31

i) Deputy Director, Civil Defense and Emergency Management Division, Daegu, Korea.ii) Assistant Professor, Department of Civil Engineering, Kyung Hee University, Yongin, Korea (jyounghoon＠khu.ac.kr).iii) Professor, Department of Civil and Environmental Engineering, Seoul National University, Seoul, Korea.

The manuscript for this paper was received for review on May 20, 2009; approved on October 1, 2010.Written discussions on this paper should be submitted before September 1, 2011 to the Japanese Geotechnical Society, 4-38-2, Sengoku,Bunkyo-ku, Tokyo 112-0011, Japan. Upon request the closing date may be extended one month.

31

SOILS AND FOUNDATIONS Vol. 51, No. 1, 31–40, Feb. 2011Japanese Geotechnical Society

A STRESS PATH METHOD FOR SETTLEMENT COMPUTATION OFNORMALLY CONSOLIDATED CLAYS

CHANG-YOUB KIMi), YOUNG-HOON JUNGii) and CHOONG-KI CHUNGiii)

ABSTRACT

A stress path method for computing the settlement of normally consolidated clays is presented. The proposedmethod introduces the characterization of the stress-strain behavior that eliminates unnecessary experimental cost dueto uncertainty in the design stress path. For arbitrary control of the eŠective stress during consolidation, a technique ofback-pressure equalization, which enforces the immediate drained condition on the soil specimen without change ofthe eŠective stress state, was used. To account for nonlinear anisotropic response in computing the consolidationstrains, the iteration procedure was designed. For the proposed method, the stress path experiments on the normallyconsolidated kaolin were conducted. An example illustrating the feasibility of proposed method in precisely calculatingthe settlement is shown for a circular footing on the clay stratum.

Key words: soil settlement, soil consolidation, clay, anisotropic soils, stress path (IGC: D5/D6/E1/E2)

INTRODUCTION

Undeniably, nonlinear and anisotropic behavior ofclay in various modes of deformation should be takeninto account to accurately estimate the consolidation set-tlement. However, most geotechnical professionals stillassume one-dimensional deformation and rely on limitedinformation obtained from standard oedometer tests tosimplify the design process of computing consolidationsettlements. In the ˆeld, however, lateral deformationsare common resulting in the actual settlement to diŠerfrom the settlement computed assuming the one-dimen-sional deformation of the clay stratum. One may attemptto simulate the whole settlement process via the ˆnite ele-ment method, in which case experimental eŠorts tend tobe devoted to identifying model parameters of a speciˆcconstitutive model rather than estimating the settlementitself.

The stress path method (Ladd and Lambe, 1964;Lambe 1967; Lambe and Marr, 1979) provides a rationalway to compute both elastic compression and consolida-tion settlement of clays on an experimental basis. Thestress path method rests on the principle of eŠectivestress, thus considering all the variations of total andeŠective stresses and the excess pore water pressure. Thegeneral procedure for the stress path method (Lambe andMarr, 1979) is, (1) select the problem mechanism an en-gineer faces, (2) select `representative' soil elements forstudy, (3) prepare stress paths for the elements selected,(4) perform the stress path tests, (5) use the results of the

stress path test to obtain the soil parameters needed forpredicting performance.

The main challenge in the stress path method is in de-termining the stress increments. Obtaining the soilparameter primarily depends on a stress path, alongwhich the soil element is supposed to be stressed. Thestress path must be set up prior to performing the stresspath test. If the designed stress path is changed, the wholeprocess of the stress path test could be meaningless. Thesame di‹culty may arise if the project engineer wants toobtain a more reˆned solution with stress paths otherthan those pre-selected. A solution will be to measure soildeformation under all probable stress paths instead of asingle designed path. However, the existing procedure ofthe stress path method, which needs separate tests for ev-ery path we want, demands enormous laboratory eŠortand time.

In addition, the di‹culty in running a stress path test inthe laboratory could be a critical issue. A designed stresspath will likely require non-standard testing procedurerather than the routine tests such as triaxial compressiontest or oedometer test. In case of the settlement problem,for example, we may need to prepare experiments for twophases of undrained loading and consolidation, wherethe eŠective stress to consolidate the soil sample can noteasily be controlled (Gangopadhyay et al., 1980).

