Strength mobilization in clays and silts P.J. Vardanega and M.D. Bolton Abstract: A large database of 115 triaxial, direct simple shear, and cyclic tests on 19 clays and silts is presented and ana- lysed to develop an empirical framework for the prediction of the mobilization of the undrained shear strength, c u , of natural clays tested from an initially isotropic state of stress. The strain at half the peak undrained strength (g M=2 ) is used to normal- ize the shear strain data between mobilized strengths of 0.2c u and 0.8c u . A power law with an exponent of 0.6 is found to describe all the normalized data within a strain factor of 1.75 when a representative sample provides a value for gM=2. Multi-linear regression analysis shows that gM=2 is a function of cu, plasticity index Ip, and initial mean effective stress p 0 0 . Of the 97 stress–strain curves for which c u , I p , and p 0 0 were available, the observed values of g M=2 fell within a factor of three of the regression; this additional uncertainty should be acknowledged if a designer wished to limit immediate founda- tion settlements on the basis of an undrained strength profile and the plasticity index of the clay. The influence of stress his- tory is also discussed. The application of these stress–strain relations to serviceability design calculations is portrayed through a worked example. The implications for geotechnical decision-making and codes of practice are considered. Key words: clays, silts, mobilized strength, correlation and normalization. Résumé : Une base de données contenant 115 essais de cisaillement triaxiaux et simples directs, ainsi que des essais cycli- ques sur 19 argiles et silts, est présentée et analysée dans le but de développer un cadre empirique pour la prédiction de la mobilisation de la résistance au cisaillement non drainé c u d’argiles naturelles testées à un état des contraintes initial iso- trope. La déformation à la demie de la résistance maximale non drainée (g M=2 ) est utilisée pour normaliser les données de déformation en cisaillement entre des résistances mobilisées de 0,2 c u et 0,8 c u . Une loi de puissance avec un exposant de 0,6 a été déterminé pour décrire toutes les données normalisées à l’intérieur d’un facteur de déformation de 1,75 lorsqu’un échantillon représentatif donne une valeur pour g M=2 . Une analyse en régression multilinéaire démontre que g M=2 est une fonction de c u , de l’indice de plasticité I p et de la contrainte effective moyenne initiale p 0 0 . Parmi les 97 courbes de contrainte–déformation pour lesquelles cu, Ip et p 0 0 sont disponibles, les valeurs observées de gM=2 sont à l’intérieur d’un fac- teur de 3 de la régression; cette incertitude additionnelle devrait être considérée si un concepteur désire limiter les tasse- ments immédiats de la fondation sur la base d’un profil de résistance non drainé et de l’indice de plasticité de l’argile. L’influence de l’historique des contraintes est aussi discutée. L’application de ces relations de contrainte–déformation pour les calculs de conception de l’utilisation est illustrée par un exemple. Les implications pour la prise de décision et les codes de pratique géotechniques sont considérées. Mots‐clés : argiles, silts, résistance mobilisée, corrélation et normalisation. [Traduit par la Rédaction] Introduction The prediction of strains and displacements is of increas- ing concern to a geotechnical engineer. The data of nonlinear stress–strain behaviour is conventionally presented in terms of shear modulus reduction curves of G/G 0 (where G is the secant shear modulus and G 0 is the linear elastic shear stiff- ness) versus the logarithm of shear strain (e.g., Hardin and Drnevich 1972). On a plot of shear stress versus shear strain, the data is usually fitted with a modified hyperbola; a recent review for clays has been undertaken by Vardanega and Bol- ton (2011). A significant practical obstacle to the application of this approach is that G 0 is rarely known. Furthermore, published data of G/G 0 often derive from resonant column (RC) tests in which strains are usually restricted to not much more than 0.1%, which is at the lower extremity of strains experienced in practical applications. The approach adopted in the current work focuses on “moderate” strains in excess of 0.1%, and normalizes stress using the undrained strength of the clay rather than its elastic stiffness. Geotechnical engineers designing structures on clay gener- ally focus on undrained strength as the key soil parameter. Ground investigations in such circumstances usually include borings from which disturbed samples are taken to determine water contents in relation to Atterberg limits. Additional probing may include standard penetration tests (SPTs). These routine tests are sometimes used to define design strengths through empirical correlations. More commonly, they are used to assess the variability of clay strength and plasticity in the region of interest, while “undisturbed” cores or in situ tests are used to define spot values of undrained strength or compressibility. The objective of this work is to enhance the foregoing by predicting the shape of the undrained stress– strain curve of clays so that this may be conveniently used in Received 15 April 2011. Accepted 19 July 2011. Published at www.nrcresearchpress.com/cgj on 29 September 2011. P.J. Vardanega and M.D. Bolton. Cambridge University, Schofield Centre, Department of Engineering, High Cross, Cambridge CB3 0EL, UK. Corresponding author: P.J. Vardanega (e-mail: [email protected]. uk). 1485 Can. Geotech. J. 48: 1485–1503 (2011) doi:10.1139/T11-052 Published by NRC Research Press Can. Geotech. J. Downloaded from www.nrcresearchpress.com by UNIVERSITY OF CAMBRIDGE on 10/05/11 For personal use only.
