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Stiffness of Clays and Silts: Normalizing ShearModulus and Shear Strain
P. J. Vardanega, Ph.D., M.ASCE1; and M. D. Bolton, Ph.D.2
Abstract: An analysis is presented of a database of 67 tests on 21 clays and silts of undrained shear stress-strain data of fine-grained soils.Normalizations of secant G in terms of initial mean effective stress p9 (i.e., G=p9 versus log g) or undrained shear strength cu (i.e., G=cuversus log g) are shown to be much less successful in reducing the scatter between different clays than the approach that uses the maximumshearmodulus,Gmax, a technique still not universally adopted by geotechnical researchers and constitutivemodelers. Analysis of semiempiricalexpressions forGmax is presented and a simple expression that uses only a void-ratio function and a confining-stress function is proposed. This isshown to be superior to aHardin-style equation, and the void ratio function is demonstrated as an alternative to an overconsolidation ratio (OCR)function. To derive correlations that offer reliable estimates of secant stiffness at any requiredmagnitude ofworking strain, secant shearmodulusG is normalized with respect to its small-strain value Gmax, and shear strain g is normalized with respect to a reference strain gref at which thisstiffness has halved. The data are corrected to two standard strain rates to reduce the discrepancy between data obtained from static and cyclictesting. The reference strain gref is approximated as a function of the plasticity index. A unique normalized shear modulus reduction curve in theshape of amodified hyperbola isfitted to all the available data up to shear strains of the order of 1%.As a result, good estimates can bemade of themodulus reductionG=Gmax 6 30% across all strain levels in approximately 90% of the cases studied. New design charts are proposed to updatethe commonly used design curves. DOI: 10.1061/(ASCE)GT.1943-5606.0000887. © 2013 American Society of Civil Engineers.
CE Database subject headings: Stiffness; Clays; Silts; Design; Deformations; Shear modulus; Statistics.
Author keywords: Stiffness; Clays; Silts; Design; Deformations; Modulus; Statistical analysis.
Introduction
Investigation of the stiffness-strain response of soils is required inmany applications within geotechnical engineering. In earthquakeengineering, the ability to predict the strain level that leads tomodulus reduction is crucial to the prediction of damping, and thefurther reduction of secant resilient modulus at larger strain ampli-tudes determines the seismic response, which is always regardedas undrained for fine-grained soils. Construction-induced groundmovements in clays and silty clays are also generally taken as un-drained, so the design engineer similarly needs to determine or es-timate the representative undrained shear stress-strain curve of thesoil to control ground movements resulting from deep excavations,or to limit the differential settlements of foundations, for example.Field and laboratory measurements of nonlinear stress-strain curvesare complex and time-consuming; they also relate only to specificlocations. Practitioners, therefore, need to make the best use of ex-isting information derived from the testing of various soils, and tounderstand how to make rational interpolations in terms of thevariable profiles that generally emerge from ground investigations.
Although a great deal of stiffness data for clays under cyclicloading has been published for the purposes of earthquake risk
evaluation, its potential as a source of information in monotonic andstatic applications has not been fully explored. Accordingly, thispaper presents a merged database for clay stiffness observed ina variety of test types (monotonic and cyclic, static and dynamic)focusing on the degree to which the nonlinear stress-strain responsecan be predicted if certain standard classification parameters areknown. The aim of this paper is to draw attention to appropriate andinappropriate correlations for undrained soil stiffness and to providean indication of the likely errors involved in such estimations.
In a monotonic test, the secant stiffness G simply reduces pro-gressively with shear strain g. This is principally because of theseparation or slippage of intergranular contacts as shear strain in-creases, thereby removing their associated contributions fromthe elastic stiffness of the assembly, as shown in discrete elementmethod (DEM) simulations byDobry andNg (1992). This reductionis generally taken to be reversible when the strain direction is re-versed because previously slipping contacts re-engage. Of course, ifthe reversed straining continues, elastic contacts will once again belost, and the stiffness will reduce as before. The rate of stiffnessreduction on the reversed loading path can be taken to be half that ofthe original loading curve because previously slipping elementsmust first recoil elastically until they are unloaded and then distortbackward by the same amount before they slip backward (Iwan1966). Together with the assumption of cyclic reversibility (Masing1926), this gives rise to the typical cyclic response, in which pairs ofvalues of cyclic stiffness, Gcyclic, and cyclic amplitude, gcyclic, aretaken to be equally representative of the monotonic response. Inother words, the monotonic curve is taken as the backbone curveof the small-strain cyclic response.
A second mechanism of stiffness reduction can arise at moderatestrains in undrained tests as the result of the buildup of positiveexcess pore pressures with a consequential reduction of effectivestress. This is because of the tendency of soils to densify as a result of
1Research Associate, Dept. of Engineering, Univ. of Cambridge, LaingO’RourkeCentre forConstruction Engineering andTechnology, CambridgeCB2 1PZ, U.K. (corresponding author). E-mail: [email protected]
2Professor of Soil Mechanics, Dept. of Engineering, Univ. of Cam-bridge, Cambridge CB2 1PZ, U.K.
Note. This manuscript was submitted on July 17, 2012; approved onJanuary 2, 2013; published online on January 4, 2013. Discussion periodopen until February 1, 2014; separate discussions must be submitted forindividual papers. This paper is part of the Journal of Geotechnical andGeoenvironmental Engineering, Vol. 139, No. 9, September 1, 2013.©ASCE, ISSN 1090-0241/2013/9-1575–1589/$25.00.
JOURNAL OF GEOTECHNICAL AND GEOENVIRONMENTAL ENGINEERING © ASCE / SEPTEMBER 2013 / 1575
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moderate granular rearrangement. This only happens beyond somethreshold shear strain, which can generally be taken for clays as0.1% [following the review by Matasovic and Vucetic (1992)]. Allsoils tend to continue compacting under moderate magnitudes ofcyclic shear stress, giving rise to excess pore pressures that increasecycle by cycle if the soil is undrained. This is most evident in loosegranular soils, which can lose all effective stress after many cycles,a phenomenon known as liquefaction. In fine-grained soils, such asclays, the phenomenon is less aggressive and is known as “cyclicmodulus degradation” (Matasovic and Vucetic 1995). As the namesuggests, stiffness reduces with continuing cycles of straining, oftenat a steady rate with respect to the logarithm of the number of cycles.
Both these mechanisms are influenced by rate effects that permitmore contact sliding by creep over longer periods and which cor-respondingly offer apparent viscous stiffening of the soil skeleton athigher shear-strain rates. This is a significant factor in merging staticand dynamic test data.