In this study, we propose a new stress path method toevaluate settlements in the clay deposit. A revision to thestress path method is made with particular considerationto attainable patterns of the stress paths in the settlement

32

Fig. 1. Typical stress path of soil elementFig. 2. Variation of total stress paths for various depths of the circular

footing

32 KIM ET AL.

problem. The work includes a systematic characterizationof overall stress-strain behavior, an iterative scheme toevaluate strains for an arbitrary stress path, and a noveltechnique for conducting the drained stress path testswith the minimum experimental eŠorts. We redesignedthe general procedure of the stress path method to copewith possible variations of the stress paths in the settle-ment problem.

The ˆrst part of the paper provides the concept of thenew stress path method and describes its detailed proce-dures. The typical stress paths expected in the settlementproblem are intensively analyzed. Secondly, the normal-ized stress-strain behavior obtained from the stress pathexperiments is presented. Finally, an illustrative exampleto compute the settlement using the proposed stress pathmethod is presented.

TYPICAL STRESS PATH IN THE SETTLEMENTPROBLEM

Figure 1 illustrates a typical stress path of a soil ele-ment in the compressible clay stratum with lowpermeability, which can be decomposed into two distinc-tive phases: (a) undrained loading (e.g., path I-U) and (b)consolidation (e.g., path U-C). Prior to loading, a soilelement in-situ remains conˆned laterally in the K0 condi-tion with the initial vertical eŠective stress, s?vi, and thehorizontal eŠective stress, s?hi, at the stress point I in Fig.1. According to Lambe and Whitman (1969), we employthe total stress invariants, p＝(sv＋sh)/2 and q＝(sv－sh)/2, for the total vertical and horizontal stresses, sv andsh, together with the eŠective stress invariants, p?＝(s?v＋s?h?)/2 and q＝(s?v－s?h)/2, for the eŠective vertical andhorizontal stresses, s?v and s?h, respectively. Note thatthere exists a diŠerent set of triaxial stress variables, p?＝(s?v＋s?h?)/3 and q＝s?v－s?h (Wood, 1990).

The ˆrst stage is the undrained loading wherein the soilelement is deformed immediately in the undrained condi-tion with an increase of the total stress by Dsv and Dsh aswell as an increase of the excess pore pressure, Due, thuschanging the total stresses along the path I–C as well asthe eŠective stresses along the path I–U. In Fig. 1, Dqu＝(Dsv－Dsh)/2 denotes the increase of q induced by theundrained loading. As the dissipation of the excess porepressure consolidates the soil element, eŠective verticaland horizontal stresses gradually increase from the stress

point, U. When total stresses do not change during con-solidation, there will be an isotropic increase of eŠectivestresses, and the eŠective stress point will follow the path-1. In actual case, the shear strength, stress-strain modulusand Poisson's ratio would gradually change due to con-solidation, which results in the decrease of total horizon-tal stress (Gangopadhyay et al., 1980), thus relocating thetotal stress point from point C to point D. Consequently,the eŠective stress path during consolidation will appearas path-2. Path-2 is likely to be nonlinear because duringthe consolidation process the eŠective stresses depend onthe excess pore water pressure, which dissipates in a non-linear fashion. In a case of the settlement problem,however, path-2 could approximate to path-3.

For the loading on a circular footing of a semi-inˆnitemass, the total stress path C–D can be examined using theelastic solutions. According to Poulos and Davis (1973),the increments of vertical and horizontal stresses, Dsv

and Dsh, induced by a uniform vertical loading on a cir-cular area can be computed by

Dsv＝p0 «1－{ 11＋(a/z)2}

3/2

$ and

Dsh＝p0

2 « (1＋2n)－2(1＋n)

(1＋(a/z)2)1/2＋1

(1＋(a/z)2)3/2$ (1)

where a is the radius of the circular footing, p0 is the pres-sure acting on the circular area, z is the depth from thesurface, and n is the Poisson's ratio of the elastic halfspace. The consolidation process will switch the soil statefrom the undrained condition to the drained condition,thus implying the change of the Poisson's ratio, n, from0.5 for the undrained condition to the drained value suchas 0.1 for clays. Figure 2 illustrates the total stress pathscomputed by Eq. (1) for the various depths along the cen-terline below the circular area of a＝25 m loaded by apressure, p0＝260 kPa. As shown in Fig. 2, the pathsC–D indicating the change in total stresses during con-solidation appear only for the shallow depth of z＝0¿30m. Figure 3 shows the ratio of the length of the stresspath C–D to the length of the path I–C, i.e., the relativemagnitude of the total stress change during consolida-tion. Even for shallow depths of z/aº1.5, the length ofthe stress path C–D reaches at most 25z or more of thelength of the path I–C. When ignoring such small non-

