GEOSYNTHETIC REINFORCED SOIL STRUCTURES

By Dov Leshchinsky,1 Associate Member, ASCE, and Ralph H. Boedeker2

ABSTRACT: An approach for stability analysis of geosynthetic reinforced earth structures over firm foundations is presented. The approach involves both internal and external stability analyses. The internal stability analysis is based on variational limiting equilibrium and satisfies all equilibrium requirements. Two extreme in­clinations of reinforcement tensile resistance are investigated: orthogonal to the radius defining the geosynthetic sheet, and horizontal, signifying the as-installed position. Although a horizontal positioning requires slightly longer anchorage to assure pullout resistance, the slip surface is shallower when compared to the or­thogonal case. As a result, the required total embedment length is longer for the orthogonal inclination. The external stability analysis is an extension of the bilinear wedge method and it allows a slip plane to propagate horizontally along a rein­forcing sheet. The results for both the internal and external stability analyses are conveniently presented in the form of design charts. Given a slope and a design safety factor, the geosynthetic sheets' profile as well as their required tensile re­sistance can be determined utilizing these charts.

INTRODUCTION

Geosynthetics (i.e., geotextiles, geogrids, etc.) are increasingly being used as reinforcing members in the construction of earth structures. In the frame­work of this paper such structures can be classified as steep embankments or retaining walls.

A major requirement in the design of reinforced structures is to assure their stability. There are a few analytical approaches, essentially extended from simplified limit-equilibrium methods (e.g., Christie and El-Hadi 1977; Ingold 1982; Murray 1982; Ruegger 1986; Schneider and Holtz 1986; Schmertmann et al. 1987), capable of dealing with the stability problem. In this work two limit-equilibrium methods of analysis are utilized, each deal­ing with a different aspect of stability. Both methods are formulated to yield closed-form solutions. Subsequently, useful design charts, produced with relative ease and providing insight into the reinforcement problem, are pre­sented. Given the strength and unit weight of the backfill soil, the height and face inclination of the earth structure, the number of reinforcement sheets to be used, and a factor of safety, one can determine from the design charts the required geosynthetic tensile resistance and the necessary embedment length to assure stability. The results presented are applicable to free-drain­ing backfill placed over a firm foundation.

It should be emphasized that a stability analysis is only one step in the design of reinforced soil structures. A comprehensive and instructive over­view of the many aspects associated with geosynthetics reinforcement is given by Bonaparte et al. (1985).

'Assoc. Prof, of Civ. Engrg., Univ. of Delaware, Newark, DE 19716. 2Geotech. Engr., Tetra Tech Richardson Inc., 910 S. Chapel St., Newark, DE

19713. Note. Discussion open until March 1, 1990. To extend the closing date one month,

a written request must be filed with the ASCE Manager of Journals. The manuscript for this paper was submitted for review and possible publication on December 21, 1987. This paper is part of the Journal of Geotechnical Engineering, Vol. 115, No. 10, October, 1989. ©ASCE, ISSN 0733-9410/89/0010-1459/$1.00 + $.15 per page. Paper No. 24009.

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STABILITY ANALYSIS OVERVIEW

Outlined in the following are the internal and external stability analyses of a geosynthetic reinforced free-draining soil structure over a firm foun­dation. Internal stability deals with the resistance to pullout failure within the reinforced soil zone resulting from the interaction between soil and re­inforcement. External stability addresses situations where a reinforced por­tion may slide horizontally as a monolithic block along one of the reinforcing sheets. In the framework of this paper, firm foundation implies that deep-seated failures are unlikely to occur and therefore are not considered in the analyses.

Internal Stability Fig. 1(a) represents a homogeneous and pore-pressure-free (i.e., u — 0)

soil mass at the verge of failure. This mass is contained between the slope and slip surfaces. The slope surface is defined by its height H and inclination l(H):m(V). Similar to Taylor (1937) and as is often used in various geo-technical problems that are based on limit-equilibrium or limit-analysis (plas­ticity), the slip surface is taken as a log spiral extending between the crest and toe.

To ensure the reinforcement capacity required to develop a prescribed de­sign tensile resistance, tj, it must be embedded beyond the slip surface so that its pullout resistance will be at least tj. Subsequently, for the state of collapse shown in Fig. 1(a), the available pullout resistance of each sheet ideally should equal its design tensile resistance

t} = 2k(tm <$>)(& -le)j (1)

where tj = pullout resistance per unit width of geosynthetic sheet j ; § = internal angle of friction of the soil; k = a parameter relating the coefficient of friction at the soil-geosynthetic interface and tan $ (typically within the

FIG. 1. (a) Definitions and Conventions in Internal Stability

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range of 0.6-1.0); CT and le = average normal stress and embedment length beyond the slip surface, respectively, of geosynthetic sheet./. Although a is unknown, one may approximate the term (a-le) in Eq. 1 as the weight of soil column above le.

