One-dimensional consolidation under generaltime-dependent loading

Enrico Conte and Antonello Troncone

Abstract: This paper presents an analytical solution for the analysis of one-dimensional consolidation of saturated soillayers subjected to general time-dependent loading. A simple calculation procedure that makes use of the Fourier seriesis proposed for practical applications. Both single loads and cyclic loads can be considered by choosing a suitable pe-riod for the Fourier series. A number of comparisons with existing theoretical solutions are shown to assess the accu-racy of the proposed procedure. Moreover, the experimental results from oedometer tests performed in the presentstudy and from a well-documented case history concerning a large embankment constructed on compressible soils areanalysed using this solution to evaluate the coefficient of consolidation of the soil.

Key words: one-dimensional consolidation, time-dependent loading, excess pore-water pressure, theoretical solution,Fourier series.

Rsum : Cet article prsente une solution analytique pour analyser la consolidation unidimensionnelle des sols saturssoumise un chargement variable avec le temps de quelque manire. Une simple procdure de calcul, qui utilise lessries de Fourier, est dveloppe pour les applications pratiques. Aussi bien les charges isoles que les charges cycli-ques peuvent tre considres, en utilisant une priode convenable de la srie de Fourier. On prsente un certainnombre de comparaisons avec dautres solutions thoriques existant dans la littrature pour tablir la prcision de laprocdure propose. En outre, cette procdure est utilise pour analyser les rsultats exprimentaux dessais oedomtri-ques effectus dans cette tude et dun cas rel bien document concernant un large remblai construit sur sols compres-sibles, et pour valuer le coefficient de consolidation du sol.

Mots cls : consolidation unidimensionnelle, chargement variable avec le temps, surpression interstitielle, solution tho-rique, srie de Fourier.

Conte and Troncone 1116

Introduction

Soil consolidation is often caused by loading as, for ex-ample, in the case of the construction of buildings or em-bankments on clayey soils. In the current state of practice,the analysis of this transient process is usually conducted us-ing Terzaghis theory (Terzaghi 1925), in which it is as-sumed that the soil is saturated, water flow and soildeformation only occur in the vertical direction, the load isinstantaneously applied and then held constant with time. Inreality, however, the loads in the construction of civil engi-neering works are generally applied gradually with time, andin many cases the loading process develops over a long pe-riod of time, so a significant part of the consolidation occursduring this time. Moreover, special structures such as silosor fluid tanks subject the soils to loading and unloadingstages that repeat themselves more or less periodically overtime. All these loads cause changes in the total stresses asthe soil consolidation develops. Consequently, Terzaghis

classical theory should be, in principle, unsuitable to analysethese situations, even when the assumption of one-dimensional conditions is sound.

Schiffman and Stein (1970) developed a general solutionfor the one-dimensional consolidation problem in which avariety of loading history, boundary, initial conditions, andmultilayered soil profiles can be considered. This solution isquite tedious to use, however, especially for practical appli-cations. In this connection, some improvements were pro-posed by Lee et al. (1992) and more recently by Zhu andYin (1999). These latter authors, in particular, dealt with thecase of a double-layered soil profile with the vertical totalstress increasing linearly with an increase in depth.

Some practical methods are available, such as the graphi-cal construction suggested by Terzaghi (1943) to predict theprogress of consolidation settlement caused by a graduallyapplied external load, the analytical expressions derived byOlson (1977) for one-dimensional consolidation of homoge-neous soil layers subject to ramp loading, or the expressionsproposed by Viggiani (1999) for loads that increase linearlywith time.

The cyclic loading case was considered by many authors(Wilson and Elgohary 1974; Baligh and Levadoux 1978;Favaretti and Mazzucato 1994; and others). In particular,Rahal and Vuez (1998) derived expressions to calculate theexcess pore-water pressure and settlement produced by aload whose rate is described by a sinusoidal function of

Can. Geotech. J. 43: 11071116 (2006) doi:10.1139/T06-064 2006 NRC Canada

1107

Received 22 December 2005. Accepted 23 May 2006.Published on the NRC Research Press Web site athttp://cgj.nrc.ca on 21 November 2006.E. Conte1 and A. Troncone. Dipartimento di Difesa delSuolo, Universit della Calabria, Ponte P. Bucci, Cubo 41b,87036 Rende (Cosenza), Italy.1Corresponding author (e-mail: conte@dds.unical.it).

time. Although these solutions deal with linear consolida-tion, they were incorporated into practical methods to ana-lyse the nonlinear response of clay layers to cyclic loads(Baligh and Levadoux 1978; Galati 2001).

