presented and discussed in the companion paper (Baker and Klein, 2003). The new features of the proposed procedure are: (1) It explicitly considers the properties of the three main elements wall–reinforcement–soils of the system. (2) Design requirements are formulated as local inequalities which are enforced at each relevant point rather then only globally. (3) The type of reinforcement and the interaction between reinforcement and soil is represented by response functions which can be established by pull-out tests. (4) Interaction between the wall and the reinforcing system is represented by a system of interaction parameters. The magnitudes of these parameters depend on the relative strengths of the wall and the anchoring system. The limiting situations of conventional reinforced soil design and classical (non-reinforced) retaining structures correspond to particular cases in which these parameters are equal to 0 and 1, respectively. Between those two limiting cases there exists a large range of intermediate design options which include different types of walls and reinforcement. (5) The design process results in distributions of tensile forces along each reinforcing layer, distribution of soil pressures acting on the wall, and distributions of shear forces and moments in the wall. These functions allow complete and rational design of all elements of the system. r 2003 Elsevier Ltd. All rights reserved. Keywords: Reinforced soil; Retaining structures; Limiting equilibrium; Integrated design procedure

*Corresponding author. Fax: +972-4-8292323. E-mail address: [email protected] (R. Baker). 1 Associate Professor. 2 Graduate student. 0266-1144/$ - see front matter r 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.geotexmem.2003.10.002

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Nomenclature Fmbd Fs Fs ðDÞ Fsj ðX Þ Fsd Fsh ðZÞ Fshd Fshb Fsfbd Ft Ftj ðX Þ Ftd Ht j; k K am

required value of Fmb local safety factor with respect to soil strength value of Fs at point D value of Fs at point X of the jth RL required value of Fs local safety against shear of the wall at depth Z required value of Fsh safety factor against shear along the base (foundation) of the wall required value of Fshb local safety factor with respect to reinforcement strength value of Ft at point X of the jth RL required value of Ft free height to be supported indices Rankine’s active earth pressure coefficient evaluated for the mobilized value fm of f Kj ðX Þ step j reinforcement interaction fuction gðX Þ normalized Coulomb–Culmann function L given reinforcement length Lj min minimum admissible length of the jth RL Lmin minimum admissible value of L MðZÞ distribution of moment in the wall Mj ðZÞ moment distribution in the jth section of the wall mj wall moments at reinforcement elevations MB moment acting on base n number of reinforcing layers N normal force on potential slip surface Pj resultant of soil stresses acting on the jth section of the wall PjR Rankine’s active force evaluated for a smooth vertical retaining structure having a height Zj supporting soil with a friction angle fm P cj force applied to the wall by jth RL Pj ðZÞ soil pressure acting on the jth section of the wall PS ðZÞ distribution of soil pressure on the wall PA ðZÞ distribution of active solid pressure on a non-reinforced wall Qj CC ðX Þ the total required horizontal force at point X of the jth RL Qj CC max maximum value of Qj CC ðX Þ Qj ðX Þ net required force function RC coverage ratio RðX ; tÞ reinforcement response function at SðZÞ shear force acting in the wall at depth Z Saj shear force in the wall differentially above the level of the jth RL S bj shear force in the wall differentially below the level of the jth RL

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SB shear force at the base of the foundation Stra reinforcement strength Strf j ðX Þ interface strength at point X of the jth RL Strws ðZÞ shear strength of the wall at depth Z Strwm ðZÞ moment capacity of the wall at depth Z Strmb moment capacity of base Tj ðX Þ tension force acting at point X of the jth RL Tj max maximal tensile force in the jth RL Vj normal vertical force acting in the wall’s cross-section at the level Zj W weight of test body fX ; tg horizontal coordinates (distance from the wall) XCC max location of the maximum point of gðX Þ Xj horizontal distance along the jth RL Xj max location of the maximum point of Qj ðX Þ Z vertical coordinate (increasing downwards) Zj depth of jth RL Greek letters D g e tj ðX Þ f fm y Zj d

distance between RL total unit weight differential distance shear stress at point X along the jth RL friction angle of soil mobilized friction angle inclination of potential slip surfaces participation factors friction angle between reinforcement and soil

1. Introduction The present work is aimed at constructing a general framework for limiting equilibrium (LE) analysis of reinforced soil retaining structures. The characteristic feature of such structures is that, in general, they include three components: soil– reinforcement–wall. In the context of the present work a wall is a structural element consisting of a stem (facing) and a foundation, while reinforcing layers (RL) are inclusions extending from the wall into the soil. Development of this type of supporting systems significantly increased the range of options available to the design engineer. One end of this range is a conventional retaining structure without soil reinforcement, while the other end consists of reinforced slopes with ‘‘wrap-around’’ geotextiles, and no physical wall. Between those two limits there exists a large range of intermediate design options which include different types of walls and reinforcements. Under certain conditions such

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intermediate designs may be optimal (in some sense). The purpose of the present work is to construct a unified design process which takes into account properties of all the components of such systems. Availability of such a procedure is a prerequisite for a rational decision with respect to the design option which is optimal under given conditions. The system soil–reinforcement–wall is highly indeterminate, requiring a complete deformation analysis for its rigorous investigation. Unfortunately such an analysis does not lead to a realistic design option due mainly to lack of reliable constitutive characterization of the components and the interfaces between them. Most practical design procedures for reinforced soil retaining structures are based on LE calculations (e.g. Department of Transport, 1978; Tensar Corporation, 1986; Schlosser, 1990; Collin, 1997; Elias and Christopher, 1997). The LE framework is ‘‘modest’’ in its input requirements (only strength properties have to be specified), thus properly reflecting the existing ‘‘state of information’’ available in most realistic projects. Rowe and Ho (1993) compared predictions of 12 different LE design procedures with performance of four experimental reinforced soil retaining structures. They concluded that ‘‘None of these methods can distinguish the differences that existed between the walls’’. The above conclusion indicates that conventional design procedures do not include significant elements characterizing the physical systems. This situation is a result of the fact that existing codes and design procedures for reinforced soils are mixtures of semi-theoretical and empirical rules, which do not combine into a coherent rational framework. The procedure presented in this work is comprehensive, in the sense that it explicitly includes all components of the system. The basic premises of the present work is that the simple LE approach is a formal theory (like elasticity) which reflect certain (not all) features of reality, deserving therefore a formal logical treatment consistent with its own framework. Consistently with this approach the present work follows a quasi-axiomatic approach, which is based on a small number of initial assumptions that are clearly stated from the outset. It is realized that a ‘‘perfect’’ design procedure should be both logically consistent and fit empirical observations. The present work emphasizes the first of those requirements (logical consistency) in the hope that such a formal approach will help to untangle some of the internal inconsistencies of conventional presentations. Results based on this approach appear to be consistent with most (but not all) available empirical observations, and if necessary, some of the basic assumptions can be modified in order to improve this correspondence.

2. The general framework 2.1. Assumptions The design procedure is formulated in the framework of slope stability LE type of analysis. The problem of reinforced soil retaining structures makes it necessary to

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introduce the following modifications to classical slope stability calculations: 1. Limiting equilibrium calculations are done for a system of ‘‘test bodies’’. Test bodies considered in the study of reinforced soil retaining structures include wall and reinforcement in addition to soil. 2. Baker and Leshchinsky (2001) introduced the notion of LE ‘‘safety maps’’ representing spatial distribution of safety factors. The present work is formulated in the safety map framework in the sense that design requirements for safety factors are imposed at every (relevant) point rather then only globally. 3. Conventional application of LE considerations to retaining structures results in the total active force acting on the structure, but it does not yield distribution of loads on the structure. Coulomb resolved this difficulty by considering a sequence of wall heights starting from the top of the wall. Using this approach he calculated not only the distribution of loads on the retaining structure but also the distributions of shear forces and moments in the structure. Leshchinsky (1992) extended this well known process to reinforced soils. Leshchinsky’s process is here called the L.E. top–down framework. In essence this framework replaces the indeterminate soil–reinforcement–wall system by a sequence of statically determinate problems. Such a replacement is obviously only an approximation, but it is probably the best that can be done in a LE framework which does not consider deformations. The validity of this approximation will have to be checked by comparing predictions of the present approach with experimental observations. The present work generalizes Leshchinsky’s top down process, including in it wall effects. 4. In a general continuum mechanics context, distribution of loads between elements of a system is governed by the relative rigidity of these elements. Rigidities play no role in LE analysis, and in the proposed design procedure the distribution of loads between the wall and the reinforcing system is governed by their relative strengths (rather then rigidities). Under usual conditions strength and rigidity are proportional to each other, and it may be expected that a procedure which distributes loads based on relative strengths leads to reasonable results. 5. Investigation into the basic structure of limiting equilibrium procedures (Baker, 2003b) showed that such procedures yield physically significant results only if clear distinction is made between ‘‘active’’ and ‘‘passive’’ stability problems. Restricting attention to conventional active problems implies the restriction pS ðZÞppA ðZÞ; where pS ðZÞ is the distribution of soil pressure along the wall when the soil is reinforced, and pA ðZÞ is the distribution of active earth pressure in the limiting case of non-reinforced soil. The inequality pS ðZÞppA ðZÞ is based on the assumption that increasing rigidity and strength of the soil by introduction of reinforcement should not increase loads acting on the wall. Practical implications of this restriction become evident only when considering systems including a large number of RL, and those implications are discussed in Baker and Klein (2003). The above five points are essential elements of the proposed procedure. In addition, the following simplifying assumptions are also introduced: (a) General variational analysis of limiting LE methods (Baker and Garber, 1978) shows that in a homogeneous setting critical slip surfaces are either straight lines or

