Laboratory measurement of strength mobilisation in kaolin: link tostress history

P. J. VARDANEGA*, B. H. LAU*, S. Y. LAM*, S. K. HAIGH*, S. P. G. MADABHUSHI* and M. D. BOLTON*

This letter presents data from triaxial tests conducted as part of a research programme into thestress–strain behaviour of clays and silts at Cambridge University. To support findings from earlierresearch using databases of soil tests, eighteen CIU triaxial tests on speswhite kaolin were performedto confirm an assumed link between mobilisation strain (cM52) and overconsolidation ratio (OCR). Inthe moderate shear stress range (0?2cu to 0?8cu) the test data are essentially linear on log–log plots.Both the slopes and intercepts of these lines are simple functions of OCR.

KEYWORDS: deformation; laboratory tests; plasticity; shear strength; stiffness

ICE Publishing: all rights reserved

NOTATION

A regression coefficientb exponent determined from regression analysisCIU consolidated isotropic undrainedcu undrained shear strengthcu consolidated isotropic undrainedd exponent determined from regression analysise0 initial void ratioG shear modulusG0 maximum shear modulusIp plasticity indexM mobilisation factor cu/tmob

m exponent determined from regression analysisn number of data points used to generate a correlationOCR overconsolidation ratiop the smallest level of significance that would lead to the

rejection of the null hypothesis, i.e. that the value of r 5 0,in the case of determining the p-value for a regression

pa atmospheric pressurep9c maximum effective consolidation pressurep90 mean stress in the triaxial after swell backq deviator stressR2 coefficient of determination of a correlation (the square of

the correlation coefficient r)S regression coefficientSE standard error in a regression, a quantification of

deviation about the fitted linewL liquid limitc shear strain, taken as 1?5 times the axial strain (ea) in this lettercM52 mobilisation strainea axial straink slope of unload–reload lineL exponent in the equation of Ladd et al. (1977)l slope of normal compression lines9v0 vertical effective stress in the grounds9vc maximum past effective vertical stress in the groundtmob mobilised shear stress

INTRODUCTIONKnowledge of soil stiffness and stress–strain behaviour isessential to the calculation of ground displacements that

may damage structures. These serviceability considera-tions, termed SLS in Eurocode 7 (BSI, 2010), should be atthe forefront of the geotechnical practitioner’s mind. Forexample, deformations are important in the design ofoffshore wind turbines, both in terms of dynamic structuralresponse under severe loads and due to the vulnerability ofthe drive and gearbox to tilting of the mast.

Research has been undertaken at Cambridge Universityto define and validate simplified mechanistic models inconjunction with soil stress–strain data to enable routinecalculations of footing settlements (Osman & Bolton, 2005;Osman et al., 2007) and the displacement of bracedretaining structures (Osman & Bolton, 2006; Lam &Bolton, 2011), for example. The calculation procedure isbased on conservation of energy and is known asmobilisable strength design (MSD).

An important feature of MSD is the need to model thestrength mobilisation of the soil. The shear stiffness of claysand silts at small strains has been shown to be empiricallydeterminable using the maximum shear modulus (G0) and aquasi-hyperbolic stress–strain relation in which the shearstrain required to halve the stiffness was seen to vary withthe liquid limit (wL) (Vardanega & Bolton, 2011a). Thisletter presents measurements of stress versus strain forkaolin clay for various stress histories and for stress levelsapproaching failure, in undrained CIU triaxial compres-sion tests. A summary of the basic kaolin parameters fromthe present study is shown in Table 1.

Previous investigations into the stiffness of reconstitutedsoils are detailed by Houlsby & Wroth (1991) and Viggiani& Atkinson (1995). Viggiani & Atkinson (1995) fittedequation (1) to laboratory data of the initial shear modulusfor a range of clays, including kaolin

G0

pa~S

p’0pa

� �d

OCRm (1)

where G0 is the maximum shear modulus, pa is atmosphericpressure, p’0 is mean stress (kPa), OCR is the over-consolidation ratio (defined as either p’c=p’0 or as s’vc=s’v0)and d, m and S are experimentally determined coefficients.Equation (1) shows the degree of overconsolidation to be akey determinant in the prediction of the small-strainstiffness of soils. The relation between degree of over-consolidation and behaviour at larger strain levels is thefocus of this letter.

