Combined Effects of Reinforcement and Pvd on Embankment Performance

Combined effects of reinforcement andprefabricated vertical drains on embankmentperformance

Allen Lunzhu Li and R. Kerry Rowe

Abstract: The behaviour of geosynthetic-reinforced embankments constructed over soft cohesive soils installed withprefabricated vertical drains (PVDs) is investigated by numerically examining an embankment constructed overdifferent foundation soils. The partial consolidation during embankment construction, the consequent shear strengthgain of the foundation soil, and the effect of the use of reinforcement on the mobilization of shear strength areexamined. It is shown that the combined use of reinforcement and PVDs can significantly increase embankmentstability and potentially allow the rapid construction of higher embankments than could be achieved with either methodof soil improvement alone. Construction rate and spacing of PVDs can significantly affect the degree of consolidationat the end of construction and the stability of the embankment. For the situation examined, the effect of well resistanceof typical vertical drains is insignificant. A relatively simple method for calculating the degree of consolidation and thestrength gain of the foundation soil during construction is evaluated based on finite element results and is shown to bereasonably conservative. A design procedure is proposed to combine the design of reinforcement and PVDs.

Key words: soft clay, prefabricated vertical drain, reinforcement, embankment stability, consolidation, strength gain.

Rsum : Le comportement de remblais arms de gosynthtiques construits sur des sols mous cohrents quips dedrains verticaux prfabriqus (PVD) est tudi au moyen dun examen numrique dun remblai construit sur diffrentssols de fondation. On examine la consolidation partielle durant la construction du remblai, le gain de rsistance au ci-saillement du sol de fondation qui en dcoule, et leffet de lutilisation de larmature sur la mobilisation de la rsis-tance au cisaillement. Il est montr que lutilisation combine de larmature et des PVD peut augmenterapprciablement la stabilit du remblai, et permettre potentiellement la construction rapide de remblais plus hauts queceux qui pourraient tre raliss avec seulement lune ou lautre des mthodes damlioration. La vitesse de construc-tion et lespacement des PVD peuvent influencer apprciablement le degr de consolidation la fin de la constructionet la stabilit du remblai. Dans le cas tudi, leffet de la rsistance des drains verticaux est ngligeable. Une m-thode relativement simple de calcul du degr de consolidation et du gain de rsistance du sol de fondation durant laconstruction est value en partant des rsultats dlments finis et sest avre tre raisonnablement conservatrice. Onpropose une procdure de calcul pour combiner la conception de larmature et des drains PVD.

Mots cls : argile molle, drains verticaux prfabriqus, armature, stabilit de remblai, consolidation, gain de rsistance.

[Traduit par la Rdaction]

Li and Rowe 1282

1. Introduction

Stability and the time required for consolidation are twomajor considerations in the design and construction of em-bankments over soft cohesive foundations having low bearingcapacity and low hydraulic conductivity. Geosynthetic rein-forcement has been widely used to improve the stability ofembankments on soft clay soils (Humphrey and Holtz 1987;Fowler and Koerner 1987; Rowe and Soderman 1987a; Rowe

1997). In parallel, vertical drains have been used to shortenconsolidation time of thick soft deposits by providing shorthorizontal drainage paths (Jamiolkowski et al. 1983). Due tothe advantages of prefabricated vertical drains (PVDs) interms of cost and ease of construction, they have almost en-tirely replaced conventional sand drains as vertical drains(Holtz 1987). The use of geosynthetic reinforcement in com-bination with prefabricated vertical drains has the potential toallow the cost-effective construction of substantially higherembankments in considerably shorter time periods than con-ventional construction methods (e.g., Lockett and Mattox1987; Bassett and Yeo 1988; Schimelfenyg et al. 1990).

There has been considerable research examining the be-haviour of reinforced embankments over soft foundations interms of field behaviour (e.g., Rowe et al. 1984; Fowler andEdris 1987), in terms of theoretical behaviour as predictedusing finite element methods (e.g., Rowe and Soderman1987b; Hird and Kwok 1990; Chai and Bergado 1993; Rowe

Can. Geotech. J. 38: 12661282 (2001) 2001 NRC Canada

1266

DOI: 10.1139/cgj-38-6-1266

Received January 10, 2001. Accepted April 25, 2001.Published on the NRC Research Press Web site athttp://cgj.nrc.ca on January 10, 2002.

A.L. Li and R.K. Rowe.1 Department of Civil Engineering,Ellis Hall, Queens University, Kingston, ON K7L 3N6,Canada.

1Corresponding author (e-mail: [email protected]).

I:\cgj\Cgj38\Cgj06\T01-059.vpFriday, December 21, 2001 1:38:56 PM

Color profile: Generic CMYK printer profileComposite Default screen

and Li 1999), and as observed in centrifuge model tests(e.g., McGown et al. 1981; Sharma and Bolton 1996). How-ever, there has been a paucity of theoretical analyses exam-ining the combined effects of reinforcement, vertical drains,and rate of construction on the embankment behaviour.Using a finite element technique, this paper examines com-bined effects of reinforcement and vertical drains on em-bankment stability and consolidation.

The design of reinforced embankments and vertical drains isusually treated separately in current design methods. A stabilityanalysis is used in the design of reinforced embankments (e.g.,Jewell 1982; Rowe 1984) and a consolidation analysis is usedin the design of PVDs (e.g., Rixner et al. 1986; Holtz et al.1991). The limit equilibrium method (Jewell 1982; Ingold1982; Fowler and Koerner 1987; Bonaparte and Christopher1987; Leshchinsky 1987; Mylleville and Rowe 1988; Koerner1994; and others) has been widely adopted in practice for cal-culating the stability of reinforced embankments under un-drained conditions. Although the assumption of undrained

conditions is conservative for embankments on typical softclays deposits, it neglects the significant effect of consolidationthat may occur at typical rates of construction when the soil isoverconsolidated during early stages of loading (Leroueil et al.1978; Rowe et al. 1995; Li and Rowe 1999a; Leroueil andRowe 2001), and especially when there are short horizontaldrainage paths due to installation of vertical drains (Li andRowe 1999b). However, the effects of partial drainage due tovertical drains on the stability of reinforced embankments havegained little attention and are not adequately addressed in cur-rent design methods. This paper shows that the effects of thereinforcement and PVDs are interrelated and should not betreated separately in design.

The present study considers the effects of both the reinforce-ment and the vertical drains simultaneously and investigatesthe combined benefits resulting from the partial consolida-tion of foundation soil due to PVDs and the basal reinforce-ment. The gain in shear strength of foundation soils due topartial consolidation during embankment construction isevaluated. The effect of the PVD spacing and well resistanceis examined using a typical range of values. The effect ofconstruction rate and stage construction sequence on stabil-ity and consolidation is also examined. The mobilized rein-forcement strain is addressed for reinforced embankmentsover soft plastic soils under partially drained conditions. Anew procedure which considers the effects of both reinforce-ment and vertical drains on the stability of embankments isproposed.

2. Problem considered and numerical model

The finite element program AFENA (Carter and Balaam1990) was modified to incorporate soil reinforcement inter-action, an elliptical cap soil model (Chen and Mizuno 1990;Rowe and Li 1999) coupled with Biot consolidation theory(Biot 1941), and a drainage element (Russell 1990) to modelthe well resistance of vertical drains.

This present study examines the construction of a highwayembankment on a 15 m deep soft clay foundation underlainby a relatively rigid and permeable layer, as shown in Fig. 1.The finite element mesh (see Li and Rowe 2000) involved1815 linear strain triangular elements, with 4003 nodes usedto discretize the embankment and foundation soils. Two-

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Li and Rowe 1267

Fig. 1. The reinforced embankment, soft foundation, and vertical drains. GWT, groundwater table.