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log spirals. It is possible to verify that a similar result holds when the structure is acted upon by a system of concentrated forces (Garber and Baker, 1979). In general, log spirals deliver more critical results then straight lines, but it is well known that results based on these two failure mechanisms converge in the case of steep slopes and/or a low ratio of cohesive to frictional strengths. Most practical reinforced soil retaining structures satisfy both of these requirements, and the present work restricts attention to straight slip surfaces. Both experimental observations and parametric studies (e.g. Bolton et al., 1978; Leshchinsky and Boedeker, 1989; Klein, 2003) show that for nearly vertical walls and c ¼ 0 materials, analysis based on straight slip surfaces produces acceptable results. (b) The inclination of reinforcement forces may vary between the initial (typically horizontal) direction, and parallel to the slip surface. Many parametric studies (e.g. Leshchinsky and Reinschmidt, 1985; Wright and Duncan, 1991), have shown that for steep granular slopes, inclination of reinforcement forces has only a limited effect on physically significant results, and it is usually assumed that these forces act in the original horizontal direction. Similar assumption is adopted in the present work. It is noted that this assumption may not be appropriate for rigid reinforcement like rock bolts where it may be necessary to estimate actual directions of reinforcement forces. Procedures allowing evaluation of these directions are available (e.g. Madhav and Umashakar, 2003). (c) As a first approximation it is assumed that the distribution of shear forces in the wall is linear between elevations of adjacent RL. In most practical situations the distance between RL is small relative to the height of the structure, and for such conditions this assumption does not introduce large errors. (d) It is assumed that the normal stress acting on a RL is equal to the overburden stress. There are some indications suggesting that these stresses may decrease slightly towards the back of the wall, but this effect is not large for steep walls. (e) Shear stresses between the wall and the soil are neglected. Under most circumstances this assumption produces conservative results. (f) In principle, each RL may have different elevation, length, and strength. However, complicated reinforcement layouts should be avoided in order to minimize execution errors. The present paper considers systems in which all RL have the same length and strength, and the distance between neighboring RL is constant. In addition, it will be assumed that the wall is vertical, the soil surface behind it is horizontal, and strength properties of compacted, retained, and base soils are the same (Fig. 1). These assumptions simplify the structure of various equations, and some of them are removed in Baker (2003a). Assumptions (a)–(f) are not essential, and they can be modified without affecting the general structure of the present approach. 2.2. Design requirements Fig. 1 represents the problem under considerations. Ht is the free height to be supported. The retaining system includes n RL, a wall, and a foundation. The RL are located at depths Zj ¼ jD; j ¼ 1; y; n; below the horizontal surface of the slope

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Fig. 1. General reinforced soil retaining structure.

where j is an index and D represents the (constant) distance between adjacent layers. The n RL define n þ 1 sections along the wall. Each section extends from Zjþ1  e to Zj  e where e is a differential distance. Each wall section j ¼ 1; y; n includes only one RL. The first wall section, which does not include a RL, is labeled Section 0 (Fig. 1). The basic design requirements for the system shown in Fig. 1 may be summarize as Fs ðDÞXFsd

at all interior points D ¼ fX ; Zg;

Faj ðX Þ  Strf j ðX Þ=jtj ðX ÞjXFad Ftj ðX Þ  Stra =Tj ðX ÞXFtd

for j ¼ 1 to n and X ¼ 0 to L;

for j ¼ 1 to n and X ¼ 0 to L;

Fsh ðZÞ  Strws ðZÞ=SðZÞXFshd Fm ðZÞ  Strwm ðZÞ=MðZÞXFmd

for Z ¼ 0 to Ht ; for Z ¼ 0 to Ht ;

ð1:1Þ ð1:2Þ ð1:3Þ ð1:4Þ ð1:5Þ

Fshb  Strshb =SB XFshbd ;

ð1:6Þ

Fmb  Strmb =MB XFmbd ;

ð1:7Þ

Fs is a local safety factor with respect to soil strength at all points D ¼ fX ; Zg in the interior of the slope (Baker and Leshchinsky, 2001, 2003). X ¼ Xj is a horizontal distance along the jth RL. Faj ðX Þ is a local safety factor against slip between the jth layer and the surrounding soil. Strf j ðX Þ is the frictional strength at point X along the interface between soil and the jth layer. tj ðX Þ is the shear stress at the same point. L is the length of all RL. Ftj ðX Þ is a local safety factor against tension failure of the jth RL. Tj ðX Þ represents the force acting at point X of the jth RL, and Stra is the strength of all RL. Z is a vertical coordinate along the wall. Fsh ðZÞ and Fm ðZÞ are the distributions of safety factors against shear and moment failure of the wall. Strws ðZÞ and Strwm ðZÞ are the shear and moment capacities of the wall at the depth Z: SðZÞ

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and MðZÞ are the distributions of shear forces and moments in the wall. SB and MB are the shear force and moment, respectively, acting at the support level B–B in Fig. 1. fStrshb ; Strmb g are the frictional resistant and moment capacity of the base (foundation), and fFshb ; Fmb g are the corresponding safety factors. fFsd ; Fad ; Ftd ; Fshd ; Fmd ; Fshbd and Fmbd g are required (design values), of the corresponding safety factors. These values are usually specified by codes, and in the present context they are considered as given input variables for the design procedure. The essential feature of the design requirements specified above is that they are all local inequalities, defined at every relevant point rather then globally. Limiting equilibrium calculations deal with a state of failure of fictitious materials having strength equal to the actual strength divided by a safety factor. In a design context safety factors are given, and the terminology ‘‘design strength’’ will be used for this reduced (mobilized) strength. 2.3. The top-down marching process Fig. 2a introduces basic notions related to the LE top down approach, specified for the particular case of straight slip surfaces. In order to consider a general step j of the top down process, assume that the functions Tk ðX Þ; k ¼ 1 to j  1; and fSðZÞ; MðZÞg; ZpjD are known, and they satisfy the design requirements specified by Eq. (1) at all points above the level of the jth RL. The purpose of step j is to establish the function Tj ðX Þ; and extend the functions fSðZÞ; MðZÞg to the level Z ¼ ðj þ 1ÞD while satisfying design requirements in Eq. (1). Baker and Leshchinsky (2001) showed that in the particular case of straight slip surfaces and non cohesive materials Fs ðDÞ ¼ tanðfÞ=tanðyÞ; where y is the inclination of the straight slip surfaces passing through the toe of the slope. This result implies that the critical condition for step j test bodies occurs on slip surfaces starting differentially above the level of the ðj þ 1Þth RL. Fig. 2a shows that such slip surfaces can be parameterized by the location X at which the section bc of the slip surface intersects the jth RL. Considering straight slip surfaces there is only one such surface passing through points b and c in Fig. 2a, and this slip surface (test body), is associated a safety factor Fsj ðX Þ: From a safety map perspective Fsj ðX Þ is the safety factor (on soil strength) at point X on the jth RL. Satisfying the design requirement (1.1) at all points along the jth RL guarantees that this requirement is satisfied also at all points D between the levels Z ¼ ðj þ 1ÞD  e and Z ¼ jD þ e: Consequently, the design requirement Fs ðDÞXFsd is equivalent to Fsj ðX ÞXFsd

j ¼ 1; y; n:

ð2Þ

Replacing Eq. (1.1) by Eq. (2) it is possible to solve the problem sequentially, starting at step 0 and ending at step n: Each step j establishes one anchor force function Tj ðX Þ; and satisfies all design requirements in section j of the wall. At the end of step n all anchor forces are known and design requirements are satisfied at all relevant locations.

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Fig. 2. Elements of the top down process. (a) Step j test bodies. (b) The reinforcement interaction function Kj ðX Þ: (c) Force polygon for step j test bodies.