Estimates of soil stiffness at any strain level are important forboth earthquake and foundation engineering practice. A key pa-rameter that must be well understood to make such predictions isthe maximum stiffness modulus Gmax. This paper also studies thenormalization of the shear-strain axis. A modified hyperbola wasadopted (e.g., Darendeli 2001), and simple correlations for referencestrain gref are proposed. The results of the analysis will be used toupdate the commonly used design curves of Vucetic and Dobry(1991). The test data in the database was adjusted for rate effects, asboth static and dynamic tests were used to obtain the original data.Two adjustments were made: the first for a typical foundation-engineering scenario and the second for a typical dynamic-loading(earthquake-engineering) scenario. Although approximate, theseadjustments are necessary to reduce some of the disparity betweenthe results of dynamic and static tests. The correlations presentedhere are useful in the low-strain region, which is common infoundation-engineering applications under working loads. For sit-uations in which failure is approached more closely, engenderinglarger strains, the mobilized-strength approach of Vardanega andBolton (2011b, 2012) is recommended.
Database
A database of stress-strain tests on fine-grained soils has beencompiled. Each set of data has been published previously by itsoriginal authors in refereed journals, refereed conference proceed-ings, or (in one case) in a research report produced by internationallyrecognized authors in this subject. The qualification for selection oftest data was that sufficient information was provided on test con-ditions to enable correlations to be performed. Table 1 summarizesthe sources of data from 21 fine-grained soils used in the study ofstiffness-strain response (digitization of the test data was undertakenunless the raw data files could be sourced). The samples were derivedfrom various countries and were tested under a variety of conditions,from normally consolidated to heavily overconsolidated, in variouslaboratories and on a variety of shear-testing devices over a period of30 years. Table 1 also details values of basic soil properties (w, e0,wL,wP, IP) that were reported in the original publications that describe thetested soils (when void ratio was not reported it was estimated usingthe stated water content). Also detailed are values of confining stress,p9, undrained shear strength, cu, and overconsolidation ratio, OCR, incases in which these were reported. For the soils in the database,the plasticity index, IP, varies from 0.10 to 1.50, with a meanvalue of 0.39 and a coefficient of variation (COV) of 0.60; voidratio, e0, ranges from 0.48 to 6.15, with a mean value of 1.40 anda COV of 0.71.
Curves of G versus strain evaluated in this paper are intendedsimply to be modulus reduction curves, assuming that no significantdegradation has been caused by continued cyclic loading in the claysand silty clays, which are the focus of this study. Any excess porepressures created in the reported tests were taken to be a validcomponent of the undrained test response; but it should also be notedthat where individual authors did measure them (e.g., Teachavor-asinskun et al. 2002) they were also found to be small relative tothe initial mean effective stress. This means that the shear modulireported here should also be relevant to the calculation of sheardistortions in drained clays tested at the same initial value of themean effective stress p9. The only additional consideration in theprediction of drained ground movements would be an allowance forvolume changes because of pore pressure dissipation.
Cyclic Data
It should be recognized that most of this data relates to cyclic testingin which the immediately preceding strain history is one of reversalof the principal strain directions. The initial behavior exhibitedwould therefore be expected to be one of maximum stiffness Gmax
(Atkinson et al. 1990). If the recent strain path in the field is known,and if the future strain path as the result of loading is similar, theengineer must anticipate that the stiffness of the future response willbe reduced. If the strain prior to future loading were known, theengineer could simply use it as a datum on the stress-strain curvederived from a cyclic test. However, a sufficient resting period maybe sufficient to wipe the memory of the previous small-strain historyof a soil, returning the strain datum to zero and the soil to itsmaximum stiffness condition (Clayton and Heymann 2001), thoughsome strain-path memory may be retained in cases in which largerstrains have previously occurred (Gasparre 2005). Furthermore,long-term aging is known to increase the elastic stiffness Gmax ofclays (Santagata and Kang 2007). Judgment will be required in theapplication of the recent-reversal stiffness data reported here.
Anisotropy
It is recognized that the various apparatuses used by the investigatorscited in Table 1 induce principal compressive strains in either thevertical direction normal to the presumed bedding (triaxial com-pression test) or at 45� to both the vertical and horizontal planes(resonant column, torsional shear, direct simple shear). Some of thescatter, which will be observed in the statistical correlation to bederived subsequently, will no doubt arise from anisotropy, espe-cially in terms ofGmax. Two studies of anisotropy in ancient, heavilyoverconsolidated clays are noteworthy. The anisotropy of GaultClay was reported by Lings et al. (2000), and a great deal of data onLondon Clay can be found in Gasparre (2005). Both of these sourcesshowed that the shear stiffness on horizontal planes was about twotimes greater than the shear stiffness on vertical planes.
Graham and Houlsby (1983) presented a mathematical frame-work that introduced an anisotropy parameter a, which is definedas Gmax,hh=Gmax,vh, and for which it was assumed that the ratio ofYoung’s moduli for changes of effective stress 5 Emax,h=Emax,v 5a2. Following a fitting of data for the aged, normally consolidatedmedium-to-highly plastic Winnipeg Clay, values of a ranged from1.14 to 1.57. Lings et al. (2000) showed a high ratio of about 4 for theYoung’smoduli forGault Clay compressed horizontally andvertically,which roughly conforms to the framework of Graham and Houlsby.
However, in Gasparre’s London Clay data, Gmax,hh=Gmax,vh �E0,h=E0,v � 2, based on four undrained compression tests for whichEumax,v=Gmax,vh � 2:7. This measured ratio only applies to particular
units of London clay fromundisturbed samples taken from the site of
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Tab
le1.