Observing Fig. 1(a) one sees that f, is inclined at an angle 9, to the hor­izontal. Two extreme angles will be examined in this paper: (1) 0, = 0, which signifies the typical as-installed inclination and produces the least re­inforcement contribution to stability and is therefore considered conservative; and (2) 9, = (3,, which signifies tj orthogonal to the slip surface radius and produces the most reinforcement contribution to stability [e.g., see Lesh-chinsky and Reinschmidt (1985)]. Physically, the orthogonal case can de­velop as the flexible geosynthetic reorients itself while its tensile resistance is being mobilized by soil movement.

To deal with a stable system, the concept of mobilized strength is intro­duced into the problem presented in Fig. 1(a). Thus, an artificial state of limit-equilibrium is attained in which

TM = o- — = cri|;m (2a)

tm. = 2k—(a-le)j = 2h]/m{a-le)j (2b) Fs

where Tm and tmj = the mobilized soil Coulomb shear strength along the slip surface and mobilized reinforcing sheet pullout resistance, respectively; o- = the normal stress distribution along the log-spiral slip surface (unknown); \\i = tan 4>; and Fs = factor of safety defining a fictitious frictional soil (i.e., a soil with reduced \\i) for which the limiting equilibrium state exists. Notice that Fs is the safety factor commonly used in slope stability problems and when it equals 1, the soil strength and reinforcement pullout resistance are fully mobilized.

It should be pointed out that with the introduction of the factor of safety, the log-spiral slip surface shown in Fig. 1(a) is defined as

R = A exp (-v|/mp) (3)

where R = the log-spiral radius; and A = an unknown constant. To enable the presentation of results in a condensed and useful format of

design charts, tj distribution with elevation ys must be known. However, there are infinite such possible distributions. The writers have chosen a linear relationship for the pullout resistance, tj versus y-, i.e., zero at the crest and a maximum value, ty, at the toe. It should be pointed out that this linear distribution is adopted from a simplified limit-state stress analysis when geo­synthetic reinforced walls are considered (e.g., Bell et al. 1975). Generally, the outcome of this distribution is believed to overestimate tx, resulting in a conservative selection of a geosynthetic based on the required tensile strength— see Interpretation for effects. Hence

h = h\\ - | j or tmj = ^ ( l - | J (4)

where y7 = the elevation of reinforcing sheet j—see Fig. 1(a). As is evident

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0\/xc,yc)

FIG. 1. (fa) Log-Spiral Analysis: Force Diagram

from Eq. 2b, for a given $, k, y}, and Fs, the pullout resistance tj can be attained by proper selection of /,,.. Thus, the specified distribution of tj stated in Eq. 4 for a selected design value of tt can be controlled by le in Eq. 2b.

For the failure mechanism expressed in Eq. 3, a selected reinforcement inclination, 6,, at the slip surface (where j = 1, 2, . . . , n), and the distri­bution of tj stated in Eq. 4, one seeks the minimum value of Fs with si­multaneous satisfaction of all three limiting-equilibrium equations for the sliding body. To achieve it, one can construct the force polygon shown in Fig. 1(b). Notice in this figure that: (1) For clarity, only one reinforcing sheet is used (expansion to n sheets is straightforward); (2) the weight of the sliding mass, W, and the reinforcement tensile resistance, t, are known in magnitude and direction; and (3) the action line of Rf (i.e., the resultant force of a and T,„ distributions) must coincide with 00' and simultaneously close the force polygon. Consequently, for assumed xc, yc, and A, one can determine by trial and error the respective Fs, which will close the polygon and produce Rf in 00' direction (note that W is also dependent on Fs). A search for a combination of xc, yc, and A that produces the minimal Fs then has to be carried out. Clearly, such a numerical (or semigraphical) process is tedious. Alternatively, one can formulate the problem in the framework of the variational calculus seeking the normal stress function that satisfies equilibria and produces the minimum Fs. Such minimization is particularly advantageous if a closed-form solution for production of design charts is desired. It should be pointed out that in principle, the two alternative ap­proaches are analogous to the Coulomb wedge used in lateral earth force computation; i.e., this lateral force can be estimated through a semigraph­ical/numerical approach or, for simple boundary conditions, through dif­ferential calculus resulting in expressions that are readily available in most geotechnical textbooks.

To facilitate results generation, the variational limiting-equilibrium ap­proach presented in a general format by Baker and Garber (1977, 1978) and more explicitly by Baker (1981) was utilized. The outline of this utilization

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in a framework of reinforced soil (though considering a different failure mechanism and a different definition of safety factor) has been presented elsewhere (Leshchinsky and Volk 1985; Leshchinsky and Reinschmidt 1985; Leshchinsky et al. 1986). Modification of the variational formulation to deal with the linear tj distribution, its two extreme inclinations and the factor of safety defined in Eq. 2 (i.e., the present problem) results in (Boedeker 1987):

A S = 1 , n,2 ( c o s P + 3,K. s i n P) e x P ( - W ) + B exp (2i|imp) (5)

where S = u/yH = a nondimensional representation of the normal stress distribution along the slip surface; A = A/H where A is defined in Eq. 3; and B = an unknown constant. It is worthwhile noting that for unreinforced slope stability problems where c-§ soils are involved, the selected failure surface coupled with Eq. 5 yields results identical to Taylor's (1937) [see Baker (1981)].