In the sequential top down formulation, results for step j are affected by all the results obtained in steps koj; however results of step j do not affect any of the results in steps koj: In other words, the top down process allows only for a one directional, rather then simultaneous, interaction between elements of the system. It can be verified (Baker, 2003a) that this restriction results in conservative estimates of reinforcement forces, but not necessarily shear forces and moments in the wall.

3. The basic Coulomb–Culmann relation Dealing with the three component system wall–reinforcement–soil it is convenient to consider test bodies including these components. abgia in Fig. 2a is a typical test body considered at step j: The following features of such bodies are relevant for the present purpose: The bottom section ab of step j test bodies is located a differential distance above the level of the ðj þ 1Þth RL. The slip surface abg of this body consists of two

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sections, a horizontal section ab along which the slip surface intersects the wall, and the section bg passing through soil and reinforcement. The forces acting on this test body are: 1. The reinforcement forces Tk ; k ¼ 1 y; j  1: At step j these forces are assumed to be known. It is noted that for the particular test body showed in Fig. 2a the first RL is internal to the test body and therefore T1 ¼ 0: 2. An unknown force Dj acting point c of the jth RL. 3. The weight W of the test body, and the soil forces N and N tanðfm Þ acting in the soil along the section bg of the slip surface. It is convenient to define a ‘‘mobilized’’ friction angle fm using the criterion value Fsd of Fs ; i.e. fm  arctanðtanðfÞ=Fsd Þ: 4. The shear and normal forces Saj þ1 ; Vjþ1 acting in the wall along the section ab of the slip surface. The reinforcement applies a system of concentrated loads to the wall, and the distribution of shear forces SðZÞ in the wall may have jump discontinuities at reinforcement elevations. Anticipating this possibility, the notation Saj  SðZ ¼ jD  eÞ and Sbj  SðZ ¼ ðjÞD þ eÞ is used for shear forces in the wall above and below the elevation of the jth RL. The difference Saj  Sbj is the force Pcj ¼ Tj ðX ¼ 0Þ which the jth RL transmits to the wall. In the technical literature the variables Pcj are frequently referred to as connection loads. 5. Neglecting shear stresses along the contact plane bh between the wall and the soil (assumption (e)), the normal vertical force Vjþ1 along ab is equal to the weight of the wall section abhi: These two forces cancel each other, and they do not enter into equilibrium equations for the test body. As a result, the ‘‘effective’’ weight W of the test body abgia is the weight of the soil triangle bghb, and along ab only the shear force Sajþ1 has to be considered. The forces Tk ; koj are all horizontal, (assumption (b)), and their combined effect can be represented by a single function Kj ¼ Kj ðX Þ which will be referred to as the reinforcement interaction function. At step j the individual force functions Tk ðX Þ; k ¼ 1yj  1; are assumed to be known. These known functions specify Tk in the kth RL in terms of the distance Xk along this layer. Fig. 2b shows a number of schematic force functions Tk ðXk Þ (thin lines). The dashed heavy line in Fig. 2b is the function Kj ðX Þ; representing the combined effect of Tk ðXk Þ; k ¼ 1yðj  1Þ on the test body abcghia. Considering the geometry of test bodies it is possible to verify that: Kj ðX Þ ¼

j1 X

Tk ðXk ¼ ðj  k þ 1ÞX Þ:

ð3Þ

k¼1

It is convenient to consider the relation Xk ¼ ðj  k þ 1ÞX as a coordinate transformation between the coordinates Xk and Xj ¼ X : Eq. (3) first transfers the known functions Tk ðXk Þ to the ‘‘current’’ coordinate system Xj ¼ X ; before adding these functions. The zero wall section is not reinforced, and this implies the identities

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K0 ðX Þ ¼ K1 ðX Þ  0: The coordinate transformation Xk ¼ ðj  k þ 1ÞX has two significant effects on the structure of the LE procedure: P (1) The maximum value of Kj ðX Þ must be less or equal to the sum j1 k¼1 Tk max where Tk max ¼ MaxX fTk ðX Þg: This is a result of the fact that the sum of maximum reinforcement forces is not relevant to any one of the step j test bodies (except in the unrealistic limiting case in which the functions Tk ðX Þ are actually constants). This observation stands in a sharp contrast to procedures based on Rankine’s pressure distribution (the tie-back design framework) which considers simultaneously all maximum reinforcement forces. (2) This transformation restricts the range of X and Z values for which the jth RL ‘‘interacts’’ with layers located above it. Inspection of Fig. 2b shows that when the vertical distance D between RL is constant, Kj ðX XL=2Þ  0; i.e. test bodies associated with large X values are not affected by RL located above the jth RL (e.g. the first RL in Fig. 2a). Fig. 2c shows a force polygon for a typical step j test body. Considering this figure, it is possible to verify that equilibrium conditions for such test body can be written in the form: Qj CC ¼ Qj CC ðX Þ  Kj ðX Þ þ Sajþ1 þ Dj ðX Þ;

ð4:1Þ

where Qj CC ðX Þ ¼ Aj gðX Þ; Aj ¼

ð4:2Þ

gðZjþ1 Þ2 gððj þ 1ÞDÞ2 ¼ ; 2 2

ð4:3Þ

X ðD  X tanðfm ÞÞ ; DðD tanðfm Þ þ X Þ

ð4:4Þ

gðX Þ ¼

where g is the total unit weight. The force Qj CC defined by Eq. (4) has two alternative physical interpretations: (i) Qj CC is the horizontal force required in order that Culmann’s safety factor Fsj ðX Þ for a test body passing through X ; is equal to the design value Fsd : (ii) In an ‘‘artificial soil’’ characterized by the mobilized value fm of f; Qj CC is Coulomb’s active force that a test body passing through X applies to a smooth vertical wall having the height Zjþ1 : The above dual interpretation is inherent to the analysis of reinforced soils, combining considerations related to both slope stability and retaining structures. This duality motivates the notation Qj CC as the step j total horizontal force implied by the Culmann–Coulomb framework. Equations (4) are valid for the particular case of uniform distribution of RL with depth (i.e. a constant D). In that case the function gðX Þ is the same for all RL. More general results, relevant for arbitrary placement of reinforcement, are presented in Baker (2003a).

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Fig. 3. The normalized Coulomb–Culmann function gðX Þ:

Fig. 3 is an example of the function gðX Þ evaluated for f ¼ 40 ; Fsd ¼ 1:5 and D ¼ 0:5 m: It is possible to verify that the maximum value gmax ¼ MaxX fgX g of gðX Þ is equal to Rankine’s active earth pressure coefficient. Kam evaluated for a system characterized by the mobilized value fm of f: The maximum value, Qj CC max ; of Qj CC ðX Þ is equal to Qj CC max ¼ MaxX fQj CC ðX Þg ¼ Aj gmax ¼

gðZjþ1 Þ2 Kam ¼ Pjþ1R : 2

ð5Þ

Formally, Pjþ1R is the active Rankine force evaluated for a smooth vertical retaining structure having a height of Zjþ1 and supporting soil with a friction angle fm : Rankine’s earth pressure theory has no ‘‘stature’’ in the present limiting equilibrium framework. However, emergence of the above result is hardly surprising; physically Qj CC max is the total horizontal force acting on a conventional retaining structure (with no anchors). The assumption that this force is horizontal (Fig. 2), corresponds to one of the few cases for which Coulomb’s and Rankine’s theories yield exactly the same results (equal upper and lower bounds).

4. Participation factors—Interaction between wall and reinforced soil The function Qj CC ðX Þ is the total horizontal force required in order to satisfy the limiting relation Fsj ðX Þ ¼ Fad : Kj ðX Þ is the part of Qj CC ðX Þ which is carried by the ðj  1Þ RL located above layer j: It is convenient to define a ‘‘net’’ required force function Qj ðX Þ and an anchor demand function Dj ðX Þ as Qj ðX Þ  Qj CC ðX Þ  Kj ðX Þ;

ð6:1Þ

Dj ðX Þ  /Qj ðX Þ  Sajþ1 S;

ð6:2Þ

where / S ¼ Maxf ; 0g: The net required force Qj is supplied partially by the shear force Sajþ1 acting in the wall, and partially by the tensile force Dj in the jth RL. Dj ðX Þ

