duncan

137

CHAPTER 8

Reinforced Slopes and Embankments

Reinforcement can be used to improve the stability ofslopes and embankments, making it possible to con-struct slopes and embankments steeper and higher thanwould otherwise be possible. Reinforcement has beenused in four distinct types of applications:

1. Reinforced slopes. Multiple layers of reinforce-ment at various elevations within ll slopes havebeen used to increase the factor of safety for slipsurfaces that cut through the reinforcement, mak-ing it possible to construct slopes steeper thanwould be possible without reinforcement.

2. Reinforced embankments on weak foundations.Reinforcement at the bottom of an embankmenton a weak foundation can increase the factor ofsafety for slip surfaces passing through the em-bankment, making it possible to construct theembankment higher than would be possible with-out reinforcement.

3. Reinforced soil walls or mechanically stabilizedearth walls. Several different proprietary systemshave been developed for reinforced soil walls,which are used as alternatives to conventional re-taining walls. Most of the companies that marketMSE walls have developed proprietary designprocedures. The stability of MSE walls can alsobe evaluated using the methods described in thischapter.

4. Anchored walls. Vertical soldier pile walls orslurry trench concrete walls can be tied backor anchored at one or more levels to provide ver-tical support for excavations or lls. Anchoredwalls have been used in both temporary and per-manent applications. The methods described inthis chapter can be used to evaluate the stabilityof anchored walls.

LIMIT EQUILIBRIUM ANALYSES WITHREINFORCING FORCES

Reinforced slopes can be analyzed using the proce-dures described in Chapter 6 by including the rein-forcement forces in the analyses as known forces.Zornberg et al. (1998a,b) have shown through centri-fuge tests that limit equilibrium analyses provide validindications of factor of safety and failure mechanismsfor reinforced slopes. Their analyses, which agreedwell with the results of their tests, were performed us-ing peak values of rather than the lower critical-state friction angle of the backll soil.

The amount of force required to achieve a targetvalue of factor of safety can be determined using re-peated trials, varying the magnitude of the force untilthe factor of safety computed is the one desired. Somecomputer programs can perform this operation auto-maticallythe input is the desired factor of safety, andthe output is the required reinforcement force. Thistype of program is better adapted to design of rein-forced slopes, since there is no need for repeated anal-yses.

FACTORS OF SAFETY FOR REINFORCINGFORCES AND SOIL STRENGTHS

Two methods have been used for limit equilibriumanalyses of reinforced slopes.

Method A. The reinforcement forces used in theanalysis are allowable forces and are not dividedby the factor of safety calculated during the slopestability analysis. Only the soil strength is dividedby the factor of safety calculated in the slope sta-bility analysis.

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Copyright 2005 John Wiley & Sons Retrieved from: www.knovel.com

138 8 REINFORCED SLOPES AND EMBANKMENTS

20 ft1

2Reinforcementforce = 10,000 lb/ft

c = 500 psf, = 0 = 100 pcf

Firm layer

10 ft

Critical circles

AB

Figure 8.1 Check problem for determining whether a com-puter program is using method A or method B for analysisof reinforced slopes.

Method B. The reinforcement forces used in theanalysis are ultimate forces, and are divided bythe factor of safety calculated in the slope stabilityanalysis. Both the reinforcing force and the soilstrength are divided by the factor of safety cal-culated in the slope stability analysis.

Method A is preferable, because the soil strengthand the reinforcement forces have different sources ofuncertainty, and they therefore involve differentamounts of uncertainty. Factoring them separatelymakes it possible to reect these differences.

When a computer program is used to analyze rein-forced slopes, it is essential to understand which ofthese methods is being used within the program, sothat the appropriate measure of reinforcing force (al-lowable force or ultimate force) can be specied in theinput for the analysis.

If the documentation of a computer program doesnot specify whether the reinforcement force should beallowable or ultimate, this can be deduced from theequations employed to compute the factor of safety.

Method A EquationsIf the factor of safety for circular slip surfaces is de-ned by an equation of the form

soil resisting momentF

overturning moment reinforcement moment(8.1)

or, more generally, if the factor of safety is dened byan equation of the form

shear strengthF (8.2)

shear stress required for equilibrium reinforcement resistance

the program uses method A, and the reinforcementforces specied in the input should be allowableforces, denoted here as Pall.

Method B EquationsIf the factor of safety for circular slip surfaces is de-ned by an equation of the form

soil resisting moment reinforcement momentF

overturning moment(8.3)

or, more generally, by an equation of the form

shear strength reinforcement resistanceF

shear stress required for equilibrium(8.4)

the program uses method B, and the reinforcementforces specied in the input should be the unfactoredlong-term load capacity of the reinforcement, denotedhere as Plim.