Following the procedure outlined by Leshchinsky and Volk (1985), one can assemble the necessary number of equations to match the number of unknown constants in the present problem (Boedeker 1987). Subsequently, for a given m, H, y, y} (j — 1 ,2 , . . . , n) and 4>,„ (i.e. <$>,„ = tan~'[(tan ()>)/ Fs]), the mobilized value tmi can be determined via a closed-form solution. Thus an adequate reinforcing geosynthetic can be selected as discussed later on. Furthermore, le. can be estimated through Eqs. 2b and 4 and since the trace of the slip surface constitutes part of the solution, the reinforcement total length at each elevation (i.e., profile) can be determined.

It should be emphasized that only the pullout mode of failure is consid­ered. However, if the available pullout resistance exceeds the tensile strength, sheet breaking may occur. For the situation where breaking is the controlling mode of failure, Leshchinsky and Volk's (1985) mechanism and formulation might be adequate. In any event, the outlined analysis and the results pre­sented are geared toward design, i.e., to dimensionalize and specify the re­inforcement so that a prescribed factor of safety is attained. As a result, one can assure by selecting an adequate geosynthetic that its allowable tensile strength equals r,—see Interpretation. Using tj and upon specification of lej

(Eq. 1), the pullout mode of failure prevails, resulting in an optimal rein­forcement (i.e., geosynthetic having the required strength which is not wastefully embedded).

External Stability Fig. 2 shows the failure mechanism assumed for the external stability

problem. Essentially, it is a bilinear planar surface extending outside the reinforcement zone between points 2 and 3 at an unknown angle £, and propagating horizontally along reinforcement sheet j , which potentially rep­resents a plane of weakness emerging at the slope face (point 1). Fig. 3(a) and 3(b) present a free-body diagram and its associated force polygons, re­spectively, for the sliding body after separating it into two wedges. Observ­ing the force polygons, it is clear that once again the concept of mobilized shear strength is being used (i.e., \\im = tan 4>/^) ' It should be pointed out, however, that the design value assigned to Fs might differ from the one used in the internal stability. Notice in Fig. 3(b) that the resultant shear force developing along points 2 and 3 is determined via \\>m, whereas along points

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©

FIG. 2. Definitions and Conventions in External Stability

1 and 2 it is estimated through b\im where k signifies the interfacial soil-reinforcement relative friction (see Eq. 1). Also notice that two conservative assumptions were implicitly included: (1) The interwedge resultant force P is horizontal; and (2) the possibility is ignored that for certain values of £ the slip plane between points 2 and 3 may intersect reinforcement sheets.

Based on the polygons shown in Fig. 3(b), one can assemble the force equilibrium equations for each wedge. It can be verified (Boedeker 1987) that by solving these equations and rearranging terms, the following expres­sions may be obtained:

(b) Wedge 2 Werigf) 1

FIG. 3. Forces in External Stability Analysis: (a) Free Body Diagram; and (b) Force Polygons

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* * =

(jHf - yjYl cot { - - + 2(H - yj)lj - ml} ml

mkli

sin { - I\I,„ cos J

cos I + 4i„, sin I

for lj s H-yj

(6a)

and

(H - yj) cot l "1 sin £ - I|J„, cos £

2/, # - yH Mcos 5 + i|»m sin £.

for /, ff-% (66)

where /, = the total length of reinforcing sheet j [see Figs. 2 and 3(a)]. Eq. 6a pertains to the case where lj lies entirely below the slope's face and Eq. 6b reflects the situation where /, extends below the crest of the structure [Fig. 3(a)].

Observing Eqs. 6a and 6b, the external stability problem can be stated as: For a given m, H, yjt lj and k, find £ that produces the maximum <\>m. Once this I is determined, max (<)>,„), which is equivalent to min (Fs), can be calculated. The numerical procedure required to determine max (4>,„) is rather straightforward: Assume values of £ and solve the nonlinear equation (i.e., either Eq. 6a or Eq. 6b) for (j>,„ until max (<)>„,) is attained. Using this pro­cedure for a given m and k, design charts relating the required ratio lj/(H — Vj) to the corresponding 4>™ were developed.

It is interesting to consider the effects of the assumed inclination of the interwedge resultant, P (Fig. 3), on the required embedment length. Schmertmann et al. (1987) showed that for the case where <)>,„ = 25°, k = 0.9, and P is inclined at an extreme angle ()),„, the required lx decreases by as much as 30% as compared to a horizontal P, depending on slope face inclination. Subsequently, the analysis using horizontal inclination yields conservative results typically affecting only a few sheets at the lower ele­vations (see Procedure and Example). Moreover, since there is a disconti­nuity in the failure mechanism when switching from internal to external sta­bility, the writers believe that this apparent conservatism is warranted.