33

Fig. 3. Relative length of total stress path

Fig. 4. Stresses of soil element during back pressure equalization

33A STRESS PATH METHODO

linearity in path-2 in the narrow range of the stress space,the linear path-3 is the nearest and most reliable approxi-mation to the nonlinear path-2 in Fig. 1. It should be not-ed that this approximation also assumes the path-in-dependent stress-strain response of soils in the stressspace between the path-1 and 3. As shown in Fig. 1, thevariations of the vertical and horizontal eŠective stressesalong path-3 are denoted by Ds?vc and Ds?hc, as well astheir stress invariants, Dp?c＝(Ds?vc＋Ds?hc)/2 and Dqc＝(Ds?vc－Ds?hc)/2.

THE STRESS PATH METHOD

A stress path method is developed to compute settle-ment using the overall stress-strain behavior obtainedfrom the systematic experiment. To increase experimen-tal productivity, we introduce a new testing conceptwhich includes: (1) normalized stress-strain behavior bythe initial stress state, (2) unique relationship between un-drained axial loading and immediate axial strains regard-less of total stress paths, (3) equalization of the back pres-sure to the pore pressure developed during undrainedloading for the active control of eŠective stresses duringconsolidation, and (4) evaluation of strains for any likelystresses by running minimum numbers of triaxial tests.

Normalized Response of the Stress-strain BehaviorIt is well-known that normally consolidated clay ex-

hibits similar stress-strain behavior when the stresses arenormalized by the initial stress on the K0 line (Atkinsonand Bransby, 1978). This normalized behavior, which hasestablished the basis for critical state soil mechanics,would particularly be useful for the stress path methodbecause the testing data of the stress path test for an ini-tial stress point could embrace the overall stress-strainresponse for the diŠerent initial stresses. Herein, thestress path tests are conducted for a representative initialstress, from which we can establish the normalized stress-strain response.

Undrained LoadingIn the previous section, we described two distinctive

phases of the representative stress path in the settlementproblem: (i) the nonlinear stress path of the undrainedloading, and (ii) the linear stress path of the consolida-tion. Herein, it is assumed that in the undrained condi-tion, the stress point of a soil element with a given stress

history, consolidated to a given initial eŠective stresspoint will follow an approximately unique eŠective stresspath for undrained conditions for any noncyclic totalstress path. Recalling the theory of pore pressure (Skem-pton, 1954), the pore pressure parameter, A, can bemeasured for the usual undrained triaxial test in whichDsh＝0, thus obtaining A as

A＝Due－Dsh

Dsv－Dsh＝

Due

Dsv(2)

where Due is the excess pore water pressure.

Back-pressure EqualizationIn conventional triaxial testing practice, it is not as

straightforward to control eŠective stresses of a specimenin such a manner that the eŠective stress point follows anarbitrary linear path (e.g., the path-3 in Fig. 1) duringconsolidation. For example, the dissipation of excesspore pressure, usually initiated by opening a drainagevalve connected to the soil specimen, always resorts to anidentical increase of eŠective vertical and horizontalstresses (e.g., the isotropic stress path, the path-1 in Fig.1). To overcome such limitation, we employ an ex-perimental technique, namely back-pressure equaliza-tion.

Figure 4 illustrates the stress state of a soil element inthe triaxial testing procedure including back-pressureequalization. In the undrained loading, the axial stress,Dsd, is applied to increase the excess pore pressure, Due,while keeping the drainage valve closed. At this time, theback pressure, ub, remains the same as at saturation ofthe soil specimen. Prior to opening the drainage valve toconsolidate the sample, we increase the back pressure bythe same amount as the excess pore pressure. This techni-cal procedure is named as the back pressure equalization,compensating for the excess pore water pressure, Due, byincreasing back pressure, thereby enforcing the immedi-ate drained condition on the soil specimen without anychange of the eŠective stresses. Subsequently, thedrainage valve is opened so that the soil specimen can fur-ther be stressed under drained condition. The back-pres-