Transcript
Strength mobilization in clays and silts
P.J. Vardanega and M.D. Bolton
Abstract: A large database of 115 triaxial, direct simple shear, and cyclic tests on 19 clays and silts is presented and ana-lysed to develop an empirical framework for the prediction of the mobilization of the undrained shear strength, cu, of naturalclays tested from an initially isotropic state of stress. The strain at half the peak undrained strength (gM=2) is used to normal-ize the shear strain data between mobilized strengths of 0.2cu and 0.8cu. A power law with an exponent of 0.6 is found todescribe all the normalized data within a strain factor of 1.75 when a representative sample provides a value for gM=2.Multi-linear regression analysis shows that gM=2 is a function of cu, plasticity index Ip, and initial mean effective stress p00.Of the 97 stress–strain curves for which cu, Ip, and p00 were available, the observed values of gM=2 fell within a factor ofthree of the regression; this additional uncertainty should be acknowledged if a designer wished to limit immediate founda-tion settlements on the basis of an undrained strength profile and the plasticity index of the clay. The influence of stress his-tory is also discussed. The application of these stress–strain relations to serviceability design calculations is portrayedthrough a worked example. The implications for geotechnical decision-making and codes of practice are considered.
Key words: clays, silts, mobilized strength, correlation and normalization.
Résumé : Une base de données contenant 115 essais de cisaillement triaxiaux et simples directs, ainsi que des essais cycli-ques sur 19 argiles et silts, est présentée et analysée dans le but de développer un cadre empirique pour la prédiction de lamobilisation de la résistance au cisaillement non drainé cu d’argiles naturelles testées à un état des contraintes initial iso-trope. La déformation à la demie de la résistance maximale non drainée (gM=2) est utilisée pour normaliser les données dedéformation en cisaillement entre des résistances mobilisées de 0,2 cu et 0,8 cu. Une loi de puissance avec un exposant de0,6 a été déterminé pour décrire toutes les données normalisées à l’intérieur d’un facteur de déformation de 1,75 lorsqu’unéchantillon représentatif donne une valeur pour gM=2. Une analyse en régression multilinéaire démontre que gM=2 est unefonction de cu, de l’indice de plasticité Ip et de la contrainte effective moyenne initiale p00. Parmi les 97 courbes decontrainte–déformation pour lesquelles cu, Ip et p00 sont disponibles, les valeurs observées de gM=2 sont à l’intérieur d’un fac-teur de 3 de la régression; cette incertitude additionnelle devrait être considérée si un concepteur désire limiter les tasse-ments immédiats de la fondation sur la base d’un profil de résistance non drainé et de l’indice de plasticité de l’argile.L’influence de l’historique des contraintes est aussi discutée. L’application de ces relations de contrainte–déformation pourles calculs de conception de l’utilisation est illustrée par un exemple. Les implications pour la prise de décision et les codesde pratique géotechniques sont considérées.
Mots‐clés : argiles, silts, résistance mobilisée, corrélation et normalisation.
[Traduit par la Rédaction]
IntroductionThe prediction of strains and displacements is of increas-
ing concern to a geotechnical engineer. The data of nonlinearstress–strain behaviour is conventionally presented in termsof shear modulus reduction curves of G/G0 (where G is thesecant shear modulus and G0 is the linear elastic shear stiff-ness) versus the logarithm of shear strain (e.g., Hardin andDrnevich 1972). On a plot of shear stress versus shear strain,the data is usually fitted with a modified hyperbola; a recentreview for clays has been undertaken by Vardanega and Bol-ton (2011). A significant practical obstacle to the applicationof this approach is that G0 is rarely known. Furthermore,
published data of G/G0 often derive from resonant column(RC) tests in which strains are usually restricted to not muchmore than 0.1%, which is at the lower extremity of strainsexperienced in practical applications. The approach adoptedin the current work focuses on “moderate” strains in excessof 0.1%, and normalizes stress using the undrained strengthof the clay rather than its elastic stiffness.Geotechnical engineers designing structures on clay gener-
ally focus on undrained strength as the key soil parameter.Ground investigations in such circumstances usually includeborings from which disturbed samples are taken to determinewater contents in relation to Atterberg limits. Additionalprobing may include standard penetration tests (SPTs). Theseroutine tests are sometimes used to define design strengthsthrough empirical correlations. More commonly, they areused to assess the variability of clay strength and plasticityin the region of interest, while “undisturbed” cores or in situtests are used to define spot values of undrained strength orcompressibility. The objective of this work is to enhance theforegoing by predicting the shape of the undrained stress–strain curve of clays so that this may be conveniently used in
Received 15 April 2011. Accepted 19 July 2011. Published atwww.nrcresearchpress.com/cgj on 29 September 2011.
P.J. Vardanega and M.D. Bolton. Cambridge University,Schofield Centre, Department of Engineering, High Cross,Cambridge CB3 0EL, UK.
Can. Geotech. J. 48: 1485–1503 (2011) doi:10.1139/T11-052 Published by NRC Research Press
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simplified deformation calculations. Shear strength mobiliza-tion with shear strain is an alternative way of looking at theconcept of engineering factors of safety. Factors of safety onundrained shear strength are often quoted in working-stressand limit-state design methods and codes of practice, butwithout a link being made to the implied strain level.A mobilization factor is specified to reduce the strains in
the soil around the structure. This paper presents a detaileddatabase of 115 triaxial, direct simple shear (DSS), and RCtests on 19 clays and silts. A novel way of normalizing theirmobilization curves is demonstrated with a view to perform-ing design calculations that deal explicitly with the service-ability criterion in limit-state design.