RESULTS

Internal Stability Based on the outlined analysis, the required t{ for a given problem [i.e.,

m, H, yj (j = 1, 2, . . . , n), 7, <(>, and Fs] was computed via a closed-form set of equations. However, to enable a condensed results presentation, the equivalent tensile-resistance parameter, introduced by Leshchinsky and Volk (1985), is used

Tm = n X Tm (7)

where Tm = the nondimensional mobilized equivalent tensile resistance; n the number of equally spaced reinforcing sheets; and

1 U 1 / 1 T =

yH2 Fs yH2 orT„, yH2 (8)

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15

T Inclination

Orthogonal Horizontal ••

m=5

1000

n (# of Reinforcing Sheets)

FIG. 4. Effect of Number of Reinforcing Sheets on T„, Approximation

Since Tm is a scalar equivalent to a distributed force, it must be taken as an approximation. To estimate the effect of using T,„ as an approximating parameter in design charts, some slopes reinforced with n < 500 were ana­lyzed (Boedeker 1987). Fig. 4 illustrates the results of one case. It was con­cluded that as long as n < 500, which is the probable case, the approxi­mation in Eq. 7 produces slightly conservative results; i.e., the required Tm based on 500 sheets is always larger than that for n < 500. Moreover, the effect of the approximation on the slip surface location for any n a 5 is negligible. Subsequently, utilization of Tm as a nondimensional approximat­ing parameter containing n, 7, H, tu and Fs, appears to be acceptable in practical terms. Although the case where 500 reinforcing sheets are used appears unfeasible, all the presented results associated with internal stability are for this case. This extends the useful range of the design charts, in a safe manner, to nearly all practical cases.

Because of restricted publication space, illustrations of typical normal stress and reinforcement tensile force distributions are not presented. These illus­trations, however, are available elsewhere (Boedeker 1987). Observing the results obtained for all analyzed cases, it was concluded that (Boedeker 1987): (1) As <|>,„ increases, the difference in the required Tm for the two extreme reinforcement inclinations (i.e., 6, = 0 and 0, = (3/ horizontal and orthog­onal, respectively) becomes increasingly negligible; and (2) for a given <j>m the trace of the slip surface for the orthogonal case extends into the em­bankment deeper than for the horizontal case. The second conclusion be-

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0 . 0 i , I i i i , I I , i i I T T - ^ . I

15 20 25 30 35 40 45

FIG. 5. Design Chart for Required Tensile Resistance

comes more apparent as ()>„, decreases, and it may have an important impli­cation in design. Although the required embedment (i.e., anchorage) length le. for the horizontal case is somewhat larger than for the orthogonal case (depending on t}—see design chart Fig. 5 and Eq. 2b), it appears that the required overall length [i.e., (ls + le)—see Fig. 1(a)] in most horizontal cases will be shorter than for the corresponding orthogonal cases. Hence, the notion that the horizontal case provides a conservative design in terms of total embedment length of reinforcement might be misleading.

Fig. 5 is the internal stability design chart for reinforced steep earth struc­tures inclined at m = 1, 1.5, 2.5, 5, 10, and °°. It constitutes a solution to the analysis outlined before. Both extreme reinforcement inclinations shown provide a range of values for the designer to consider. Its utilization is straightforward; i.e., for a given <j> and selected Fs, determine Tm using the chart. Upon the designer's selection of the number of reinforcing sheets n, and for a given 7 and H, one can determine tx and subsequently ts through

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Eq. 4. Based on Eq. 1, the embedment length beyond the slip surface, le., can be estimated.

Figs. 6(a-f) provide the critical slip surface traces complementing Fig. 5. These figures should be viewed in the framework of design charts; i.e., for a selected 4>m (same as used in Fig. 5) one can determine the embedment length between the slope and slip surface [lSJ—see Fig. 1(a)] and thus the overall reinforcement length. The traces, however, correspond to the or­thogonal case only. Traces for the horizontal case are not presented here because they are shallower than the orthogonal ones and essentially require shorter reinforcement, and because of publishing space restrictions.

It is interesting to note that for m = °°, the values of Tm in Fig. 5 are identical to the coefficients of active lateral earth pressure Ka as determined using Coulomb theory, where 8 = 0 and 5 = 8; are the horizontal and or­thogonal inclinations, respectively. Also, for m = °o the log-spiral mecha­nism degenerates to planar surfaces [e.g., see Fig. 6(f)] identical to those predicted by Coulomb theory.

External Stability Figs. l(a-f) are the external stability design charts for earth structures

inclined at m = 1, 1.5, 2.5, 5, 10, and °°. These charts are for k = 0.6, 0.8, and 1.0, where k relates the soil-reinforcement interface friction to tan § (see Eq. 1). Notice that in some charts an inflection point is apparent; this is a consequence of a change in the prevailing equation, either Eq. 6a or Eq. 6b.

To utilize these charts for a given k, §, and m, one has to select a design value for Fs, compute c|>,„ = tan - 1 (tan §/Fs), and determine the required ratio [lj/(H — yj)] from the chart. The necessary total embedment length lj (see Fig. 2) for each sheet,/ at elevation ys can then be determined and com­pared with the required length for the internal stability (i.e., lSj + le). The longer of the two lengths should be selected.