Fig. 2. Preconsolidation pressure and initial vertical effectivestress profiles of soils A and B.



noded bar elements were used for modeling the reinforcementand two-noded joint elements were used for both the embank-ment fill reinforcement interface and the embankment fill foundation interface (Rowe and Soderman 1987a). Thecentreline of the embankment and the far-field lateral bound-ary were taken to be smooth and rigid, with the lateral bound-ary located 100 m from the centreline. The bottom boundaryof the finite element mesh was assumed to be rough and rigid.Consideration was given to the effect of a series of fully pen-etrating prefabricated vertical drains in a square pattern atthree different spacings, S = 1, 2, and 3 m, equivalent to spac-ings of 1.07, 2.14, and 3.21 m, respectively, for a triangularpattern. Zero excess pore pressure was assumed along thedrains to simulate ideal drains without well resistance. Two-noded drainage elements were used for vertical drains whenthe well resistance was considered. Embankment constructionwas simulated by turning on the gravity of the embankmentin 0.75 m thick lifts at a rate corresponding to an embank-ment construction rate.

3. Model parameters

The parameters used in the finite element analyses re-ported in the subsequent sections of the paper are describedin the following subsections.

3.1. Selection of foundation soil propertiesTwo soft foundations, denoted soils A and B, were exam-

ined. Soil A has a liquid limit of 76% and a plasticity indexof 40%, and soil B has a liquid limit of 48% and a plasticityindex of 30%. The vertical preconsolidation pressure, p ,profiles are shown in Fig. 2. Both clays were slightlyoverconsolidated, with overconsolidation ratios (OCRs) of2.61.1 and 2.91.1, respectively, below the first 2 m. Soil Bhas a 2 m thick crust. The soil model parameters are summa-rized in Table 1. The first soil profile has an undrained shearstrength, suo, of 5 kPa at the surface, increasing with depth ata rate, c, of 1.5 kPa/m; the second soil profile has an un-drained shear strength, suo, of 20 kPa at the surface, decreas-ing to 10 kPa at 2 m depth, and then increasing at a rate, c,

of 2.0 kPa/m. The stress ratio of undrained shear strength(under plane strain conditions) to preconsolidation pressure, = su/p , is 0.33 and 0.31 for soils A and B, respectively,in normally consolidated states.

The vertical hydraulic conductivity, kv, of soft clays wastaken to be a function of void ratio, e:

[1] k ke eCk

v voo=

exp

where kvo is the reference hydraulic conductivity at the refer-ence void ratio, eo; and Ck is the hydraulic conductivitychange index. The values for these parameters and the ratioof horizontal hydraulic conductivity, kh, to vertical hydraulicconductivity, kv, are summarized in Table 1. The initial aver-age consolidation coefficient for soils A and B, respectively,is about 1.3 and 3.4 m2/a in a normally consolidated stateand 13.4 and 20.3 m2/a in an overconsolidated state.

3.2. Embankment fill parameters and construction ratesThe embankment fill was assumed to be a purely fric-

tional granular soil with a friction angle of 37 and a bulkunit weight of 20 kN/m3. The dilatancy angle was takento be 6 based on the equation proposed by Bolton (1986).The nonlinear elastic behaviour of the fill was modelled us-ing Janbus (1963) equation:

[2]EP

KP

m

a a

=

3

here E is the Youngs modulus of the soil; Pa is the atmo-spheric pressure; 3 is the minor principal stress; and K andm are material constants selected to be 300 and 0.5, respec-tively, for typical fill materials.

A range of embankment construction rates (0.58 m/month)is examined in this paper.

3.3. Interface parameters and reinforcement stiffnessRigidplastic joint elements (Rowe and Soderman 1987a)

were used to model the fillreinforcement and fillfoundation

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1268 Can. Geotech. J. Vol. 38, 2001

Soil A Soil B

Failure envelope slope, MN / C* 0.874 0.910

Friction angle (normally consolidated) () 27 28Failure envelope slope, MO/C

* 0.63 0.75

Cohesion intercept for overconsolidated clay, ck (kPa)* 2.74.7 3.46.3Aspect ratio, R 0.70 1.25Compression index, 0.3 0.15Recompression index, 0.030 0.025Coefficient of earth pressure at rest, K0 0.6 0.6Poissons ratio, 0.35 0.35Average unit weight (kN/m3) 15.2 16.7Reference hydraulic conductivity, kvo (m/s) 110

9 1109

Reference void ratio, eo 2.5 1.5Hydraulic conductivity change index, Ck 0.5 0.5Ratio of horizontal to vertical hydraulic conductivity, kh/kv 3 3

*In m 2 2J stress space, where m is the mean effective stress and J2 is the second invariant of thedeviatoric stress tensor.

Table 1. Soil model parameters.



interfaces. The fillreinforcement interface was assumed tobe frictional, with = 37 and cohesion intercept c = 0. Thefillfoundation interface had the same shear strength as thatof the foundation soil at the ground surface. Reinforcementwith tensile stiffness, J, varying from 250 to 2000 kN/m wasexamined.

3.4. Prefabricated vertical drainsThe prefabricated drains modelled had a typical rectangular

cross section of 100 mm 4 mm (Holtz 1987) and weretaken to be equivalent to a circular drain having a diameter dwof 66 mm based on Kjellman (1948): dw = 2(b + t)/, where band t are the width and thickness of the drain, respectively.The drain spacing, S, of 1, 2, and 3 m examined is within thetypical spacing range used in practice (Holtz 1987). The ef-fective diameter of drain influence was taken to be De = 1.13Sfor a square configuration and De = 1.05S for a triangularconfiguration (Rixner et al. 1986). The diameter of the smearzone, ds, was assumed to be ds = 2dm (Rixner et al. 1986),where dm is the diameter of a circle with an area equal to thecross-sectional area of the mandrel. The mandrel dm was as-sumed to be two times the drain diameter, dw. The hydraulicconductivity of soil in the smear zone was assumed to be iso-tropic and the same as vertical hydraulic conductivity (i.e.,33% of the normal horizontal hydraulic conductivity). Bothan ideal drain condition without well resistance and non-idealdrains with well resistance were examined.

Typical values of the discharge capacity, qw, of many pre-fabricated drains are from 500 to 100 m3/a (Holtz et al.1991). As long as the discharge capacity is greater than 100150 m3/a under the confining pressures acting on the drain,there should be no significant decrease in the consolidationrate (Holtz 1987; Holtz et al. 1991). However, discharge ca-pacity will decrease due to drain bendingfolding and deteri-oration of the drain filter. Based on a review of test resultsfor PVD products, the discharge capacity of some PVDproducts can be as low as 5100 m3/a under high confiningpressures and low hydraulic gradients (Rixner et al. 1986;Holtz et al. 1991). Within this range, well resistance valuesof 5, 10, 25, 50, and 100 m3/a for PVDs were examined.