If it is not clear which method is used by a computerprogram, this can be determined by analyzing the re-inforced slope problem shown in Figure 8.1. This slopeis 20 ft high and is inclined at 2.0 vertical on 1.0 hor-izontal. The soil within the slope is uniform, with 100 pcf, 0, and c 500 psf. There is a rm layerbeneath the toe. A reinforcing force of 10,000 lb/ftacts horizontally at midheight, 10 ft above the toe ofthe slope. Results for two analyses are shown in Figure8.1. The analyses considered only circular slip surfacestangent to the top of the rm layer. The critical circles,located as shown in Figure 8.1, exit slightly above thetoe of the slope.

The method used by any computer program can bedetermined based on the computed factor of safety:

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REINFORCEMENT FORCES 139

If the program computes F 2.19, the programuses method A. Allowable reinforcement forcesshould be used with the program.

If the program computes F 1.72, the programuses method B. Ultimate reinforcement forcesshould be used with the program.

Slight deviations from F 2.19 or F 1.72 can beexpected, depending on the number of slices used bythe program, and the method used to locate the criticalcircle. The differences should be no more than 1 or2%, however.

TYPES OF REINFORCEMENT

The principal types of reinforcing materials that havebeen used for slopes and embankments are geotextilefabrics, geogrids, steel strips, steel grids, and high-strength steel tendons. Geotextiles are manufactured byweaving polymeric bers into a fabric or by mattingthe bers together to form a continuous nonwoven fab-ric. Woven geotextiles are stiffer and stronger thannonwoven geotextiles and more useful for reinforcedslope applications. Geogrids are manufactured bystretching sheets of polymer plastic in one or both di-rections to form a high-strength grid. Stretching thepolymeric materials makes them stiffer and stronger.Galvanized or epoxy-coated steel strips have been usedfor slope reinforcement. The strips usually have raisedribs to increase their pullout resistance. Welded mildsteel mats or grids have also been used for reinforcingslopes and embankments.

Key sources of information about geosynthetic re-inforcing for slopes are the book by Koerner (1998),which contains a great deal of information on the fun-damental characteristics and properties of polymers,geotextiles, and geogrids, and the FHWA (2000) pub-lication entitled Mechanically Stabilized Earth Wallsand Reinforced Soil Slopes: Design and ConstructionGuidelines, which covers a wide range of subjects re-lated to geotextiles, geogrids, steel strips, and steelgrids, and their use in reinforced walls and slopes.

REINFORCEMENT FORCES

The long-term capacity of reinforcement, denoted hereas Tlim, depends on the following factors:

Tensile strength. For steel, the tensile strength isthe yield strength. For geosynthetics, the tensile

strength is measured using short-term wide-widthtensile tests.

Creep characteristics. Steel does not creep appre-ciably, but geosynthetic materials do. The tensileloads used for design of geotextile- and geogrid-reinforced walls must be reduced to values lowerthan those measured in short-term tensile tests, tostresses that are low enough so that little or nocreep deformation will occur over the design lifeof the structure.

Installation damage. Geotextiles and geogrids aresubject to damage during installation that resultsin holes and tears in the material. Epoxy-coatedand PVC coatings on steel are subject to damageduring installation, and galvanization is thereforepreferred for corrosion protection.

Durability. The mechanical properties of geosyn-thetics are subject to deterioration during serviceas a result of attack by chemical and biologicalagents. Steel is subject to corrosion.

Pullout resistance. Near the ends of the reinforce-ment, capacity is limited by the resistance to pull-out, or slip between the reinforcement and the soilwithin which it is embedded.

Reinforcement stiffness and tolerable strain withinthe slope. To be useful for slope reinforcement,the reinforcing material must have stiffness as wellas strength. A very strong but easily extensiblerubber band would not provide effective reinforce-ment, because it would have to stretch so much tomobilize its tensile capacity that it would not beable to limit the deformation of the slope.

Values of Tlim, the long-term capacity of reinforcingmaterials, must satisfy the following three criteria:

1. Tlim capacity determined by short-term tensilestrength, creep, installation damage, and deteri-oration of properties over time.

2. Tlim capacity determined by pullout resistance.3. Tlim capacity determined by stiffness and tol-

erable strain.

Methods of applying these requirements to geosyn-thetics and steel reinforcing are described in the fol-lowing sections.

Criterion 1: Creep, Installation Damage, andDeterioration in Properties over Time

Geotextiles and geogrids. The effects of creep, in-stallation damage, and long-term deterioration ongeosynthetic materials can be evaluated using theexpression

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Table 8.2 Corrosion Rates for Steel Reinforcementin Mildly Corrosive Backll

Materialcorroding

Period oftime

Corrosionrate

(m/yr)Corrosion ratea

(in. /yr)Zinc First two years 15 5.9 104Zinc Thereafter 4 1.6 104Carbon steel Thereafter 12 4.7 104

aThese corrosion rates are applicable for steel reinforce-ment in backll with these electrochemical properties: Re-sistivity greater than 3000 cm, pH between 5 and 10,chlorides less than 100 ppm, sulfates less than 200 ppm,organic content less than 1%.