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is the ‘‘demand’’ imposed on the jth RL by the design requirement Fsj ðX ÞXFsd : Negative parts of Dj ðX Þ are obviously not relevant, corresponding to situations in which the jth RL has to ‘‘push’’ the test body down in order to satisfy Fsj ðX Þ ¼ Fsd : The design requirement in Eq. (2) is an inequality, and test bodies resulting with negative values of Dj do not impose any demand on the jth RL. The operation /Qj ðX Þ  Sajþ1 S Eq. (6.2) ‘‘cuts off’’ the non relevant, negative, parts of Qj ðX Þ  Sajþ1 ; replacing them with zero. It is not difficult to verify that points X at which Qj ðX Þ  Sajþ1 o0 (and Dj ðX Þ ¼ 0) are associated with Fsj ðX Þ > Fsd ; and such situations are obviously legitimate. Qj CC ðX Þ has a maximum located at XCC max ¼ D=tanð45 þ fm =2Þ (for homogeneous reinforcing systems D is constant and the location of this maximum does not depend on j). Considering the definition of Qj ðX Þ (Eq. (6.1)), it is clear that Qj ðX Þ must have a maximum, Qj max ; located at some point Xj max : Xj max defines the critical slip surface for step j: In general the maximum points XCC max and Xj max of Qj CC ðX Þ and Qj ðX Þ do not occur at the same place. The Coulomb–Culmann framework does not provide a criterion for separation of Qj into the ‘‘components’’ Dj and Sajþ1 carried by the jth RL and wall, respectively. In order to quantify the interaction between the wall and the reinforcing system it is convenient to introduce a set of variables Zj ; j ¼ 1; y; n; and write Eq. (6.2) in the form Sajþ1 ¼ Zj Qj max ;

ð7:1Þ 

Qj ðX Þ  Zj Dj ðX j Zj Þ ¼ /Qj ðX Þ  Zj Qj max S ¼ Qj max Qj max   Qj ðX Þ  Zj ; ¼ Qj max Qj max Dj max ðZj Þ ¼ Qj max ð1  Zj Þ;



ð7:2Þ ð7:3Þ

Eq. (7.1) replaces the unknown Sajþ1 by another unknown, Zj ; and as such it is merely a variable transformation with no ‘‘physical content’’ . The first two forms of Eq. (7.2) result from substituting Eq. (7.1) into (6.2). This equation shows that demand functions depend on Zj ; and the notation Dj ðX j Zj Þ signifies that Eq. (7.2) is valid for a given value of Zj Zj separates the maximum value of the net required force Qj max ; between the ‘‘components’’ Dj max and Sajþ1 ; and it must be in the range 0pZj p1: This observation allows dropping the operation / S in the last form of Eq. (7.2). Eq. (7.3) results from application of Eq. (7.2) at the maximum point Xj max (which is common to both Qj ðX Þ and Dj ðX ÞÞ: The parameters Zj specify the extent to which the wall participates in the task of carrying the maximum values of the net required loads Qj max : It appears appropriate therefore to call these numbers ‘‘participation factors’’ or ‘‘wall interaction parameters’’. Fig. 4 illustrates the relations between various step j force functions. abc is the function Qj CC ðX Þ defined in Eq. (4). def is the reinforcement interaction function Kj ðX Þ defined and discussed in Eq. (3). ghi is the net required net force

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Fig. 4. Relations between step j force functions.

Qj ðX Þ ¼ Qj CC ðX Þ  Kj ðX Þ (Eq. (6.1)). In the particular case of Zj ¼ 0 this function is also the demand function for the jth RL, i.e. Dj ðX j Zj ¼ 0Þ ¼ /Qj ðX ÞS (Eq. (7)). Fig. 4 shows that Dj ðX j Zj ¼ 0Þ ¼ 0 in the ranges X oXg and X > Xi : This is the effect of RL located above layer j: jkl is the function Dj ðX j Zj ¼ 0:3Þ: This function is a shifted down version of Dj ðX j Zj ¼ 0Þ with the negative parts being cut off. The vertical distance between Dj ðX j Zj ¼ 0Þ and Dj ðX j Zj ¼ 0:3Þ is the effect of the wall on the demand imposed on the jth RL. Increasing Zj decreases both the maximum demand Dj max (Eq. (7.3)) and the range of X values in which the demand function Dj ðX j Zj Þ is different from zero. Point m is the limiting case associated with Dj ðX j Zj ¼ 1Þ: In this case the wall carries the entire load and no demand is imposed on the jth RL.

5. The general perspective-design options Before continuing with formal derivations it is convenient to consider the role of participation factors from a design perspective. In general, conventional LE considerations do not provide unique criteria for determination of participating factors. Consequently, in this framework any set of participation factors 0pZj p1; j ¼ 1; y; n is, in principle, legitimate. In the remaining part of the present work it will be verified that each given set of Zj (and L) implies unique distributions of tensile forces Tj ðX Þ; j ¼ 1; y; n in the reinforcing system (Sections 6 and 7), and unique distributions of shear forces and moments fSðZÞ; MðZÞg in the wall (Section 8). As a result, any choice of Zj ; j ¼ 1; y; n has engineering consequences with respect to the design of the supporting system. Stated differently, it is possible to consider participation factors as ‘‘design, variables’’, which can be chosen at will, but each such choice has a ‘‘price tag’’ attached to it. The particular case in which all Zj are equal to 1 is equivalent to the classical situation of conventional retaining structure without soil reinforcement. The limiting case in which all Zj are equal to zero corresponds to the assumption that the wall does not participates in the task of

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carrying the maximum values of the net required loads Qj max : Most conventional design procedures for reinforced soil retaining structures neglect the contribution of the wall during the anchor design stage. In the present framework such practice is equivalent to the limiting case Zj ¼ 0; j ¼ 1; y; n: In general, increasing the values of participation factors is associated with transfer of loads from the reinforcing system to the wall. Consequently large participation factors imply weak reinforcing system but substantial wall, and vice versa. Schematic illustration of the role of participation factors as design variables is shown at the top of Fig. 5. This design option is called here D.O. —0 (design option zero). The significant feature of this design option is that each given set of Zj implies a particular design, consisting of fL; Stra g and Strw (where Strw stands for wall strength; including both shear and moment capacity). Design option zero is not really practical, requiring the designer to make n decisions (choices) with respect to the magnitude of participation factors. However, utilizing the fact that there is a unique relation between the sets Tj max and Zj ; for j ¼ 1; y; n; makes it possible to ‘‘reverse’’ the direction of the arrows along the top branch of D.O.—0, and establish participation factors associated with a given choice of reinforcement strength. This relation is established in Appendix A. Following the evaluation of Zj it is possible to evaluate the functions fSðZÞ; MðZÞg and design the wall. This design option will be referred to as D.O.—1 (design option 1) and it is illustrated in the middle part of Fig. 5. In essence D.O.—1 establishes the requirements that the wall has to satisfy for a given choice of reinforcing system. Alternatively, reversing the arrows along the bottom branch of D.O.—0, it is possible to establish participation factors for a given wall, resulting with the design process shown in the bottom part of Fig. 5. Such a design process establishes the requirements that the anchoring system should satisfy in order to be compatible with a given wall. This process is referred to as D.O.—2. Under most situations, moments control the structural design of the wall, and the significant characteristics of a ‘‘given’’ wall is its moment capacity. Appendix B presents a procedure for establishing participation factors for the particular case in which the wall is

Fig. 5. Design options.

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constructed from pre-cast elements, using the assumption that interfaces between elements have zero moment capacity, i.e. they act as ideal hinges.

6. Interaction between RL and soil 6.1. Local slip between RL and soil For the present purpose it is important to make a clear distinction between the demand function Dj ðX Þ and the response Tj ðX Þ of the jth RL to this demand. Restricting attention to horizontal reinforcing systems and soil surface, and using the assumption that the normal stress acting on the jth RL equals the overburden (assumption (d)) it is not difficult to verify that the frictional strength Strf j ðX Þ at interface j is Strf j ðX Þ ¼ gD tanðdÞj ¼ Const:; where d is the friction angle between the soil and the RL material. The corresponding expressions for inclined anchors and/or arbitrary soil surface are given in Baker (2003a). Considering horizontal equilibrium of a differential RL element (Fig. 6a), shows that tj ðX Þ ¼ 1=2Rc dTj ðX Þ=dX where Rc is the ‘‘coverage ratio’’ which specifies the reinforcement horizontal area per unit total area ðRc ¼ 1 for two dimensional reinforcing systems like geotextiles). Introducing the above relation into the definition of Faj ðX Þ (Eq. (1.2)) results in: Faj ðX Þ ¼

Strf j ðX Þ aj ¼ ; jtj ðX Þj jdTj =dX j

ð8:1Þ

where aj ¼ 2gRc Zj tanðdÞ ¼ 2gRc D tanðdÞj:

ð8:2Þ

The constants aj represent the local bond strength between the jth RL and the surrounding soil. Imposing the design requirement Faj ðX ÞpFad (Eq. (1.2)) results with the following restriction on the slope of the force functions Tj ðX Þ:   dTj ðX Þ aj   ð9Þ  dX pF : ad