34

Fig. 5. Evaluating strains for consolidation loading

Fig. 6. Flow of the proposed stress path method

34 KIM ET AL.

sure equalization makes it possible to perform the testfollowing any stress paths including anisotropic stresspath such as the path-3 in Fig. 1. Additionally, it pro-vides a methodology to measure eŠective stress-strainrelationship at every stress point along the given consoli-dation path. The back-pressure equalization has an ad-vantage over the conventional measurement of eŠectivestress in the triaxial specimen equipped with a pore pres-sure transducer. The eŠective stresses computed from thepore pressure measurement will not be reliable becausethe pore pressure is locally measured at the top and bot-tom boundary of the specimen during consolidation. Inthe consolidation of the triaxial test, the eŠective stress isunlikely to be well deˆned due to non-uniform distribu-tion of the excess pore pressure within the specimen.

Characterizing Stress-strain Behavior of ClaysTo characterize the stress-strain response, we assume

that the constitutive relationship is anisotropic under thetriaxial stress condition as:

{evc

ehc}＝«AC BD${Ds?vc

Ds?hc} (3)

where evc and ehc are the vertical and horizontal strains in-duced by the vertical and horizontal stress incrementsduring consolidation, Ds?vc and Ds?hc, and A, B, C, and Dare the components of the secant compliance matrix. Thesubscript c indicates the consolidation stage. The plasticdeformation may contribute to the asymmetric compli-ance matrix so that the value of B is not necessarily equalto C. Determining four unknown parameters, A, B, C,and D, requires at least two sets of the stress-strain data,thereby establishing a linear equation as:

« (evc)I

(ehc)I

(evc)II

(ehc)II$＝«AC BD$« (Ds?vc)I

(Ds?hc)I(Ds?vc)II

(Ds?hc)II$ (4)

where the subscript I and II indicate the two diŠerent setsof the stress-strain data. Since we also assume a linearstress path during consolidation, the combination of Ds?vcand Ds?hc can be replaced with the stress ratio, Kc＝Ds?hc

/Ds?vc, and the stress invariant, Dp?c, as

«(evc)I

(ehc)I(evc)II

(ehc)II$＝«AC B

D$

21＋(Kc)I

(Dp?c)I

2(Kc)I

1＋(Kc)I(Dp?c)I

21＋(Kc)II

(Dp?c)II

2(Kc)II

1＋(Kc)II(Dp?c)II

(5)

For a target stress path with the constant stress ratio, Kc,between (Kc)I and (Kc)II, the strain components are com-puted by Eq. (3) with pre-determined values of A, B, C,and D. For clays exhibiting mild nonlinearity, one mayestimate strains by extrapolating strains for the stresspath beyond the region between (Kc)I and (Kc)II.

The compliance parameters, A, B, C, and D, however,are not constant due to nonlinear stress-strain behaviorof clays. The parameters could be estimated via an itera-tion procedure, when the strain energy density is chosenas the basis of expressing the non-linearity of soil stiŠness

(Hird and Pierpoint, 1997). For the secant modulus ofthe nonlinear stress-strain curve in the triaxial stress con-dition, the strain energy density per unit volume, W, canbe deˆned as

W＝12

(Ds?vcevc＋2Ds?hcehc) (6)

Equation (6) can alternatively be expressed as

W＝[A＋(B＋2C)Kc＋2DK 2c]

2(Dp?c)2

(1＋Kc)2 (7)

Figure 5 demonstrates the concept of the computing thecompliance parameters, A, B, C, and D, and the consoli-dation strains, evc and ehc. For the given sets of the stress-strain data for the path I and II, the iterative computa-tion of strain energy densities, WI, WII, and W, as well asthe compliance parameters, continues via Eqs. (5) and

35

Table 1. Summary of index properties and consolidation characteris-tics

Unit weight (kN/m3) 17.0Speciˆc gravity 2.59Liquid limit (z) 44.4Plasticity index (z) 23.3Percentage passing through #200 sieve (z) 98.0Uniˆed classiˆcation CLNatural water content (z) 37.6¿40.2Max. past pressure (kPa) 98¿116Compression index 0.253¿0.286Recompression index 0.110¿0.126Permeability (10－9 m/s) 5.24¿29.3