Statistical analysisWhen performing a regression analysis, the coefficient of
determination (R2) value alone does not give sufficient infor-mation to determine the validity of the correlation. In addi-tion to a scatter plot showing the original data, the followingpertinent statistical measures have been used in the analysespresented later in the paper:
• n, number of data points used in the regression• p-value (or p), probability of a correlation not existing• SE, standard error.This methodology is similar to that used in Kulhawy andMayne (1990).
Undrained shear strengthThe soil mechanics literature on undrained shear strength
has two distinct perspectives. Many early papers were con-cerned with empirical correlations that would allow practis-ing engineers to estimate strength based on elementaryclassifications or probings (e.g., Atterberg limits, vane sheartests, and SPTs). In the 1960s and thereafter, however, theemergence of critical state soil mechanics (CSSM) (Schofieldand Wroth 1968) fostered a fundamental understanding thatclarified the relationship between undrained and drainedshear strength and that provided theoretical relationships be-tween undrained strength and overconsolidation ratio (OCR),for example. Subsequent authors have done much to rational-ise soil test and classification data within the broader CSSMframework, e.g., Muir Wood (1990). In this way, the earlierempirical findings have been generalized to cover most typesof element test, and have therefore become more widely ap-plicable.
Normally consolidated clayThe undrained shear strength, cu, is the obvious parameter
to normalize the mobilized shear strength, tmob. It can bemeasured directly or predicted using established correlations.Skempton’s correlation (Skempton 1954, 1957) for the shearstrength of normally consolidated soils as a function of plas-ticity index is often used
½1� cu
s 0v;0
¼ 0:11þ 0:37Ip
where s 0v;0 is the in situ vertical effective stress and Ip is the
plasticity index. Muir Wood (1990) shows that there is appre-ciable scatter around eq. [1] for a wider variety of clays.
Overconsolidated clayOCR has a significant effect on undrained shear strength.
Ladd et al. (1977) on empirical grounds and Muir Wood(1990) additionally from theoretical relations based on crit-ical state soil mechanics both show
½2� cu=s0vi
ðcu=s 0viÞnc
¼ ðOCRÞL
where s 0vi is the vertical effective stress, nc indicates normal
consolidation, OCR is the overconsolidation ratio (or, morestrictly, yield stress ratio), and L varies from 0.85 to 0.75 asOCR increases.
Correlations with liquidity index (IL)Muir Wood (1983) gives a correlation for undrained shear
strength (cu) based on liquidity index, which can implicitlyallow for the reduction of water content by overconsolidation,but is more convenient as it is available through disturbedsoil samples.
½3� cu ¼ 170 e�4:6IL kPa
Correlations with SPT N60 valuesFor standard site investigation the SPT test is often con-
ducted, allowing estimates to be made of cu varying withdepth. Hara et al. (1974) gives a correlation for cu with SPTblowcount for a database of cohesive soils. The majority ofthe soils in the database were reported to have void ratiosranging from 1.0 to 2.0. The OCR for the soils in the data-base was reported to vary from 1.0 to 3.0.
½4� cu ¼ 29ðN60Þ0:72 kPa OCR < 3:0
where N60 is the SPT blowcount. Stroud (1974) showed thatplasticity index influences cu/N60 for stiff clays. Reid andTaylor (2010) comment that Stroud’s chart does not show astatistical analysis of the data. The optimum power curve(eq. [5]) is fitted to the data (reproduced as Fig. 1) whichconfirms that there is a correlation, but with a flatter curvethan that proposed by Stroud (1974).
½5� cu ¼ 10N60ðIpÞ�0:22 kPa
R2 ¼ 0:37; n ¼ 53; SE ¼ 1:14; p < 0:001
AnisotropyIt is well-known that the undrained strength of clay de-
pends on the mode of shearing, e.g., Mayne (1985). Data onthe small strain stiffness of some clays is now also known todisplay anisotropy, e.g., Graham and Houlsby (1983); Lingset al. (2000); Gasparre (2005). However, there is as yet nodatabase available that permits the generalization of degreeof anisotropy at different strain magnitudes for differentclays. The approach adopted in this paper is to use the dataof shear strength to normalize the shear stresses consistentwith moderate strains. In applying the results, engineersshould ideally seek data for undrained shear strength ob-tained in a test mode appropriate to the problem, or coulduse the correlations between test types presented in Mayne(1985).
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Database of strength mobilizationData was found for triaxial, RC, and DSS tests on natural
clay specimens subjected to consolidated isotropic undrained(CIU) shearing. In all cases the sample was taken from zeroshear stress to failure. Additional Ko-consolidated tests arediscussed later. Table 1 summarizes the publications, claytypes, test apparatus, and number of tests available for inclu-sion in the dataset after digitization of the original test plots,or the input of filtered raw data of London clay in the case ofYimsiri (2002) and Gasparre (2005). Some data in the 16publications was not used in the study due to the publishedcurves being unreadable for the purposes of digitization.Some tests (seven out of 122) were available, but were never-theless excluded from the database for a variety of reasons,which are outlined in Table 1. Figure 2 shows a plot of theexcluded test data.The variety of test types included in the database in Table 1
might have been thought to be a drawback to the creation ofuseful correlations. This will be shown not to be the case. Nostatistical difference was found between the values of the keycurve-fitting parameter determined for different test catego-ries; see Table 2. Rather than a drawback, the merging of dif-ferent test data is a significant advantage as the results of thecorrelations will be more generally applicable to the data ofundrained strength, cu. The use as a normalizing parameterof cu, determined from the peak strength in any given test, isassumed to automatically filter out anisotropic effects fromthe correlations.Engineers may wish to make judgements about the strain
that would be experienced at some mobilized shear stress,tmob, in relation to the peak undrained shear strength, cu.The strength tmob mobilized at shear strain g was identifiedas Gg. Many of the tests show deviator stress, q, versus axialstrain, 3a. For the purposes of this paper, shear strain and mo-bilized shear strength are defined as, respectively
½6� g ¼ 1:53a
½7� tmob ¼ 0:5q
BSI (1994) describes the quantity cu/tmob as the mobiliza-tion factor, M, which is equivalent to a factor of safety onshear strength.