4. The equivalent vertical drain in a planestrain problem

Strictly speaking, the analysis of a system involving dis-crete vertical drains should be conducted with a fully three-dimensional analysis, whereas most embankments are mod-elled for plane strain conditions. To avoid the need for a fullthree-dimensional analysis, some approximations are re-quired to consider the vertical drains in a plane strain analy-sis. A number of authors (e.g., Cheung et al. 1991; Hird etal. 1992; Chai et al. 1995) have shown that vertical drainscan be effectively modelled by using appropriate approxi-mate methods to represent the typical arrangement of verti-cal drains in plane strain finite element analyses. Thetechnique for matching a plane strain vertical drain systemwith an axisymmetric vertical drain system proposed byHird et al. (1992, 1995) was adopted for the analyses re-ported in this paper. The theories on radial consolidation arewell developed for different boundary conditions (Barron1948; Kjellman 1948; Yoshikuni and Nakanodo 1974;

Hansbo 1981; Zeng and Xie 1989). Hansbos (1981) solu-tion provides a relatively simple means of considering theeffect of smear and well resistance, compares well with thesolutions of Barron (1948) and Zeng and Xie (1989), andhas gained wide acceptance (Jamiolkowski et al. 1983). Asdescribed here, this solution was used to check the finite ele-ment approximation adopted in the present study. For a cy-lindrical unit cell of soil influenced by a single drain underan instantaneous loading, the average degree of consolida-tion Uh on a horizontal plane at depth z and time t is givenby Hansbo as

[3] UT

hh=

1

8exp

where Th = Cht /4R2; = ln(n /s) + (k /ks)ln(s) 3/4 + z(2l

z)k /qw; ratio n = R/rw; ratio s = rs /rw; qw = kwrw2; Ch is thehorizontal consolidation coefficient; k, ks, and kw are the hy-draulic conductivity of horizontal direction, soil in the smearzone and vertical drains, respectively; rw, rs, and R are theradius of the vertical drains, smear zone, and influence zone,respectively; and l is the length of the vertical drain. Forplane strain conditions, Hird et al. (1992) show that Th =Cht /4B

2 and = 2/3 + 2z(2l z)k /BQw, where Qw is theequivalent discharge capacity, and B is the half width of theinfluence zone for the plane strain unit cell. To match the av-erage degree of consolidation at any time and any depth inthese two cells, one can simply set Uhpl for a plane straincondition equal to Uhax for an axisymmetric condition (i.e.,Uh in eq. [3]):

[4] Uhpl = Uhax

This can be achieved by any one of three methods: geomet-ric matching, permeability matching, and a mix of geometricand permeability matching. Permeability matching involvingadjusting the hydraulic conductivity of soil for plane strainconditions (kpl) and putting B = R is adopted in this paper.To satisfy eq. [4], the hydraulic conductivity and well resis-tance matching requirements are

[5] kk

ns

kk

s

plax

ax

s

=

+

2

334

ln ln( )

where kax is the hydraulic conductivity for the axisymmetriccondition, and

[6] QR

qw w=

2

For the vertical drain used in this paper having width b =100 mm, thickness t = 4 mm, and ratio s = 4, the calculatedratio kpl /kax is equal to 0.137, 0.120, and 0.112, respectively,for the influence radius R of 0.563, 1.125, and 1.688 m (i.e.,S = 1, 2, and 3 m for a square configuration). Although thematching technique using Hansbos (1981) solution is basedon the assumption of an elastic soil and constant hydraulicconductivity, Hird et al. (1995) show that the matching tech-nique also works well for consolidation involving plastic soiland variable hydraulic conductivity. To examine the accu-racy of the matching procedure, a finite element analysis

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Li and Rowe 1269



was conducted using both soil profiles for a unit cell ofvertical drains with R = 1.125 m, and it was found that therewas an excellent match between the plane strain and axi-symmetric results (Li 2000).

5. Results and discussion

5.1. Effects of PVDs and reinforcement on embankmentstability

The soil improvement involving the combined use of bothPVDs and geosynthetic reinforcement has numerous advan-tages. The benefits of the use of PVDs are twofold (Holtz etal. 1991). The first is to accelerate the consolidation byshortening the drainage path and taking advantage of anynaturally higher horizontal hydraulic conductivity of founda-tion soils. The second is to improve embankment stabilitydue to the strength gain in foundation soils from the increasein effective stress associated with consolidation. Geo-synthetic reinforcement provides additional embankmentstability and tends to force the potential failure surfacethrough soil with higher shear strength at depth in founda-tion deposits exhibiting a shear strength profile increasingwith depth (Rowe and Soderman 1987b).

To study the combined effect of PVDs and reinforcement,finite element analyses were conducted to simulate embank-ment construction over soils A and B at a construction rate,CR, of 2 and 4 m/month, respectively. Figure 3 shows thevariation of net embankment height (i.e., embankment fillthickness minus maximum settlement) with fill thickness. It

is assumed here that the reinforcement could sustain thestrains developed without breakage (the implication of thisassumption will be examined shortly). The foundation soilwas taken to be rate insensitive, isotropic, and not strainsoftening. For soil A, the unreinforced embankment couldonly be constructed to a height of 2.85 m. If reinforcementwith tensile stiffness J = 250 kN/m was used, the failureheight increased to 3.38 m. If reinforcement stiffness wasgreater than 500 kN/m, the embankment did not fail becausethe reinforcement provided enough resistance to allow therate of shear strength gain of the foundation soil duringconstruction to be fast enough to facilitate the rate of em-bankment loading. For soil B the unreinforced embankmentfailure height was 5.18 m. If reinforcement stiffness, J, wasequal to or greater than 500 kN/m, the embankment did notfail due to bearing capacity failure at a construction rate of4 m/month.

Even though, for reinforcement with J 500 kN/m, thefoundation did not experience a bearing capacity failure atthese construction rates, a large yielded zone (i.e., where thesoil was close to critical state) was found to be progressivelygrowing beneath the embankment during construction. Con-sequently, the large plastic deformations, called progressiveplastic deformations here, were developed in foundationsoils during embankment construction. Due to these progres-sive plastic deformations, the reinforcement strain developedduring construction could be much higher than the allowablestrain of 47% (after reduction factors for creep, installationdamage, and durability are applied; Industrial Fabrics Asso-ciation International 1999). Under these circumstances, themaximum height to which the embankment could be con-structed would be limited by the allowable reinforcementstrain for a given type of reinforcement being considered.The slippage between soil and reinforcement was not ob-served in the finite element simulations for the cases exam-ined. The cases with significantly weak soil-reinforcementplanes where the slippage occurs before the allowable rein-forcement strain is mobilized are not considered herein.

Figure 4 shows the embankment height corresponding tothe tensile force mobilized at allowable reinforcement strainsranging from 4 to 7% for reinforcement with stiffnessesranging from 250 to 2000 kN/m and embankments con-structed over both foundation soils A and B, where zeroforce corresponds to unreinforced cases. The embankmentheight that is limited by the allowable reinforcement strainsis called the maximum height or maximum embankmentheight here. For the unreinforced cases, the height corre-sponds to failure height. The use of reinforcement signifi-cantly increased the maximum embankment heightcompared with the unreinforced case for both soil profiles.For example, using a reinforcement with J = 2000 kN/m andan allowable strain of 6%, the maximum heights, H, of 4.75and 6.88 m were 67 and 33% above the failure height for anunreinforced embankment for soil profiles A and B, respec-tively. Figure 4 also shows the maximum height of an em-bankment over a foundation under undrained conditions forreinforcement having an allowable strain of 6% and stiffnessranging from 500 to 2000 kN/m. Comparing the undrainedresults with the partially drained results, it is evident that theshear strength gain during embankment construction arisingfrom the use of vertical drains can result in a significant in-

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Fig. 3. The variation of net embankment height with fill thickness.



crease in maximum embankment height. For example, usinga reinforcement with J = 2000 kN/m and an allowable strainof 6%, the maximum heights, H, for the partially drainedcases with PVDs were 36 and 45% above that of the un-drained cases for soils A and B, respectively.