This study presents an analytical solution for one-dimensional consolidation problems in soils subjected toloading described by an arbitrary function of time. Both sin-gle loads and cyclic loads can be considered. The accuracyof the proposed solution is assessed by comparing the resultswith those provided by other theoretical methods. Moreover,the solution is used in conjunction with laboratory and fieldobservation of pore pressure or settlement to achieve a sim-ple evaluation of the coefficient of consolidation of the soil,which is a difficult parameter to determine in practice, espe-cially for in situ conditions (Leroueil 2001).

Solution method

The equation governing one-dimensional consolidation insaturated soils is provided by the mass conservation equationfor porous media. According to the linearized theory and un-der the assumptions of incompressible solid particles and nochange of temperature occurring, this equation can be writ-ten as follows (Lancellotta 2004):

[1] n ut t

v

zz z

+

= 0

where z is the spatial coordinate, t is time, u is the excesspore-water pressure depending on both z and t, z is the ver-tical strain of the soil, vz is the rate of water flow across aunit area of soil in the z direction, n is the soil porosity, and is the pore-water compressibility. Under the additional as-sumptions that soil behaves as an isotropic, linearly elasticmaterial governed by the effective stresses, water flow is de-scribed by Darcys law with constant coefficient of perme-ability, and creep and inertial effects are ignored, it can beshown that eq. [1] leads to

[2] c uz

u

t tv

dd

=

2

21

where

[3] c k mv w w v= /[4] = +m m nv v/( ) is a parameter accounting for compressibility of the soiland pore fluid, is the total vertical stress that is a functionof time, kw is the coefficient of permeability of the soil, mv isthe coefficient of volume change of the soil, w is the unitweight of water, and cv is the coefficient of consolidation.Overall, the assumptions on which eq. [2] is based are thesame as those of Terzaghis basic theory (Terzaghi 1925),except for the assumptions of water incompressibility, in-stantaneous loading application, and no change in total verti-cal stress with time. Moreover, it should be observed that,because of the presence of , eq. [2] could also be used toanalyse one-dimensional consolidation in unsaturated soilswith a degree of saturation close to 100%. In these circum-stances, in fact, the air contained in the pores is occludedand cannot flow as a continuous fluid, so air and water be-have as a homogeneous compressible fluid flowing under the

pore-water pressure gradients (Barden 1965). As aconsequence, the consolidation is described by the samegoverning equations as those for saturated soils, providedthat the fluid compressibility is accounted for (Verruijt1969; Ghaboussi and Wilson 1973; Chang and Duncan1983; Conte 1998; and others). Under one-dimensional con-ditions, these equations reduce to eq. [2]. On the contrary, ifpore-fluid compressibility is ignored, = 1.0.

For the purposes of the present study, it is convenient toconsider a uniformly distributed load that varies harmoni-cally with time. It is assumed that this load is applied to thetop surface of a soil layer of thickness H (Fig. 1) and is de-scribed by the following equation:

[5] ( ) cos ( ) sin ( )t A t B t= +where A and B determine the load amplitude, and is thecircular frequency. As a result, the last term on the right-hand side of eq. [2], expressing the loading rate, is

[6] dd

t

A t B t= +sin( ) cos( )

To achieve the solution to this problem, the followingboundary conditions are considered (Fig. 1): the base of thelayer is impervious, i.e.,

[7]

=u t

z

( , )0 0

whereas the upper surface is fully permeable, i.e.,

[8] u H t( , ) = 0It should be noted that for the situations in which the lowersurface of the soil layer is also permeable like the upper sur-face (double-drainage condition), the boundary conditiongiven by eq. [7] has to be imposed at the middle of the layer,as in Terzaghis theory. In other words, the results for thesingle drainage condition can be adopted to determine thesolution for the double drainage condition by interpreting Has the drainage height.