Analysis of databaseThe collected database comprises 115 stress–strain curves
from 16 publications describing a variety of test types. Thisvariety will be an advantage in the application of the empiri-cal correlations that follow, as the same framework is shownto fit irrespective of test method. Figure 3 shows an exampleof the Todi clay stress–strain data at various confiningstresses (Burland et al. 1996). Plots were made of tmob/cu (=1/M) versus shear strain for the 19 clays (115 tests) beingconsidered. Power laws were fitted to the data points that cor-responded to 1.25 ≤ M ≤ 5 for each test curve in the data-base. This region is referred to by the authors as themoderate-strain region. The reason for excluding the data inthe low-strain region (M > 5) is partly because it is difficultto resolve low-strain measurements, and partly because suchdeterminations are best made in relation to the small-strainshear modulus, G0 (Vardanega and Bolton 2011). Data in thehigh-strain region (M < 1.25) was excluded as the shapes ofthe test curves immediately pre- and post-peak display an ex-ceptionally high degree of variability, presumably due toyielding, softening, and strain localization. In this region, theprediction of settlements is almost irrelevant as the clay is ap-proaching failure.Power curves are useful for curve-fitting to engineering
data as they have only two regression constants, are straightlines on log–log plots, and pass through the point (0,0),which is a necessary condition for many physical phenom-ena. The power law model used in the subsequent analysis is
½8� tmob
cu¼ AðgÞb
where log(A) is the intercept of the best-fit linear line throughthe stress–strain data plotted on log–log axes and b is theslope. The Todi clay data from Fig. 3 is shown again inFig. 4, fitted with power curves through the moderate-strain
Fig. 1. Relationship between cu/N and Ip for a variety of clays (re-plotted from Stroud 1974).
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region. The curve-fitting parameters for the database are sum-marized in Appendix A1.The exponent b determined for each test (CIU, DSS, cyclic
triaxial, and RC) is given in Fig. 5, plotted against plasticity
index, Ip, for the 115 test curves. It is clear that the scatter isexceptionally high and no correlation is present. No correla-tion was found using liquid limit, wL; plastic limit, wP; Ip orIL; water content, w; or initial mean effective stress, p00. It can
Table 1. Summary of database of strength mobilization.
Source Clay type Test typeNo. of tests includedin the database Comments
Bjerrum and Landva (1966) Manglerud quick DSS 3 Data from Fig. 8 excluded as only twopoints in moderate strain regionavailable after digitization
Moh et al. (1969) Bangkok (weathered,soft, and stiff)
CIU 12 Test at confining stress of 51 kPaexcluded as only two points inmoderate strain region available afterdigitization
Koutsoftas (1978) Plastic CU cyclic–static 3 Inorganic marine clays from NewJersey. Only the pseudo-static testshave been used in this database.
Silty CU cyclic–static 2 —Clough and Denby (1980) San Francisco Bay
mudCIU 3 —
Lefebvre and LeBoeuf (1987) Grande Baleine OC CIU 5 —Grande Baleine NC 5 —Olga OC 4 —Olga NC 4 —
Díaz-Rodriguez et al. (1992) Mexico City CIU 2 —Shibuya and Mitachi (1994) Hachirōgata Cyclic triaxial 6 Test T1 of Hachirōgata clay was
rejected as only one data point abovecu/5 was present after digitization
Burland et al. (1996) Todi CIU 7 —Yimsiri (2002) London CIU 6 Raw data files provided
Tests E1 and E2 were not included asthe specimen was cored horizontally.The power-law fitting described laterin the paper was, however, found tobe equally applicable. Data not shownon Fig. 2.
Callisto and Rampello (2004) Vallericca (Italy) CIU 9 —Futai et al. (2004) Ouro Preto (Brazil) CIU 8 Test curves from Fig. 3 unable to be
digitized so only the tests that wereconducted on the sample that wastaken from a depth of 5 m areincluded in the database (Fig. 4).