Source: After FHWA (2000).

Table 8.1 Reduction Factors for Tensile Strengths ofGeotextiles and Geogrids for Use in Eq. (8.5)

Reduction for: Factor PolymerRange ofvaluesa

Creep RFCR Polyester 1.62.5Polypropylene 4.05.0Polyethylene 2.65.0

Installationdamage

RFID Any polymer 1.13.0

Deteriorationin service

RFD Any polymer 1.12.0

aThese values (from FHWA, 2000) are applicable toreinforcement in granular soils with maximum particlesizes up to 19 mm, values of pH from 4.5 to 9.0, and in-service temperatures below 30C. Geotextiles weighingless than 270 g/m2 are subject to greater damage duringinstallation and should not be used for reinforcement.

TultT (8.5)lim (RF )(RF )(RF )CR ID D

where Tlim is the long-term limit load (F/L); Tult theshort-term ultimate strength, measured in a wide-striptension test (F/L); RFCR the strength reduction factorto allow for creep under long-term load; RFID thestrength reduction factor to allow for installation dam-age; RFD the strength reduction factor to allow for de-terioration in service. Values of RFCR, RFID, and RFDrecommended by the FHWA are given in Table 8.1.The units of Tlim and Tult are force per unit length ofreinforced slope.

Steel reinforcement. Steel reinforcing does notcreep appreciably and is not subject to installationdamage. Epoxy and PVC coatings are subject to in-stallation damage, but galvanized coating is not. Theeffects of long-term deterioration of steel due to cor-rosion can be evaluated using the expression

T A (8.6)lim C y

where Tlim is the allowable long-term reinforcementtension load (F/L); Ac the cross-sectional area of re-inforcement after corrosion, calculated by reducing themetal thickness by the loss expected during the life ofthe installation [Ac is the area per unit length of slope(L2/L)]; and y the yield strength of steel (F/L2). Cor-rosion rates for steel in mildly corrosive backll ma-terials are given in Table 8.2.

Criterion 2: Pullout ResistanceTo develop tensile capacity, reinforcement must be re-strained sufciently by friction in the soil. The maxi-mum possible resistance (Tpo) is proportional to theeffective overburden pressure. Tpo begins from zero atthe end of the reinforcement, where the embeddedlength is zero and increases with distance from the end,as shown in Figure 8.2. The slope of the curve repre-senting the variation of Tpo with distance can be ex-pressed as

dTpo 2zF* (8.7)dL

where Tpo is the pullout resistance (F/L); L the lengthof embedment, or distance from the end of the rein-forcement (L); the unit weight of ll above the re-inforcement (F/L3); z the depth of ll above thereinforcement (L); the adjustment factor for exten-sible reinforcement (dimensionless); and F* the pull-out resistance factor (dimensionless).

Values of and F* recommended by the FHWA(2000) are listed in Table 8.3. These values are con-servative estimates. Larger values may be applicableand can be used if they are supported by tests per-formed on the specic soil and reinforcing material.

Equation (8.7) gives the slope of the pullout resis-tance curve at any location. If the thickness of llabove the reinforcement is constant, the slope of thepullout curve is constant, and the pullout resistance canbe expressed as

T 2zF*L (8.8)po ewhere Le is the distance from the end of the reinforce-ment or length of embedment (L).

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ALLOWABLE REINFORCEMENT FORCES AND FACTORS OF SAFETY 141

v = z = overburden pressure

(a)

(b)

(c)

(d)

Reinforcementz

Rei

nf. F

orce

Slope varies Slope constant

Tlim

TallTall =

Tlim or Tpo from aboveFR

Tpo Tpo

Figure 8.2 Variation of Tlim and Tall with distance along re-inforcement.

Table 8.3 Pullout Resistance Factors and F* forUse in Eq. (8.7)

Pulloutresistance

factoraType of

reinforcementResistance

factor value

Geotextiles 0.6Geogrids 0.8Steel strips and steel grids 1.0

F* Geotextiles 0.67 tan Geogrids 0.8 tan Steel strips and steel grids 1.0 tan

aHigher values of F* generally apply at depths shal-lower than 6 m. Larger values of both and F* may beapplicable and can be used if they are supported by testsperformed on the specic soil and reinforcing material.

Source: FHWA (2000).

If the overburden pressure increases with distancefrom the end of the reinforcement, as it does beneaththe slope on the left in Figure 8.2, the slope of the Tpocurve also increases with distance, and the pullout re-sistance diagram is a curve. In this case the pulloutresistance can be expressed as

2T tan F*(L ) (8.9)po e

where is the slope angle in degrees, as shown inFigure 8.2.