Fig. 7. Consolidation and stress probing paths

Table 2. Summary of triaxial stress path experiments

Test Dqu Kc＝Ds?hc/Ds?vc

U1-C1 0 0.57U1-C2 0 0.75U1-C3 0 1.0U2-C1 4.5 0.57U2-C2 4.5 0.75U2-C3 4.5 1.0U3-C1 9 0.57U3-C2 9 0.75U3-C3 9 1.0U4-C1 13.5 0.57U4-C2 13.5 0.75U4-C3 13.5 1.0U5-C1 18 0.57U5-C2 18 0.75U5-C3 18 1.0

35A STRESS PATH METHODO

(7), until three energy densities converge to a single value.Finally, the strains can be computed using the convergedset of the parameters, A, B, C, and D, via Eq. (3). Figure6 presents the ‰ow chart of the iteration procedure.

STRESS PATH EXPERIMENTS

SoilsTriaxial testing samples were reconstituted using slurry

consolidation technique. Dried Kaolin was pulverizedand dry-sieved through a 0.425-mm sieve to removecoarse and medium particles. The slurry with a water con-tent of 75z was placed into a large consolidometer(350-mm diameter by 600-mm height) and loaded inseveral stages until the applied pressure reached the con-solidation pressure of 100 kPa. Index properties and con-solidation characteristics of the samples from standardoedometer tests are summarized in Table 1.

Results of ExperimentsTests were performed using a CKC e/p Cyclic Loader,

an automated, feedback-controlled triaxial testing system(Chan, 1981). Triaxial specimens with a nominal di-ameter of 71 mm and a height-to-diameter ratio between2.1 and 2.3 were hand-trimmed from the reconstitutedblock. The specimens were saturated using a back pres-sure of 250 kPa over 24 hours until the B-value reached0.97. Each specimen subsequently was reconsolidated un-til the axial stress reached twice the maximum past pres-sure (i.e., 200 kPa), subjected to a undrained loading fol-lowed by the back pressure equalization, and then con-solidated with a selected stress path of Kc value. Axialstrains of the specimen were measured via the externalLVDT. Water volume expelled from the saturated speci-men is equal to the volume change of the specimen, thusproviding the horizontal strains from the relationship be-tween axial and volumetric strains. During K0 reconsoli-dation, the stresses were controlled such that the K0 valuewas at a constant 0.5 determined from preliminary K0

consolidation tests. Under undrained loading, the axialstress was incrementally increased by 3.0 kPa, and sus-tained until the rate of the axial strain was less than 0.002z/h. During consolidation (i.e., the drained loadingwith constant stress ratio Kc), mean consolidation stressesof axial and horizontal stresses were applied at a rate of

0.8 kPa/h to prevent accumulation of excess pore waterpressure within the specimen.

Figure 7 summarizes the applied stress paths. Eachstress path test was initiated at an initial stress point of s?vi

＝200 kPa and s?hi＝100 kPa. For the undrained loading,ˆve diŠerent values of Dqu, denoted by U1 to U5 in Fig.7, were chosen, implicitly simulating various shear stresslevels induced by the rapid increase of the superstructureloads on the clay deposit. At each Dqu values, three stresspath tests, denoted by C1, C2 and C3, of which Kc valuesare 0.57, 0.75, 1.00, respectively, were conducted. A totalof 15 stress path tests were performed, as listed in Table2. Note that the data for paths C1 and C3 were used todetermine the constitutive relationship of Eq. (3), whilethe strains measured along the paths denoted by C2 werecompared with the computed data for veriˆcation.

Figure 8 shows how the vertical strains, evu, and thepore pressure parameter, A's, developed during un-drained loading. Note that Dqu is normalized by the ini-tial vertical stress, s?vi. A unique relationship between evu

and Dqu/s?vi as well as between A and Dqu/s?vi was ob-served, which can provide a basis for computing the verti-cal strains and the eŠective stresses for an arbitrary levelof the undrained loading. This experimental data alsosupport the generalization of the unique eŠective stress

36

Fig. 8. Vertical strain and pore pressure parameter during undrained loading

Fig. 9. Stress-strain responses during consolidation

36 KIM ET AL.

37

Fig. 10. Contour of the strain energy density during consolidation

Fig. 11. Strain energy, vertical and horizontal strains for Kc path

37A STRESS PATH METHODO

path for undrained loading (Lambe and Marr, 1979).Figure 9 compares the measured vertical and horizon-

tal strains for the drained consolidation for Kc＝0.57,0.75, and 1.0, for the stress paths presented in Fig. 7.Even for diŠerent initial stresses, the vertical strains forthe same value of Kc are rather uniform, whereas somehorizontal strains decrease initially, then increase later.For Kc＝0.57, the highest value of Dq/Dp?, the sampleexhibits horizontal expansion with negative values of ehc,except for U1-C1. As the value of Kc increases towardsthe isotropic stress state, the horizontal strains becomepositive, thus laterally contracting the soil specimen.