Sum
maryof
DatabaseforStiffnessDegradatio
nof
Fine-Grained
Soils:S
oiland
Curve-FittingParam
etersfortheTwoRepresentativeStrainRates
Studied
Pub
lication
Testin
gapparatus
Testlabela
Soilp
roperties
Static
adjustment
Dyn
amic
adjustment
we 0
wL
wP
I P
p9(kPa)
c u(kPa)
bG
max
(MPa)
cOCRd
na
R2
gref
na
R2
gref
And
erson
andRichart
1976
Reson
antcolum
nLedaClayI
0.79
2.19
0.69
0.25
0.44
5223
70.76
30.97
70.00
121
70.80
60.97
70.00
214
DetroitClay
0.46
1.30
0.55
0.25
0.30
5944
50.59
10.94
60.00
075
50.87
60.94
50.00
145
FordClay
0.30
0.82
0.37
0.18
0.19
6371
50.73
80.98
40.00
024
51.12
80.93
50.00
060
SantaBarbara
Clay
0.80
2.28
0.83
0.39
0.44
1122
50.58
30.92
50.00
051
51.04
30.91
90.00
084
Eaton
Clay
0.27
0.72
0.40
0.20
0.20
6379
40.55
20.98
50.00
054
40.95
70.96
20.00
082
Kim
and
Nov
ak19
81Reson
antcolum
nWindsor
SiltyClay
0.51
1.36
0.51
0.21
0.30
267
2330
2.7
100.74
40.97
20.00
053
101.16
40.99
30.00
091
Windsor
SiltyClay
0.51
1.36
0.51
0.21
0.30
404
2337
2.7
100.74
40.96
30.00
052
101.24
80.97
70.00
094
Wallacebu
rgSiltyClay
0.38
1.05
0.42
0.18
0.25
267
3946
5.1
90.72
10.98
20.00
046
91.29
70.98
80.00
063
Wallacebu
rgSiltyClay
0.38
1.05
0.42
0.18
0.25
404
3957
5.1
90.77
70.98
20.00
044
91.18
00.98
70.00
072
SarniaSiltyClay
0.23
0.59
0.30
0.15
0.14
267
7610
21.8
80.93
50.99
20.00
028
81.27
70.99
80.00
042
SarniaSiltyClay
0.23
0.59
0.30
0.15
0.14
404
7612
71.8
80.82
00.99
30.00
033
81.36
90.97
20.00
048
Ham
ilton
ClayeySilt
0.17
0.48
0.25
0.13
0.12
267
127
128
5.8
90.85
80.99
60.00
024
91.19
30.99
30.00
037
Ham
ilton
ClayeySilt
0.17
0.48
0.25
0.13
0.12
404
127
159
5.8
90.86
00.99
10.00
026
91.24
70.99
50.00
039
Chatham
ClayeySilt
0.28
0.75
0.29
0.15
0.14
267
4676
2.1
90.79
10.99
30.00
027
91.12
50.99
60.00
043
Chatham
ClayeySilt
0.28
0.75
0.29
0.15
0.14
404
4694
2.1
90.79
30.99
30.00
030
91.18
40.99
40.00
046
PortS
tanley
SiltyClay
0.23
0.58
0.35
0.16
0.20
404
5512
96.8
80.85
60.99
20.00
032
81.42
70.97
70.00
046
Iona
SiltyClay
0.20
0.62
0.27
0.14
0.13
404
268
120
6.4
90.88
50.99
20.00
038
91.31
30.98
60.00
060
Georgiann
ouet
al.19
91Reson
ant
column,
triaxial,and
torsionalshear
Pietrafitta
ClayI-RC
0.42
1.20
0.62
0.32
0.30
320
164
130.63
60.93
30.00
082
121.34
00.99
50.00
107
Pietrafitta
ClayI-TX
0.42
1.20
0.62
0.32
0.30
320
164
210.96
00.97
50.00
102
201.13
80.90
60.00
176
VallericcaClayI-RC
0.29
0.84
0.53
0.22
0.31
6082
110.66
60.92
60.00
036
91.26
60.99
80.00
058
VallericcaClayI-TX
0.29
0.84
0.53
0.22
0.31
6082
220.71
90.96
80.00
069
210.82
10.92
70.00
123
VallericcaClayI-TS
0.29
0.84
0.53
0.22
0.31
6072
90.70
80.98
70.00
066
42.16
40.93
20.00
065
Tod
iClay-RC
0.17
0.50
0.48
0.20
0.28
200
159
160.73
80.97
60.00
033
141.24
50.96
30.00
063
Ram
pello
and
Silv
estri19
93Reson
antcolum
nVallericcaClayII
0.29
0.80
0.59
0.28
0.32
5041
4.4
110.78
60.96
10.00
048
111.31
80.98
90.00
083
VallericcaClayII
0.29
0.80
0.59
0.28
0.32
100
119
4.4
150.67
10.96
10.00
039
120.99
60.99
90.00
073
Pietrafitta
ClayII
0.42
1.13
0.87
0.35
0.53
100
934.0
120.70
10.97
30.00
069
121.18
70.98
70.00
108
Pietrafitta
ClayII
0.42
1.13
0.87
0.35
0.53
130
113
4.0
80.58
20.93
20.00
093
81.58
30.97
10.00
075
Shibu
yaand
Mitachi19
94Torsion
alshear
Hachir� ogata
ClayT19
0.90
2.34
1.16
0.41
0.75
131
3517
1.0
100.63
00.98
60.00
146
61.04
60.98
50.00
255
Hachir� ogata
ClayT24
0.97
2.55
1.22
0.44
0.78
115
4516
1.0
110.64
00.99
00.00
152
61.03
70.97
10.00
292
Hachir� ogata
ClayT11
1.28
3.40
1.37
0.52
0.85
7725
121.0
90.79
30.99
40.00
135
61.00
90.98
20.00
229
Hachir� ogata
ClayT15
1.13
3.01
1.12
0.51
0.61
6930
81.0
100.77
20.99
50.00
214
71.13
50.98
90.00
393
Hachir� ogata
ClayT10
1.31
3.48
1.40
0.51
0.89
4522
51.0
80.59
30.99
00.00
283
50.96
80.99
20.00
504
Hachir� ogata
ClayN33
1.64
4.31
1.65
0.58
1.07
3720
31.0
110.55
50.99
10.00
231
80.98
20.97
50.00
400
Hachir� ogata
ClayT1
2.50
6.15
2.39
0.89
1.50
2320
21.0
80.54
80.98
00.00
335
41.29
10.98
50.00
447
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Tab
le1.
(Con
tinued.)