5.2. Effects of embankment construction rate andpartial consolidation during construction

Figure 5 shows the variation of maximum height withconstruction rate for both reinforced (with J = 1000 kN/m)and unreinforced embankments. It is evident that the em-bankment stability was sensitive to the construction rates ex-amined. It was found that there was a threshold constructionrate for the unreinforced embankment below which the em-bankment would not fail due to bearing capacity failure offoundation soil. As shown in Fig. 5, the threshold rate wasabout 0.5 and 2 m/month for soils A and B, respectively.This implies that if construction was controlled at a rateslower than the threshold rate, reinforcement was not neededto maintain embankment stability. The use of reinforcementcould efficiently increase embankment stability only whenthe construction rate was greater than the threshold rate. Theincrease in stability with the decrease in construction rate re-sulted from the rate of consolidation and the consequentstrength gain at the end of construction. The rate of consoli-dation of the foundation soil at the end of construction canbe represented in terms of the average degree of consolida-

tion, U , below the embankment centre. Figure 5 shows thevariation of U at the end of construction with constructionrate for two unreinforced embankments (i.e., H = 2.6 and4.5 m) over soils A and B. The degree of consolidation atthe end of construction significantly increased with a de-crease in construction rate when the rate was approaching orslower than the threshold rate.

Figure 5 also shows the unreinforced embankment failureheights: (i) Hfa, using the initial undrained shear strengthprofiles of the soils (see Fig. 2); and (ii) Hfb, using thestrength profiles of soils in normally consolidated states (as-suming that current effective vertical stresses were equal topreconsolidation pressures), calculated by su = p , wherep is the initial preconsolidation pressure and is 0.33 and0.31 for soils A and B, respectively (see Sect. 3.1.). Both Hfaand Hfb were calculated using limit equilibrium methods. Itis noted that when the construction rate increased, the de-crease in failure height of the unreinforced embankment di-minished to an asymptotic value Hfb corresponding to theundrained shear strength of soils in normally consolidatedstates, su = p .

For reinforced embankments, the maximum height fol-lowed the same trend with construction rate as that of thefailure height for unreinforced embankments; however, themaximum height was also governed by allowable reinforce-ment strains. At lower construction rates, the embankmentstability was governed by the partial consolidation duringconstruction. At higher construction rates, the reinforcementdid significantly increase the maximum embankment height.

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Li and Rowe 1271

Fig. 4. The maximum height H of reinforced embankments withdifferent reinforcement stiffnesses and allowable strains.

Fig. 5. The effect of construction rate on the stability ofunreinforced and reinforced (J = 1000 kN/m) embankments.



Figure 6 shows the variation of foundation deformationand reinforcement strain with construction rate for a 3.5 mhigh embankment constructed over foundation soil A. Aslower construction rate resulted in less foundation shear de-formations and reinforcement strains, especially when theconstruction rate was slower than 2 m/month for this partic-ular case. Figure 6 indicates that the excessive horizontal de-formations of foundation soils, heave at the embankmenttoe, and reinforcement strains can be prevented by control-ling the construction rate.

5.3. Effects of reinforcement stiffness for embankmentsconstructed at different rates

The reinforcement stiffness can be an important factor af-fecting embankment stability under partially drained conditionswhen the construction rate is faster than the threshold rate, asshown in Fig. 7. The maximum height corresponding to an al-lowable maximum reinforcement strain of 5% is plotted againstreinforcement stiffness for different construction rates. The re-sults plotted for J = 0 correspond to the unreinforced case. It isevident that above construction rates of 0.5 and 2 m/month forsoils A and B, respectively, stiffer reinforcement will permit theconstruction of a higher embankment.

5.4. Effects of spacing of vertical drainsFrom the classical consolidation theory, it is well known

that the rate of consolidation is inversely proportional to thesquare of the length of the drainage path. Three configura-

tions of PVD system with spacing of 1, 2, and 3 m were ex-amined to investigate the effect of spacing on embankmentstability over soil A. The threshold construction rate forunreinforced embankments was calculated to be 2, 0.5, and0.2 m/month for PVD spacings of 1, 2, and 3 m, respec-tively. Figure 8 shows the variation of maximum embank-ment height for an allowable strain of 5%, H = 5%, withreinforcement stiffness at different construction rates for soilA with PVD spacings of 1 and 3 m. Comparing the embank-ment heights for S = 1, 2, and 3 m (Figs. 7, 8), the improve-ment in stability due to a change in spacing from 2 m to 1 mwas much greater than that observed where the spacing waschanged from 3 m to 2 m. For example, at 4 m/month andJ = 2000 kN/m, H = 5% is 5.71, 4.12, and 3.96 m at spacingsof 1, 2, and 3 m, respectively. The reason for this is that thevertical drain spacing significantly influences the degree ofconsolidation at the end of embankment construction, andconsequently the shear strength gain during construction.For example, at the end of construction of a 3.75 m high em-bankment on foundation soil A at a rate of 4 m/month, theaverage degree of consolidation under the embankment cen-tre was about 37, 24, and 21% for spacings of 1, 2, and 3 m,respectively.

5.5. Effects of well resistance of vertical drainsFigure 9 shows the variation in maximum height for the

range of discharge capacities of PVDs with a 2 m spacing

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Fig. 6. The effect of construction rate on the deformations offoundation soil A and the maximum reinforcement strain.

Fig. 7. The variation of the maximum height H = 5% with rein-forcement tensile stiffness at different construction rates and re-inforcement stiffnesses.



for embankments constructed at 2 m/month over soil A and4 m/month over soil B. When the discharge capacity was10 m3/a (or greater), for soil A, the effect of well resistanceon embankment stability was insignificant. The effect wassomewhat greater for the stiffer soil B, but still small for dis-charge capacities as low as 10 m3/a. For a discharge capacitygreater than 50 m3/year, there was no practical effect of wellresistance on the maximum embankment height for eithersoil A or soil B. The effect of well resistance on embank-ment stability can be explained by examining the degree ofconsolidation at the end of construction. Figure 9 also showsthe average degree of consolidation at the end of construc-tion of 3.75 and 5.5 m high embankments over soils A andB, respectively. The variation of maximum height with dis-charge capacity is reflected by the variation of consolidation.Typical values of the discharge capacity of many PVDs areover 50 m3/a and around 100500 m3/a (Rixner et al. 1986;Holtz et al. 1991). Therefore, the well resistance of manyPVD products theoretically does not affect the stability ofembankments for the typical soil profiles examined as indi-cated in Fig. 9. However, the effect of well resistance needsto be further investigated when drains are very long and lat-eral stresses are high, or if the drains are susceptible to fold-ing and (or) deterioration.

5.6. Effects of two-stage constructionSince shear strength gain may be significant in relatively

short periods of time after construction due to the fast rate ofconsolidation when vertical drains are used, stage construc-

tion methods are often employed to construct an embank-ment to the design grade. In this section, stage constructionmethods were simulated for reinforced embankments (withreinforcement stiffness values between 250 and 2000 kN/m)constructed over soils A and B. During the first stage, thereinforced embankments were constructed to a height of 2 mover soil A at a rate of 2 m/month and a height of 3.5 mover soil B at a rate of 4 m/month. Figure 10 shows the vari-ation in maximum embankment height H at the secondstage with mobilized tensile force at different allowablestrains for soils A and B.

For soil A, the second stage was initiated after 4 monthsof consolidation to reach an average degree of consolidationof 62%. The average increase in maximum height of rein-forced embankments related to one-stage construction wasabout 0.64 m, which was not particularly significant becausethe increase of shear strength of the foundation soil couldnot be fully mobilized due to the constraint of the allowablereinforcement strains. However, if one was to build a higherembankment at the first stage for the case with a stiffer rein-forcement, the benefit of two-stage construction would bemore pronounced due to the greater shear strength gain ofthe foundation soil under the higher embankment load. Forexample, if a reinforced embankment (with J = 2000 kN/mand all = 5%) was constructed to H = 3.5 m over soil A dur-ing the first stage, after 4 months of consolidation, the em-bankment could be constructed to a maximum height of5.63 m during the second stage, with a net increase of1.22 m (an increase of 28%) relative to the maximum heightof one-stage construction.