Criterion 3: Reinforcement StiffnessReinforcing materials must be stiff enough so that re-inforcement forces can be mobilized without excessivestrain. The value of Tlim should not exceed the productof the long-term secant modulus of the reinforcementmultiplied by the tolerable strain for the slope:

T E (8.10)lim secant tolerable

where Esecant is the secant modulus of reinforcing ataxial strain tolerable (F/L) and tolerable is the strainwithin the slope at the location of the reinforcing thatcan be tolerated without excessive slope deformationor failure (dimensionless).

Steel reinforcing is often described as inextensiblebecause it stiffness is very high compared to its yieldstrength. With a modulus equal to 500 to 1000 timesits yield strength, the yield strength of steel is mobi-lized at a strain of only 0.1 to 0.2%, far less than thetolerable strains for soil reinforcing applications. As aresult, criterion 3 never governs the value of Tlim forsteel reinforcing. The stiffness of geosynthetic mate-rials, on the other hand, may be low enough so thatcriterion 3 governs the value of Tlim for applicationswhere the tolerable strain is small.

As shown in Figure 8.3, Esecant is the slope of a lineextending from the origin to the point on the T curvewhere the strain is equal to tolerable. Note that the unitsof Esecant, like the units of Tlim, are force per unit length.

Values of tolerable strain are based on the results ofnite element analyses (Rowe and Soderman, 1985);and on experience (Fowler, 1982; Christopher andHoltz, 1985; Haliburton et al., 1982; Bonaparte et al.,1987). A summary of published recommendations isgiven in Table 8.4.

ALLOWABLE REINFORCEMENT FORCES ANDFACTORS OF SAFETY

The preceding section is concerned with the long-termcapacity of reinforcement (Tlim). These values of Tlim

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Tensile Strain -

Tens

ile L

oad

- (F/L

)

1Esecant

Geosyntheticload-strain curve

tol

Figure 8.3 Denition of Esecant for geosynthetic reinforce-ment.

Table 8.4 Tolerable Strains for Reinforced Slopesand Embankments

Application tol (%)Reinforced soil walls 10Reinforced slopes of embankments on rm

foundations10

Reinforced embankments on nonsensitiveclay, moderate crest deformationstolerable

10

Reinforced embankments on nonsensitiveclay, moderate crest deformations nottolerable

56

Reinforced embankments on highlysensitive clay

23

Source: Compiled from Fowler (1982), Christopher andHoltz (1985), Haliburton et al. (1982), Rowe and Soder-man (1985), and Bonaparte et al. (1987).

Table 8.5 Recommended Values of FR

Consequencesof failure

Uncertainties inTlim and load inreinforcement

Appropriatevalue of FR

Minimal Small 1.5Minimal Large 2.0Great Small 2.0

reect consideration of long-term loading, installationdamage, deterioration in properties over time, pulloutresistance, and tolerable strains, but they do not includea factor of safety.

The allowable load assigned to reinforcing materialsshould include a factor of safety, as indicated by

TlimT (8.11)all FR

where Tall is the allowable long-term reinforcementforce (F/L) and FR is the factor of safety for reinforce-

ment force. The value of FR should reect (1) the de-gree of uncertainty involved in estimating the value ofTlim, (2) the degree of uncertainty involved in estimat-ing the load that the reinforcement must carry, and (3)the consequences of failure. Recommended values ofFR are given in Table 8.5.

ORIENTATION OF REINFORCEMENT FORCES

Various orientations of reinforcement forces have beensuggested (Schmertmann et al., 1987; Leshchinsky andBoedeker, 1989; Koerner, 1998; FHWA, 2000). Theextremes are (1) reinforcement forces that are alignedwith the original orientation of the reinforcement, and(2) reinforcement forces that are parallel to the slipsurface. The latter assumption, which results in largerfactors of safety, has been justied by the concept thatthe reinforcement will be realigned where the slip sur-face crosses the reinforcement. This is more likely ifthe reinforcement is very exible. The assumption thatthe orientation of the reinforcement force is the sameas the orientation of the reinforcement is more conser-vative, is supported by the ndings of Zornberg et al.(1998a), and is the more logical, reliable choice. Thisis the approach recommended here.

REINFORCED SLOPES ON FIRM FOUNDATIONS

Reinforcement in embankments can be used to con-struct slopes steeper than would be possible withoutreinforcing. Usually, several layers of reinforcing areused, spaced more closely near the base of the slopeand farther apart near the top, as shown in Figure 8.4.Secondary shorter lengths of lower-strength reinforce-ment, between the primary reinforcement layers, canbe used to improve surcial stability (FHWA, 2000).Zornberg et al. (1998a) showed that such layers nearthe bottom of the slope signicantly enhance stabilityif they are wrapped around at the face and overlap theadjacent layers. In this conguration they are anchoredrmly and not subject to pullout failure.