Thus, design requirement (1.2) restricts the rate of change (slope), of legitimate reinforcement force functions. 6.2. Reinforcement response functions and the basic maximization problem Each point, t; on a demand function Dj ðtÞ represents the demand imposed on the jth RL by a test body bounded by a slip surface passing through t: This demand results with a distribution of tensile forces RðX ; tÞ at points X > t of the jth RL (Fig. 6b). In essence, the test body associated with a slip surface passing through t (point g), performs a ‘‘pull-out test’’ on the section L  t of the jth RL, and RðX ; tÞ is the response generated at X due to a demand applied at t: The pull-out test analogy is obviously not perfect; in a conventional pull-out test the soil is stationary and the reinforcement is pulled-out by an external load applied to one of its end point. In

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Fig. 6. Interaction between reinforcement and soil. (a) Differential equation for Tj ðX Þ: (b) The pull-out test analogy.

reinforced soil structures the soil is moving (probably not uniformly) relative to a deforming RL; this relative movement results in shear stresses being transferred to the layer along the section fg of Fig. 6b, and the demand at t is the resultant of these stresses. The functions RðX ; tÞ will be referred to as ‘‘reinforcement response functions’’. Reinforcement response functions describe the interaction between reinforcement and soil. These functions may be established based on pull-out tests (Baker, 2003a). Reinforcement response functions provide a natural tool for incorporating various physical aspects of the reinforcing system into the formal LE framework. All legitimate response functions must, however, satisfy the following consistency requirements: RðX ¼ tÞ ¼ Dj ðtÞ;

ð10:1Þ

Rðt j X ¼ LÞ ¼ 0;

ð10:2Þ

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RðX ; t j Dj ðtÞ ¼ 0Þ ¼ 0;

ð10:3Þ

  @RðX ; tÞ aj    @X pF : ad

ð10:4Þ

Eq. (10.1) is an identity following from the action–reaction principle. Eq. (10.2) is an obvious boundary condition. Eq. (10.3) means simply that zero demand induces zero response. Eq. (10.4) is a result of the observation that, physically, reinforcement response functions represent the force distribution in a RL for a given demand Dj ðtÞ; and this distribution must satisfy the inequality in Eq. (9). Fig. 7 shows schematic response functions RðX ; tÞ corresponding to different points t of a given demand function Dj ðtÞ: At every point X the jth RL must satisfy simultaneously the demands associated with all test bodies corresponding to t values in the range tpX : Therefore the step j force function, Tj ðX Þ; is a solution of the following constrained maximization problem: Tj ðX Þ ¼ MaxfRðt j X Þg tpX

subject to the restriction of Eq. (9):   dTj ðX Þ aj    dX pF :

ð11:1Þ

ð11:2Þ

ad

It is convenient to consider first the un-constrained maximization problem defined by Eq. (11.1) (temporarily ignoring the constraint 11.2). Inspection of Fig. 7 makes it possible to derive the following general conclusions: 1. In the range X oXj max the solution of the unconstrained maximization problem (11.1) is Tj ðX Þ ¼ Dj ðX Þ: This is a consequence of the fact that in that range Dj ðtÞ is an increasing function while RðX j tÞ decreases with X (so that for each fixed value of X the maximum of Rðt j X Þ occurs at t ¼ X ). At this stage the ‘‘solution’’ Tj ðX Þ ¼ Dj ðX Þ is only formal, since it may violate the constraint (11.2).

Fig. 7. The maximization problem.

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2. In the range X oXj max demand functions are increasing, and the result Tj ðX Þ ¼ Dj ðX Þ implies that in this range also Tj ðX Þ is increasing. This is possible only if shear stresses tj ðX Þ acting on the jth RL are negative (pointing towards the wall). It is possible to verify that tj ðX Þ must be positive in the range X > Xj max ; and continuity of tj ðX Þ at X ¼ Xj max implies that  dTj ðX Þ ¼ 0: dX X ¼Xj max This relation guaranties that the constraint (11.2) is satisfied at Xj max ; and at that point the solution Tj ðX Þ ¼ Dj ðX Þ of the unconstrained problem is physical, not only formal. This observation implies the following general result which is valid for all reinforcement response functions: Tj max ¼ Dj max :

ð12Þ

The result Tj max ¼ Dj max is natural. The ‘‘anchors’’ in reinforced soils are ‘‘passive’’, and forces in these anchors are reactions generated by the tendency of test bodies to move down and out. A reaction can never exceed the action (demand) generating it; so Tj ðX Þ must satisfy Tj ðX ÞpDj max (a RL will not ‘‘voluntarily’’ supply more than the maximum demand imposed on it). The maximum demand must however be met, and this implies Tj max ¼ Dj max : The formal maximization problem defined by Eq. (11.1) automatically generates this intuitive result. 3. Fig. 7 shows that the solution Tj ðX Þ ¼ Dj ðX Þ is valid in the part cd of section cf in which the constraint dTj ðX Þ=dX ¼ dDj ðtÞ=dtpaj =Fad is satisfied automatically and inequality (11.2) is not ‘‘active’’. At point d this inequality is ‘‘engaged’’, preventing Tj ðX Þ from continuing along the steep section df of Dj ðtÞ: In the range X oXd the solution of the constrained problem is realized along the boundary dTj ðX Þ=dX ¼ aj =Fad of inequality (11.2), resulting in the linear section d’e: It is noted that Dj ðX Þ ¼ Dj ðX j Zj Þ; i.e. Dj ðX Þ depends on Zj ; therefore also Tj ðX Þ ¼ Tj ðX j Zj Þ and Pcj ¼ Tj ðX ¼ 0 j Zj Þ ¼ Pcj ðZj Þ: In the particular case when jdDj ðtÞ=dtjf oaj =Fad ; the section d’e of Tj ðX Þ does not exists, and this function follows Dj ðtÞ from c’ through f to g: In such situations Pcj ¼ 0; i.e. the RL is practically ‘‘detached’’, and it does not apply force to the wall. 4. The above discussion shows that in the range X pXj max the solution of the maximization problem 11 is trivial, and this problem has to be formally solved only in the range X > Xj max : In that range, response functions RðX j tÞ for different values of t may cross each other (e.g. the maximum demand at point a of Fig. 7, is ‘‘generated’’ by the response function RðX j Dj ðta ÞÞ shown dashed in that figure). Consequently it is not known a priori which value of t generates the critical result for a given X : Solving the complete maximization problem (11) in the range Xj max oX oL results with a function Tj ðX Þ which is an upper envelope of individual response functions.

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6.3. Inextensible reinforcement There exist various models describing the behavior of flexible reinforcing systems (e.g. Sobhi and Wu, 1996; Madhav et al., 1998; Gurung and Iwao, 1999a,b). Reinforcement response functions based on these models are discussed in Baker (2003a). In the present work attention is restricted to inextensible (horizontally rigid) reinforcing systems. Both elastic analysis and experimental results appear to indicate that for this particular case the functions RðX j tÞ are approximately linear (e.g. Abramento and Whittle 1995a,b). Utilizing this observation it is possible to show that functions RðX ; tÞ appropriate for this particular case are: RðX ; tÞ ¼

Dj ðtÞðL  X Þ : Lt

ð13Þ

The discussion in the previous section showed that for X > Xj max the function Tj ðX Þ may be an envelope of response functions (Fig. 7). Each point on this envelope must satisfy Eq. (11.2), so it is necessary that jdTj ðX Þ=dX j ¼ [email protected]ðX ; tÞ[email protected] jpaj =Fad : Applying this requirement with the function RðX ; tÞ given in Eq. (13) results in jdTj ðX Þ=dX j ¼ @RðX ; tÞ[email protected] ¼ Dj ðtÞ=L  tpaj =Fad ; or equivalently, LXt þ Dj ðtÞ=ðaj =Fad Þ: This relation should be satisfied for all values of t; resulting with the following ‘‘minimum length criterion’’:  Dj ðtÞ Lj min ¼ Max t þ : ð14Þ t aj =Fad It is possible to verify that LXLj min implies satisfaction of the no-slip design criterion Faj ðX ÞXFad (Eq. (1.2)) in the range X XXj max : The following additional comments are relevant with respect to Eq. (14): (a) Eq. (14) is valid for the linear model defined in Eq. (13). Minimum length criteria for non linear models are derived in (Baker, 2003a). (b) Eq. (14) is valid for the case in which jdDj ðtÞ=dtji > aj =Fad (Fig. 7). If this inequality is not satisfied then the minimum admissible length of the jth RL is given by point i at which the demand function is equal to zero. (c) The magnitude of Lj min depends on the interaction function Kj ðX Þ; and through it, on the given value of L; i.e. Lj min ¼ Lj min ðLÞ: (d) Eq. (14) defines the minimum required length for the jth RL. Considering systems in which all the RL have equal length L; the minimum acceptable value of L is Lmin ¼ Maxj fLj min ðLÞg: This non-linear relation for Lmin has to be solved numerically. Fig. 8a shows a system of linear response functions (dashed lines), superimposed on a given demand function Dj ðt j Zj Þ (solid line), and a numerical solution of the constrained maximization problem 11 (dotted line abcde), which is the force function Tj ðX j Zj Þ: The numerical solution of the maximization problem was obtained by applying Eq. (11) at 100 equally spaced points in the range Xj max pX pL: Experimentally established anchor force functions Tj ðX Þ frequently have the