Figure 10 shows the contours of the strain energy den-sity, W, for diŠerent levels of undrained loading. As canbe seen in Fig. 10, the patterns of the strain energy con-tour for the diŠerent levels of undrained loading are simi-lar. The wide contour lines become narrow for W higherthan 0.3 kJ/m3 so that signiˆcant nonlinearity of thestress-strain response takes place after the initial portionof the loading. The contours of strain energy density ex-hibit modest variation with the radial pattern of the equaldensity lines, which implies occurrence of mild non-

linearity in stress-strain behavior for the small change ofeŠective stresses in the consolidation process.

Figure 11 presents the contour plots of the strainenergy density as well as vertical and horizontal strainsfor the diŠerent drained loadings, which could simplifythe iteration procedure for determining stiŠnessparameters. As shown in Figs. 11(a) and (b), twoseparate sets of the contour plots for Kc＝0.57 and 1.0 arepresented. In the iteration procedure in Fig. 6, the con-tour plots can be used to adjust stress increments, Dp?c,satisfying newly calculated W, as well as choosing inputstress-strain data against the corresponding level of Dp?c.

The iteration procedure for computing consolidationstrains can be veriˆed by selecting the data set for Kc＝0.57 and 1.0 as the input, and subsequently computingthe strains for Kc＝0.75 via the procedure in Fig. 6.Figure 12 compares computed and measured values ofstrains for Kc＝0.75. Computed strains agree surprisinglywell with measured strains for every level of the un-drained loading. The iteration procedure provides a sim-ple but practical tool to compute consolidation strainswithout either a complicated constitutive model or a com-plicated stress probe test.

AN EXAMPLE OF EVALUATINGCONSOLIDATION SETTLEMENT

An example of evaluating settlement of a circular foot-ing with the radius of 5 m is presented. As illustrated in

38

Fig. 12. Comparison of computed and measured strains for C2 paths

Fig. 13. Circular footing on the clay stratum

38 KIM ET AL.

Fig. 13, it is assumed that a 13 m-deep clay stratum isfully saturated and the water table is located on theground surface. The properties of the clay are assumed asthe values in Table 1. In the middle of the clay stratum atthe depth of 6.5 m, the in-situ eŠective stresses for K0＝0.5, s?vi and s?hi, are 46.7 and 23.4 kPa, respectively. Theapplied pressure of the circular footing on the groundsurface, p0, is 15 kPa. Elastic solutions of Eq. (1) yieldthe stress increments of (Dsv)u＝7.55 kPa and (Dsh)u＝0.90 kPa, for the undrained Poisson's ratio of 0.5 in themiddle of the stratum (i.e., a/z＝0.77 in Eq. (1)). Thesubscript u in the stress increments denotes the undrainedcondition. Note that it is a common practice to use theelastic solution of Eq. (1) even for a layer with ˆnitethickness (Terzaghi et al., 1996). Similarly, assuming thePoisson's ratio of 0.3 for the fully consolidated clay, the

39

Table 3. Iteration procedure in computing consolidation strain

Iteration Input stress increment Normalized stress increment Computed strain (z)