Manuscript Received 19 January 2012; first decision 13February 2012; Accepted 9 March 2012Published online at www.geotechniqueletters.com on 30 March2012.*Department of Engineering, University of Cambridge,Cambridge, UK

Vardanega, P. J. et al. (2012) Geotechnique Letters 2, 9–15, http://dx.doi.org/10.1680/geolett.12.00003

9

EXPERIMENTAL METHODOLOGYDuring triaxial compression, the axial stress is increasedwhile keeping the cell pressure constant. An undrained testmaintains constant volume, allowing excess pore pressuresto develop. Conventional triaxial testing methodology isoutlined in Bishop and Henkel (1957).

In the triaxial tests, an external linear variable differ-ential transformer (LVDT) measures the overall movementof the sample (used to capture the strain data) to anaccuracy of 0?125 mm. A strain accuracy of 1023 issufficient to capture the influence of the OCR on themoderate stress region (defined in the next section), whichis the aim of this letter.

Details of the effective stress history of the triaxial testsperformed are displayed in Table 2, where p’c is themaximum effective consolidation pressure, p’0 is the meaneffective stress after swelling back and OCR denotes theratio between the two. A backpressure of 350 kPa wasmaintained during the tests.

STRESS–STRAIN BEHAVIOURFigure 1 shows the stress–strain curves measured in theeighteen CIU triaxial tests, four of which were reported byXu (2011). The data are plotted on log–log axes. The rangeof OCR studied is from 1 to 20. The data are presented interms of shear strain c in the triaxial test, which is taken inthis letter as

c~1:5ea (2)

This is done on the premise that (e.g. Terzaghi et al.,1996)

c

2~

e1{e3

2

e3~{e1

2

(3)

where e1 and e3 are the principal strains.

The stress–strain curves from triaxial testing are sensiblylinear over a range of moderate stresses when the data areplotted on log–log axes.

Verification of cu values at varying stress levelsThe undrained shear strength ratio (cu=s’v0)nc for normallyconsolidated soils can be estimated from Skempton’scorrelation (1954, 1957)

(cu=s’v0)nc~0:11z0:37Ip (4)

From Table 1, the plasticity index Ip for the clay testedin the present study is 0?33. Using equation (4), this givesan empirical value of (cu=s’v0)nc 5 0?23, which correspondsreasonably well with the average of the test values of(cu=p’0)nc of 0?19 (at p’c 5 500 kPa) and 0?29 (at p’c 5 180kPa); see Table 3. Equation (5) gives the relationship for

Table 1. Summary of basic properties of the tested kaolin(numbers in square brackets indicate the number of tests todetermine the parameter)

Plastic limit wP: % 29?6 [4]Liquid limit wL: % 62?6 [1]Slope of normal compression line l 0?250 [4]Slope of unload–reload line k 0?039 [6]

Table 2. Stress history of triaxial tests performed

Test ID p’c: kPa p’0: kPa OCR

180(1) 180 180 1180(2) 180 90 2180(5) 180 36 5180(10) 180 18 10180(20) 180 9 20300(2) 300 150 2300(5) 300 60 5300(10) 300 30 10300(15) 300 20 15300(20) 300 15 20500(2) 500 250 2500(10) 500 50 10500(15) 500 33?3 15500(20) 500 25 20500(1)_Xu 500 500 1500(2)_Xu 500 250 2500(5)_Xu 500 100 5500(10)_Xu 500 50 10