Marques et al. (2004) St-Roch-de-l’Achigan CIU 3 Testing at three temperaturesGasparre (2005) London CIU 5 Raw data files provided
Test t19 was removed as the authorreported the existence of a pre-existingfissure
Test t33 was removed due to an anom-alous double peak in the stress–straincurve
Lunne et al. (2006) Osnøy DSS 6 CAUC tests in the paper were not usedin the database
Drammen DSS 3 Two tests on Drammen clay from 17.58and 17.85 m depths were excluded asthe fitting of a power law provedinvalid in the strain region of interest.
be concluded that b is more likely to be explained by struc-ture, fabric, the presence or absence of fissures, samplingtechnique, and general sample condition.Table 2 shows the average b-values for the three test cate-
gories in the database. The standard deviation and number oftests is also shown. The average b-value is 0.60 and the rangeplus or minus one standard deviation of the mean is approx-imately 0.45 to 0.75. This range captures the b-value of mostclays. The collected data fitted with power curves in themoderate-strain region yield A-values ranging from 2.79 to455.9 with an average A of 16.9 and a range of exponents bfrom 0.3 to 1.2. Figures 6a, 6b, and 6c show this range of b-values for A = 1, 10, and 100, respectively. This demon-strates the variability that can exist between different claysand tests. Figure 7 shows all the moderate-strain region dataplotted for the entire database.
Mobilization strainA variety of b-values describe the stress–strain data of the
individual clays in the database, the average value being b =0.6. It was decided to accept this as the best value for predic-tion. In addition, a pivot strain was used to normalize thestrain axis. This pivot point was taken as the strain levelwhen M = 2, denoted as gM=2. This strain level is referred toby the authors as the mobilization strain. Equation [8] istherefore modified, and becomes
½9� tmob
cu¼ 0:5
g
gM¼2
� �0:6
where tmob/cu is the inverse of the mobilization factor (1/M).Figure 8 shows the measured values of tmob/cu plotted
against those predicted using eq. [9]. The resulting regressionis
½10� tmob
cu
� �¼ 0:49
g
gM¼2
� �0:6
R2 ¼ 0:90; n ¼ 1365; p < 0:001
The coefficient in eq. [10] is 0.49, rather than the 0.5 asdefined in eq. [9], because of the decision to lock the b-valueat 0.60. Equations [9] and [10] are operationally identical andit is evident that eq. [9] successfully normalizes the shearstrain data in the database. The regression model has a coef-ficient of determination R2 of 0.90; in other words 90% ofthe variation in the data can be explained using the best-fitmobilization strain gM=2 and the average b-value of 0.60.Equation [9] effectively offers a one-parameter model fornonlinear kinematic hardening inside the volumetric yieldsurface. Jardine (1992) describes this as behaviour lying be-tween the Y2 and Y3 yield surfaces in a nested yield surfacevisualization. The need, in Fig. 4 and subsequently, to im-pose the lower limit tmob/cu > 0.2 on the chosen moderate-strain range must partly reflect the initially linear elastic be-haviour at small strains within what Jardine describes as theY1 yield surface. The upper limit tmob/cu < 0.8 of the chosenrange, within which eq. [9] has been shown to be useful, istaken to reflect the onset of nonlinear plastic behaviour, de-scribed by Jardine in terms of approaching the Y3 yield sur-face. These limits are shown in Fig. 9a to be useful indefining the moderate-strain region for the database. Theymust obviously be taken as approximations as the shapesand relative locations of the Y1, Y2, and Y3 yield surfacesmust be soil and stress-history dependent.Figure 4 is typical of a set of stress–strain curves at differ-
ent confining pressures, in that the variation in gM=2 is muchmore significant than the variation in exponent b. Use of themobilization strain has been shown to be effective in reduc-ing the error in prediction of tmob/cu as shown in Fig. 8. Thenormalized stress–strain data are shown for the whole data-
Fig. 2. Tests excluded from the database.
Table 2. Analysis of test categories in the database.
base in Fig. 9a, and again in Fig. 9b using log–log axes. Thesmall scatter in the vicinity of the pivot point tmob/cu = 0.5 isdue to random error introduced either by digitizing thestress–strain curves published by the authors listed in Table 1or noise in their original test data. The factor error incurredby using eq. [9] is seen in Fig. 8 to be generally no morethan a factor 1.4 on stresses at a given normalized strainwithin the chosen mobilization interval and, correspondingly,no more than a factor 1.75 on normalized strain at a givenstress, as seen in Fig. 9b. Although four out of 19 clays andsilts have at least one point on their stress–strain curve lyingoutside these bounds, this only applies to about 1% of the to-tal number of digitized data points. It is also evident thatmost of these troublesome points lie on the conservative sideof prediction (eq. [9]), and none of them refer to low mobili-zation factors M < 4.
Predicting mobilization strainMultiple regression analysis was used in an attempt to dis-
cover the significant parametric influences on the referencestrain gM=2, and to arrange the key parameters in appropriategroups for the purposes of prediction. Some of the tests inthe database were found to be atypical in that they werefound to have gM=2 values that remained as outliers which-ever correlation was attempted. The Manglerud quick clay isbest characterized as highly structured inorganic clayey siltwith a very low plasticity index of 8%; it was also excludedfrom the regression analysis. Some of the publications didnot give sufficient information to determine appropriate val-ues for p00 (San Francisco Bay mud, Osnoy clay, Drammenclay, St-Roch-de-l’Achigan clay); these were necessarily ex-cluded from the analysis. The subsequent analysis relates to14 of the original 19 clays in the database.
Fig. 3. Todi clay data (digitized and re-plotted from Burland et al. 1996).
Fig. 4. Power laws fitted to Todi clay data from Fig. 3.