1st

(Dp?c)I (Dp?c)II(Dp?c)I

s?vi

(Dp?c)II

s?vi(evc)I (ehc)I (evc)II (ehc)II

4.188 4.188 0.090 0.090 0.406 －0.067 0.123 0.034

StiŠness parameters (MPa－1) Strain energy density (kJ/m3)A B C D W WI WII

1.381 －1.234 －0.397 0.438 0.283 0.725 0.200

2nd

(Dp?c)I (Dp?c)II(Dp?c)I

s?vi

(Dp?c)II

s?vi(evc)I (ehc)I (evc)II (ehc)II

2.617 4.982 0.056 0.107 0.254 －0.043 0.149 0.043

StiŠness parameters (MPa－1) Strain energy density (kJ/m3)A B C D W WI WII

1.378 －0.736 －0.416 0.603 0.931 0.487 1.260

10th

(Dp?c)I (Dp?c)II(Dp?c)I

s?vi

(Dp?c)II

s?vi(evc)I (ehc)I (evc)II (ehc)II

3.190 4.469 0.068 0.096 0.310 －0.052 0.132 0.037

StiŠness parameters (MPa－1) Strain energy density (kJ/m3)A B C D W WI WII

1.380 －1.015 －0.408 0.510 0.563 0.568 0.552

20th

(Dp?c)I (Dp?c)II(Dp?c)I

s?vi

(Dp?c)II

s?vi(evc)I (ehc)I (evc)II (ehc)II

3.211 4.458 0.069 0.095 0.312 －0.052 0.132 0.037

StiŠness parameters (MPa－1) Strain energy density (kJ/m3)A B C D W WI WII

1.380 －1.022 －0.408 0.508 0.555 0.554 0.555

Finalstrains

evc＝ADsvc＋BDshc＝1.380 MPa－1×4.49 kPa－1.022 MPa－1×3.89 kPa＝0.22zehc＝CDs?v＋DDs?h＝－0.408 MPa－1×4.49 kPa＋0.508 MPa－1×3.89 kPa＝0.01z

39A STRESS PATH METHODO

elastic solutions yield an increase of vertical and horizon-tal eŠective stresses, (Dsv)d＝7.55 kPa and (Dsh)d＝0.30kPa. The subscript d in the stress increments denote thedrained or fully consolidated condition. The settlement atthe center of the circular footing can be computed via thefollowing steps:(1) For Dqu/s?vi＝((Dsv)u－(Dsh)u)/2s?vi＝0.071, the data

in Fig. 8 yield the vertical strain, evu＝1.12z, and thepore pressure parameter, A＝ 0.54. Total increase ofexcess pore pressure during undrained loading is

Due＝A((Dsv)u－(Dsh)u)＋(Dsh)u＝4.49 kPa (8)

(2) The change in the eŠective stresses during consolida-tion, Dsvc＝4.49 kPa and Dshc＝3.89 kPa are com-puted via the following relationships as:

Ds?vc＝(Dsv)d－(Dsv)u＋Due

＝7.55－7.55＋4.49＝4.49 kPa (9)Ds?hc＝(Dsh)d－(Dsh)u＋Due

＝0.30－0.90＋4.49＝3.89 kPa (10)

Thus, Kc＝Ds?hc/Ds?vc＝0.87 and Dp?c＝(Ds?vc＋Ds?hc)/2＝4.19 kPa

(3) For Kc＝0.87 and Dp?c/s?vi＝4.19/46.7＝0.090, thevertical strain developed during consolidation, evc, inthe middle of the stratum is estimated to be 0.22z viathe iteration procedure illustrated in Fig. 6. Duringiteration, the input strain data can be obtained fromthe contours of Fig. 11 for Dqu/s?vi＝0.071. The input

stress increments, (Dp?c)I and (Dp?c)II, are updated ateach iteration via

(Dp?c)I＝W(1＋(Kc)I)2

2(A＋(B＋2C)(Kc)I＋2D(Kc)2I)

and

(Dp?c)II＝W(1＋(Kc)II)2

2(A＋(B＋2C)(Kc)II＋2D(Kc)2II)

(11)

where W is calculated for the target value of Kc＝0.87using Eq. (7). Table 3 provides the details to computeevc. The values of WI and WII are calculated using Eq.(7) with (Kc)I＝0.57 and (Kc)II＝1.00, respectively.The iteration continues until maximum error betweenW, WI and WII is less than 0.1z.

(4) The settlement, r, is ˆnally computed by integratingvertical strains, evu and evc, as,

r＝fH

0evdz§(evu＋evc)H＝(1.12z＋0.22z)

×13 m＝17 cm (12)

where ev is the total vertical strains induced by both un-drained and consolidation loadings, and H is the depth ofthe stratum.

For further precise computation, the clay stratum canbe divided into several layers. In the original stress pathmethod, such division requires separate tests to estimatethe strains for each layer. Due to the characterization ofthe stress-strain behavior, herein, the proposed method

4040 KIM ET AL.

has a certain advantage over the original method. Thestrains in a divided layer can simply be estimated from theexisting experimental data of the stress path tests.