Pub
lication
Testin
gapparatus
Testlabela
Soilp
roperties
Static
adjustment
Dyn
amic
adjustment
we 0
wL
wP
I P
p9(kPa)
c u(kPa)
bG
max
(MPa)
cOCRd
na
R2
gref
na
R2
gref
Sog
a19
94Triaxial
San
Francisco
Bay
Mud
3m
0.92
2.44
0.88
0.48
0.40
3014
1.5
230.50
10.96
40.00
082
140.75
00.93
50.00
156
San
Francisco
Bay
Mud
3m
0.92
2.44
0.88
0.48
0.40
3014
1.5
280.55
20.93
40.00
136
160.80
20.95
30.00
285
San
Francisco
Bay
Mud
5m
0.92
2.44
0.88
0.48
0.40
5013
1.5
270.64
90.94
20.00
134
150.72
10.98
30.00
278
PisaClay4m
0.30
0.80
0.39
0.29
0.10
5036
4.5
260.70
50.98
10.00
039
160.92
90.98
20.00
078
PisaClay10
m0.63
1.66
0.92
0.44
0.48
8529
1.3
250.70
80.97
80.00
056
170.83
30.99
40.00
106
PisaClay19
m0.62
1.65
0.96
0.41
0.55
135
291.3
230.62
80.97
70.00
126
181.20
50.96
80.00
245
PisaClay14
m0.40
1.05
0.52
0.31
0.21
116
491.3
220.74
20.91
10.00
049
160.76
10.98
70.00
086
PisaClay14
mH
0.40
1.05
0.52
0.31
0.21
116
551.3
220.87
30.93
20.00
040
150.83
80.99
00.00
061
Dorou
dian
andVucetic
1999
Directsim
ple
shear
HighlyPlasticSilt
9.5m
0.62
1.79
0.92
0.54
0.38
163
6414
0.97
00.91
40.00
071
81.00
60.99
60.00
140
HighlyPlasticSilt
20.7m
0.52
1.42
0.83
0.50
0.33
240
100
180.82
00.99
40.00
076
101.27
40.98
20.00
133
HighlyPlasticSilt
31.0m
0.47
1.35
0.82
0.51
0.31
320
126
1728
0.76
70.98
80.00
110
191.22
50.99
10.00
200
HighlyPlasticSilt
64.6m
0.46
1.36
0.81
0.51
0.30
570
179
150.72
50.99
10.00
094
111.03
40.99
60.00
188
Yim
siri20
01Triaxial
Lon
donClayI(B-1)
0.24
0.70
0.81
0.23
0.58
270
202
9433
0.67
90.99
60.00
061
310.80
30.97
30.00
127
Lon
donClayI(B-2)
0.24
0.70
0.81
0.23
0.58
270
199
9026
0.85
60.98
80.00
127
230.91
80.98
30.00
209
Lon
donClayI(C-1)
0.25
0.57
0.69
0.22
0.47
310
365
114
270.64
50.98
10.00
063
250.78
30.96
40.00
145
Lon
donClayI(C-2)
0.25
0.59
0.69
0.22
0.47
310
336
127
180.78
30.98
30.00
103
160.93
70.91
60.00
203
Lon
donClayI(D
-1)
0.20
0.57
0.54
0.19
0.36
410
348
105
240.77
10.97
50.00
227
210.96
30.96
70.00
439
Lon
donClayI(D
-2)
0.20
0.55
0.54
0.19
0.36
410
407
9724
0.68
60.99
10.00
169
200.82
00.97
50.00
332
Teachavorasinskun
etal.20
02Triaxial
Bangk
okClaySite
10.57
1.50
0.63
0.30
0.34
5027
117
0.76
80.99
50.00
114
170.89
60.98
60.00
213
Bangk
okClaySite
10.57
1.50
0.63
0.30
0.34
5027
119
0.87
60.98
30.00
116
190.96
30.99
20.00
181
Bangk
okClaySite
10.57
1.50
0.63
0.30
0.34
150
381
130.79
50.99
40.00
071
130.94
30.99
60.00
122
Bangk
okClaySite
10.57
1.50
0.63
0.30
0.34
150
381
160.85
90.99
00.00
085
160.97
10.99
60.00
138
Bangk
okClaySite
10.57
1.50
0.63
0.30
0.34
250
451
230.68
00.98
80.00
078
230.80
10.99
70.00
147
Bangk
okClaySite
10.57
1.50
0.63
0.30
0.34
250
451
210.78
20.97
10.00
079
210.89
40.99
00.00
132
Bangk
okClaySite
20.65
1.72
0.83
0.43
0.40
5022
111
0.72
50.99
10.00
136
110.85
90.97
20.00
242
Bangk
okClaySite
20.65
1.72
0.83
0.43
0.40
5022
116
0.63
90.98
90.00
095
160.73
50.99
60.00
170
Bangk
okClaySite
30.65
1.72
0.83
0.43
0.40
6024
116
0.73
30.99
40.00
061
160.82
60.98
20.00
108
Bangk
okClaySite
30.63
1.66
0.83
0.43
0.40
6025
118
0.82
30.99
00.00
061
160.94
90.97
50.00
103
Gasparre20
05Triaxial
Lon
donClayII(t36
)0.26
0.72
0.66
0.29
0.37
260
158
4436
0.81
40.99
80.00
212
340.92
00.98
90.00
394
Lon
donClayII(t52
)0.26
0.70
0.65
0.28
0.37
257
187
5655
1.08
40.99
80.00
336
331.00
70.97
20.00
417
Lon
donClayII(t33
)0.26
0.70
0.72
0.28
0.44
395
290
8131
1.12
60.95
70.00
080
221.39
50.90
60.00
134
Lon
donClayII(t13
)0.25
0.69
0.59
0.26
0.33
502
250
105
420.83
20.94
90.00
206
320.68
90.97
20.00
172
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Heathrow Terminal 5. Such data are not available for other clays inthe database. Therefore, when dealing with triaxial compressiondata, the isotropic condition Eu
max=Gmax 5 3 has been assumedthroughout. Fig. 1 shows a plot of G versus shear strain for the 67tests detailed in Table 1.
ExaminationofThreeExistingNormalizationMethods
We now turn our attention to the normalization of the severelyscattered data of shear modulus versus shear strain for the 67 tests on21clays in the study (seeFig. 1). Three commonmethods of stiffnessnormalization are used for clays: G=p9, G=cu, and G=Gmax.
Normalization with Mean Effective Stress p9
NormalizingGwith p9 is a technique used by some researchers (e.g.,Pantelidou and Simpson 2007; Hight et al. 2007; Grammatiko-poulou et al. 2008). Fig. 2 shows the plot ofG=p9 versus strain for theclays considered in the database. Much of the scatter from Fig. 1remains in Fig. 2, though the data does appear to converge at higherstrains.
Normalization with Undrained Shear Strength (cu)
Another possible normalization method for G or E (taken to be 3Gwhen Poisson’s ratio is 0.5) is to divide it by cu. Butler (1975) andHewitt (1989), for example, use the E=cu ratio to develop empiricalrelationships to estimate the settlement of structures. Five of the tenpublications consulted here gave values of cu. Fig. 3 shows the plotof G=cu versus strain for the available data. The scatter betweendifferent clays has not been appreciably reduced from Fig. 1.
Normalization with Gmax
UsingGmax to normalize the reduction of shear modulus with strainis common, especially in earthquake engineering literature, and wasused in the 10 publications listed in Table 1. Fig. 4 shows the plot ofG=Gmax versus log shear strain. It is evident that the use ofG=Gmax isa much more effective way of reducing scatter than either p9 or cu.
Maximum Shear Modulus
It is clear from Fig. 4 that the maximum shear modulus, Gmax,is successful as a normalizer for shear modulus data and that thecommonly used surrogates are not acceptable. It follows that Gmax
should ideally be estimated or measured when studies of soilstiffness degradation are undertaken. In this section, commonly usedsemiempirical expressions for Gmax are reviewed.