Lastly, the initial condition is

[9] u z( , )0 0=As suggested by Carslaw and Jaeger (1959), the solution

to eq. [2] with the loading, boundary, and initial conditionsgiven previously can be obtained using Duhamels theorem,which for the case under consideration is expressed by thefollowing equation:

[10] u z t u z tt

t( , ) ( ) ( , )= d d d 0

where d d / is provided by eq. [6] and u denotes the solu-tion to eqs. [2] and [7][9] when the loading rate is kept atunity. As shown in Appendix A, this latter solution is

[11] u z tc H

z c tj

jjj j( , ) ( ) cos ( ) exp( )=

=

2 131

2

v

v

+ 1

22 2

cH z

v

( )

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in which j = (2j 1) /(2H). Therefore, substituting for tin eq. [6] and t for t in eq. [11] and performing the par-tial derivative of u z t( , ) with respect to t, eq. [10] leads tothe following expression that allows us to calculate the ex-cess pore pressure at any time and depth, u(z, t), induced bythe harmonic load described by eq. [5]:

[12] u z t Y zH

j

jj

( , ) ( ) cos= +

=

2 1 2 51

where

[13a] = (2j 1) /2[13b] Yj = {(A + B2)[cos (t) exp(2Tv)]

(A2 B) sin (t)}[13c] = cv/(H2)[13d] Tv = cvt/H2

Moreover, under the assumption that the coefficient of vol-ume change mv is constant, the settlement at any time s(t) isprovided by the following equation:

[14] s t m t u z t zH ( ) [ ( ) ( , )]= v d0which, after substituting eqs. [5] and [12], yields

[15] s t m H A t B t ( ) cos ( ) sin ( )= +v

+

=

2 2 2 61

Yjj

Based on the use of the derived equations in conjunctionwith a Fourier series, an analytical solution to eq. [2] when ageneral time-dependent loading (t) is applied to the topsurface of the soil layer is presented in this section. It isknown, in fact, that if (t) is a periodic function satisfyingDirichlets conditions in the interval (0, T), it can be ex-panded in harmonic components using the Fourier series

[16] ( ) [ cos ( ) sin( )]t A A t B tkk

k k k= + +=

o2 1in which the series amplitudes Ak and Bk associated with thefrequency k = 2k /T (with k = 1, 2, ) are provided, re-spectively, by the following equations:

[17] AT

t t tkT

k= 2 0 ( ) cos ( ) d[18] B

Tt t tk

Tk= 2 0 ( ) sin( ) d

where T is the period of (t); and Ao can be obtained fromeq. [17] setting = 0, i.e.,

AT

t tT

o d= 2 0 ( )It should be noted that the expansion of (t) in Fourier com-ponents may in practice be achieved by summing a finitenumber of terms in eq. [16]. The series appearing in thisequation, in fact, converges quite rapidly, especially when(t) is a continuous function over the period T. Generally, afew tens of terms are enough to provide satisfactory resultsfor practical purposes. On the contrary, however, the conver-gence is slower when (t) is sectionally continuous andpresents sudden jumps. In these latter cases, a larger numberof harmonic components of the Fourier series is therefore re-quired. In addition, it should be observed that when (t) isrepresented at intervals by constant or linear functions,which often is the case, closed-form expressions can bereadily obtained by solving the integrals in eqs. [17] and[18]. This makes the use of these equations very attractivefrom a practical point of view. Some examples are presentedin the next sections.

Taking into account the linearity of the problem and tak-ing advantage of the previous expressions, a simple solutionprocedure can be developed. Specifically, this procedure firstrequires that the prescribed loading function (t) is ex-panded in a number M of harmonic components usingeqs. [16][18]. Then, for each of these components, the cor-responding pore pressure is calculated using eq. [12], inwhich u(z, t) is replaced by uk (z, t); A and B are replacedby Ak and Bk, respectively; and is replaced by k. In thisconnection, it should be noted that the term Ao correspond-ing to = 0 yields u(z, t) = 0. Lastly, once the functionsuk(z, t) have been determined for all the values of k consid-ered, the actual excess pore-water pressure at a given depthand time is obtained by superimposing all the calculatedterms, i.e.,

[19] u z t u z tkk

M

( , ) ( , )==

1

Likewise, the settlement as a function of time, s(t), can beevaluated using the following expression:

[20] s t m H s tkk

M

( ) ( )= +=

v o1

where

[21] o d= 1 0T t tT ( )and sk(t) is provided by eq. [15], in which s(t) is replacedby sk(t); A and B are replaced by Ak and Bk, respectively; and is replaced by k.