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Li and Rowe 1273

Fig. 8. The effect of PVD spacing on the embankment stability. Fig. 9. The effect of discharge capacity on the embankment sta-bility and consolidation at the end of construction.



For soil B, the second stage was initiated after 1 monthof consolidation to reach an average degree of consolida-tion of 55%. The maximum height of reinforced embank-ments during the second stage is shown in Fig. 10. Forreference, the failure height of the unreinforced embank-ment was 6.4 m and represented the lowest maximumheight. The increase in embankment height relative to thatfor one-stage construction was in the range of 0.81.2 mfor the reinforcement stiffness values and allowable strainsconsidered. An alternative to the use of stiff reinforcementto allow construction to a greater embankment heightwould be to achieve a greater strength gain of the founda-tion soil by extending the time for consolidation betweenconstruction stages. For example, if the stoppage at H =3.5 m was extended from 1 month to 4 months, the calcu-lated unreinforced embankment failure height was over9.5 m during the second stage of construction. Therefore,reinforcement was not necessary to achieve a design gradeequal to or below 7.0 m in terms of stability (with a mini-mum factor of safety of 1.3). Nevertheless, the use of rein-forcement was found to reduce the shear deformation offoundation soils.

Similar to the case of one-stage construction (see Fig. 4),Fig. 10 shows that during two-stage construction the higherthe reinforcement force, the higher the allowable strain, andthe greater the maximum embankment height that can beachieved.

From the foregoing findings, it is evident that for a two-stage construction sequence it is desirable to build the em-bankment as high as possible during the first stage, by usingreinforcement, so that the maximum amount of strength gainof the foundation soil can occur during the construction stop-page. Furthermore, as indicated by Li and Rowe (1999a), theuse of reinforcement can also possibly reduce the number ofstages required for a multistage construction sequence. Tosafely construct an embankment to a design grade andachieve a maximum degree of consolidation within a limitedperiod of time, it is necessary to have a design method thatcan readily accommodate the change of the configuration ofboth reinforcement and PVDs and construction sequences.

5.7. Reinforcement strainFigure 11 shows the variation of maximum reinforce-

ment strain with embankment fill thickness for both one-stage and two-stage construction, assuming that the rein-forcement could sustain the strain developed without break-age. In Fig. 11, H1 is the fill thickness at the first stage ofconstruction (i.e., 2 and 3.5 m over soils A and B, respec-tively). During the early stage of construction, the maxi-mum reinforcement strain for all cases was small andwithin 1% when the foundation deformed, mainly elasti-cally. But then it increased significantly when the embank-ment loading caused large plastic deformations of thefoundation as the fill thickness exceeded the failure heightof the unreinforced embankment (i.e., 2.85 and 5.18 m for

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Fig. 10. The variation of embankment maximum height with re-inforcement tensile force for two-stage construction.

Fig. 11. The variation of maximum reinforcement strain with fillthickness.



soils A and B, respectively). After the initial, mainly elas-tic, deformations were developed, the stiffer reinforcementresulted in smaller reinforcement strains for a given fillthickness. This effect of reinforcement stiffness was moresignificant for embankments over soil B (the stronger soil)than for embankments over soil A. In all cases, the em-bankment did not fail due to bearing capacity failure offoundation soil, even though very high reinforcement strainwas developed. The increase in embankment height at veryhigh reinforcement strain levels was attributed to the in-crease in shear strength of the foundation soil due to partialconsolidation during construction.

Comparing two-stage construction with one-stage con-struction for the cases with reinforcement stiffness J =2000 kN/m, it is evident that the additional strength gainresulting from consolidation during the stoppage causedless reinforcement strain in the second stage for a givenembankment height. It was also found that the higher em-bankment during the first stage and the longer the stop-page, the greater the stiffening effect and the lower thestrain that was developed when a significant amount of ad-ditional fill was placed. For clarity, the results from thecases with a higher embankment and longer stoppage at thefirst stage are not plotted in Fig. 11.

It is evident from Fig. 11 for both one-stage and two-stageconstruction that the stiffer the reinforcement and (or) thehigher allowable reinforcement strain, the greater the fillheight that can be achieved. Figure 11 also shows that dur-ing the stoppage between the two construction stages theconsolidation settlement of the foundation soils under therelatively low embankments has an insignificant effect onthe reinforcement strain. However, Li (2000) has shown thatthe long-term differential settlement of foundation soils un-der relatively high embankments can significantly increasethe mobilized reinforcement strain and force.

5.8. Consolidation during embankment constructionand shear strength gain

Significant consolidation of foundation soils may occur dur-ing embankment construction because of the high consolida-tion coefficient of initially overconsolidated soils (Leroueilet al. 1978; Rowe and Li 1999; Leroueil and Rowe 2001)and the short horizontal drainage path provided by PVDs (Liand Rowe 1999b). To evaluate the consolidation and conse-quent shear strength gain during construction, two embank-ments with heights H = 4.41 and 6.48 m over soils A and B,respectively, and reinforcement with tensile stiffness J =2000 kN/m and allowable strain all = 5% were examined.The average excess pore pressure dissipation in foundationsoils under the embankment centre at the end of constructionwas 31 and 40% for embankment heights H = 4.41 and6.48 m, respectively.

Figure 12 shows the contours of the increase in undrainedshear strength su of the foundation soil during constructionfor both embankments. For the sake of clarity, Fig. 12 doesnot include the increase in undrained shear strength near thetop and bottom layers, where the gradient of shear strengthincrease is high due to the drainage boundary effects. Theundrained shear strength, su, was calculated at each integra-tion point in the finite element analysis based on the effec-tive stresses at the end of construction, and then su wasaveraged along the horizontal direction within the influencezone of each PVD. Due to the presence of the PVDs, the in-crease of undrained shear strength su (i.e., su at the end ofconstruction minus initial undrained shear strength, suo) wasrather uniform throughout the thickness of the deposit, eventhough the initial undrained shear strength increased withdepth at a rate of c (see Sect. 3.1.). The increase in shearstrength of soil under the embankment centre (5 kPa for soilA and 11 kPa for soil B) was higher than that under the em-bankment slope (i.e., about 3 kPa for soil A and 6.5 kPa forsoil B). Figure 12 also shows the calculated potential failuresurface using the limit equilibrium program (REAP). It isevident that the increase in shear strength along the failuresurface had the highest magnitude at locations below theembankment crest and gradually dropped to practically noincrease (

for the cases having vertical drains installed in foundationsoils. The significant consolidation during construction is aresult of two factors, namely the high consolidation coeffi-cient of foundation clay in its initial overconsolidated state,and the time-dependent loading. Classical consolidation the-ories are unable to predict the effect of these two factors dueto their limitations, as discussed by Mesri and Rokhsar(1974), Olson and Ladd (1979), and Holtz et al. (1991). Totake these two factors into account, the conventional consol-idation theories should be modified without making themtoo complicated for practical use.

The theory proposed by Mesri and Rokhsar (1974) canconsider the change of compressibility of clay fromoverconsolidated states to normally consolidated states forone-dimensional consolidation problems. However, the gov-erning equations must be solved using a numerical tech-nique, which may not be readily available for design inmany cases. As the consolidation of vertical drains is usuallycalculated in terms of average degree of consolidation, theauthors suggest the following simple approximate method tocalculate the average degree of consolidation to consider thefast dissipation of excess pore pressure during the earlystage of embankment construction when the soil is in anoverconsolidated state. All assumptions of Terzaghis con-solidation theory are preserved except for the change incompressibility as a soil moves from the overconsolidated tonormally consolidated state and the time-dependent loading.It is assumed that the soil becomes normally consolidatedwhen the average degree of consolidation at a particular timeis such that the average vertical effective stress of soils alongthe drainage paths is equal to the preconsolidation pressure.At this time, the change of soil compressibility is a stepchange (i.e., from recompression index Cr to compressionindex Cc). The time-dependent loading is taken to be a linearramp loading.