Hardin and Black (1968, 1969) provide a commonly used em-pirical relationship for Gmax in kPa
Gmax ¼ Cð2:9732 eÞ2
ð1þ eÞ ðOCRÞKðp9Þ0:5 (1)
where e5 voids ratio; OCR5 overconsolidation ratio; p9 5 initialmean effective stress (kPa);K5 0, 0:18, 0:30, 0:41, 0:48, and 0:5 (forIp values of 0, 0:2, 0:4, 0:6, 0:8, and . 1:0, respectively); C 5 con-stant 3,2301=2 kPa (or 1,230, if both Gmax and p9 are measured in psi).
This equationwas developed from a similar expression for sands,butwith the factorOCRK included for clays (Hardin and Black 1968,1969). This extension was based on the stiffness data of two recon-stituted clays, a flocculated kaolinite and a dispersed Boston BlueClay. Further refinements to the Hardin equation were published byT
able
1.(Con
tinued.)
Pub
lication
Testin
gapparatus
Testlabela
Soilp
roperties
Static
adjustment
Dyn
amic
adjustment
we 0
wL
wP
I P
p9(kPa)
c u(kPa)
bG
max
(MPa)
cOCRd
na
R2
gref
na
R2
gref
Lon
donClayII(t19
)0.23
0.62
0.60
0.28
0.32
518
266
9671
0.71
00.97
10.00
127
610.90
40.95
30.00
304
Sum
maryof
statisticsforeach
column
Maxim
umvalue
2.50
6.15
2.39
0.89
1.50
570
407
179
1771
1.12
60.99
80.00
3356
612.16
40.99
90.00
5041
Minim
umvalue
0.17
0.48
0.25
0.13
0.10
2311
21
40.50
10.91
10.00
0239
40.68
90.90
60.00
0374
Meanvalue,m
0.52
1.40
0.70
0.32
0.39
209
126
683
170.74
60.97
50.00
0972
141.05
50.97
50.00
1658
Num
berof
datapo
ints,n
6767
6767
6762
3567
4267
6767
6767
6767
67Stand
arddeviation,
s0.39
0.99
0.35
0.14
0.23
149
119
473
120.12
20.02
20.00
19
0.24
60.02
30.00
1COV
0.75
0.71
0.50
0.45
0.60
0.71
0.95
0.70
1.10
0.67
10.16
30.02
30.73
10.64
80.23
30.02
40.70
2
Note:Som
epo
intswerelostwhentheadjustmentfor
rateeffectswas
madeas
Gvalues
increasedabov
eG
max
becauseof
theincrease
instiffnessresulting
from
stress
reversal.
a The
shortacrony
msin
theTestLabel
column[e.g.t36,
B-1,T24
,N33
,and
RC]aredescriptorsused
intheoriginal
publications.
bValuesin
italicswereconservativ
elyestim
ated
asthree-qu
arters
oftherepo
rted
value.
c Valuesin
italicswereestim
ated
usingaHardin-styleequatio
n.dBangk
okClayOCRvalues
areapprox
imate(fieldvalues
may
reveal
light
overconsolidation).
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Hardin and Blandford (1989), but the form of the equation remainedessentially the same.
Shibuya et al. (1997) introduced a simplified voids-ratio functionusing specific volume
FðeÞ ¼ ð1þ eÞx (2)
This avoided the sharp reduction in stiffness given by Eq. (1), ase→ 2:973, and it is based on a sounder physical parameter thatis equivalent to dry density. The investigators fitted exponent
x5 22:4 to the data that was available to them, and accordinglyrecommended
Gmax
pr9¼ B
ð1þ eÞ2:4�p9pr9
�0:5
(3)
where pr9 5 reference pressure, taken as 1 kPa.It is recognized that elastic contact mechanics provides that the
appropriate maximum stiffness of an assembly of grains is Gmax }ðp9ÞnðGgÞ12n, where Gg is the shear stiffness of the grain material,
Fig. 1. Collected secant shear-stiffness versus shear-strain data (shear strain expressed numerically and not as a percentage)
Fig. 2. Secant shear modulus normalized with confining stress versus shear strain (shear strain expressed numerically and not as a percentage; key asfor Fig. 1)
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ll ri
ghts
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erve
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and n is an exponent that can be taken as 0.50 for smooth sphericalcontacts and 0.33 for conical asperities (Richart et al. 1970; Goddard1990). The physically meaningful dimensionless groups involvedin presenting data of Gmax varying with p9 would accordingly beGmax=Gg and p9=Gg, plotted on log-log axes. But because clayplatelets are variously anisotropic because of their crystalline nature,an appropriate value for Gg appears to be unattainable. For thisreason, the alternative dimensionless form of Eq. (3) has beenadopted, using an arbitrary reference pressure, pr95 1 kPa.
In Shibuya et al. (1997), factor B for soft clays in Eq. (3) rangedfrom 18,000 to 30,000, with an average of about 24,000. This ex-pression is used to construct Fig. 5 for that portion of the newdatabase described in Table 1 for which data for Gmax were available
(83 estimates). Some of the publications consulted reported extrameasurements of Gmax, confining stress, and void ratio that were notaccompanied by a complete shear-modulus reduction curve, and thesewere used in the analysis of Gmax. It is evident that a central body ofdata fits with factor B� 20,000 within the range 15,000–25,000,thereby confirming the findings of Shibuya et al. (1997). The outliersshowing high measured values (B� 50,000) were from high-qualitysamples of overconsolidated, aged clays from Italy, i.e., Pisa, Val-lericca, Pietrafitta. The highly plastic silt from Santa Barbara was alsofound to exhibit high B values. Low measured values (B, 15,000)were associated with similar high-quality tests on London Clay fromHeathrow Terminal 5, which was described as highly fissured(Gasparre 2005) and in tests from Kennington Park (Yimsiri 2001).
Fig. 3. Secant shear modulus normalized with undrained shear strength versus shear strain (shear strain expressed numerically and not as a percentage;key as for Fig. 1)
Fig. 4. Secant shear modulus normalized with small strain shear modulus versus shear strain (shear strain expressed numerically and not asa percentage; key as for Fig. 1)
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If possible, therefore,Gmax should be measured in situ during siteinvestigation, in order to gain evidence of the stiffness of the naturalclay structure and to reduce the scatter in B-values implicit in usingEq. (3). Plate dilatometer tests may be used with the correlationsestablished in Hryciw (1990). Shear wave-speed (Vs) measurements(e.g., Stokoe et al. 2011) may provide more accurate estimates(Gmax 5 rV2
s ) from Rayleigh wave or refraction surveys, such as byseismic cone profiling (Abbiss 1981; Heymann 2003) or by usingcross-hole methods (e.g., Yoshimi et al. 1977). Alternatively, Gmax
can be found from high quality rotary cores tested in the laboratoryusing bender elements in triaxial tests (Atkinson 2000). In each case,Gmax should bemeasured at some location inwhich both p9 and e canbe inferred, so that a corresponding value ofB can be calculated fromEq. (3).