The described solution procedure allows us to analyse thesoil consolidation induced by general time-dependent loads

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Fig. 1. Soil layer subject to harmonic loading.

such as those shown schematically in Fig. 2, where both cy-clic loads (Fig. 2a) and single loads (Figs. 2b, 2c) are indi-cated. It should be noted, however, that owing to the factthat the procedure deals with periodic functions, when a sin-gle load (Fig. 2b) or a load applied gradually in time(Fig. 2c) is considered, the period T must be long enough tonullify the effect of the other loads acting whenever thefunction (t) periodically repeats itself. Another limitationof the present solution could derive from the assumption thatthe involved soil parameters are held constant during theconsolidation process, as in Terzaghis theory. Neverthelessand as previously pointed out, this assumption is often ac-cepted for practical purposes, especially to evaluate thepore-water pressure variations with time and depth (Balighand Levadoux 1978; Laflamme and Leroueil 2003). Thesevariations may be then incorporated into practical methodsto predict settlement accounting for the nonlinear behaviourof the soil (Baligh and Levadoux 1978; Galati 2001).

Comparison with theoretical solutions

To assess the accuracy of the proposed procedure, com-parisons were first carried out with existing theoretical solu-tions. Some of these comparisons are presented in Figs. 35in terms of isochrones. In all the examples considered, the

soil is saturated and water compressibility is ignored (i.e., = 1.0).

The first case examined refers to the classical theory de-veloped by Terzaghi (1925) for one-dimensional consolida-tion. To use the procedure presented in the previous section,the periodic load plotted in Fig. 3a is considered, where q isthe load intensity, T is the period, ta is a time used to accountfor the initial condition, and tb is a time chosen appropriatelyso that the consolidation process is expected to develop inpractice before this time. Specifically, it was assumed forthis example that tb = 2.5H2/cv, ta = 0.05tb, and T = 1.05tb.Expressing the load under consideration of an even function,eqs. [17], [18], and [21] yield

[22a] A qk

t tk k k=

[sin( ) sin( )]b a

[22b] Bk = 0

[22c] o b a= q t tTThe comparison between the results obtained usingTerzaghis theory and the present solution is shown inFig. 3b for different values of the time factor Tv.

A second comparative study concerns a soil layer sub-jected to a cyclic square load of intensity q and period T(Fig. 4a). For this load, the amplitudes of the Fourier series

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1110 Can. Geotech. J. Vol. 43, 2006

Fig. 2. Examples of time-dependent loads that can be consideredusing the proposed solution: (a) cyclic load; (b) single load;(c) load applied gradually over time. T, period of the load.

Fig. 3. Comparison of the present solution with that of Terzaghi(1925): (a) loading scheme used in the present solution;(b) isochrones for different values of the time factor, Tv.

are again provided by eqs. [22a][22c], in which ta = T/4and tb = 3T/4. This case was considered by Baligh andLevadoux (1978), who proposed an analytical solution tocalculate the excess pore pressure at any depth and at theend of a given number, N, of half-cycles of loading. The re-sults calculated using this latter solution and those obtainedin this study are compared in Fig. 4b for several values of N.In the calculations, it was assumed that T = 0.1H2/cv.

The last example considered in this section is that of aload that first increases linearly from zero up to q and thenremains constant with time. In the present analysis, the peri-odic function plotted in Fig. 5a is used to represent this load.As a result, the following expressions can be obtained usingeqs. [17], [18], and [21]:

[23a] A qTk t

t t tk k k k= + 212 2

c

c c b[cos ( ) sin( ) ]

[23b] B qTk t

t t tk k k k= 2 2 2

cc c b[sin( ) cos ( )]

[23c] o b c=

qT

tt2

where the time tc defines the end of the loading stage, and tbis a time chosen so that the consolidation is expected to bepractically completed in the time interval (0, tb) (Fig. 5a). Inthis study, the values assumed for these parameters are tb =2.6H2/cv, tc = 0.13tb, and T = 1.1tb. For the case under con-sideration, the excess pore pressure at any time and depthmay be evaluated using the analytical expressions providedby Olson (1977). Some results achieved using Olsons solu-tion are compared with those calculated using the proposedprocedure in Fig. 5b.

As shown in Figs. 35, the agreement between the presentsolution and existing theoretical solutions is excellent for allthe cases examined.

Analysis of experimental results

In this section, the proposed solution is used to evaluatethe coefficient of consolidation of the soil from laboratoryand field observations for situations where the customaryprocedures based on Terzaghis classical theory (Taylor1948) are in principle unsuitable because of the applicationof time-dependent loads.