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A multiple linear regression (MLR) was performed usingthe data-analysis package in Microsoft Excel. The best modelthat could be found is given as eq. [11]. Figure 10 shows thelogarithm of the measured mobilization strains log10(gM=2)plotted against the values predicted from eq. [11]). An errorup to a factor of three still remains in the prediction, althoughthe coefficient of determination, R2, is 0.44 and the p value isexceptionally low.
R2 ¼ 0:44; r ¼ 0:66; n ¼ 97; SE ¼ 0:236; p < 0:001
where patm is the atmospheric pressure (101.3 kPa), cu is theundrained shear strength of the clay, and C is the regressionconstant (= 0.0109).Rearranging
½12� gM¼2 ¼ CðIpÞ0:45 cu
p00
� �0:59p00patm
� �0:28
Figure 11 shows tmob/cu values from the database plottedagainst predicted tmob/cu values using eqs. [9] and [12]. Useof the mobilization strain gM=2 as predicted using routineground information (cu, p00, and Ip), together with the averageb-value of 0.6, and in the absence of any stress–strain test,creates a factor error of up to 2.0 in the prediction of tmob/cuvalues in the moderate-strain region, or correspondingly afactor of error of 3.2 in the strains estimated at a given stress.This study has not, therefore, negated the need for laboratorytesting of the stress–strain behaviour of clays, but it does of-fer a framework within which shear strength mobilization canbe estimated within different margins of probable error, de-pending on what soil testing data are available.As eq. [12] allows the mobilization strain gM=2 to vary
from less than 0.1% for low-plasticity silty clays to greaterthan 3% for high-plasticity clays, at least one test to actually
determine a value of coefficient C in eq. [12], for the partic-ular clay of interest, is strongly advised.
Link to OCREquation [12] suggests that mobilization strain is a func-
tion of undrained shear strength, plasticity index, and presentconfining stress. However, using eqs. [1] with eq. [2], and re-arranging
½13� OCR ¼ cu=p00
0:11þ 0:0037ðIpÞ� �1:25
Therefore, from eqs. [12] and [13] we can alternatively saythat
½14� gM¼2 ¼ f ðOCR; Ip; p00ÞMany of the publications used to compile the database do
not explicitly state OCR. Equation [13] has inherent errorsdue to the use of eq. [4] to compute (cu/p00)nc and thereforeback-calculation of OCR was not attempted for all the claysin the database. However, given there is greater confidencein the relationship for London clay it was decided to computevalues of the mobilization strain with increasing OCR as anexample. Equation [14] suggests that gM=2 should vary withstress history, and should therefore vary with depth in anoverconsolidated deposit.Depth-related data of high quality cores of London clay is
available in the database, from Yimsiri (2002) and Gasparre(2005). Two additional tests on intact samples from Cannon’sPark in London, were reported in Jardine et al. (1984). Sev-enteen tests on London clay were therefore available to plotmobilization strain (gM=2) data against sample depth (seeFig. 12). A logarithmic trend results with mobilization straindecreasing with depth (eq. [15]). The correlation has a coef-ficient of determination R2 of 0.46, which means that(eq. [15]) explains 46% of the variation of mobilization strainfor the three London clay sites studied. This could be evi-dence of reduced OCR reducing the mobilization strain. It is
Fig. 5. Derived b-values versus Ip for database of clay data.
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Fig. 6. charts showing eq. [8] with various values of A and b: (a) A = 1; (b) A = 10; (c) A = 100.
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acknowledged that only three sites in a single deposit (Lon-don clay) are described on Fig. 12 and by eq. [15], and thatany pattern of variation may be due to some other soil pa-rameter that varies between the different London clay geolog-ical groups. Despite this, decreasing OCR seems a credibleexplanation for (eq. [15]).
½15� 1000gm¼2 ¼ �2:84 lnðdÞ þ 15:42
R2 ¼ 0:46; r ¼ �0:67; n ¼ 17;
p ¼ 0:003; SE ¼ 1:79
½12bis� gM¼2 ¼ 0:0109ðIpÞ0:45 cu
p00
� �0:59p00patm
� �0:28
Substituting eq. [2] in
½16� gM¼2 ¼ 0:0109ðIpÞ0:45 cu
p00
� �nc
OCR0:8
� �0:59p00patm
� �0:28
Taking a representative Ip = 0.39 for London clay (averageof the tests quoted in this paper) we get, from eq. [1]
Fig. 7. Moderate strain region data (115 tests, 19 clays and silts).
Fig. 8. tmob/cu data plotted against predicted tmob/cu from eq. [9].
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Now, writing
½19� OCR ¼ Ds þ s 0v;0
s 0v;0
¼ Ds
g 0zþ 1
where Ds is the apparent past overburden pressure, g′ is thebuoyant unit weight ∼10 kN/m3, and z is the depth in theLondon clay. Substituting in eq. [18] we obtain
½20� gM¼2 ¼ 0:000872 1þDs
10z
� �0:47
ð10zÞ0:28
Using eq. [20], and taking an assumed bandwidth of Ds =300 to 1000 kPa for London clay, the predicted profiles withdepth of gM=2 can be computed; these are plotted on Fig. 12.The fit to the scattered observations is not unreasonable.
K0-consolidated test data
In K0-triaxial tests the test curves do not start at zero shearstress. Jardine et al. (1984, 1986) reported the data of high-quality triaxial tests performed on reconstituted low-plasticityclay. Figure 13 shows the original data re-plotted for tests
Fig. 9. tmob/cu data versus normalized strain: (a) shear stress mobilization versus normalized shear strain: natural axes; (b) shear stress mobi-lization versus normalized shear strain: logarithmic axes.