1

1

2

2

Region of power-law fit5

5

20

20

OCR = 1OCR = 2OCR = 5OCR = 10OCR = 20

10

10

11

0.1Shear strain

Mob

ilise

d sh

ear s

treng

th/u

ndra

ined

shea

r stre

ngth

tm

ob/c

u

0.1

0.010.001 0.01

(a)

1

1

2

2

Region of power-law fit

5

5

15

1520

20

OCR = 2OCR = 5OCR = 10OCR = 15OCR = 20

10

10

0.1Shear strain c

Mob

ilise

d sh

ear s

treng

th/u

ndra

ined

shea

r stre

ngth

tm

ob/c

u

0.1

0.010.001

(b)

0.01

1

1

1

2

2

Region of power-law fit

Reported strain valuesshould be used cautiously

in this region

5 15 20

OCR = 1 (Xu, 2011)

OCR = 10 (Xu, 2011)OCR = 5 (Xu, 2011)

OCR = 2

OCR = 2 (Xu, 2011)

OCR = 15OCR = 20

OCR = 10

1010

0.1Shear strain c

Mob

ilise

d sh

ear s

treng

th/u

ndra

ined

shea

r stre

ngth

tm

ob/c

u

0.1

0.010.0001

(c)

0.001 0.01

Fig. 1. Triaxial test data for tests with (a) 180 kPa pre-consolidation, (b) 300 kPa pre-consolidation and (c) 500 kPapre-consolidation

10 Vardanega, Lau, Lam, Haigh, Madabhushi and Bolton

overconsolidated soils taken from Ladd et al. (1977); inFig. 2 it is fitted to the triaxial tests data, taking (cu=p’0)nc

as 0?23. While not identical, the use of cu=p’0 and cu=s’v0

interchangeably in Ladd’s equation (as has been done inFig. 2) was explored by Muir Wood (1990) who concludedthat no significant difference results.

cu=s’v0

(cu=s’v0)nc

~OCRL (5)

where s’v0 is the vertical effective stress, OCR is theoverconsolidation ratio defined as s’vc=s’v0 and L is anempirical exponent, which may decrease with increasingOCR (Ladd et al., 1977; Muir Wood, 1990) from 0?85 to0?75; nc denotes normal consolidation.

From critical state soil mechanics (Schofield and Wroth,1968), Muir Wood (1990) showed that L should be given by

L~l{k

l(6)

Based on data collected by Mayne (1980) and presentedby Muir Wood (1990), L varies between 0?2 and 1?0 with amean value of 0?63 and a standard deviation of 0?18. This isa significantly greater range than would be implied byequation (5).

Using the values from Table 1 in equation (6), it wouldbe expected that L 5 0?84, although Muir Wood cautionsthat it is difficult to determine a reliable value of k from themean slope of a swelling line. From Fig. 2, the value of L isshown to be 0?68 (when the regression is forced through theorigin, as implied by equation (5)), which is slightly lower

than the theoretical value; however, it is similar to the meanof previously collected experimental data (Mayne, 1980).

The general form of Ladd’s relationship is shown to fitthe test data well. This allows one to conclude that the cu

values computed from the test data are not unreasonable.

Mobilisation strain frameworkA recent review of strength mobilisation in clays and silts(Vardanega and Bolton, 2011b) presented a large databaseof 115 tests on natural samples of 19 clays compiled from arange of publications. It was shown that a power law fitsthe moderate stress region (0?2,tmob/cu,0?8) of the stress–strain curves very well. The power law has the followingelementary form (Vardanega & Bolton, 2011b)

tmob=cu~Acb (7)

In the previously published database, b for the 115 testsin the database was shown to range from 0?3 to 1?2 with anaverage of 0?6.