The results from some oedometer tests performed in thepresent study are first considered. The apparatus used for thetests is not conventional and operates on specimens of100 mm diameter and 29 mm height. A hydraulic system

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Fig. 5. Comparison of the present solution with that of Olson(1977): (a) loading scheme used in the present solution;(b) isochrones for different values of the time factor, Tv.

Fig. 4. Comparison of the present solution with that of Balighand Levadoux (1978): (a) loading scheme used in the present so-lution; (b) isochrones for different values of the number of load-ing cycles, N.

provides the load whose magnitude is controlled by a pres-sure regulator. This allows loading and unloading with gen-eral time history to be readily applied to the soil specimen.The tests were carried out on two prepared soil mixtures ofdifferent particle-size distribution, herein denoted soil A andsoil B. Soil A is a sandy silt with clay, and soil B is a clayeysilty sand. The grading curves are shown in Fig. 6. Samplesof these soils were compacted at the optimum water contentaccording to the standard Proctor procedure. Table 1 sum-marizes the main index properties of the tested soils andthose after compaction. The organic content was about 1%for both the soils. After compaction, the specimens werecarefully trimmed and placed in the ring of the oedometercell, where soil saturation was achieved by applying a back-pressure in a way similar to that usually adopted in thetriaxial tests. Consolidation tests under time-dependent load-ing were then carried out. During these tests, water flow wasprevented at the base of the specimen and drainage was al-lowed at the top.

Some experimental results are presented in Figs. 7 and 8in terms of loading process and settlement progress versustime. The average values of the coefficient of volume changemv deduced from the compression readings are 7.4 105and 8.0 105 kPa1 for soils A and B, respectively. The co-efficient of consolidation was evaluated by matching themeasured results with those predicted using the proposed so-lution on a trial and error basis. These latter results are alsoshown in Figs. 7 and 8 for thoroughness. In the theoreticalanalyses, water was assumed to be incompressible. A fairlyclose agreement can be observed between simulation andobservation, although constant values of mv and cv wereadopted in the calculations. The back-calculated values of cvare 1.7 108 and 1.8 106 m2/s for soils A and B, respec-tively.

Lastly, a well-documented case history presented originallyby Bilotta and Viggiani (1975) is analysed. It concerns a largeembankment constructed on a soil deposit in which a consid-erable number of Casagrande piezometers and vibrating-wirepiezometers were installed at various depths for monitoringthe pore-water pressure changes occurring during and afterthe construction. Measurements were performed for over3 years. The layout of the embankment is shown in Fig. 9,where the verticals equipped with piezometers are also indi-

cated. The loading time history is shown schematically inFig. 10. The embankment was 7 m high and 200 m wide.Moreover, as can be seen in Fig. 10, its construction began inMarch 1971 and finished in May of the same year. In October1973, a partial unloading was produced by removing a 1 mthick soil layer.

The subsoil essentially consists of an 8 m thick layer ofsilty sand that overlies a thick layer of silty clay extending toa depth of about 30 m below the ground surface (Fig. 11).Below this depth, a sand formation with interbedded silty orclayey levels is present. Fossil fragments and organic mate-rial were found almost everywhere throughout the soil de-

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1112 Can. Geotech. J. Vol. 43, 2006

Property Soil A Soil BLiquid limit (%) 37 22Plasticity limit (%) 16 15Plasticity index (%) 21 7Specific gravity 2.69 2.66Optimum water content (%) 14 11Dry unit weight (kN/m3) 16.9 18.3

Table 1. Index and after-compaction properties of the testedsoils.

Fig. 6. Particle-size distribution of the tested soils.

Fig. 7. Theoretical and measured results from the oedometertests on soil A: (a) loading process; (b) settlement versus time.

posit. In addition, drilling revealed the presence of gaseousinclusions in the lower part of the silty clay layer, at depthsgreater than 22 m from the ground surface. The silty clay isnormally consolidated and is characterized by the presenceof very thin layers of silty sand. It can be classified as an in-organic clay of medium or high plasticity according toAtterberg limits and Casagrandes classification. The profilededuced from vane tests and triaxial tests shows that the un-drained cohesion increases linearly with an increase in depthfrom about 20 to 80 kN/m2. Conventional oedometer testswere also conducted. They provided values of the consolida-tion coefficient cv ranging from 0.5 107 to 1.8 107 m2/sand an average value of mv = 4.2 104 kPa1 (Bilotta andViggiani 1975).