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REINFORCED SLOPES ON FIRM FOUNDATIONS 143

Critical circle

F = 1.40

F = 1.43

F = 1.33

F = 1.37

(a)

(b)

(c)

Figure 8.4 Limit equilibrium analyses of a reinforced slopeusing circular, wedge, and smooth noncircular slip surfaces:(After Wright and Duncan, 1991). (a) critical circular slipsurface; (b) critical two-part wedge; (c) critical noncircularslip surfaces.

AssumptionsCohesionless fill (c' = 0)No pore pressure (u = 0)Soil-reinforcement interface friction angle = 0.9 '0.6

0.5

0.4

0.3

0.2

0.1

0

K =

2(Tall)/(

)(H)

2

30 40 50 60 70 801.5:1 1:1 0.75:1 0.5:1

SLOPE ANGLE, (degrees)

'f = 15

' f = 20

' f = 25

' f = 30

' f = 35

(a)

Figure 8.5 (a) Reinforcement force coefcients; (b) rein-forcement length coefcients. (After Schmertmann et al.,1987.)

The stability of reinforced slopes can be evaluatedusing the procedures outlined in Chapter 6. An ex-ample is shown in Figure 8.4. It can be seen that thefactor of safety varies slightly with the shape of theslip surface, from F 1.43 for the most critical two-part wedge slip surface, to F 1.33 for the most crit-ical noncircular slip surface, a difference of about 7%.The most critical circle gives a factor of safety F 1.40, which is sufciently accurate for practical pur-poses.

Schmertmann et al. (1987) ChartsBefore an analysis of the type illustrated in Figure 8.4can be performed, the strength, length, and spacing ofthe reinforcement must be estimated. Determiningthese by trial and error can be time consuming becausemany trials can be required to determine strength,length, and spacing.

Designing reinforced slopes is facilitated greatly byslope stability charts of the type developed bySchmertmann et al. (1987), which are shown in Figure8.5a and b. Figure 8.5a can be used to determine thetotal reinforcing force, and Figure 8.5b can be used todetermine the length of reinforcing required for a givenfactor of safety. The terminology used in these chartsis:

K dimensionless force coefcient2Tall 2(H)(8.12)

H H effective height, including effect ofq

surcharge (L) (8.13)

arctan factored friction angletan

F(degrees) (8.14)

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0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

LH'

30 40 50 60 70 801.5:1 1:1 0.75:1 0.5:1

SLOPE ANGLE, (degrees)

'f = 15

'f = 20

'f = 25

'f = 30

'f = 35LT = LB

LT

LB

AssumptionsCohesionless fill (c' = 0)No pore pressure (u = 0)Soil-reinforcement interface friction angle = 0.9 '

LB/H'LT/H'

(b)

Figure 8.5 (Continued )

Figure 8.6 Potential modes of failure of reinforced embank-ments: (a) block sliding outward along reinforcement withslumping of the crest; (b) foundation failure with rotationalsliding through embankment; (c) excessive elongation of re-inforcement. (Modied from Haliburton et al., 1978).

LB required length of reinforcement at the bottomof the slope (L)

LT required length of reinforcement at the top ofthe slope (L)

slope angle (degrees)

unit weight of soil (F/L3)

q surcharge pressure (F/L2)

u pore pressure, assumed to be zero throughoutthe slope

Example. As an example of the use of these charts,consider the slope shown in Figure 7.27. Pertinent pa-rameter values from Figure 7.27 are H 24 ft, 130 pcf, 37, arctan(1.25) 39, q 0,c 0, and u 0. The factor of safety computed in

the STABGM users manual and Chapter 7 is F 1.71.

The charts in Figure 8.5a and b can be used to de-termine the total reinforcement force and the length ofreinforcement required for a factor of safety F 1.71using the following steps:

1. Compute arctan tan /F arctan 0.75/f1.71 24.

2. Compute H H q / 24 ft 0/130 pcf 24 ft.

3. From Figure 8.5a, determine K 0.11.4. Compute Tall (0.11)( )(130 pcf)(24 ft)2 410012

lb/ft.

In Figure 7.27 there are ve active layers of rein-forcement. The sixth layer shown in the STABGMmanual and Figure 7.27, at the elevation of the toe,does not cut across any of the slip surfaces shown inFigure 7.28 and therefore does not inuence the factorof safety. The factors of safety computed usingSTABGM and UTEXAS4 are the same whether thisbottom layer of reinforcement is included or not. Thetotal reinforcement force from step 4, 4100 lb/ft,agrees well with the total of 4000 lb/ft provided bythe ve active layers of reinforcement.

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EMBANKMENTS ON WEAK FOUNDATIONS 145

1.4 2.0 3.0 5.0 10.0 20.0

5.16.07.0

10.017.030.0

Ratio B/T Value of N (NAVFAC, 1986)

qult = cNc (F/L2)

q = H (F/L2) = total unit weight of embankment fill (F/L6)

H = embankment height (L)

F =qultq

Firm layer

Weak foundationSaturated claysu = c, u = 0

Internally stablereinforced embankment

Equivalent uniform load

T

B

H

c

c = average undrained strength of foundation (F/L2)

Figure 8.7 Bearing capacity mode of failure of a strongly reinforced embankment on aweak foundation.