Procedure and Example For a given inclination and height (m and H), the soil's unit weight and

internal angle of friction (y and <(>), and the ratio k between the coefficient of friction at the soil-reinforcement interface and tan <J>, the following steps are needed to utilize the internal and external stability charts:

1. Select a factor of safety for internal stability, Fs. 2. Compute (j)m = tan -1 (tan §/Fs). 3. Use Fig. 5 to estimate Tm for the given 4>m and m. Since the actual re­

inforcement inclination is unknown, the user may select a value bracketed by the two extreme possibilities; i.e. horizontal (most conservative) and orthogonal (least conservative) inclinations.

4. Select the number of equally spaced reinforcing sheets, n, and compute h = TmFsyH2/n.

5. Choose the proper chart from Figs. 6(a) through 6(f) and determine ls. [see Fig. 1(a)].

6. For each reinforcing sheet located at yjt use Eqs. 4 and 1 to estimate the required anchorage length, /„., beyond the potential sliding mass. The term (<r • le)j in Eq. 1 may be replaced by the weight of the soil column over le); if part of lej is under the slope face, however, such an approach may require a simple trial and error solution since le. is unknown to start with.

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1.0

0.8

0.6

i 0.4

0.2

0.0

-0.2

-0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.i (a) x/H

-

m

:

UVAW/

=1

/ /

Xh

/ /

' •§/

/ 1

P/Q

/ •

/ /

/ *

/W/SU&SPZiSl,

/

1.0

0.8

0.6

0.4

0.2

0.0

-0.2

m=

•

: *

WAVM

1.5

/S/

/ \

/

^

v /

/ /

V /

' /

1 / y ' /

/

i

Z?/

I sw-v •/•n^m,

-0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

(") x/H

(c)

1.0

0.8

0.6

0.4

0.2

0.0

0.2

m=2.5

W/WM

-0.2 0.0 0.

~1

1 y /

/ /

,?/ V

/ /

V/ ^ ' /

/ ' / V^

' / / ' /

y

/ / / ,

{•§/

/ / A *>/

/

/ " 'm>v& •smug?

2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8

x/ 'H

FIG. 6. Design Chart for Slip Surface Trace and Embedment Length: (a) m = 1.0; (b) m = 1.5; and (c) m = 2.5

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1.0

0.8

0.6

0.4

0.2

0.0

-0.2

1.0

0.8

0.6

• i °-4

• m = 5

W A V * -

/ /

/

{} / /

/ y [/ V

• sWJBSWasT

(d) -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8

x/H

•m=10

*WAWA

U IA

$ **•?

b y

/

/ y A V

/ • $ /

« /

- ^M wy/A\ 7W" "

0.2

0.0

-0.2

-0 .2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8

(e) x /H

1.0

0.8

0.6

0.4

0.2

0.0

-0.2

m=«>

«*»»!

/ /

^ ft

,

y Y/ /

V <§/<

/

} /

-#m, /SvVA'o' 7c?

(0

-0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8

x/H

FIG. 6. Design Chart for Slip Surface Trace and Embedment Length: (d) m = 5; (e) m = 10; and (0m = «

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—,

'-5

i;

«=»r

1.0

\l

:\

\

m=

^ <^

1.0 N \

\

^

I

\ ^ : \

i V

^

•m=

V ^

=1.5

^ ^ ^

^

2.0

. ^

I

m-

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<r

^ ^ ^

2.5

^ ^

^

15

20

25

30

35

40

45

, ,

, 15

20

25

30

35

40

45

15

20

25

30

35

40

45

(a)

<*

W

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FIG

. 7.

D

esig

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t fo

r E

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bedm

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Len

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: (a

) m

= 1

.0;

(b)

m =

1.5

; an

d (

c) m

= 2

.5

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X

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5 ~<

ro

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2.0

X

1.0

^ :^

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\vp

1

m=

x

IDQ

^5 ^

Sa

15

20

25

30

35

40

45

15

20

25

30

35

40

45

15

20

25

30

35

40

45

W

0m

(e

) 0

m

(0

<t> m

FIG

. 7.

D

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(e)

m =

10;

and

(f)

m =

»

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TABLE 1. Required Embedment Length in Example

j (1)

1 2 3 4 5 6 7 8 9

10

>!/ (m) (2)

0.0 0.3 0.6 0.9 1.2 1.5 1.8 2.1 2.4 2.7

'., (m) (3)

0.58" 0.30" 0.18d

0.14 0.14 0.14 0.14 0.14 0.14 0.14

L (m) (4)

0.00 0.36 0.66 0.90 1.11 1.26 1.41 1.50 1.56 1.62

// (m) (5)

0.58 0.66 0.84 1.04 1.25 1.40 1.55 1.64 1.70 1.76

l,b (m) (6)

2.22 2.00 1.78 1.55 1.33 1.11 0.89 0.67 0.44 0.22

// (m) (7)

2.22 2.00 1.78 1.55 1.33 1.40 1.55 1.64 1.70 1.76

"Based on internal stability (i.e., /, = ls. + /„.). bBased on external stability. "Design value. dDetermined by trial and error.