Pore-water pressures measured in the silty clay layer atthree depths (i.e., 11.0, 17.5, and 24.5 m below the groundsurface) along the vertical axis through the centre of the em-bankment (Fig. 9) are plotted in Fig. 12 as a function of time.Pore pressure is expressed in Fig. 12 in terms of elevation ofthe water level, hw, above the average sea level. Because ofthe large lateral extent of the embankment compared with thethickness of the silty clay layer (Fig. 9), it is reasonable to as-

sume that, directly under the centre of the embankment, thesoil strains and water flow occurred essentially under one-dimensional conditions. Therefore, the three-dimensional ef-fects should not have significantly affected the pore pressuremeasurements shown in Fig. 12. In the light of this, simula-tion was performed using the proposed solution along withthe function indicated in Fig. 12a to represent the loadingprocess. It was assumed to be = 1.0. The best agreementbetween calculated and observed pore pressures was ob-tained using a value of cv = 3.8 107 m2/s, which is overthree times higher than the average value of the coefficientof consolidation deduced from the oedometer tests. Thisagrees with the general trend that emerged from similar ex-

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Conte and Troncone 1113

Fig. 8. Theoretical and measured results from the oedometertests on soil B: (a) loading process; (b) settlement versus time.

Fig. 9. Layout of the embankment, indicating the location of theverticals equipped with piezometers: (a) plan (, verticals withpiezometers); (b) cross section AA (, Casagrande piezometer;, vibrating-wire piezometer) (adapted from Bilotta and Viggiani1975).

Fig. 10. Loading time history during and after construction ofthe embankment (adapted from Bilotta and Viggiani 1975).

perience documented in the literature, according to whichthe consolidation parameters obtained from field perfor-mance are generally higher than those from laboratory testson small specimens because the latter cannot account for themacrofabric features of the natural soil deposits (Bilotta andViggiani 1975; Burghignoli and Calabresi 1975; Leroueil etal. 1990).

The computed results are also displayed in Fig. 12 forcomparison with the measured values. As can be seen, thepore pressures calculated at depths of 11.0 m (Fig. 12b) and17.5 m (Fig. 12c) are in good agreement with those observedat the same depths during the loading stage and after the em-bankment construction. The theoretical curves also accountfor the decrease in pore pressure produced by the partial un-loading that occurred in October 1973. The comparison be-tween simulation and observation is not equally favourablefor the pore pressure evolution at a depth of 24.5 m(Fig. 12d), with the calculated results that generally overesti-mate the experimental values. This could be ascribed to thegaseous inclusions found in the lower part of the silty claylayer, which could have affected both the measurements andthe pore pressure variations with time that occurred in thiszone. It should be observed from Fig. 12d that a betteragreement between measured and computed pore pressuresmay be obtained, especially at the early stage of consolida-tion, when a value of = 0.8 is used instead of = 1.0, toapproximately account for the pore-fluid compressibility dueto the presence of the gaseous inclusions.

Another comparison of theoretical and experimental re-sults is presented in Fig. 13 in terms of isochrones at the endof the embankment construction (28 May 1971) and just be-fore the partial removal of the load (16 October 1973). Fig-ure 13 reports the measurements performed in all thepiezometers installed in the silty clay layer (Fig. 9). The cal-culations were carried out using the value of cv indicatedpreviously and assuming = 1.0. As shown in Fig. 13, theagreement between predicted and observed behaviour is sat-isfactory. The greatest discrepancies are at depths greaterthan 22 m where the gaseous inclusions were found. Further

calculations were thus carried out considering a value of =0.8, as previously specified, and the results are plotted inFig. 13 only for the lower part of the soil layer. These re-sults essentially confirm what has already been stated, al-though in this case the three-dimensional effects should haveexerted a significant influence on the experimental data.

Concluding remarks

A simple to use analytical procedure has been proposedfor the analysis of one-dimensional consolidation of soilssubjected to time-dependent loading. The procedure makesuse of the Fourier series and allows consideration of bothsingle loads and cyclic loads by a suitable choice of the se-ries period. Pore-fluid compressibility can be also accountedfor. Excellent agreement has been found between the resultsobtained using the present solution and those derived fromexisting theoretical solutions. Moreover, the analysis of ex-perimental results from laboratory and field measurementshas proved the usefulness of the proposed solution for ob-

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1114 Can. Geotech. J. Vol. 43, 2006

Fig. 11. Soil profile at the site where the embankment was con-structed (adapted from Bilotta and Viggiani 1975).