Significance of Stress History
Viggiani and Atkinson (1995) proposed a formulation for Gmax,which, assuming that p9p is taken as being equivalent to p9max (in otherwords, taking OCR as equal to the yield-stress ratio), can be writtenas
Gmax
pr9¼ A
�p9pr9
�n
ðOCRÞm ¼ A
�p9pr9
�n�p9max
p9
�m
(4)
where n and m5 constants that depend on clay type, e.g., plasticityindex Ip; A 5 a factor that accounts for clay structure in a fashionsimilar to that of parameter B in Eq. (3); R0 5 overconsolidationratio, as defined by pp9=p9 (often termed the yield-stress ratio); pp95 effective stress at the intersection of a swelling line with thenormal compression line; and p9 5 mean effective stress.
This paper shows an alternative form
Gmax
pr9¼ B
va
�p9pr9
�k
(3a)
where v5 11 e.Butterfield (1979) demonstrated that compression and swelling
lines of clays were best seen as straight on logðvÞ2 logðp9Þ axes,rather than conventional v2 logðp9Þ axes. New compression in-dexes l� (plastic) and k� (elastic) were used (see Fig. 6). On theswelling line we can say
lnðvÞ ¼ lnðvminÞ þ k�ln�p9max
p9
�(5)
lnðvÞ ¼ ln
"vmin
�p9max
p’
�k�#(6)
v ¼ vmin
�p9max
p9
�k�
(7)
Therefore, by raising Eq. (7) to a power denoted here as a, weobtain
va ¼ ðvminÞa�p9max
p9
�ak�
(8)
Substituting Eq. (8) into Eq. (3) yields
Gmax
pr9¼ B
ðvminÞa�p9pr9
�k�p9
p9max
�ak�
(9)
On the normal compression line
lnðvminÞ ¼ lnðvnÞ2 ðl�Þln�p9max
pr9
�(10)
vmin ¼ vn
�p9max
pr9
�2l�
(11)
Substituting Eq. (11) into Eq. (9) gives
�Gmax
pr9
�¼ B
van
�p9pr9
�k�p9
p9max
�ak��p9max
pr9
�al�
(12)
which can be written as
�Gmax
pr9
�¼ B
van
�p9pr9
�kþal��p9max
p9
�aðl�2k�Þ(13)
Fig. 5. Determination of B values for the database, using Eq. (3)
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This has the same form as Eq. (4), in which A, n, and m are soilconstants given as A5B=van , n5 k1al�, and m5aðl� 2 k�Þ.Houlsby and Wroth (1991) showed theoretically that of the threevariables e, p9, and OCR in Eq. (1) one is redundant (under isotropicstress conditions). Rampello et al. (1997) remarked that the void-ratio function FðeÞ is unnecessary. The Viggiani and Atkinson(1995) expression Eq. (4) has become popular, and is exploredfurther in a recent experimental study by Choo et al. (2011).However, in practice, the advantage of Eq. (3) over Eq. (4) is that thevoids ratio can be found from the natural water content in a standardsite investigation, whereas OCR cannot be estimated so easily.Nevertheless, either Eq. (3) or Eq. (4) is an improvement on Eq. (1),which involves a redundant mixture of all of these parameters,notwithstanding the algebraic relationship revealed previously.
Rate Effects
Foundation Analysis
Some of the scatter in Figs. 1–4 is because of the influence of rateeffects, as the original researchers employed different test appara-tuses and test frequencies. It has been known for many years that thestiffness and strength of clays is rate-sensitive (e.g., Richardson andWhitman 1963). Vardanega and Bolton (2011a) suggested thatresonant-column and static test data could be merged within a da-tabase using a simple rate-effect adjustment. The selection of anappropriate rate-effect adjustment for static situations, hereaftercalled the static adjustment, is outlined in subsequent sections.
A carefully conducted undrained triaxial test that achieves peakstrength at an axial strain of about 2% (and therefore a shear strain ofabout 3%) after 8 h would have a shear-strain rate of 1026=s. On theother hand, a resonant column vibrating under maximum excitationwith a cyclic shear-strain amplitude of 0.1% at 50 Hz would havea peak shear-strain rate of 0.3/s, which is 5:5 log10 cycles faster thanthe triaxial test. The focus of this paper is to evaluate stiffness at smallstrains. Accordingly, all stiffness data will be normalized to astandard test rate of 1026=s, by assuming a strain-rate effect of 5%per log10 cycle, which is consistent with the findings of Lo Prestiet al. (1997) and d’Onofrio et al. (1999). In doing so, it is acceptedthat the stiffness of very low plasticity clays at low cyclic-strainamplitudes in resonant-column tests is likely to be underestimated,and that the stiffness of high-plasticity clays at large strain ampli-tudes in resonant-column tests may remain overestimated. Never-theless, the disparity in stiffness between dynamic and static testresults should have been reduced.
It is assumed that the onset of grain slippage (and thefirst instanceofG,Gmax) occurs at 1025 strain, and that only strains greater thanthis will lead to rate effects. The maximum shear strain rate duringvibration at a frequency f and a shear-strain amplitude of g is
_g ¼ 2p fgs
(14)
That part of the strain rate that can give rate effects is
_gR ¼ 2p f�g2 1025
�s
(15)
The frequency values for the various test apparatuses that were usedin the publications evaluated for this analysis are shown in Table 2.The rate effect is taken to be a 5% increase in stiffness per factor of 10increase in plastic strain rate _gZ . So the rate-linked reduction factor Zon the stiffness measured in a test is taken as
Z ¼"1þ 0:05 log10
�_gZ
1026
�#(16)
such that G5Gmeasured=Z. Unless otherwise stated, resonant-column test data (for example) is assumed to have been taken atf 5 50 Hz, therefore
_gZ ¼ 100p�g2 1025
�s
(17)
Z ¼(1þ 0:05 log10
�100p
�g2 1025
�1026
��(18)
where strain amplitude g for each test was taken from the publishedpapers.
Dynamic Analysis
Using similar equations to those shown previously, for a rate effectadjustment that is simulative of a typical earthquake situation,
Fig. 6. Definition of terms
Table 2. Assigned Test Frequencies for Rate Correction
Publication Apparatus Assigned test frequency
Anderson andRichart 1976
Resonant column Assumed 50 Hz
Kim and Novak 1981 Resonant column Assumed 50 HzGeorgiannouet al. 1991
Resonant column Assumed 50 Hz
Triaxial Assumed 0.1 HzTorsional shear Assumed 0.025 Hz
Rampello andSilvestri 1993
Resonant column Assumed 50 Hz
Soga 1994 Triaxial Given in thesisShibuya andMitachi 1994
Torsional shear Given in paper
Doroudian andVucetic 1999
Direct simple shear Assumed 0.025 Hz
Yimsiri 2001 Triaxial Strain rates quotedin thesis
Teachavorasinskunet al. 2002
Triaxial Given in paper
Gasparre 2005 Triaxial Strain rates quoted in thesis
JOURNAL OF GEOTECHNICAL AND GEOENVIRONMENTAL ENGINEERING © ASCE / SEPTEMBER 2013 / 1583
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a typical earthquake frequency is taken as 1 Hz, hereafter calledthe dynamic adjustment. A simplistic shear strain rate is therefore1022=s. Whereas the specific values are a matter of judgment, theincrease in stiffness that is implied when moving from 1026 (static)to 1022 (dynamic) should allow us to see the relative differencebetween the shear-modulus reductions with strain in the twosituations.