The proposed method is shown conceptually in Fig. 13. Astress of is applied during a period of time tC (from pointO to point A in Fig. 13). During the period to time tO/C,when the soil is overconsolidated, the consolidation is gov-erned by consolidation coefficient CO/C; after tO/C, when thesoil becomes normally consolidated, the consolidation isgoverned by consolidation coefficient CN/C. For a foundation

deposit under a two-way drainage condition, the average de-gree of consolidation at any time is defined as

[7] UH t u z

H

H

= 2

20

2

( ) d

where H is the drainage distance (i.e., the half thickness ofthe foundation deposit), (t) is the applied stress at time t,and u is the excess pore pressure at time t. At time tO/C, theapplied stress (t) is equal to tO/C/tC, the average de-gree of consolidation is U CO/ for a total stress of , andthe average change in effective stress at this time isU CO/ . The excess pore pressure that will need to dissi-pate after application of the full stress is (1 U CO/ ),and it is assumed that this excess pore pressure is devel-oped over a period of time tC = tC tO/C. After time tO/C, theaverage consolidation, U CN / , is calculated using CN/C for theramp load of (1 U CO/ ). The proposed approximation isshown graphically in Fig. 13, with the linear load function0A being replaced by two linear load functions: 0B and0A for soil in overconsolidated and normally consoli-dated states, respectively. It is assumed that the average de-gree of consolidation under the load 0A after tO/C isequivalent to the average degree of consolidation under theload 0A plus the average degree of consolidation underthe load 0B at time tO/C. Namely, the total average degreeof consolation at time t ( tO/C) is

[8] U U U U= + O C O C N C/ / /( )1

To consider the consolidation of soil under a time-dependentloading, a number of methods have be proposed (e.g., Taylor1948; Schiffman 1958; Olson 1977; Zhu and Yin 1998,1999). Olson (1977) derived relatively simple solutions con-sidering both vertical and radial drainage for a linear ramploading problem based on the assumptions of the classicconsolidation theories except time-dependent loading. Theequations for vertical consolidation are as follows:

[9a] T T UTT T M

M T =

c vc

; [ exp( )]12 1

14

2

[9b] T T c ;

UT M

M T M Tvc

c= 1 2 1 14 2 2[exp( ) ]exp( )

where T is the time factor for vertical consolidation; Tc is thetime factor at the end of construction; and M = (2m + 1)/2,for m = 0, 1, 2, 3, until the sum of all remaining terms isinsignificant. The equations for horizontal (radial) consolida-tion are as follows:

[9c] T T UT

TA

ATh hc hhc

h h =

; [ exp( )]1 1

1

[9d] T T UAT

AT ATh hc hhc

hc h = ; [exp( )]exp( )11

1

where Th is the time factor for horizontal consolidation; Thcis the time factor at the end of construction; and A = 8/(where is defined in Sect. 4) using Hansbos (1981) solu-tion for vertical drains.

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Fig. 13. Linear ramp load function for the consolidation analysisconsidering the soil in its overconsolidated and normally consoli-dated states.



The equation for combined vertical and radial consolida-tion using the method of Carrillo (1942) is as follows:

[10] U U U= 1 1 1( )( )h v

To examine eqs. [8][10], Fig. 14 shows the predicted aver-age degree of consolidation for a unit cell of a vertical drainsystem in a 15 m thick foundation soil A (including bothvertical and horizontal drainage) under a one-dimensionalloading condition using the finite element method and theproposed analytical method. Also shown is the calculatedaverage degree of consolidation based on the consolidationcoefficient, CN/C, for soil in its normally consolidated stateand based on the average consolidation coefficient, Cavg, attime t which is an average value of CO/C and CN/C weightedby the increase in effective stress at time t. The finiteelement results are calculated from a simulation of an axi-symmetric vertical drain. It is evident that the prediction ofconsolidation using eq. [8] is conservative after time tC andcompares well with finite element results, especially at theend of construction. The method using CN/C significantly un-derestimates the consolidation during the early stage of load-ing, and the method using an average of CO/C and CN/Csignificantly overpredicts the consolidation, hence thesemethods are not suitable for use.

Equations [8][10] can be used to estimate the average de-gree of consolidation of the foundation soil and the conse-quent average shear strength gain of foundation soils. Asshown in Fig. 12, the variation of the strength gain of foun-dation soil with depth is relatively uniform for a relativelyuniform foundation soil, and the magnitude of strength gainfor soil along the failure surface gradually decreases from amaximum below the embankment crest to a minimum infront of the embankment toe. The following simple proce-dure is used to predict the strength gain in foundation soilswith PVDs as shown by finite element results. Since the av-erage degree of consolidation is calculated over the depth ofthe soil stratum through which vertical drains penetrate, thesoil parameters can be averaged over the depth of foundationin the following calculations.

Assuming that the embankment loading is one-dimensional, the increase in undrained shear strength of soilbelow the embankment centre, suc, can be estimated basedon the SHANSEP method (Ladd and Foott 1974) by

[11] suc vo fill uo= + [ ( )] H U s

where the ratio = su/p is constant for a given soil; voand suo are the initial vertical effective stress and undrainedshear strength, respectively, prior to embankment construc-tion; and fill is the unit weight of the embankment fill.

In the SHANSEP method, the undrained shear strength ofclay is normalized to the vertical effective stress, andstrength gain is usually estimated by evaluating the increasein vertical effective stress (Ladd 1991). For locations belowthe embankment centre, this normalization works reasonablywell, since a K 0 consolidation condition is representative forthese locations, where K 0 is the coefficient of lateral earthpressure at rest. However, for the locations along the failuresurface below the embankment slope, the loading imposedby the embankment is at least two-dimensional and the con-solidation may not be well approximated by K 0 consolida-tion. The proposed method uses the undrained shear strengthnormalized to the effective mean stress, m , for locationsalong the failure surface, assuming that the ratio = su/m isconstant for a given soil in its normally consolidated state.This is based on the fact that the shear strength of soil isgoverned by three-dimensional effective stresses. For sim-plicity, the strength gain is evaluated only by examining thechange in effective mean stress, and the effect of change indeviatoric stresses on strength gain is neglected.

When soil is in its normally consolidated state, the effec-tive mean stress m = [(1 + 2K 0)/3]p . Therefore, is calcu-lated by

[12] =+

3

1 2 0K

where K 0 is the coefficient of lateral earth pressure at restfor soil in its normally consolidated state.

The increase in three-dimensional stress due to the em-bankment loading can be estimated using an influence factor,Iq, for total mean stress m based on elastic solutions (e.g.,Poulos and Davis 1974), and Iq can be averaged over the po-tential slip surface. The average degree of consolidation ofsoils along the potential failure surface, Uf , is calculated sepa-rately from that of soil below the embankment centre basedon the fact that the degree of consolidation of soil below theembankment centre and shoulder is different due to three-dimensional loading conditions. Since the increase of excesspore pressure is equal to the increase in total mean stress basedon an elastic assumption, the consolidation problem along afailure surface can be considered as if the total mean stress,m = fillHIq, is applied over the construction period. Thedifferent approaches given to one- and three-dimensionalloading conditions are applying the total embankment load,fillH, to the soil below the embankment centre versus apply-ing the partial embankment load, fillHIq, in terms of meanstress to the soil along the potential slip surface. Therefore,the average degree of consolidation, Uf , can be calculated us-ing eqs. [8][10] and the consequent strength gain, suf, canbe calculated by

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Fig. 14. Comparison of calculated average degree of consolida-tion for a unit cell of vertical drain system in soil A using dif-ferent methods (including both vertical and horizontal drainage).FEM, finite element method.