Analysis of new kaolin dataFigure 3 shows power curves fitted to the 18 triaxial testson reconstituted kaolin. It is observed that power curves donot fit the test data as well at low OCRs as they do atmoderate to high OCRs. Figure 4 shows that the b valuefrom equation (7) is related to the OCR via the followingregression equation

b 5 0?011(OCR) + 0?371 (8)

Table 3. Curve-fitting and normalisation parameters

Test ID A b R2 n cu: kPa e0 cM52

180(1) 5?861 0?484 0?850 59 51?7 1?15 0?00521180(2) 3?531 0?425 0?938 79 38?4 1?27 0?00880180(5) 2?825 0?443 0?999 102 32?0 1?12 0?01969180(10) 3?882 0?595 1?000 115 19?9 1?29 0?03201180(20) 3?572 0?530 0?999 139 15?9 1?21 0?02506300(2) 3?311 0?356 0?980 52 68?6 1?23 0?00445300(5) 3?034 0?389 0?998 110 45?6 1?17 0?00988300(10) 3?681 0?540 1?000 140 30?6 1?21 0?02488300(15) 3?733 0?489 0?991 194 34?2 1?08 0?01737300(20) 5?082 0?580 0?995 131 27?5 1?19 0?01924500(2) 4?227 0?423 0?926 50 105?1 1?13 0?00530500(10) 3?732 0?460 0?991 131 57?5 1?18 0?01337500(15) 3?750 0?584 0?998 159 36?4 1?14 0?03265500(20) 3?350 0?602 0?997 200 32?8 1?20 0?04389500(1)_Xu 3?236 0?311 0?987 45 93?4 n/a 0?00261500(2)_Xu 2?972 0?377 0?991 96 88?1 n/a 0?00815500(5)_Xu 1?879 0?291 0?998 96 69?7 n/a 0?01121500(10)_Xu 3?192 0?470 0?998 133 56?0 n/a 0?01978

1.0

0.8

0.8

0.6

0.4

0.4

0.2

1.2 1.6

0

0–0.2

OCR 1

OCR 2

OCR 5

OCR 10 OCR 15OCR 20

log 1

0c u

/p' 0

(cu/

p' 0) n

c

log10cu/p'0

(cu/p'0)nc= 0.68log10 (OCR)

R2 = 0.94n = 18

p'c = 180 kPap'c = 300 kPap'c = 500 kPap'c = 500 kPa (Xu 2011)

log10 (OCR)

Fig. 2. Fitting the equation of Ladd et al. (1977) equation to the test data

Laboratory measurement of strength mobilisation in kaolin: link to stress history 11

for which R2 5 0?591, n 5 18, SE 5 0?064 and p , 0?001.The mobilisation strain is the magnitude of shear strain

at which half the shear strength is mobilised (Vardanega &Bolton, 2011b), hence

ðcM~2Þ~0:5

A

� �1=b

(9)

Table 3 lists the values of the curve-fitting parameters Aand b together with measured values of undrained shearstrength cu and mobilisation strain cM52 for the 18 tests onkaolin. The undrained shear strength is used to normalisethe shear stress and the mobilisation strain is used tonormalise the shear strain. The resulting predictionequation for shear strength mobilisation has the form

0

0

y = 0.356x + 0.520R2 = 0.980

y = 0.389x + 0.482R2 = 0.998

y = 0.540x + 0.566R2 = 1.000

y = 0.489x + 0.572R2 = 0.991

y = 0.580x + 0.706R2 = 0.995

–4.0 –3.0 –2.0 –1.0 0log10 c

–0.8

–0.7

–0.6

–0.5

–0.4

–0.3

–0.2

–0.1

log 1

0 (t

mob

/cu)

OCR=2

OCR=5

OCR=10

OCR=20

OCR=15

M = 2

(b)

OCR

OCR

2 1552010

(a)

–0.8

–0.7

–0.6

–0.5

–0.4

–4.0

–0.3

–3.0

–0.2

–2.0

–0.1

–1.0 0

y = 0.484x + 0.768R2 = 0.850

y = 0.425x + 0.548R2 = 0.938

y = 0.443x + 0.451R2 = 0.999

y = 0.595x + 0.589R2 = 1.000

y = 0.530x + 0.553R2 = 0.999

log10 c

log 1

0 (t

mob

/cu)