7. Determine the total embedment length /,• (= lSj + lej) required for each reinforcing sheet.

8. Select a factor of safety for external stability, Fs. 9. Compute 4>m.

10. Select the appropriate chart from Figs. 7(a—f) and determine at all ele­vations yj the required length /, for external stability.

11. For each reinforcing sheet j = 1,2, . . . , n, choose the longer length found in steps 7 and 10.

To demonstrate this procedure, consider the following problem: m = 2.5, H = 3.0 m, 7 = 18 kN/m 3 , 4> = 35°, and k = 0.8. If a safety factor of Fs

= 1.5 is selected for internal stability than <)),„ = 25°. From Fig. 5 it follows that Tm varies between 0.227 and 0.245. Selecting an intermediate value T,„

i.O

2.4

1.8

1.2

0.6

0.0

0.6 - 0

^Aw^

6 0.0

2.5

0. 6 1.2 1.8 2.4 3.0

X [m]

i^yaftv*-1*^

7=18 kN/m3

0=35°

k=0.8

F,=1.5 t =5.74 kN/m

3.6 4.2 4.8 5.4

FIG. 8. Example Problem: Required Geosynthetic Profile

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= 0.236 and 10 equally spaced reinforcing sheets (i.e., one every 0.3 m) yields t} = 5.74 kN/m. The length ls is measured from Fig. 6(b) and based on Eqs. 4 and 1, le. is computed. These values are presented in Table 1. Choosing the same safety factor for external and internal stability and using Fig. 7(b), it follows that IJ(H - y,) = 0.74. Thus, /, required for external stability is computed at all elevations and presented in Table 1. Fig. 8 shows the reinforcement profile satisfying both the internal and external stability requirements. It is worthwhile pointing out that the lj computed based on the internal stability criterion coupled with the trace of the slip surface for hor­izontal reinforcement inclination (not presented in this paper) will result in shorter reinforcing sheets than for the orthogonal case. The required tensile resistance, however, will be slightly larger.

INTERPRETATION

General The factor of safety of a given reinforced structure depends on, among

other qualities, the pullout resistance/breakage strength of each geosynthetic sheet. For an arbitrary structure this safety factor may correspond to a com­plex failure mechanism that is controlled by local conditions. It is clear that the results presented are inadequate for varying local conditions. These re­sults, however, are design oriented; i.e., a tool is provided that enables the designer to specify a reinforced structure for which prescribed safety margins are attained or exceeded.

By extending the reinforcing sheets to the slope face (Fig. 8), the potential for an internal slip surface emerging above the toe is eliminated. This can be verified for a given problem by using Figs. 5 and 6 [where the slope height is taken as (H — yj) rather than H] and showing that the existing tj exceeds the required one. There is a possibility, however, of a superficial failure developing at, or near, the slope surface. To prevent such a failure each reinforcing geosynthetic sheet at the face can be folded back over the exposed soil portion and reembedded. This and other options are outlined elsewhere (e.g., Leshchinsky and Perry 1987; Carroll and Richardson 1986).

The internal and external stability analyses resulted in the dimensions of the reinforcing sheets as well as their required tensile resistance. Based on the required tensile resistance for stability, one may select an appropriate geosynthetic using as an additional guide, for example, suggestions by Koer-ner and Hausmann (1987). It should be pointed out, however, that because of limited knowledge at present there are two major concerns when using geosynthetics in reinforced soil problems: durability (aging) and creep (or stress relaxation). Information regarding in-soil durability of geosynthetics is scarce and, therefore, serious consideration is required before its appli­cation to critical structures. Some factors to be considered are discussed by Schneider and Groh (1987), Whelton and Wrigley (1987), and Hodge (1985). The potential for developing the geosynthetic tensile resistance needed for internal stability must be ensured. When subjected to tensile force, however, most polymers tend to creep. This tendency increases rapidly with the level of tensile force. Specifying a geosynthetic possessing tensile strength equal to the required tensile resistance tx, therefore, may result in excessive creep or load transfer to sheets positioned above. Ideally, the allowable tensile force, which must be greater than the required resistance, would be deter-

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mined from a creep test where in-soil conditions are simulated (e.g., Murray and McGown 1987). Note that recommended allowable values stated by Den Hoedt (1986), which resulted from simple creep tests, are on the order of 25-50% of the geosynthetic's ultimate tensile strength, depending on its polymer type. It may be pointed out, however, that creep restrictions stem­ming from laboratory tests are based on at least one order of time-magnitude extrapolation.