[13] s HI U suf mi fill q f uo= + [ ( )]

where mi is the initial effective mean stress.It is worth noting that U CO/ in eq. [8] is the average degree

of consolidation at time tO/C when effective mean stress mis equal to mp = (1 + 2K0)p /3 (e.g., the mean of pre-consolidation stresses). The above approximation has provedto be reasonably accurate and conservative, as shown in fol-lowing sections.

6. Proposed method

6.1. Procedure for combining consideration ofembankment reinforcement and vertical drains

The method of analysis is based on an undrained strengthanalysis (USA) method suggested by Ladd (1991) and alimit state design philosophy. The steps in the procedure areoutlined as follows and an example is given in the Appen-dix.

(1) Select the design criteria, including height (H), width(B), and slope (n) of the embankment; average degree ofconsolidation (U) required; available time (t) to achieve U ;and construction rate (CR).

(2) Select soil parameters for the embankment fill andfoundation, including undrained shear strength (su) profile;preconsolidation pressure (p ) and current vertical effectivestress (v ) with depth; coefficient of lateral earth pressureat rest (K0) profile; normalized shear strength for soil in itsnormally consolidated state (su/m , where m is the effec-tive mean stress for soil in its normally consolidated state);coefficient of consolidation of soil in overconsolidated(CO/C) and normally consolidated (CN/C) states; ratio of hy-draulic conductivity in the horizontal direction (kh) to thatin the vertical direction (kv) for undisturbed soil; ratio ofhydraulic conductivity of undisturbed soil (kh) to that ofdisturbed soil (ks); length (Hd) of the longest drainage pathin the vertical direction; and friction angle () and unitweight (fill ) of the embankment fill.

(3) Select a prefabricated vertical drain system whoseconfiguration is described by the drain spacing (S); the ef-fective diameter (De) of the drain influence zone (De = 1.05Sfor a triangular pattern and 1.13S for a square pattern); thelength (L) of a single drain (equal to the thickness of clayeydeposits in most cases); the diameter of the smear zone (ds)caused by drain installation; and the equivalent diameter(dw) and discharge capacity (qw) of a single drain.

(4) Calculate the average degree of consolidation at theavailable time t using eqs. [8][10]. If the calculated averagedegree of consolidation is less than the required U , repeatstep 3 to select a new PVD configuration (e.g., spacing Sand length L) until U is met.

(5) Estimate the average influence factor Iq (m =fillHIq) for the increase in total mean stress of the founda-tion soil along the potential slip surface using the followingelastic solutions (e.g., Poulos and Davis 1974) and Fig. 15:

[14a]

xxa

yR

x by

aRR

= + + +

22

1

0

2( ) ln

[14b]

yxa

yR

x b= +

22

( )

where x is the distance from the embankment toe and y isthe depth; and for an elastic material under plane strain con-ditions, R0, R1, and R2 are distances as shown in Fig. 15.

[14c] z x y= +( )

where is Poissons ratio; therefore,

[14d]

m =+ +x y z

3

[14e] Iqm=

(6) Calculate the average degree of consolidation alongthe potential slip surface at the end of the embankment con-struction, Uf .

(7) Estimate the average strength increase, suf, of soilalong the potential failure surface at the end of constructionusing eq. [13].

(8) Factor the strength of soils using partial factor fc forthe undrained shear strength of foundation soil (su* = su/ fc)and f for fill material (tan * = (tan )/f), and f for theunit weight of fill fill(fill

* = fill f) as appropriate.(9) Factor suf using partial factor fc (suf* = suf/ fc).(10) Using a limit equilibrium method, calculate the equi-

librium ratio (ERAT) of restoring moment to overturningmoment for the embankment without reinforcement usingthe factored soil parameters of embankment fill and factoredundrained shear strength profile, including the strength gainduring construction, i.e., su* + suf* ; reinforcement is notneeded if ERAT 1.0 and is needed if ERAT < 1.0, in whichcase continue with steps 11 and 12.

(11) Use a limit equilibrium program designed for theanalysis of reinforced embankments (e.g., REAP; Myllevilleand Rowe 1988) to calculate the required reinforcement ten-sile force, Treq, using the new factored undrained shearstrength profile obtained in step 10 (Treq is the force required

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Fig. 15. Vertical embankment loading on a semi-infinite mass(modified from Poulos and Davis 1974).



to give an overturning moment = restoring moment based onfactored soil properties, i.e., ERAT = 1).

(12) Choose an allowable strain, all , for the reinforcementand then the required reinforcement tensile stiffness is calcu-lated as

[15] JT

reqreq

all

=

In this procedure, the limit states examined involve failure ofthe embankment, foundation, and reinforcement. All calcula-tions except the slope stability analysis can be done by handor using a spreadsheet program. This approach can be madeequally applicable to a stage construction sequence by add-ing the consolidation during stoppage in steps 4 and 6, keep-ing other steps the same. Design parameters (e.g., S, L, Treq,and Jreq) can be obtained iteratively from steps 112. In thedesign iteration, the construction rate and stage sequencescan be varied such that the design grade can be achieved inan optimum time schedule. To assure embankment stabilityduring construction, it is essential to monitor the develop-ment of reinforcement strains, excess pore pressures, settle-ment, and horizontal deformations and to confirm thatmeasured values are consistent with the design values.

6.2. Comparison of the results using finite elementanalyses and the proposed method

Using the proposed method, the equilibrium ratio (ERAT)of restoring moment to overturning moment at the end of con-struction for the embankments analyzed in the previous sec-tion can be calculated. A number of cases with differentreinforcement construction rates and construction stages arereanalyzed using the proposed method and are given in Ta-ble 2, where T is the tensile force developed in the reinforce-ment at an allowable strain all . The embankment should havean ERAT of 1 at height H, which was calculated from finiteelement analyses in previous sections. ERAT in Table 2 is inthe range of 0.890.96. It is shown that the proposed method

is conservative for both one-stage construction and two-stage construction and compares well with finite elementanalysis results for the cases examined. Compared with thatpredicted by the finite element analyses, the shear strengthgain along the failure surface predicted using eq. [13] is con-servative. For example, the average shear strength gains pre-dicted using eq. [13] for the embankments H = 4.41 m oversoil A and H = 6.48 m over soil B are 1.92 and 4.62 kPa, re-spectively (see Table 2). These values are less than thosecalculated using finite element analyses (see Fig. 12).

7. Recommendations and conclusions

The performance of embankments constructed using pre-fabricated vertical drains (PVDs) and geosynthetic reinforce-ment has been theoretically examined. A tentative designprocedure has been developed to consider the benefits arisingfrom the use of both PVDs and reinforcement. The followingconclusions are based on the analyses reported herein.

Partial consolidation of the foundation soil during em-bankment construction is significant because of the drainageimprovement provided by prefabricated vertical drains andfoundation soils usually become normally consolidated atthe end of embankment construction. Consequently, theshear strength gain of soft foundation soil due to partial con-solidation can substantially improve the stability of embank-ments such as those examined. The rate of increase in shearstrength of foundation soil is a function of construction ratefor a particular reinforced embankment and PVD system.Where PVDs are installed, the strength gain is relatively uni-form throughout the thickness of the deposit and can be rea-sonably predicted using the proposed method. For afoundation installed with PVDs, the embankment designbased on assuming undrained conditions and neglecting theshear strength increase in the foundation soil during con-struction is too conservative.