OCR=1

OCR=2

OCR=5

OCR=10

OCR=20

M = 2

OCR1 2155

15201010

22

51

10

0 y = 0.423x + 0.626R2 = 0.926

y = 0.460x + 0.572R2 = 0.991

y = 0.584x + 0.574R2 = 0.998

y = 0.602x + 0.525R2 = 0.997

y = 0.311x + 0.510R2 = 0.987

y = 0.377x + 0.473R2 = 0.991

y = 0.291x + 0.272R2 = 0.998

y = 0.470x + 0.504R2 = 0.998

–4.0 –3.0 –2.0 –1.0 0log10 c

–0.8

–0.7

–0.6

–0.5

–0.4

–0.3

–0.2

–0.1

log 1

0 (t

mob

/cu)

OCR=2

OCR=1(Xu, 2011)OCR=2(Xu, 2011)OCR=5(Xu, 2011)OCR=10(Xu, 2011)

OCR=10

OCR=15

OCR=20

M = 2

(c)

Fig. 3. Fitting power curves to the test data: (a) 180 kPa data (five tests); (b) 300 kPa data (five tests); (c) 500 kPa data (eight tests)

0·7

0·6

0·5

0·4

0·3

0·2

0·1

00 5 10

OCR15

b = 0·011 OCR + 0·371R2 = 0·591

20 25

b

n = 18SE = 0·064

p'C = 180 kPa

p'C = 300 kPa

p'C = 500 kPa

p'C = 500 kPa (Xu, 2011)p < 0·001

Fig. 4. b versus OCR

12 Vardanega, Lau, Lam, Haigh, Madabhushi and Bolton

tmob=cu~0:5c

cM~2

� �b

in the range 0:2vtmob=cuv0:8

(10)

Figure 5 shows a plot of tmob/cu as predicted usingequation (8) for the exponent and equation (10) for themobilised shear strength ratio, versus correspondingmeasurements. The R2 on the plot is 0?96 and the slope is

very close to 1?0 (0?99), which validates the use of the twoequations in tandem. This level of accuracy is only attainedif the mobilisation strain (cM52) is known precisely.

INFLUENCE OF STRESS HISTORY ON MOBILISATIONSTRAINVardanega & Bolton (2011c) showed that the mobilisationstrain is related to depth of sample for a database of testson natural London clay samples. The observation of cM52

increasing with decreasing depth is akin to suggesting thatcM52 increases with OCR. Figure 6 shows the mobilisationstrain cM52 plotted against OCR. A good coefficient ofdetermination is observed (R2 5 0?815) and the p-value isvery small (,0?001). For kaolin, the following regressionrelationship is available

log10(cM~2) ~ 0:680 log10(OCR){2:395 (11)

for which R2 5 0?815, n 5 18, SE 5 0?151 and p , 0?001.Rearranging equation (11) gives

cM~2ð Þ~ 0:0040(OCR)0:680 (12)

Figure 7 shows the predicted versus measured plot whenequations (8), (10) and (12) are used to predict tmob/cu. Theerror bands widen to around ¡40% due to the scatterabout the trend line in Fig. 6. Using equations (8), (10) and

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

1.0

Plus 20%

R2 = 0.96

Minus 20%

180 kPa tests

300 kPa tests

500 kPa tests

0.90.80.7Predicted tmob/cu

Mea

sure

d t m

ob/c

u

Measured tmob/cu = 0.99 predicted tmob/cu

0.60.50.40.30.20.1

1.0

01.10

Fig. 5. Predicted versus measured tmob/cu using equations (8) and (10)