Effects of tj Distribution Limit-equilibrium analysis deals with the global stability of a system as­

sumed to be at the verge of collapse. At working conditions, however, there is no guarantee that uniform mobilization of pullout resistance will occur; i.e., it is possible that while some sheets are underutilized, the pullout re­sistance of others is fully mobilized, resulting in excessive structural defor­mations. In the context of the presented paper, the design of pullout resis­tance for each sheet (i.e., its anchorage length) is controlled by the assumed linear distribution of tj. Since the linear distribution is frequently employed in the design of geosynthetic-retained earth walls, the writers preferred to present their results in the same framework. However, in designing rein­forced steep slopes, tt = allowable geosynthetic tensile strength (i.e., uni­form tj distribution if same reinforcement is used) is often introduced into the stability analysis (e.g., Koerner 1986). Consequently, it seems important to assess the effects of the assumed distribution on the output of the limit-equilibrium analysis. These effects have been studied (Boedeker 1987) using the distributions expressed by tj = f, (1 - yj/H)a where a was varied be­tween 1 (i.e., linear distribution—Eq. 4) and 1/4 (i.e., nearly uniform dis­tribution at the bottom and highly nonuniform at the top). The following was observed: (1) The scalar sum of the required tensile resistance (i.e., S"=i tj) is nearly independent of the assumed distribution; and (2) the poten­tial slip surface is also nearly independent of the assumed distribution. These observations imply, for example, that if one assumes a case where tj = con­stant, the required value of this constant is about half the maximum tensile resistance, /,, as determined based on the linear distribution (i.e., based on Fig. 5). It follows that the required anchorage length for tj — constant will be longer at the upper half of the structure and shorter at the lower half. The required strength (based on tt), however, is only half the value needed for the linear case. Since the linear distribution is typically the extreme value assumed in design, it seems that t, obtained from Fig. 5 results in a con­servative selection of geosynthetic. However, to ensure that the upper half of the reinforcement sheets are firmly anchored against potentially higher pullout forces, one can use the other extreme distribution assumed in design: tj = constant. In this case this constant value approximately equals t,/2 and the required lej in the upper half can be calculated using Eq. 1. For example, modifying lej according to an assumed uniform distribution [t = (5.74/2) kN/m], will result in the following changes in Table 1: /ei0 = 0.70 m (0.14), 4, = 0.35 m (0.14), /„8 = 0.23 m (0.14), and lei = 0.18 m (0.14), where the numbers in parentheses are those in the table and ls. remains essentially unchanged.

SUMMARY AND CONCLUSIONS

The stability problem of geosynthetic reinforced earth structures is divided

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into two separate aspects: internal and external. To study each aspect, sta­bility analysis methods were modified. The external stability analysis is based on the bilinear wedge method. The internal stability analysis is based on the variational limiting equilibrium approach, and it is rigorous in the sense that all equilibria requirements are satisfied. The analyses results are presented in a format of design charts. Based on these charts, the reinforcing sheets' profile as well as the geosynthetic tensile resistance, required for stability can be determined.

The results of the presented analysis imply that:

1. Assumed horizontal reinforcement's tensile force produces values of re­quired tensile resistance that are slightly larger than those for orthogonal geo­synthetic force.

2. Although a horizontal positioning of geosynthetic tensile force requires slightly longer anchorage length, the corresponding slip surface makes a shallower cut into the slope compared with the orthogonal inclination. As a result, the total geosynthetic embedment length for the horizontal case turns out to be shorter than for the orthogonal case.

3. Based on an internal stability analysis for a given earth structure, the po­tential slip surface and the scalar sum of the required reinforcing forces are nearly independent of the assumed distribution of the geosynthetic tensile resistance. The design charts may be generalized to deal with other distributions of tensile resistance.

ACKNOWLEDGMENTS

The writers appreciate some of the fundamental issues raised by the re­viewers. Modification of the text in response to those issues was partially based upon knowledge and insight acquired from projects in the general area of slope stability supported by the National Science Foundation under Grant Nos. ECE-8503572 and CES-8722818.

APPENDIX I. REFERENCES

Baker, R. (1981). "Tensile strength, tension cracks and stability of slopes." Soils and Founds., Journal of the Japanese Society of Soil Mech. and Found. Engrg., 21(2), 1-17.

Baker, R., and Garber, M. (1977). "Variational approach to slope stability." Proc, 9th Int. Conf. on Soil Mech. and Found. Engrg., Tokyo, Japan, 2, 2-12.

Baker, R., and Garber, M. (1978). "Theoretical analysis of the stability of slopes." Geotechnique, London, England, 28(4), 395-411.

Boedeker, R. H. (1987). "Analysis and design of geotextile reinforced granular em­bankment over firm foundations," thesis presented to the University of Delaware, at Newark, Del., in partial fulfillment of the requirements for the degree of Master of Civil Engineering.

Bell, J. R., Stilley, A. N., and Vandre, B. (1975). "Fabric retained earth walls." Proc, 13th Annual Engrg. Geology and Soils Engrg. Symp., Univ. of Idaho, 271-287.

Bonaparte, R., Holtz, R. D., and Giroud, J. P. (1985). "Soil reinforcement design using geotextiles and geogrids." Geotextile testing and the design engineer (STP 952), J. E. Fluet, ed., Am. Soc. for Testing and Materials, Philadelphia, Pa., 69-116.