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with various OCRs marked as R1, R1.4, etc. It is possible todefine a new parameter t0, which is the initial shear stressafter one-dimensional swelling. This can conveniently betaken in Fig. 13 as the stress mobilized at 3a = 10–5.Test R4 begins approximately at K0 = 1, t0 = 0, where K0
is the initial coefficient of earth pressure. This is used to ob-tain a fitting to eq. [9]. The stress–strain prediction of anyother K0 test is then achieved by scaling for the actual un-drained strength achieved in that test, and then by shiftingthe scaled curve vertically so that it starts at shear stress t =t0. Figure 14 shows all the test curves accompanied by the
predictions achieved using this procedure. The performanceis generally satisfactory for t0 > 0, K0 < 1, but less so forthe test from the largest yield stress ratio R8 for whicht0 = –7 kPa. Updating eq. [9] accordingly we obtain
½21� tmob � t0
cu¼ 0:5
g
gM¼2
� �0:6
where gM=2 refers to the mobilization strain of test R4, t0,and cu refer to the start and finish of any other K0 test, and(tmob, g) represents the predicted stress–strain curve.
Fig. 10. Logarithm of the measured mobilization strains plotted against values predicted from eq. [11].
Fig. 11. tmob/cu values from the database versus predicted tmob/cu values using eqs. [9] and [12].
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DiscussionEngineers generally begin designs for clay by establishing
the undrained strength profile, and then assigning a safetyfactor that is thought to safeguard against material variability.Some form of penetrometer probing is usually conducted tofix a design line for cu. If SPTs have been conducted, Fig. 1suggests that the undrained strength of a clay of known plas-ticity index Ip could be estimated within an error factor of1.4, even allowing for uncertainties in energy transmission.This is also a typical partial factor on cu adopted in codes ofpractice (e.g., Eurocode 7 Geotechnics; CEN 2003). Manyengineers assume that the standard safety factors on materialstrength and loads are also effective in preventing excessivedeformations. Even where deformation calculations are car-ried out, they usually rely on linear elastic calculations withan estimated value of soil stiffness.
This paper sets out an explicit understanding of soil strainsin relation to mobilized stresses. Taking the example of asimple circular footing on clay, Osman and Bolton (2005) in-troduced the notion of a mobilized shear strength, tmob, suffi-cient to hold in equilibrium the vertical bearing pressure, q,arising from working loads. Applying the usual symbol Nc,originally defined as an ultimate bearing capacity factor, butnow used as an equilibrium factor at working loads
½22� tmob ¼ q
Nc
Eason and Shield (1960) established an upper bound ofNc = 6.05 for a rough circular foundation, and a value of 6will be used here. Osman and Bolton (2005) used a continu-ous deformation field within a Prandtl bearing mechanism to
Fig. 12. Mobilization strain data for London clay samples plotted against depth from Jardine et al. (1984); Gourvenec et al. (1999, 2005);Yimsiri (2002); Gasparre (2005).
Fig. 13. Original triaxial data from Jardine et al. (1986).
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relate the average strains, gmob, to the ratio of undrained foot-ing settlement, w, to diameter, D
½23� w
D¼ gmob
1:35
Associating cmob from eq. [22] with gmob from eq. [23], byusing the power curves (eq. [9]), we predict
½24� q
6cu¼ 1:35w
gM¼2D
� �0:6
Furthermore, using mobilization factor M (which is func-tionally equivalent to a safety factor on soil strength)
½25� M ¼ 6 cu
q
and rearranging, we obtain
½26� w
D¼ gM¼2
1:35M1:67
Equation [26] demonstrates that the material parametergM = 2 is required in addition to the mobilization ratio M ifengineers are to make reliable estimates of footing settlement.Osman and Bolton (2005) showed a close correspondencebetween this method, termed mobilizable strength design(MSD), and a fully nonlinear finite element analysis of circu-lar footings.In limit-state codes of practice, and in load and resistance
factor design (LRFD), the overall safety factor is split intovarious partial factors. For example, if partial factors of 1.1and 1.3 were applied to the characteristic dead load W andlive load V, respectively, and a partial factor of 1.4 were ap-plied to the undrained shear strength cu, then for the samecircular footing the following limitation on load would apply:
½27� 1:1W þ 1:3V � 6ðcu=1:4ÞðpD2=4ÞIn the particular example where V = 0.25W we can rewriteeq. [27] as
W � 6=2 cuðpD2=4ÞSo the net effect of the three partial factors is equivalent to
the application of a single mobilization factor M = 2 on un-drained shear strength in relation to dead load. Using M = 2in eq. [26] we can determine the range of likely proportionalsettlements
½28� w
D¼ gM¼2
4:3
The range of gM=2 for various natural clays found in thedatabase, and shown in Fig. 10, is from 0.0015 (Grande Ba-leine normally consolidated clay) to 0.044 (Manglerud quickclay) with a mean value of 0.0088. It is that the range coversa factor of 30, however, that is most significant, as eq. [28]shows that the provision of a single mobilization factor re-sults in the same uncertainty factor of 30 in settlements.This corresponds to a range of settlements from 0.7 to20.5 mm for a 2 m diameter foundation (or equivalently asquare foundation). The adoption of a strength-reduction fac-tor M = 2 should therefore lead to the design of foundationsthat would generally settle by a tolerable amount in relationto building damage.If, on the other hand, an engineer was permitted to adopt a
partial factor of unity on applied loads, such as in the designof storage tanks where the maximum working loads areclosely predictable, the consequence for undrained settle-ments would be significant. Allowing M to fall from 2.0 to1.4 in eq. [26] might seem acceptable from a conventionalreliability perspective, but the settlements would increase bya factor of about 1.431.67 ≈ 1.8, corresponding to a rangefrom 1.3 to 37 mm for a 2 m diameter foundation. Therecould well be serviceability issues at the upper end of thisrange, for the most compliant soils, even where the probabil-ity of soil failure was considered acceptably small.An engineer who wants to limit deformations can use the
information provided in this paper, in conjunction with anysite-specific stress–strain data, to narrow the range of ex-pected settlement values. If the reference strain gM=2 for a
Fig. 14. Predicted stress–strain curves using eq. [21] (test data from Jardine et al. 1986).