5. From Figure 8.5b determine LB /H 0.88, LT /H 0.55.

6. Compute LB (0.88)(24 ft) 21 ft and LT (0.55)(24 ft) 13 ft.

In Figure 7.27, LT LB 20 ft. The results fromFigure 8.5b indicate that the reinforcement could besomewhat shorter at the top of the slope.

EMBANKMENTS ON WEAK FOUNDATIONS

Reinforcement near the base of an embankment can beused to improve stability with regard to spreading ofthe embankment and with regard to shear failurethrough the embankment and foundation. With rein-forcement at the bottom of the embankment, the slopescan be made as steep as for an embankment con-

structed on a rm foundation. The volume of the em-bankment and the total load it imposes on thefoundation can be reduced and its height can be in-creased.

Reinforced embankments have been used at a num-ber of sites where weak foundations posed difcult sta-bility problems, including Almere in the Netherlands(Rowe and Soderman, 1985); Mohicanville Dike 2 inOhio (Duncan et al., 1988; Franks et al., 1988, 1991);St. Alban in Canada (Busbridge et al., 1985; Schaeferand Duncan, 1986, 1987); Hubrey Road in Ontario,Canada (Rowe and Mylleville, 1996); and Sackville,New Brunswick, Canada (Rowe et al., 1996).

Modes of failure. Potential modes of failure of re-inforced embankments on weak foundations have beendiscussed by Haliburton et al. (1978) and by Bonaparteand Christopher (1987). Three possible modes of fail-ure are shown in Figure 8.6.

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2.0

1.8

1.6

1.4

1.2

1.0

0.8

0.6

0.4

0.2

00.020 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20

2.0

1.8

1.6

1.4

1.2

1.0

0.8

0.6

0.4

0.2

00.020 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20

= 18.4 (3:1)' = 30

0.40.2 0.1 0

D /

H

su

(F)()(H)for 3 on 1 slope

Tall()(H2)

= 26.6 (2:1)' = 30

D /

H

Tall()(H2)

su

(F)()(H)

-0.06

0.3

TallH2

for 30TallH2

for 2 on 1 slopeTallH2

00.10.20.30.4

0.14

0.12

0.10

0.08

0.06

0.04

0.02

0

-0.02

-0.04

Adjus

tmen

t Fac

tor,

Tall

15 40352520

30

DH 0.5

DH 1.0

Tall = (Tall) = 30 + Tall

H

D

Embankment

Foundation

(',)

(cu)

(a)

(c)

(b)

Figure 8.8 Stability charts for embankments on weak foundations. (After Bonaparte andChristopher, 1987.)

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1000

980

960

940

920

900

880

150 100 50 0 50 100

Distance from Center Line, ft

Elev

atio

n in

Fee

t

Foundation Clay

Su = 160 psf

31

250 psf360 psfPeat500 psf

360 psf

Slurry Trench Seepage CutoffNew

EmbankmentSteel Mat

= 30, c = 180 psf

250 psf

Su = 1000 psf

Peat

Su = 600 psf

Su = 800 psf

Su = 400 psf

Figure 8.9 Cross section through Mohicanville Dike 2. (After Collins et al., 1982.)

0 10 20 30 40 50 60 70 80Distance from Centerline, ft

0

20

40

Rei

nfor

cem

ent F

orce

, kip

s/ft

30

10

Top of dike at el. 983 ft.

Reinforcement force required for F = 1.3 at end of construction,

Figure 8.10 Required reinforcement force for MohicanvilleDike 2. (After Fowler et al., 1983.)

Figure 8.6a shows the embankment sliding acrossthe top of the reinforcing. This mode of failure is mostlikely if the interface friction angle between the em-bankment and the reinforcement is low, as it may bewith geotextile reinforcement. A wedge analysis canbe used to assess the safety of the embankment withregard to this mode of failure.

Figure 8.6b shows a shear surface cutting across thereinforcement and into the weak foundation. Thismode of failure can occur only if the reinforcementruptures or pulls out. Safety with regard to this modeof failure can be evaluated using circular, wedge, ornoncircular slip surfaces, including reinforcementforces in the analysis as discussed previously.

Figure 8.6c shows large settlement of the embank-ment resulting from excessive elongation of the rein-forcement. This mode of failure can occur if the strainin the reinforcement required to mobilize the reinforce-ment load is too large. Satisfying limit load criterion3, discussed previously, will prevent this type of fail-ure.