The construction rate significantly affects embankmentstability and deformation. For each case, there is a unique

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J (kN/m) all (%) T (kN/m) CR (m/month) H (m) Uf (%) su (kPa) ERAT (-)Soil AOne-stage construction1000 5 50 2 3.70 30.80 1.37 0.9141000 7 70 2 4.18 29.10 1.70 0.8932000 5 100 2 4.41 28.70 1.92 0.9202000 7 140 2 5.04 27.40 2.42 0.9031000 5 50 1 4.17 35.20 2.80 0.9322000 5 100 1 4.96 34.80 3.94 0.955Two-stage construction (H1 = 3.5 m, 4 months stoppage)2000 5 100 2 5.63 38.20 5.82 0.960Soil BOne-stage construction1000 5 50 4 5.75 38.00 3.76 0.9452000 5 100 4 6.48 37.10 4.62 0.9551000 5 50 8 5.28 35.00 2.39 0.9372000 5 100 8 5.87 33.80 2.93 0.951Two-stage construction (H1 = 6 m, 1 month stoppage)2000 5 100 4 8.58 40.30 8.92 0.904

Table 2. Calculated ERAT of reinforced embankments obtained using the proposed method.



threshold construction rate below which reinforcement is notneeded and the foundation will not experience bearing ca-pacity failure, even for a relatively high embankment. Thespacing of PVDs significantly influences the degree of con-solidation at the end of embankment construction, but theeffect of well resistance of typical vertical drains is insignifi-cant. The proposed eq. [8] reasonably well predicts the aver-age degree of consolidation when the soil moves from itsoverconsolidated state to a normally consolidated state underan embankment loading.

Reinforcement can substantially increase the stability ofembankments under partially drained conditions when theconstruction rate is greater than the threshold rate. The benefitof the increase in shear strength due to partial consolidation isenhanced by the use of reinforcement. For a foundationunder partially drained conditions, very high reinforcementstrains may be developed due to plastic deformations offoundation soils under relatively high embankments that canbe constructed because of the beneficial effects of both rein-forcement and partial consolidation during construction. Itwas found that the embankment failure height (based onlimit state design considerations) is generally governed bythe allowable strain of reinforcement when the bearing ca-pacity of foundation soil ceases to be a problem due to shearstrength gain during construction.

With two-stage construction, construction of the embank-ment as high as possible by the use of reinforcement in thefirst stage results in the best performance due to the highereffective stress increase in foundation soil that is achievedduring consolidation. Therefore, the use of reinforcementhas the potential to allow embankment construction in ashorter period of time than that what might be required for acase without reinforcement. Since stability and consolidationare interrelated, reinforcement and PVDs should be treatedtogether in design.

The proposed method for estimating the consolidation andconsequent strength gain of foundation soil during and afterconstruction under embankment loading proved to be con-servative and reasonably accurate for the relatively uniformfoundation deposits examined. The proposed method canconsider both prefabricated vertical drains and reinforcementand appears to warrant further investigation as a means ofpreliminary design of embankments involving the use of re-inforcement and PVDs. It is recommended that any suchdesign be combined with monitoring to confirm that the per-formance (e.g., excess pore pressures) is consistent with thatdeduced in the design. A parametric study can be carried outusing the proposed method to optimize the configuration ofreinforcement and vertical drains.

Acknowledgement

The research reported in this paper was funded by the Nat-ural Sciences and Engineering Research Council of Canada.

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Appendix: A worked example

(1) Select design criteria A 4.5 m high four-lane high-way embankment with a 28 m wide crest and side slope of2:1 (h:v) is constructed over a soft foundation deposit. Theaverage degree of consolidation U of the foundation soilmust reach 90% within a 9 month period of time by the useof prefabricated vertical drains. The construction conditionsallow a construction rate of 4 m/month.

(2) Soil parameters Foundation soil B and the embank-ment fill described in Sect. 3.1. are used for this example.The undrained shear strength, vertical effective stress, andpreconsolidation pressure profiles are shown in Fig. 2b. Theaverage foundation soil parameters are vo = 50.8 kPa, m =37.3 kPa, p = 73.6 kPa, mp = 54 kPa, suo = 20.9 kPa,CO/C = 2.32 10

3 m2/h, and CN/C = 3.86 104 m2/h. The

other required soil parameters are K0 = 0.6, = su/p = 0.31,= su/mp = 0.423, kh/kv = 3, kh/ks = 3, and Hd = 7.5 m. Theembankment fill material has = 37 and fill = 20 kN/m3.

(3) Select prefabricated vertical drain system The con-figuration of the PVD system is given as S = 2 m (a squarepattern), De = 2.26 (De = 1.13S for square pattern), L =15 m, dw = 0.066 m, qw > 100 m

3 /a, and ds = 0.264 m (thediameter of the smear zone caused by drain installation isfour times the equivalent diameter of the vertical drain).

(4) Calculate the average degree of consolidation in theavailable time, t = 9 months, using eqs. [8][10] The timeat the end of construction for the 4.5 m high embankment at arate of 4 m/month is 810 h (34 days). Assuming an embank-ment load of = 4.5 20 = 90 kPa, the average degree ofconsolidation required to bring the vertical stress to thepreconsolidation pressure (from 50.8 to 73.6 kPa) is UO C/ =(73.6 50.8)/90 = 0.253. Using eqs. [9] and [10] and CO/C,the back-calculated tO/C is 472 h for soil to reach UO C/ =25.3%. Then tC = 810 472 = 338 h; using eqs. [9] and [10]and CN/C, the calculated UN C/ is 88.7% at t = 9 months. Usingeq. [8], the total average degree of consolidation is U = 0.253 +(1.0 0.253) 0.887 = 0.916 = 91.6%, which is greater thanthat required. Therefore, the configuration of PVDs is satis-factory. Continue to the next step.

(5) Estimate the average influence factor Iq The influ-ence factor Iq for total mean stress at a number of locationsalong the potential slip surface in the foundation soil is cal-culated using eq. [14]. The average Iq is 0.48.

(6) Calculate the average degree of consolidation alongthe potential slip surface (Uf ) at the end of construction us-ing eqs. [8][10] This step is similar to step 4. The totalmean stress load, m = fillHIq = 20 4.5 0.48 =43.2 kPa, is applied in a time of 810 h. The required averagedegree of consolidation U CO/ is 39% to bring the initialmean stress m = 37 kPa to the mean preconsolidation pres-sure mp = 54 kPa. Using eqs. [9] and [10] and CO/C, theback-calculated tO/C is 186 h. Using eqs. [9] and [10] and CN/C,the calculated UN C/ is 5.8% at t = 9 months. Using eq. [8], thecalculated Uf is 42.5%.

(7) Calculate the average strength increase of soil along thepotential failure surface at the end of construction usingeq. [13] The average strength increase suf = [(m +fillHIq Uf )] suo = 0.423 (37.3 + 20 4.5 0.48 0.425) 20.9 = 2.65 kPa.

(8) Factor strength of soils using partial factor fc = 1.3 forundrained shear strength profile (s u* ) and f = 1.2 for fillmaterial (* = 32) and f = 1.0 for unit weight of fill(fill* = 20 kN/m3).

(9) Factor suf using partial factor fc = 1.3, giving s uf* =2.04 kPa.

(10) Using the limit equilibrium program REAP(Mylleville and Rowe 1988) and the new factored undrainedshear strength profile (s u* + s uf* ), the calculated minimumERAT is 0.74 for the embankment without reinforcement.Since ERAT < 1.0, reinforcement is needed and thereforecontinue to steps 11 and 12.

(11) Using REAP, the calculated required reinforcementtensile force, Treq, is 160 kN/m to maintain a minimumERAT of 1.0.

(12) Choose an allowable strain of all = 5% for the rein-forcement. The required reinforcement tensile stiffness is cal-culated using eq. [15] to be Jreg = 3200 kN/m (or two layersof reinforcement that will develop 160 kN/m at 5% strain).



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Combined Effects of Reinforcement and Pvd on Embankment Performance