The proposed method would possess some limitationsfor its further application: (i) The characterization of thestress-strain behavior and its application would only bevalid for the normally consolidated clays, because there isno concept of pre- and post-yield response. However, thepresent method may marginally be applied to lightlyoverconsolidated clays, which usually exhibit minor pre-yield deformation. (ii) The stress invariants, p?, p and q,are chosen for the triaxial stress condition, thus ignoringthe eŠect of the intermediate stress in ˆeld. (iii) The stressincrements for consolidation loading are computed viaelastic solutions, which might diŠer from actual stressvariations in ˆeld. Even though we still have several issueslisted above, it could be emphasized that the proposedmethod will certainly enhance the practical use of triaxialtesting data that appears abandoned in the routine geo-technical design in the settlement problem.

CONCLUSIONS

This paper described a new stress path method for eval-uating settlements in saturated clay based on the originalconcept of Lambe (1967). The revision of the method wasmade with the special consideration of possible patternsof the stress paths in the settlement problem. Theproposed method introduces the characterization of thestress-strain behavior conducted prior to computing thesettlement, thereby reducing experimental cost in applica-tion of the stress path method. To control the eŠectivestresses during consolidation, a new experimental tech-nique, namely back-pressure equalization, was suggested.To account for nonlinear anisotropic response of thestress-strain relationship during consolidation, the itera-tion procedure taking advantage of characterized soilresponses was designed.

The stress path experiments on artiˆcially sedimentednormally consolidated kaolinite were conducted, whichprovided basic information on nonlinear and anisotropicstress-strain response to assess strains for an arbitraryvariation of eŠective stresses during consolidation. Theiteration procedure in computing consolidation strains

was veriˆed by comparing the computed and measureddata. The example of calculating the settlement for animaginary circular footing on the clay deposit has beenshown, which illustrates applicability and versatility ofthe proposed method in precise calculation of the settle-ments in the saturated clay deposit.

ACKNOWLEDGEMENT

This work was supported by the National ResearchFoundation of Korea (NRF) grant funded by the Koreagovernment (MEST) (No. 2009–0068085) and the En-gineering Research Institute of Seoul National Univer-sity. The authors would like to thank the reviewers fortheir valuable comments.

REFERENCES

1) Atkinson, J. H. and Bransby, P. L. (1978): The Mechanics of Soils:an Introduction to Critical State Soil Mechanics, McGraw-HillBook Co., London, New York.

2) Chan, C. K. (1981): An electropneumatic cyclic loading system,Geotechnical Testing Journal, 4(4), 183–187.

3) Gangopadhyay, C. R., Das, S. C. and Som, N. N. (1980): Stress-path in‰uence on drained deformations of clay, Journal of the Geo-technical Engineering Division, ASCE, 106(11), 1243–1260.

4) Hird, C. C. and Pierpoint, N. D. (1997): StiŠness determinationand deformation analysis for a trial excavation in Oxford Clay,Geotechnique, 47(3), 665–691.

5) Ladd, C. C. and Lambe, T. W. (1964): Strength of ``undisturbed''clay determined from undrained tests, Symposium of LaboratoryShear Testing of Soils, ASTM, 361, 342–371.

6) Lambe, T. W. (1967): The stress-path method, Journal of SoilMechanics and Foundation Engineering, ASCE, 93(GT6), 309–331.

7) Lambe, T. W. and Whitman, R. V. (1969): Soil Mechanics, Wiley,New York.

8) Lambe, T. W. and Marr, W. A. (1979): Stress-path method: Sec-ond edition, Journal of Geotechnical Engineering, ASCE,105(SM6), 727–738.

9) Poulos, H. G. and Davis, E. H. (1973): Elastic Solutions for Soiland Rock Mechanics, Wiley, New York.

10) Skempton, A. W. (1954): The pore pressure coe‹cient A and B,Geotechnique, 4, 143–147.

11) Terzaghi, K., Peck, R. B. and Mesri, G. (1996): Soil Mechanics inEngineering Practice, John Wiley & Sons, New York.

12) Wood, D. M. (1990): Soil Behaviour and Critical State SoilMechanics, Cambridge University Press.