Even if an embankment is completely stable inter-nally, it may still be subject to bearing capacity failure,as shown in Figure 8.7. This mode of failure can beanalyzed using bearing capacity theory. If the foun-dation thickness (T) is small compared to the width ofthe equivalent uniform embankment load (B), the valueof the bearing capacity factor Nc increases, as shownin the tabulated values in Figure 8.7 and the factor of

safety with respect to bearing capacity failure also in-creases. Therefore, the shallower is the weak founda-tion, the less likely is the bearing capacity mode offailure.

Bonaparte and Christopher (1987) charts. Prelim-inary estimates of the reinforcement force required fora given factor of safety can be made using the stabilitycharts shown in Figure 8.8, which were developed byBonaparte and Christopher (1987). The terminologyused in these charts is:

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Distance from Centerline, ft

Rei

nfor

cem

ent F

orce

, kip

s/ft

DownstreamUpstream

80 40 0 40 800

8

16

24

32

40

Station 6+55

Computed byfinite-elementanalyses

Figure 8.11 Reinforcement force distribution about the centerline for Station 655. (AfterDuncan et al., 1988.)

Tall

2Hdimensionless reinforcement force coefcient

(8.15)

su

FHdimensionless factored stability coefcient

(8.16)

Tall allowable reinforcement force (F/L)

Tall change in Tall for embankment 30

total unit weight of embankment (F/L3)

H height of embankment (L)

su undrained shear strength of foundation soil(foundation 0)

D foundation depth (L)

slope angle (degrees)

Example. As an example of the use of these charts,consider the Mohicanville Dike No. 2 embankmentshown in Figure 8.9. The Mohicanville project is de-scribed in more detail in the next section. Pertinentparameter values for the embankment and foundation

are H 24 ft, 136 pcf, 32 for the em-bankment ll, embankment slope angle arctan(0.33) 18, average foundation shear strength 700psf, D 80 ft, su /FH 700/(1.3)(136)(24) 0.16for factor of safety on soil shear strength F 1.3.

The cohesion of the embankment ll (c 200 psf)is neglected because the charts in Figure 8.8 were de-veloped for cohesionless embankment ll. The chartsassume that the reinforcement is placed at the bottomof the embankment, where it is most effective in im-proving stability. The charts can be used to determinethe magnitude of the reinforcement force required fora given factor of safety using the following steps:

1. Compute D /H (80 ft) /(24 ft) 3.3.2. Compute

s 700 psfu 0.16FH (1.3)(136 pcf)(24 ft)

3. From Figure 8.8a, estimate 0.4 (the2T /Hallvalue of D /H is above the top of the chart, and

0.4 was estimated by extrapolation).2T /Hall4. From Figure 8.8c, determine 0.01.2T /Hall5. Compute 0.40 0.01 0.39.2T /Hall

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6. Compute Tall (0.39)(136)(24)2 31,000 lb/ft.7. Compute Tlim TallFR (31,000 lb/ft)(1.5)

47,000 lb/ft for FR 1.5.

As shown in the next section, this result is in rea-sonable agreement with the results of detailed studies,indicating that the charts in Figure 8.8 can be used forpreliminary assessment of reinforcement force.

Case history. Mohicanville Dike No. 2 is a rimdike on the Mohicanville Reservoir in Wayne County,Ohio (Duncan et al., 1988; Franks et al., 1988, 1991).Constructed on a weak peat and clay foundation, thedike failed during construction, and for many years thecrest was 22 ft below its design elevation. A cross sec-tion through the dike is shown in Figure 8.9.

After evaluation of a number of alternatives for rais-ing the dike to its design height, it was concluded thatconstruction of a reinforced embankment afforded thebest combination of cost and reliability. Limit equilib-rium analyses and nite element analyses were pre-formed to determine the reinforcing force required forstability of the embankment. It was found that toachieve a factor of safety F 1.3, a reinforcing forceof 30,000 lb/ft was required. The results of equilib-rium analyses are shown in Figure 8.10.

A heavy steel mesh was selected for reinforcement.This mesh has No. 3 mild steel bars spaced 2 in. apartperpendicular to the axis of the dike, welded into amesh with No. 2 bars spaced 6 in. apart parallel to theaxis of the dike. This mesh provided a cross-sectionalarea of about 1 in2 of steel per foot of embankmentlength and a yield force (Tlim) equal to 48,000 lb/ft.This provides a factor of safety on reinforcement ca-pacity, FR Tlim /Tall 48,000/30,000 1.6.

The steel mesh was rolled up after fabrication intorolls containing strips 8 ft wide and 320 ft long. Thesteel yielded in bending, deformed plastically as it wasrolled up, and stayed rolled up without restraint. Therolled strips of mesh were transported on trucks andwere unrolled at the project site using the same equip-ment as that used to roll up the mesh in the fabricatingplant. Each strip was cut into two 160-ft-long piecesthat reached across the full width of the embankment,from upstream to downstream. The strips were draggedinto position on the embankment using a front-endloader and a bulldozer. They were laid on, and werecovered by, about 1 ft of clean sand.