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Fig. 8. Solution of the maximization problem for non-extensible anchors. (a) Construction of reinforcement force functions. (b) Distribution of safety factors along the jth RL.

character of the line abcde resulting from solution of the maximization problem (11) (e.g. Bolton et al., 1978; Tatsuoka, 1993, and many others). Response functions associated with the linear model have convenient properties making it possible to establish Tj ðX Þ by simple geometrical considerations without actually solving the formal maximization problems (11) or (14). The function Tj ðX Þ in Fig. 8a consists of three sections ab, bcd and de. Section ab is a tangent line to Dj ðX Þ through the given end point a ¼ fX ¼ L > Lj min ; Tj ðX ¼ LÞ ¼ 0g of the RL. Section de is a tangent line to Dj ðX Þ having the slope aj =Fad ; and Tj ðX Þ ¼ Dj ðX Þ along the central section bcd. The line hgcde is a solution of Eq. (11) for the particular case in which L ¼ Lj min : The section hg of this solution is a straight tangent line to Dj ðX Þ having the slope aj =Fad ; and the location of h defines Lj min : In order to verify that the above geometrical construction results with a legitimate solution of the problem it is convenient to consider the safety factor functions fFsj ðX Þ; Faj ðX Þ; Ftj ðX Þg shown in Fig. 8b. This figure is plotted for the particular

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case of Fsd ¼ Fad ¼ Ftd ¼ 1:5: The following observations are relevant with respect to Fig. 8b: (a) Fsj ðX Þ is equal to the required design value Fsd along the central section bcd where Tj ðX Þ ¼ Dj ðX Þ; and Fsj ðX Þ > Fsd along ab and de where Tj ðX Þ > Dj ðX Þ: Consequently, MinX fFsj ðX Þg ¼ Fsd ; i.e. the derived function Tj ðX Þ guaranties satisfaction of design requirements with respect to soil strength (Eq. (2) and (1.1)). (b) Eq. (8.1) shows that the safety factor function Faj ðX Þ is constant along the straight sections ab and de of Tj ðX Þ: Along de this constant is equal to the required design value Fad : Starting with the limiting value Fad at d the function Faj ðX Þ increases to infinity at X ¼ Xj max where the slope of Tj ðX Þ is equal to zero (point c). Faj ðX Þ decreases from infinity at c to some value Faj ðXb Þ; remaining constant between b and a. Eq. (8.1) shows that the safety factor Faj ðX Þ is inversely proportional to the slope of Tj ðX Þ; therefore Lj XLmin implies Faj ðXbÞXFad (Fig. 8a). As a result, the solution for Tj ðX Þ guaranties satisfaction of the local design requirement with respect to interface stability, i.e. MinX fFa ðX Þg ¼ Fad (Eq. (1.2)). (c) Eq. (A.1) defines a lower bound on legitimate values of Zj : This bound guaranties that the jth RL is not over-stressed, (i.e. Tj max pStra =Ftd Þ: Consider first a case in which Dj max ¼ ð1  Zj ÞQj max ¼ Tj max ¼ Stra =Ftd (a ‘‘fully utilized’’ RL). In that case Ftj ðX ¼ Xj max Þ ¼ Ftd (point c). Obviously Tj ðX ÞpTj max (definition of a maximum), thus Ftj ðX ÞXFtd ; resulting in satisfaction of the design requirement with respect to the strength of the RL (Eq. (1.3)). Tj max oStra =Ftd in under-utilized RL. In that case Ftj ðX ÞpFtj ðX ¼ Xj max Þ > Ftd and the design requirement in Eq. (1.3) is obviously satisfied. The above observations verify that in the particular case of rigid reinforcement (Eq. (13)), the solution of the maximization problem (11) can be established by simple geometrical considerations, and this problem need not be solved numerically. Such a ‘‘geometrical solution’’ of the problem is not possible for general, non-linear, response functions, and the general case is considered in Baker (2003a). Fig. 8b illustrates why it is necessary to adopt the local approach embedded in Eq. (1). Dealing with a multicomponent system like reinforced soil, it is impossible to satisfy equilibrium requirement in terms of constant (global) safety factors. This difficulty is the source of most of the conceptual inconsistencies in conventional presentations of the subject. The local definition of safety factors generates the additional degrees of freedom required in order to satisfy static requirements for both test bodies and reinforcement, while still maintaining safety requirements (at the minima of the functions fFsj ðX Þ; Faj ðX Þ; Ftj ðX Þg). It is noted that the minima of these functions do not occur at the same place (the minimum of Fsj ðX Þ occurs along the section bcd, the minimum of Faj ðX Þ occurs along the section de, the minimum of Ftj ðX Þ occurs at point c), and there is no single point at which Fsj ðX Þ ¼ Fsd ; Faj ðX Þ ¼ Fad and Ftj ðX Þ ¼ Ftd : The solution for Tj ðX Þ shown in Fig. 8a has the following interpretation in terms of ‘‘failure modes’’: (a) The function Tj ðX Þ increases in the range X oXj max ; and in that range shear stresses acting on the anchor are negative (pointing towards the wall). This

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implies that test bodies associated with small values of X tend to move out more than the anchor. The failure mode associated with such X values corresponds to a situation in which the test body (and the wall), are falling down, leaving the reinforcement behind, stuck in the ground. (b) In the range X > Xj max the function Tj ðX Þ is decreasing, and shear stresses acting on the RL are positive (pointing away from the wall). Consequently test bodies associated with large X values tend to fall down by extracting the reinforcement from the ground and L  Xj max is the anchoring zone of the jth RL. The constants aj (Eq. (8.2)) represent the bond strength between the jth RL and the surrounding soil. Having identified the term L  Xj max as the anchoring zone, the total resistance against pull–put is aj ðL  Xj max Þ and it is possible to define a global safety factor Fej against pull-out of the jth RL as F ej 

aj ðL  Xj max Þ aj XFaj ðXb ÞXFad : ¼ Tj max ðTj max =ðL  Xj max ÞÞ

ð15Þ

The term Tj max =ðL  Xj max Þ in Eq. (15) is the slope of the line ac in Fig. 8a. This slope is smaller than the slope of the section ab in the same figure, resulting in the inequality Fej XFaj ðXb ÞXFad (the second of these inequalities is satisfied if Lj XLmin Þ: Eq. (15) shows that satisfaction of the local requirement Faj ðX ÞXFad (Eq. (1.2)) guaranties that the no slip condition is satisfied also in the conventional global sense. For design purposes it is possible to replace the local no-slip requirement (Eq. (1.2)) by the more conservative pull-out requirement Fej XFad :

7. Wall variables 7.1. Shear forces Considering horizontal equilibrium of a wall section between the levels Z ¼ jD þ e and Z ¼ ðj þ 1ÞD  e; a linear approximation of the shear forces function Sj ðZÞ in section j of the wall is given by Sajþ1  Sbj Pj ðZ  jDÞ ¼ Sbj þ ðZ  jDÞ; D D jD þ epZpðj þ 1ÞD  e;

Sj ðZÞ ¼

ð16:1Þ

where Sbj ¼ Saj  Pcj ;

ð16:2Þ

Pj ¼ Sajþ1  Sbj ¼ Zj Qj max  Sbj ðZj Þ:

ð16:3Þ

In this equation Pj is the resultant of soil stresses acting on the jth section of the wall. It is noted that the procedure described in Sections 5 and 6 yields all the information required for complete specification of Sj ðZÞ for a given value of Zj :

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The zero section of the wall (Fig. 1) is under special conditions, and in order to be consistent with other wall sections it is convenient to use the approximation S0 ðZÞ ¼ gDKam Z=2 (rather then the ‘‘exact’’ relation gKam Z2 =2Þ: The distribution of shear forces, SðZÞ; along the entire depth 0pZpHt of the wall is a union of the functions Sj ðZÞ: This function has jump discontinuities at anchor elevations for which Pcj a0: The maximum shear force Sj max acting in the jth section of the wall is the larger one between the boundary values jSj ðZ ¼ Zj þ eÞj ¼ jSbj j ¼ jSaj  Pcj ðZj Þj and Sj ðZ ¼ Zjþ1  eÞ ¼ Sajþ1 ¼ Zj Qj max : Distributions of shear forces and moments in systems involving large number of anchors is presented and discussed in Baker and Klein (2003). Fig. 9 shows the effect of Z1 on the distribution of shear forces in the first two sections of the wall. The results in this figure are derived for the input f ¼ 40 ; g ¼ 20 ½kN=m3 ; d ¼ 2=3f; D ¼ 0:5 ½m ; L ¼ 0:7Ht and Fsd ¼ Fad ¼ Ftd ¼ 1:5: The following comments are relevant with respect to Fig. 9: (1) The distribution of shear force, S0 ðZÞ; in the zero section of the wall does not depend on participation factors, and Sa1 is equal to the Rankine force for the elevation Z ¼ D: (2) Fig. 9a shows the shear force function associated with Z1 ¼ 0: This distribution corresponds to the conventional reinforced soil design process which disregards the wall contribution during anchor design. Fig. 9f shows the shear force function associated with the other limiting case of Z1 ¼ 1: This limiting case corresponds to a classical situation in which the soil is not reinforced ðT1 max ¼ 0Þ:

Fig. 9. Effect of participation factors on distributions of shear forces in the first two sections of the wall. (a) Z1 ¼ 0; T1max ¼ 3:44: (b) Z1 ¼ 0:2; T1max ¼ 2:75: (c) Z1 ¼ 0:4; T1max ¼ 2:06: (d) Z1 ¼ 0:6; T1max ¼ 1:36: (e) Z1 ¼ 0:8; T1max ¼ 0:69: (f) Z1 ¼ 1; T1max ¼ 0:

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(3) Increasing Z1 from zero to 1 is associated with gradual increase of shear forces in the wall and a corresponding decrease in T2 max : (4) Pcj ¼ Saj  Sbj is the force which the jth RL applies to the wall (Eq. (16.2)). Fig. 9 shows that this force decreases gradually with Z1 ; becoming zero when Z1 is between 0.4 and 0.6. Combination of Eq. (16.2) and (16.3) yields Pcj ¼ Saj  Zj Qj max þ Pj ðZj Þ: This relation makes it possible to discuss the particular case of detached anchors (i.e. a system in which the anchors are not attached to the wall). In such systems, Pcj ¼ 0; and the above relation results in the following system of anchor interaction parameters: Zj ¼

Saj þ Pj ðZj Þ : Qj max

ð17Þ

Eq. (17) is a non-linear equation for Zj : This relation is useful in two different contexts: (a) Constructing a physical system in which the anchors are not attached to the wall is a legitimate design option which under certain conditions may be appropriate. (b) The relation in Eq. (17) may be used in order to study the consequences resulting from a situation in which a connection between the wall and the jth RC breaks down (a case of ‘‘insufficient connection strength’’), and this layer becomes detached. The implications of such a scenario are discussed in Baker and Klein (2003).

7.2. Distribution of soil pressure on the wall The distribution of soil pressure, pj ðZÞ; acting on the jth section of the wall is the derivative of the shear force function Sj ðZÞ; and Eq. (16.1) implies: pj ðZÞ ¼

dSj ðZÞ Pj ¼ ¼ pj ; dZ D

jD þ epZpðj þ 1ÞD  e:

ð18Þ

The distribution of soil pressure along the entire wall, pS ðZÞ; is the union of the functions pj ðZÞ: Linear approximation of the shear force functions is, obviously, associated with a constant approximation of soil pressures at each wall section. Consequently, the entire function pS ðZÞ is approximated as a ‘‘stair case function’’. p0 ðZÞ ¼ p0 ¼ gDKam =2 along the zero section of the wall. Assumption 5 in Section 2.1 introduced the restriction pS ðZÞ ¼ pA ¼ ðZÞ; which distinguishes between active and passive stability problems. Eq. (18) provides the tool required in order to apply this restriction. The implications of this requirement are presented and discussed in Baker and Klein (2003).

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7.3. Wall moments The distribution of moment Mj ðZÞ in the jth section of the wall is the integral of the shear force function Sj ðZÞ: Performing this integration yields: Mj ðZÞ ¼ C þ ðSbj  jPj ÞZ þ

Pj 2 Z ; 2D

jD þ epZpðj þ 1ÞD  e;

ð19Þ

where C is an integration constant. The global moment distribution function, MðZÞ; is value continuous at anchor elevations and the magnitude of the integration constant C can be established based on the requirement that Mj ðZ ¼ jDÞ ¼ mj ; where the constants mj ; j ¼ 1; y; n are wall moments at anchor elevations. Using this requirement, it is possible to write Eq. (19) in the form Mj ðZÞ ¼ mj þ Sbj ðZ  jDÞ þ

Pj ðZ  jDÞ2 ; 2D

jD þ epZpðj þ 1ÞD  e:

ð20Þ

Applying Eq. (20) at Z ¼ ðj þ 1ÞD results in the following relation between wall moments at adjacent anchor elevations: mjþ1 ¼ mj þ ðSbj þ Pj =2ÞD;

j ¼ 1; y; n:

ð21Þ

In the 0th section of the wall, moments are given by M0 ðZÞ ¼ gDKam Z2 =4 resulting in m0 ¼ MðZ ¼ 0Þ ¼ M0 ðZ ¼ 0Þ ¼ 0 and m1 ¼ m1 R ¼ gD3 Kam =4 where m1 R is the Rankine moment at the elevation Z ¼ D: The slightly unusual expressions for p0 ðZÞ; S0 ðZÞ and M0 ðZÞ result from use of the linear approximations of all shear force functions (including S0 ðZÞ). At a general step j of the top down process, the constants fmj ; Sbj ; Pj g are known quantities, and Eq. (20) defines a unique distribution of moments in the jth section of the wall. Eq. (21) is used in order to calculate the m value needed for evaluation of moments in the ðj þ 1Þth step. Eqs. (16) and (20) define unique functions fSðZÞ; MðZÞg for each given set of legitimate Zj ; j ¼ 1; y; n (and L). This observation forms the basis for the design option D.O. —2 shown in the bottom part of Fig. 5. The functions fSðZÞ; MðZÞg are ‘‘demands’’ imposed on the wall, and using the design requirements in Eqs. (1.4) and (1.5) it is possible to design a structure satisfying these demands. In the general fSB ¼ Sanþ1 ¼ Zn Qn max a0; and MB ¼ mnþ1 ¼ mn þ ðSbn þ ðSbn þ Pn =2ÞDa0g: The quantities fSB ; MB g have to be carried by the foundation of the system, and it is necessary to design the foundation in accordance with usual rules of foundation engineering. The common practice in the design of reinforced soil retaining structures is to consider the base of the structure merely as a working or leveling pad. This practice is justified in cases when fSanþ1 ; mnþ1 g are small, but not in general. If the base of the wall is not properly designed, the structure will yield, transferring loads to the reinforcement, and making it necessary to use a stronger anchoring system. In that case the magnitude of Zn is controlled by the ‘‘given’’ foundation, and this value dictates the required strength of the nth RL. This observation illustrates the fully integrated nature of the proposed design process in which all elements of the system are explicitly accounted for.

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Fig. 10. Effect of participation factors on distributions of moments in the first two sections of the wall. (a) Z1 ¼ 0: (b) Z1 ¼ 0:2: (c) Z1 ¼ 0:4: (d) Z1 ¼ 0:6: (e) Z1 ¼ 0:8: (f) Z1 ¼ 1:0:

Fig. 10 shows the effect of Z1 on the distribution of moments in the first two sections of the wall. The results in Fig. 10 are evaluated for the same input data as in Fig. 9. The practically significant results in Figs. 9 and 10 are summarized in Fig. 11. The following comments are relevant with respect to this figure: (1) The most important result illustrated in Fig. 11 is that T1 max decreases and jM1 max j increases with Z1 : It can be verified (Baker and Klein, 2003), that a similar effect exists for a general step j: Tj max and jMj max j control the required strength of the jth RL, and the jth wall section. As a result, large values of Zj are associated with situations in which the wall is strong relatively to the anchoring system (and vice versa). This observation supports the statement made in Section 2.1 that the present design procedure distributes loads between the wall and the anchoring system in proportion to the relative strengths of these elements. (2) Fig. 11 shows that Tj max decreases linearly with Zj : This result follows directly from combination of Eqs. (7.3) and (12) ðTj max ¼ Dj max ¼ ð1  Zj ÞQj max Þ: The increase of jMj max j with Zj is non-linear. In general jSj max j increases linearly with Zj ; and increasing the magnitude of participation factors results with transfer of loads (both jMj max j and jSj max j) from the anchoring system to the wall. The deviation from the linear variation of jSj max j; occurring at small values of Z1 in Fig. 11, is associated with situations resulting in jSbj j > Sajþ1 (e.g. Fig. 9a). Under most conditions, the dependence of jSj max j on Zj is not practically significant since structural design of the wall in controlled by moments rather than by shear forces. (3) The results in Fig. 11 show that all legitimate values of Z1 result with a non zero jMj max j and jSj max j: Consequently, the wall is never stress free, and, in principle,