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clay were predicted solely on the basis of Atterberg limitsand effective stress levels, using eq. [12], then Fig. 10 showsthe possibility of an error up to a factor of 3. If a sufficientnumber of stress–strain tests is conducted to obtain a reliablemean value for gM=2, Fig. 9 suggests that a factor error up to1.75 on strains might occur towards the extremes of thechosen range 5 < M < 1.25 due to the inaccuracy of describ-ing all clays using the same power exponent b = 0.6. Even inthe vicinity of a measured value for gM=2, the factor error instrain predictions from one test to another can apparently beas large as 1.3, as observed around the pivot point in Fig. 9b.Equation [26] confirms that the error in nonlinear settlementprediction should mirror the error in gM=2.All the foregoing relates to the undrained foundation settle-
ment. However, the ratio of fully drained to undrained settle-ment of shallow foundations on soils in their quasi-elasticrange of behaviour, as described here, should fall in the range1.4 to 1.6 as the secant Poisson’s ratio rises from about 0.2 to0.3 (see Burland et al. 1977). The methodology set out inthis paper therefore offers a design engineer an order of mag-nitude improvement in settlement control compared with theuse of codified safety factors.
ConclusionsA database of the undrained stress–strain behaviour of nat-
ural silts and clays was compiled from 16 publications byvarious authors. A total of seven of the 122 tests were ex-cluded either because they were found to display inherentlyerratic features or due to the data falling outside the range ofinterest corresponding to moderate-strain levels and typicalsafety factors. A method of estimating the undrained shearstress–strain curves of clays is recommended, using a nor-malization based on their undrained shear strength cu and areference strain gM. This can conveniently be discussed interms of mobilization factor M = cu/tmob. Plots of tmob/cu =1/M versus shear strain were obtained for the 19 clays andsilts in the database. It was discovered that for the range ofgreatest practical interest (1.25 ≤ M ≤ 5) these curves couldreasonably be described as power curves whose apexes lie atthe stress–strain origin.This observation led to the adoption of a reference strain
gM=2 for each test, defined as the shear strain required to mo-bilize one-half of the peak strength. An average exponent of0.6 was used to describe the normalized power function for5 < M < 1.25. The undrained stress–strain equations of alarge database of clays, variously overconsolidated, therebycame to fit eq. [9] in the moderate strain range. The use ofeq. [9] to derive a mobilization factor consistent with anymoderate strain level, based on the measurement of referencestrain gM=2, does not generally result in an error exceed-ing ±40%, see Fig. 8. This error is largely due to the expo-nent b being taken at a standard value of 0.6, whereas it wasfound to range from 0.3 to 1.2; see Fig. 3. The correspondingerror factor on the strain predicted at a given mobilized stressratio is 1.75; see Fig. 9.If eq. [12] is used to predict gM=2, based only on a routine
ground characterization instead of actual stress–strain tests,then the possible error in the prediction of mobilized stressfor a given strain increases to a factor of 2; see Fig. 11. Andthe possible error factor on the strain predicted for a given
mobilization of stress correspondingly increases to 3.2. Useof eq. [12] is not recommended for highly structured quickclays or residual soils, which were excluded from the regres-sion analysis.Although the database, and eq. [9], was based on standard
undrained triaxial compression, DSS, and RC tests for whichthe initial shear stress was zero, one set of tests on reconsti-tuted low plasticity reported by Jardine et al. (1986) had beenallowed to swell one-dimensionally prior to being tested incompression from an initial K0 ≠ 1. Some success was dem-onstrated, at least for cases with K0 ≤ 1, by simply shiftingthe standard power curve vertically so that it started at an ini-tial shear stress t0 corresponding to its K0 value (eq. [21]).Prescribed geotechnical factors of safety cannot be used to
achieve undrained settlement targets let alone ultimate settle-ments. The use of a single mobilization factor for the clays inthe current database leads to the settlement of a notional 2 mfooting varying over a factor of 30. The information pre-sented here allows an engineer to reduce this variability byan order of magnitude.
AcknowledgementsThe authors thank the Cambridge Commonwealth Trust
and Ove Arup and Partners for financial support to the firstauthor. Thanks are also due to Dr Brian Simpson, Dr PaulMorrison, and Dr Stuart Haigh for their helpful advice andsuggestions; as well as Dr A. Gasparre for the provision ofher triaxial test data for analysis.
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