The reinforcing mat was placed at elevation 960 ft,approximately 4 ft above the original ground elevation.In most areas, about 6 to 8 ft of old embankment llwas excavated to reach elevation 960 ft. In one 100-ft-long section of the embankment, where the foun-dation soils were thought to be exceptionally weak, asecond layer of reinforcing was placed at elevation961 ft.

The steel mat was not galvanized or otherwise pro-tected against corrosion. Although the steel reinforce-ment will probably corrode in time, it is needed onlyfor the rst few years of the embankments life. Afterthe foundation gains strength through consolidation,the reinforcement will no longer be required for sta-bility.

The embankment was designed using limit equilib-rium analyses and nite element analyses that modeledconsolidation of the foundation soils as well as inter-action between the embankment and the steel reinforc-ing. The embankment was instrumented to measurereinforcement forces, settlements, horizontal move-ments, and pore pressures. Computed and measuredreinforcement forces at the end of construction areshown in Figure 8.11. It can be seen that the calculatedvalues agree quite well with the measured values. It isworthwhile to note that the nite element analyseswere performed before the embankment was con-structed, and the results shown in Figure 8.11 thereforeconstitute a true prediction of performance, not anafter-the-fact matching of analytical results and eldmeasurements.

This case history indicates that both limit equilib-rium analyses and nite element analyses can be usedto design reinforced embankments on weak founda-tions and to anticipate their performance. In most caseslimit equilibrium analyses can be used as the sole de-sign tool. However, in precedent-setting cases, as Mo-hicanville Dike No. 2 was in the mid-1980s, it isprudent to perform more thorough analyses using thenite element method.

Recapitulation

Reinforcement can be used to improve the stabil-ity of slopes and embankments, making it possibleto construct slopes and embankments steeper andhigher than would otherwise be possible.

Reinforced slopes and embankments can be ana-lyzed using the procedures described in Chapter6 by including the reinforcement forces as knownforces in the analyses. The amount of force re-quired to achieve a target value of factor of safetycan be determined using repeated trials.

Two methods have been used for limit equilibriumanalyses of reinforced slopes: method A, in whichallowable reinforcing forces are specied, andmethod B, in which ultimate reinforcing forces arespecied. Method A is preferable, because it pro-vides a means of applying different factors ofsafety to soil strength and reinforcing force, whichhave different sources of uncertainty and differentamounts of uncertainty associated with their val-ues.

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The principal types of reinforcing materials thathave been used for slopes and embankments aregeotextile fabrics, geogrids, steel strips, steelgrids, and high-strength steel tendons.

The long-term capacity of reinforcement, denotedhere as Tlim, depends on tensile strength, creepcharacteristics, installation damage, durability,pullout resistance, and stiffness.

The allowable load assigned to reinforcing mate-rials should include a factor of safety, as indi-cated by the expression Tall , whereT /Flim RTall is the allowable force, Tlim is the capacity ofthe reinforcement to carry long-term loads,and FR is the reinforcement factor of safety.The value of FR should reect the level of uncer-tainty in the analyses and the consequences offailure.

Designing reinforced slopes is facilitated greatlyby slope stability charts of the type developed bySchmertmann et al. (1987), which are shown inFigure 8.5a and b.

Potential modes of failure of reinforced embank-ments on weak foundations include sliding acrossthe top of the reinforcing, shear through the re-inforcement and into the weak foundation, largesettlement of the embankment resulting from ex-cessive elongation of the reinforcement, and bear-ing capacity failure.

Preliminary estimates of the reinforcement forcerequired for a given factor of safety can be madeusing the stability charts shown in Figure 8.8.

The Mohicanville Dike No. 2 case history showsthat both limit equilibrium analyses and nite el-ement analyses can be used to design reinforcedembankments on weak foundations and to antic-ipate their performance. Finite element analysesshould be performed for precedent-setting appli-cations.

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Front MatterTable of Contents8. Reinforced Slopes and Embankments8.1 Limit Equilibrium Analyses with Reinforcing Forces8.2 Factors of Safety for Reinforcing Forces and Soil Strengths8.2.1 Method A Equations8.2.2 Method B Equations

8.3 Types of Reinforcement8.4 Reinforcement Forces8.4.1 Criterion 1: Creep, Installation Damage, and Deterioration in Properties over Time8.4.2 Criterion 2: Pullout Resistance8.4.3 Criterion 3: Reinforcement Stiffness

8.5 Allowable Reinforcement Forces and Factors of Safety8.6 Orientation of Reinforcement Forces8.7 Reinforced Slopes on Firm Foundations8.7.1 Schmertmann et al. (1987) Charts

8.8 Embankments on Weak Foundations

Appendix: Slope Stability ChartsReferencesIndex

duncan

Documents

reinforced soil walls

desired factor of safety

concrete walls

marketmse walls

stabilizedearth walls

reinforcement forces

reinforced embankments

slopes steeper thanwould