Carroll, R. G., and Richardson, G. N. (1986). "Geosynthetic reinforced retaining

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walls." Proc, 3rd Int. Conf. on Geotextiles, Vienna, Austria, Austrian National Committee of the Int. Society for Soil Mech. and Found. Engrg., 2, 389-394.

Christie, I. F., and El-Hadi, K. M. (1977). "Some aspects of the design of earth dams reinforced with fabric." Proc, Int. Conf. on the Use of Fabrics in Geo-technics, Paris, France, 1, 99-103.

Den Hoedt, G. (1986). "Creep and relaxation of geotextile fabrics." Geotextiles and Geomembrs., Elsevier Applied Science Publishers Ltd., England, 4(2), 83-92.

Hodge, J. (1985). "Durability testing." Geotextile testing and the design Engineer (STP 952), J. E. Fluet, ed., Am. Soc. of Testing and Materials, Philadelphia, Pa., 119-121.

Ingold, T. S. (1982). "An analytical study of geotextile reinforced embankments." Proc, 2nd Int. Conf. on Geotextiles, Las Vegas, Nev., Industrial Fabrics Asso­ciation Int., 3, 683-688.

Koerner, R. M. (1986). Designing with geosynthetics. Prentice-Hall, Englewood Cliffs, N.J.

Koerner, R. M., and Hausmann, M. R. (1987). "Strength requirements of geosyn­thetics for soil reinforcement." Geotech. Fabrics Report, 5(1), 18-26.

Leshchinsky, D., and Reinschmidt, A. J. (1985). "Stability of membrane reinforced slopes." J. Geotech. Engrg., ASCE, 111(11), 1285-1300.

Leshchinsky, D., and Volk, J. C. (1985). "Stability charts for geotextile reinforced walls." Transp. Res. Record, 1031, 5-16.

Leshchinsky, D., Volk, J. C , and Reinschmidt, A. J. (1986). "Stability of Geo-textile-Retained Earth Railroad Embankment." Geotextiles and Geomembranes, Elsevier Applied Science Publishers Ltd., England, 3(2 & 3), 105-128.

Leshchinsky, D., and Perry, E. B. (1987). "A design procedure for geotextile rein­forced walls." Geotechn. Fabrics Report, 5(4), 21-27.

Murray, R. T. (1982). "An analytical study of geotextile reinforced embankments and cuttings." Proc, 2nd Int. Conf. on Geotextiles, Las Vegas, Nev., Industrial Fabrics Associated Int., 3, 707-713.

Murray, R. T., and McGown, A. (1987). "Geotextile test procedures: background and sustained load testing." Application Guide 5, Transport and Road Research Laboratory, Dept. of Transport, Ground Engrg. Div., Structures Group, Crow-thorne, Berkshire, England.

Ruegger, R. (1986). "Geotextiles reinforced soil structures on which vegetation can be established." Proc, 3rd Int. Conf. on Geotextiles, Vienna, Austria, 2, 453-458.

Schmertmann, G. R., et al. (1987). "Design charts for geogrid-reinforced soil slopes." Proc, Geosynthetics '87 Conf, New Orleans, La., Industrial Fabrics Association Int., 1, 108-120.

Schneider, H. R., and Holtz, R. D. (1986). "Design of slopes reinforced with geo­textiles and geogrids." Geotextiles and Geomembranes, Elsevier Applied Science Publishers Ltd., England, 3(1), 29-51.

Schneider, H., and Groh, M. (1987). "An analysis of the durability problems of geotextiles." Proc, Geosynthetic 87 Conf., New Orleans, La., Industrial Fabrics Association Int., 2, 434-441.

Taylor, D. W. (1937). "Stability of earth slopes." J. of the Boston Society of Civ. Engrs., XXIV(3), 197-246.

Whelton, W. S., and Wrigley, N. E. (1987). "Long-term durability of geosynthetics soil reinforcement." Proc, Geosynthetics '87 Conf, New Orleans, La., Industrial Fabrics Association Int., 2, 442-455.

APPENDIX II. NOTATION

The following symbols are used in this paper:

A,B = unknown constants; Fs = factor of safety; H = height of structure;

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k = soil-geosynthetic friction normalized with respect to tan (<|>); / = total length of embedded reinforcing sheet (= le + ls); le = embedment length beyond slip surface; ls = embedment length between slope and slip surfaces; m = slope inclination; n = total number of reinforcing sheets; R = radius of log spiral; S = normalized stress (= cr/yH); T = normalized pullout resistance (= t/yH2); t = pullout (tensile) resistance; y = elevation of reinforcement sheet (y = 0 is toe elevation); P = independent variable (angle) in polar system; y = backfill unit weight; £ = inclination of failure plane. 8 = inclination of t to a horizontal plan; (j = stress normal to slip surface; CT = average stress normal to the geosynthetic sheet; T = shear stress along slip surface;

cj> = internal angle of friction; and i|/ = tan (<j>).

Subscripts j = reinforcing sheet number; and

m = mobilized value.

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