Influence of Strain
Konder (1963), Duncan and Chang (1970), and Hardin and Drne-vich (1972) used hyperbolae to model shear stress-strain curves,which were asymptotic to Gmax at zero strain and to tmax at infinitestrain. By defining a reference strain (gref 5 tmax=Gmax), it was po-ssible to rewrite the equation of a hyperbola as a normalized secantshear modulus (G=Gmax) that reduces with normalized shear strain(g=gref )
GGmax
¼ 1�1þ g
gref
� (19)
Darendeli (2001) and Zhang et al. (2005) both raised the normalizedshear strain (g=gref ) to a power of a to better fit the data of smallstrains [Eq. (20)]. This definition retains the feature that secant shearstiffness reduces to half of its initial maximum value when g5 gref .The current study adopted the same family ofmodified hyperbolae tofind an optimum fit for each soil. Thea term in Eq. (20) is referred toas the curvature parameter
GGmax
¼ 1
1þ�
g
gref
�a (20)
The best-fit values of parameters a and gref for each of the soilsstudied are listed in Table 1 for both of the typical strain rate levelsdiscussed in the previous section together with the correspondingcoefficients of determination, R2. The statistical fit for Eq. (20) isvery good (R2 ranges from0.911 to 0.998 for the static corrected dataand from 0.906 to 0.999 for the dynamic corrected data). Furtherempirical correlations must now be obtained for gref and a in termsof the readily available soil properties (eo, Ip,wL,wP, p9, OCR) fromTable 1.
Fitting a Model for Secant Stiffness Reductionwith Strain
A hyperbolic model was fitted to the entire dataset, normalizing thestrain g with the reference strain gref appropriate to each of the soilsfrom Table 1. No significant correlation could be found between thecurvature parameter and any of the basic soil properties, listed inTable 1.
Therefore, for the data with the static adjustment applied, themodified hyperbola used to characterize the database is
log10½ðGmax=GÞ2 1� ¼ 0:736 log10
�g
gref
�(21a)
where R2 5 0:946, n5 1,164, SE5 0:168, and p, 0:001. This canbe rearranged as
GGmax
¼ 1
1þ�
g
gref
�0:736 (21b)
Fig. 7(a) shows the data (with the static adjustment applied) ofG=Gmax plotted against normalized shear strain, and Fig. 7(b) showsEq. (21) fitted to the dataset. A very good fit to the data was obtained,with the deviation from Eq. (21a) and (21b) most apparent at lowerstrains, g, 0:1gref .
For the data with the dynamic adjustment applied, the best-fitmodified hyperbola used to characterize the database was
log10½ðGmax=GÞ2 1� ¼ 0:943 log10
�g
gref
�(22a)
where R2 5 0:942, n5 959, SE5 0:170, and p, 0:001. This can berearranged as
GGmax
¼ 1
1þ
g
gref
!0:943 (22b)
Similarly, very good fit to the data are again obtained, with de-viation from Eq. (22a) and (22b) most apparent at lower strains,g, 0:1gref .
A regression analysis was first performed on individual soilproperties in the database to discover the strength of their relation-ship to reference strain gref at which the initial linear elastic stiffnesshad halved. Fig. 8 shows the plot of reference strain versus plasticityindex (following the work of Vucetic and Dobry 1991) for the grefvalues derived when the data with the static adjustment was applied,which gives rise to
gref ¼ J
IP1000
(23)
where Ip 5 plasticity index (expressed numerically, not as a per-centage); for data with the static adjustment applied, J5 2:2,R2 5 0:75, n5 62, SE5 0:00031, and p, 0:001, with five outliers;for data with the dynamic adjustment applied, the best fit J-valueis J5 3:7, R2 5 0:65, n5 62, SE5 0:00061, and p, 0:001, withfive outliers.
These coefficients of determination are reasonable, and thep-values are very low, but nevertheless Fig. 10 shows that there is a650% uncertainty in the predicted value of gref for its entire range(the same error exists for the data adjusted dynamically).
Discussion of London Clay Outliers
The outliers in Fig. 8 chiefly comprise five triaxial tests on high-quality cores of stiff, fissured London Clay, tested independently intwo different laboratories, but each employing sensitive strain-measurement techniques over an internal gauge length. Othertests conducted by the same investigators on similar samples fellwell within the main regression zone for gref . The outliers havea reference strain about three times greater than normal; that is, theyretain their original linear elastic stiffness under much larger strains.The reason for this is unknown, but it may relate to the high degreeof fissuring in these samples referred to in Gasparre (2005). Possi-ble fissuring of core samples is, no doubt, one of the reasons forpreferring field tests for Gmax, as recommended by Stokoe et al.
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(2011). If there was fissure opening in these samples from the outset,then thewhole samplewould have appearedmore compliant, and themeasurement of strainwithin the gauge lengthwould not correspondwith the intergranular slippage that causes loss of stiffness in anunfissured material. If fissure opening contributed an extra 0.15% ofmeasured shear strain, the location of the London Clay outliers inFig. 8 would be understandable.
Accuracy of Prediction Model
ThepredictionofG=Gmax nowdepends on two equations, Eq. (23) forgref , and Eq. (21b) or Eq. (22b) for the shape of the hyperboliccurve. The comparative success of these correlations (for thestatically corrected data) is shown in Fig. 9, in which 90% of thedata falls within a 630%margin, except those pertaining to certainLondon Clay tests in which measurements of G=Gmax at the upperend of the strain region considered in this paper can exceedpredictionsby a factor of up to 2.5. A loss of accuracy at low values ofG=Gmax is
particularly noticeable. Very similar levels of accuracy in the pre-diction also were observed for the dynamically corrected data.