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Fig. 11. Trade-off between required strengths of wall and reinforcement.

it should not be considered only as a decorative element. Fig. 11 shows that the conventional reinforced soil assumption Zj ¼ 0 is associated with small demands ðjMj max j; jSj max jÞ being imposed on the wall, and this design is consistent with a ‘‘weak’’ wall. However, Zj ¼ 0 implies that Tj max ¼ ð1  Zj ÞQj max ¼ Qj max is large, and it is necessary to use a strong anchoring system. Consequently, there exists a natural ‘‘trade-off’’ between the strengths of the wall and the anchoring system. The conventional reinforced soil design methodology corresponds to one ‘‘end point’’ ðZj ¼ 0Þ of the range of possible solutions. This limiting case is obviously legitimate; however it does not necessarily lead to the most economic design (e.g. Baker, 2003a; Klein, 2003; Baker and Klein, 2003). (4) Figs. 10a and b show that m2 ðZ1 ¼ 0Þo0 and m2 ðZ1 ¼ 0:2Þ > 0: Consequently there is a value of Z1 between zero and 0.2 yielding m2 ðZ1 Þ ¼ 0: It can be verified (Baker and Klein, 2003) that similar situation exists at all steps j: This observation provides the basis for determination of wall interaction parameters appropriate for hinged walls presented in Appendix B. It is noted that shear forces and moments at Z ¼ D are fixed at their Rankine values, and there cannot be a hinge at that elevation. In addition, the interfaces between pre-cast elements (hinges) must have sufficient shear strength to transmit the shear forces fSaj ; jSbj jg; for j ¼ 1–n:

8. Summary The present work presents the elements of a fully integrated LE process for design of reinforced soil retaining structure (design examples based on this procedure are presented and discussed in Baker and Klein, 2003). The main features of the

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proposed procedure are: (1) It explicitly considers strength properties of the main three elements wall– reinforcement–soil of the system. (2) Design requirements are formulated as local inequalities, and they are enforced at each relevant point rather then only globally. (3) Interaction between reinforcement and soil is represented by reinforcement response functions which can be established by pull-out tests. These functions characterize both reinforcement type and properties of the retained soil. (4) The interaction between the wall and the anchoring system is represented by a system of interaction parameters. The magnitudes of these parameters depend on the relative strength of wall and anchoring system. The classical situations of conventional reinforced soil design, and non-reinforced retuning structures, correspond to the limiting values 0 and 1 of these parameters. Explicit expressions for interaction parameters are derived for a number of practically important situations (given anchors strength, detached anchors, hinged walls, etc.). (5) The design process results with distributions of tensile forces in each RL, distribution of soil pressures on the wall, and distributions of shear forces and moments in the wall. These functions allow complete and rational design of all components of the system. (6) In general the wall has a structural function and it cannot be considered merely as a decorative skin, and it requires a proper foundation (not only a leveling pad). The required size of this foundation depends on the relative strengths of the wall and the reinforcing system. In certain limiting situations (small enough values of fMaxfjMðZÞjg; MaxfjSðZÞjg; MB ; SB g) the required wall thickness and foundation length may become negligible, justifying the common design approach. Such an example is discussed in Baker and Klein (2003). However, in the present context such situations are derived results, valid for particular situations, rather than a priori assumptions.

Appendix A. Participation factors for a given system of anchors Consider, first, a case in which Qj max is larger than the design strength Stra =Ftd of the reinforcement. In this case the jth RL cannot supply the demand imposed on it, and the excess of Qj max above the design strength must be carried by the wall. This results in a lower bound on legitimate values of participation factors. Combining Eq. (12), (7.3), (1.3) and solving for Zj results in Zj X1 

Stra =Ftd Qj max

If Qj max XStra =Ftd :

ðA:1Þ

Eq. (A.1) defines only a lower bound on legitimate values of Zj : However, the present work deals with inextensible reinforcement and cantilevers are flexible, not ‘‘attracting’’ more loads then necessary. Consequently it is reasonable to use the lower bound in Eq. (A.1) for evaluation of Zj when Qj max XStra =Ftd :

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When Qj max oStra =Ftd the jth RL can carry the full value of Qj max : In such cases it is reasonable to set Zj ¼ 0: The motivation for this setting can be establish as follows. The relation Tj max oStra =Ftd implies that the jth RL is under-utilized and its strength may, in principle, be reduced to Straj ¼ Ftd QJ max : Recalling that participation factors cannot be negative; and introducing Straj ¼ Ftd Qj max into Eq. (A.1) gives Zj ¼ 0: Construction considerations exclude complex lay-outs with variety strengths. Considering a constant strength reinforcing system, minimal safety factors for under-utilized RL are larger than the required value of Ftd ; but the lower bound setting Zj ¼ 0 is still legitimate. Combining results; the above considerations lead to the following settings of participation factors for a given reinforcement strength Stra : 8 > if Qj max oStra =Ftd ; <0 Zj ¼ ðA:2Þ Stra =Ftd > if Qj max XStra =Ftd : :1  Qj max The following properties of Eq. (A.2) are significant: (1) Eq. (A.2) results from a consistent use of the lower bond defined by Eq. (A.1). This is equivalent to the assumption that ‘‘a RL will carry as much of Qj max as it can’’. This assumption results with minimum ‘‘allocation’’ of loads to the wall, and maximum allocation of loads to the reinforcement. Conventional design procedures completely disregard wall effects during design of the anchoring system, thus failing to allocate any loads to the wall, and Eq. (A.2) provides a consistent improvement over that very conservative assumption. (2) The top part of Eq. (A.2) corresponds to situations in which the jth RL can carry the entire load without violation of the design requirement in Eq. (1.3). In such cases the RL is under-utilized; resulting with Tj max ¼ Dj max ¼ Qj max : This result implies that in under-utilized RL, maximal tensile forces increase with depth. The bottom part of Eq. (A.2) is relevant for fully utilized RL. This setting results in Tj max ¼ Dj max ¼ ð1  Zj Þ Qj max ¼ Stra =Fad ¼ Const: which implies that the distribution of maximal tensile forces in fully utilized RL do not depend on depth. The experimental results reported by Rowe and Ho (1993) show both types of behavior; in certain cases measured maximal tensile forces are approximately constant, independent of depth, while in other cases, these forces increase, almost linearly, with depth. Based on the present analysis these two types of behavior are associated with fully utilized and under-utilized RL.

Appendix B. Participation factors for hinged walls Walls used in practical reinforced soil retaining structures are usually constructed from precast elements. The design of such systems is based on the assumption that interfaces between elements have zero moment capacity (i.e. the interfaces are considered as hinges). It is instructive therefore to establish participation factors

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associated with this assumption. Two arrangements of interface elevations relative to anchors elevations are common: (a) Interfaces between elements coincide with anchor elevations. This arrangement is frequently used in segmental block walls. (b) Interface between elements are located mid-way between anchor elevations (Terre Armee configuration). The present appendix presents participation factors for arrangement (a) only. The corresponding expressions for arrangement (b) are presented in Baker (2003a). Consider a general step j > 1: In such a step both mj and mjþ1 are equal to zero and Eq. (21) is reduced to Sbj þ Pj =2 ¼ 0;

j ¼ 2; y; n:

ðB:1Þ

Using Eq. (15.2), (15.3) and (7.1), the above expression can be written in the form Zj ¼

Sb ðZj Þ Pcj ðZj Þ  Saj ¼ ; Qj max Qj max

j ¼ 2; y; n:

ðB:2Þ

Eq. (B.2) is a non-linear equation for Zj m1 ¼ gD3 Kam =4a0 and in this particular case, Eq. (20) results in Z1 ¼ ðPc1 ðZ1 Þ  Sa1  2m1 =DÞ Q1 max ¼ 2ðPc1 ðZ1 Þ=gD2 Kam Þ  1Þ: The following comments are relevant with respect to Eq. (B.2): (a) Eq. (B.2) shows that participation factors for a hinged wall are determined uniquely by the requirement that wall moments at hinge elevations are equal to zero. This result is conceptually simpler then the corresponding result for a ‘‘given’’ anchoring system discussed in Appendix A. However, computationally, Eq. (B.2) is more awkward than Eq. (A.2), requiring numerical solution of a non-linear relation which involves the constructions discussed in Section 6. (b) Participation factors are non-negative, and the first form of Eq. (B.2) shows that this equation has a legitimate solution only if Sbj is negative. Recalling that Saj ¼ Zj1 Qj1 max > 0; it follows that the shear forces function, SðZÞ; associated with hinged walls, fluctuates about zero, and the maximal shear force in the wall remains small (Figs. 9a and b). More important however, the distribution of moments in the wall is the integral of the shear force function, and the fluctuating shear force function is associated with relatively small maximal moments in each section of the wall.

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