Fig. 12. (a) Loading process considered in the calculations. (bd)Calculated and measured elevation of water level above the aver-age sea level, hw, versus time at 11.0, 17.5, and 24.5 m below theground surface (adapted from Bilotta and Viggiani 1975).

taining representative values of the coefficient of consolida-tion of the soil, whose evaluation in practice is generally nota simple operation.

References

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Bilotta, E., and Viggiani, C. 1975. Una indagine sperimentale invera grandezza sul comportamento di un banco di argillenormalmente consolidate. In Proceedings of the 12th ItalianGeotechnical Conference, Cosenza, 1821 September 1975. Ital-ian Geotechnical Society (AGI), Rome, Italy. Vol. 1, pp. 223240. [In Italian.]

Burghignoli, A., and Calabresi, G. 1975. Determinazione delcoefficiente di consolidazione di argille tenere su campioni digrandi dimensioni. In Proceedings of the 12th Italian Geo-technical Conference, Cosenza, 1821 September 1975. ItalianGeotechnical Society (AGI), Rome, Italy. Vol. 3, pp. 239255.[In Italian.]

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Galati, A. 2001. Previsione dei cedimenti di banchi di terreni agrana fina indotti da variazioni cicliche sul contorno dellepressioni interstiziali a dei carichi applicati. Ph.D. thesis, Uni-versity of Catania, Catania, Italy. [In Italian.]

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Appendix A: Solution to the equationgoverning one-dimensional consolidationwhen the loading rate is unity

Under all the assumptions specified in a previous section,the differential equation governing one-dimensional consoli-dation when the rate of the external load is kept at unity is asfollows:

[A1] c uz

u

tv

=

2

21 1

where z and t are the independent variables, u is the un-known function depending on both z and t, and cv and areconstant quantities. The boundary conditions considered inthis study are

2006 NRC Canada

Conte and Troncone 1115

Fig. 13. Calculated and measured excess pore pressures versusdepth at the end of the embankment construction (28 May 1971)and just before the partial removal of the load (16 October 1973)(adapted from Bilotta and Viggiani 1975).

[A2]

=u

z0 at z = 0, t > 0

and[A3] u = 0 at z = H, t > 0and the initial condition is[A4] u = 0 at t = 0, 0 z H

To reduce eq. [A1] to a separable partial differential equa-tion, we set[A5] u z t U z t Y z( , ) ( , ) ( )= +where U is a function of z and t, and Y is a function only ofz. Because of this equation, eq. [A1] is transformed into

[A6] c Uz

cY

z

Ut

v vdd

+ =

2

2

2

21 1

Now, we notice that if

[A7] dd v

2

21 0Y

z c+ =

we have only to integrate the partial differential equation

[A8] c Uz

Ut

v

=

2

21

subject to the initial and boundary conditions expressed interms of U(z, t).

The function Y(z) satisfying eq. [A7] with the spatialboundary conditions of the problem is

[A9] Y zc

H z( ) ( )= 12

2 2

v

As a result, because of eq. [A5], the boundary and initialconditions for U(z, t) are

[A10]

=Uz

0 at z = 0, t > 0

[A11] U = 0 at z = H, t > 0and

[A12] Uc

H z= 12

2 2

v

( ) at t = 0, 0 z H

Lastly, using a solution procedure similar to that developedby Taylor (1948) to achieve the solution to the equation gov-erning Terzaghis one-dimensional consolidation, the solu-tion to eq. [A8] with eqs. [A10][A12] can be written as

[A13] U z tc H

z c tj

jjj j( , ) ( ) cos ( ) exp( )=

=

2 131

2

v

v

where j = (2j 1) /(2H). Consequently, substitutingeqs. [A9] and [A13] into eq. [A5] yields the complete solu-tion to eq. [A1], i.e.,

[A14] u z tc H

z c tj

jjj j( , ) ( ) cos ( ) exp( )=

=

2 131

2

v

v

+ 1

22 2

cH z

v

( )

Reference

Taylor, D.W. 1948. Fundamentals of soil mechanics. John Wiley &Sons, Inc., New York.

2006 NRC Canada

1116 Can. Geotech. J. Vol. 43, 2006