New Design Charts
Vucetic and Dobry (1991) presented design charts that are com-monly used for seismic engineering purposes. They emphasize theimportance of the plasticity index. A shortcoming of these charts isthat they do not give a mathematical formulation for the degradationcurves that they indicate. Fig. 10(a) shows new design charts forstatic situations, based on the use of the plasticity index in Eq. (23),with the shape of the degradation curve given by Eq. (21b); Fig. 10(b)shows new design charts for dynamic situations, based on the use ofthe plasticity index in Eq. (23), with the shape of the degradationcurve given by Eq. (22b). The resulting charts show that the range ofexpected behavior is narrower than that suggested by the originalcurves of Vucetic and Dobry (1991), which are also shown. It is
Fig. 7.Twin axis normalization of statically adjusted test data and fitted hyperbola (key as for Fig. 1): (a)G=Gmax with static adjustment applied versusnormalized shear strain, g=gref ; (b) modified hyperbola fitted to statically adjusted data
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interesting that the past stress historywas not identified as significantin the original work by Vucetic and Dobry (1991) and also in theanalysis presented in this study.
Summary
A database of the secant shear stiffness of 21 clays and silts wascompiled from 67 tests from 10 publications. Three commonmethods of normalizing secant shear stiffness were examined inrelation to strain data. Plots of G=p9 versus shear strain and G=cuversus shear strain were shown to be relatively unsuccessful in
reducing the scatter of the collected data from different soils. Incomparison, G=Gmax versus shear strain was clearly seen to be thebest normalizer for shear modulus.
Gmax was normalized using a reference stress pr95 1 kPa, andwasshown to be best predicted as a power function of two easily esti-mated variables, specific volume (11 e) and mean effective stress( p9), and to a parameter B that may relate to soil structure and rangedfrom 15,000 to 50,000, with a typical value of 20,000. In anisotropicclay, the value of B reflects the shear conditions posed by the test.Engineers could request a test of Gmax in the plane of shear thatcorresponds with the mode of deformation expected in the designapplication.
Fig. 8. Correlation of reference strain with plasticity index (with static adjustment applied; key as for Fig. 1)
Fig. 9. Measured versus predicted G=Gmax
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Eq. (3) was shown to be functionally similar to a prediction basedsolely on OCR and p9, but specific volume is much more easilyobtained than OCR, so Eq. (3) may be preferred in practice. Gmax isthe best normalizer of secant stiffness G, and it cannot successfullybe substituted with undrained shear strength cu, confining stress p9,or even p9 together with a function of void ratio.
Previously published test data were normalized to two standardstrain rates by applying a stiffness adjustment factor of 5% per log10cycle of strain rate, which brought all tests to a rate equivalent to astandard triaxial test completed in a working day or a 1-Hz earth-quake. This adjustmentwas in accordancewith previously publisheddata for moderate strain amplitudes in clay.
A modified hyperbolic model was fitted to 67 sets of data forG=Gmax versus shear strain. A reference strain for elastic deter-ioration, gref , was first defined as the shear strain for which G=Gmax
drops to 0.5. This was shown to be reasonably well predicted byEq. (23) in terms of the plasticity index IP (expressed numerically,not as a percentage), gref 5 JðIP=1000Þ where J5 2:2 for the da-tabase with the static adjustment applied, and J5 3:7 for the da-tabase with the dynamic adjustment applied.
All the normalized data ofG=Gmax versus g=gref plotted in a verynarrow band. The curvature parameter a of a modified hyperbola,Eqs. (21b) and (22b), was fixed at one of two values to obtain a fitagainst the database, depending on the strain rate applicable forthe engineering application under consideration. Eqs. (21b) or (22b)and (23) together predicted over 90% of the G=Gmax ratios withinamargin of 630%across the full range of values from 0 to 1.0 for allsoils, with the exception of certain London Clay data, which issignificantly underpredicted. The influence of fissures may be thecause of certain London Clay outliers.
Reduced stiffness at intermediate strain levels can be estimatedbased only on knowledge of the plasticity index of the soil, and itgenerally falls within630% of predictions irrespective of the strainlevel of interest. Newdesign charts have been presented to update thecommonly used Vucetic and Dobry (1991) curves.
Acknowledgments
The authors thank the Cambridge Commonwealth Trust and OveArup and Partners for financial support to the first author during his
Fig. 10. New design charts for stiffness degradation of clays and silts: (a) static applications; (b) dynamic applications
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doctoral studies. Thanks are also due to Dr. Brian Simpson and Pro-fessor Mark Randolph for their helpful advice and suggestions.Thanks also to Dr. A. Gasparre for the provision of her triaxial testdata for analysis.
Notation
The following symbols are used in this paper:A 5 nondimensional factor linking Gmax to p9;B 5 coefficient linked to soil structure that along
with p9 and v relates to G0;C 5 coefficient in theHardinandBlack (1968)model;
COV 5 coefficient of variation (ratio of the standarddeviation to the mean);
cu 5 undrained shear strength;E 5 elastic Young’s modulus equal to 3G when
Poison’s ratio is equal to 0.5;Emax 5 elasticYoung’smodulus at very small strains;
e 5 voids ratio;e0 5 initial voids ratio;f 5 cyclic-test frequency;G 5 secant shear stiffness;
Gcyclic 5 secant shear stiffnessmeasured in a cyclic test;Gg 5 shear stiffness of grain material;
Gmax 5 shear stiffness at very small strains(sometimes referred to as G0);
Gmax,hh 5 shear stiffness at very small strains in thehorizontal plane;
Gmax,vh 5 strict definition ofG0 5 shear stiffness at verysmall strains in the vertical plane;
Ip 5 plasticity index;J 5 regression coefficient linking IP with gref ;K 5 exponent onOCRdependent onplasticity index;m 5 exponent on R0;n 5 exponent on normalized initialmean effective
stress;n 5 statistical term indicating number of data
points used to generate a correlation;OCR 5 overconsolidation ratio;
p 5 a statistical term indicating the smallest levelof significance that would lead to rejection ofthe null hypothesis, i.e., the value of r5 0, inthe case of determining the p-value fora regression;
p9 5 mean effective stress;p9max 5 maximum past mean effective stress;
pp9 5 effective stress at the intersection of a swellingline with the normal compression line;
pr9 5 reference value of mean effective stress(1 kPa);
R0 5 overconsolidation ratio defined as pp9=p9(sometimes called the yield-stress ratio);
R2 5 coefficient of determination of a correlation(the square of the correlation coefficient, r);
SE 5 standard error in a regression, a quantificationof deviation about the fitted line;
Vs 5 shear-wave speed;v 5 specific volume (11 e);
wL 5 liquid limit;wP 5 plastic limit;x 5 exponent on specific volume;
Z 5 rate-linked stiffness reduction factor;a 5 ratio of Young’s moduli when used in
reference to the framework from Graham andHoulsby (1983) or the curvature parameter ina modified hyperbolic equation;
g 5 shear strain;gcyclic 5 shear strain amplitude measured in a cyclic
test;gref 5 reference strain equal to the shear strain at
0.5Gmax;_g 5 shear-strain rate;_gZ 5 plastic shear-strain rate; andr 5 density of medium (soil).
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