0.1

0.01

log10(cM=2) = 0.680log10(OCR)–2.395R2 = 0.815

n = 18SE = 0.151p < 0.001

OCR1 10 100

0.001

p'c = 180 kPap'c = 300 kPap'c = 500 kPap'c = 500 kPa (Xu, 2011)

cM=2

Fig. 6. Logarithm of mobilisation strain versus logarithm ofOCR

1.0

1.11.0

0.9

0.9

0.8

0.8

0.6

0.6

0.5

0.5

0.7

0.7

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.10

0Predicted tmob/cu

Mea

sure

d t m

ob/c

u

Measured tmob/cu = 0.95 predicted tmob/cu

R2 = 0.70

Plus 40%

Minus 40%

500 kPa tests

300 kPa tests

180 kPa tests

Fig. 7. Measured versus predicted tmob/cu using equations (8), (10) and (12)

Laboratory measurement of strength mobilisation in kaolin: link to stress history 13

(12), the predicted stress–strain curves are drawn in Fig. 8.Similar behaviour is shown in the data of Todi claypresented by Burland et al. (1996) and analysed inVardanega & Bolton (2011b).

Implications for designThe implication for geotechnical design is that less strain isneeded to mobilise the same proportion of shear strength thedeeper a geotechnical structure is built. For a bored pile inoverconsolidated clay, for example, the soil in contact withthe shaft at the top of the pile is likely to be significantlymore compliant than the soil in contact with the base.

If the pile head settlement at working load is to becalculated using a t–z analysis, one must make assumptionsabout the variation of t–z spring behaviour with depth. If adesigner assigns a single value of G/cu for the soil, thisimplies a single strain to failure at all depths. This letter hasshown that such an assumption would be unwarranted.

SUMMARY AND CONCLUSIONSThis letter has focused on establishing a link betweenmobilisation strain and stress history. The followingsummary points and conclusions are made.

1. Data from 18 CIU triaxial tests on reconstituted kaolinsamples confirm that the stress–strain curves (in themoderate stress region) are roughly linear on log–logplots.

2. The mobilisation strain framework presented byVardanega & Bolton (2011b, 2011c) is verified forreconstituted kaolin. A simple stress–strain model forkaolin is

2.

tmob=cu~0:5c

cM~2

� �b

in the range 0:2vtmob=cuv0:8

where b 5 0?011(OCR) + 0?371. The average exponentb recorded by Vardanega & Bolton (2011b), fornatural clays of unknown OCR, was 0?6 within arange of 0?3–1?2. This is not inconsistent with thecurrent data of these tests on reconstituted kaolin.

3. The mobilisation strain cM52 is shown to increasestrongly with the logarithm of overconsolidation ratiovia the following relationship for kaolin

3.cM~2 ~ 0:0040(OCR)0:680 (12)

4. Just as OCR has been previously found (equation (1))to influence small-strain stiffness, so it has now beendemonstrated to influence both the position and slopeof stress–strain curves of clay in the region ofmoderate strength mobilisations, when plotted onlog–log axes.

ACKNOWLEDGEMENTSThanks are due to Mr Chris Knight for his technicalservices, to Miss X. Xu (former MEng student atCambridge University) for providing her triaxial data foranalysis, to the Cambridge Commonwealth Trust and OveArup and Partners for their financial support of the firstauthor, and to the EPSRC for financial support from grantEP/H013857/1 ‘Cyclic behaviour of monopile foundationsfor offshore windfarms’.

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0·90

0·80

0·70

0·60

0·50

0·40

0·30

0·20

0·10

0·0700·0600·0500·0400·0300·0200·0100

Moderatestressrange

mob

/cu

OCR 1

OCR = 1 y = 4·121x 0·382

y = 3·638x 0·393

y = 3·296x 0·426

y = 3·352x 0·481

y = 3·394x 0·536

y = 3·920x 0·591

OCR 2

OCR = 2

OCR 5

OCR = 5

Increasing OCR

OCR 10

OCR = 10

OCR 15 OCR 20

OCR = 15

OCR = 20

0

1·00

0·080

Shear strain

Fig. 8. General stress–strain curves drawn using equations (8), (10) and (12)

14 Vardanega, Lau, Lam, Haigh, Madabhushi and Bolton

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Laboratory measurement of strength mobilisation in kaolin: link to stress history 15