viggiani, g., and atkinson, j. h._1995_geotechnique

Vi&b, G. & Atkinson, J. H. (1995). Gdotechnique 45, No. 2, 249-265

Stiffness of fine-grained soil at very small strains

G. VIGGIANI* and J. H. ATKINSON*

The stiffness of soil at very small strain G, is a useful parameter for characterizing the non-linear stressstrain behaviour of soil for monotonic loading and it is required for analyses of the dynamic and small strain cyclic loading of soils. Tests were carried out on fine-grained soils in a hydraulic triaxial cell fitted with bender elements and with local axial gauges. From the results of these tests simple expressions were obtained which describe the variation of GO with current state in terms of the current stress and overconsolidation ratio. The parameters in these expressions were found to depend on plasticity index. The simple expressions for G, were found to apply generally at larger strains, with the values for the parameters also depending on the current strain. Values of G, measured in laboratory tests on reconstituted London clay agree well with values measured in tests on undisturbed samples and in field tests which make allowance for the different states in the various tests.

KEYWORDS: clays; dynamics; elasticity; laboratory equipment; laboratory tests; stiffness.

La rigidite G, d’un sol, sous tres faible deformation, est un parametre interessant qui permet de caractiriser la non-linearite du comportement en contrainte-deformation de ce sol lors d’un chargement monotone et d’analyser les cycles de chargement dynamique a faible deformation. Des essais ont iti! realises en celhde triaxiale hydrau- lique sur des sols finement genus equip&s de cap teurs en flexion et de jauges axiales locales. Les resultats obtenus au tours de ces essais ont permis de definir des relations simples donnant la variation de GO en fonction de la contrainte courante et du degre de surconsolidation. Les parametres de- pendent de I’indice de plasticite. Une expression simple de G, est applicable a de plus fortes deformations, les parametres &ant alors en plus fonction de la deformation courante. Les valeurs de GO mesurees en laboratoire sur de l’argile de Londres reconstituQ sont en bon accord avec celles obtenues sur des Cchantillons intacts ou lors d’essais in-situ et rendent compte des differents &tats rencontres lors des differents essais.

INTRODUCTION The shear stiffness of soil measured in dynamic field and laboratory tests is generally significantly greater than the shear stiffness measured in conventional triaxial tests, assuming that the stress-strain behaviour is linear. As a result it was generally believed that stiffnesses measured in dynamic tests did not represent the stiffness of soil in monotonic loading and were applicable only to dynamic loadings such as earthquakes, shocks or machine vibrations. Dynamic tests investigating the variation of shear modulus with shear strain amplitude showed that the stiffness reduced with increasing strain as in Fig. l(a) (Anderson & Richart, 1976). Conventional triaxial tests, often with local measurement of strain, showed a similar reduction of shear modulus with strain, as in Fig. l(b) (Jardine, Symes & Burland, 1984). The non-linearity of the stress-strain

Manuscript received 22 March 1993; revised manuscript accepted 23 March 1994. Discussion on this Paper closes 1 September 1995; for further details see p. ii. * City University, London.

behaviour of soils was also inferred from back- analysis of field observations (Simpson, Calabresi, Sommer & Wallays, 1979).

Figure l(c) is an idealization of soil stiffness over a large range of strains, from very small to large, and approximately distinguishes strain ranges. At very small strains the shear modulus reaches a nearly constant limiting value G,. For reconstituted soils the strains at which the stiffness starts to decrease varies with plasticity from about 0.001% for low-plasticity soils to about 0.01% for plastic clays (Georgiannou, Rampello & Silvestri, 1991; Lo Presti, 1989). At strains exceeding about 1% the stiffness is typically an order of magnitude less than the maximum, and it continues to decrease as the state approaches failure. In the intermediate small strain range the stiffness decreases smoothly with increasing strain. Strains in the ground near structures in stiff soils are generally in the small strain and very small strain regions (Burland, 1989).

Non-linear numerical analyses have been used successfully to predict movements around engineering structures (Jardine, Potts, St. John & Hight, 1991) where the introduction of non-

250 VIGGIANI AND ATKINSON

linearity alone considerably improved the quality of the prediction. Non-linear models for soil behaviour may be developed using the theory of

10-Z

Shear ytryin’ %1’-’

a

10-l 1 Shear strain: %

(b)

Small strains

\ / ( Shear strain (log scale)

(4

IL :e

-L

10

.arger ;trains

-

Fig. 1. Typical variation of stiBoess with strain for soil: (a) dynamic tests on fine-grained soils (after Anderson 81 Richart, 1976); (b) triaxial tests on London clay (after Jardine et al., 1991); (c) idealization for a wide range of strains (after Atkinson & Sallfors, 1991)

elasticity with empirical non-linear stress-strain curves obtained from laboratory stress path tests (Jardine & Potts, 1988) or using kinematic hard- ening elasto-plastic models with multiple plastic potential surfaces (Stallebrass, 1990). In the former case the initial value of shear modulus is useful for defining the starting point of an empirical stress-strain curve; in the latter case it is needed to define the stiffness inside the innermost yield surface.

There is considerable evidence that soil behaviour within the region of very small strain is linear and elastic. In both slow and dynamic cyclic loading tests stress-strain loops show little or no hysteresis, which means that the behaviour is conservative and little or no energy is dissi- pated (Papa, Silvestri & Vinale, 1988; Silvestri, 1991). Volumetric and shear deformations are fully recoverable (Lo Presti, 1989) and uncoupled so that no pore pressures are generated during undrained shear (Georgiannou et al., 1991; Sil- vestri, 1991).

Values of the shear modulus at very small strains G, can be measured using dynamic techniques in field and laboratory tests (Atkinson & Sallfors, 1991), in which the deformation properties of the soil are related to elastic shear wave velocities. In the laboratory, the most common is the resonant column test in which the response of a cylindrical sample subjected to forced harmonic torsional vibrations is measured. The resonant column test can be used to evaluate the stiffness of soils at shearing strains ranging from O-00001 % to 1%. However, since analyses of resonant column tests are based on the assumption that the behaviour of the soil is linear and elastic, analyses of the test data are strictly valid only in the region of very small strain (Isenhower, 1979). An alternative laboratory technique involves transmitting and receiving shear waves using small electro-mechanical transducers known as bender elements (Shirley & Hampton, 1977). In bender element tests the strains are not constant throughout the sample because of both material and geometric damping. The maximum shear strain generated in the soil is, however, very small and was estimated to be less than 10m3% (Dyvik & Madshus, 1985). In field tests the velocities of shear waves can be measured from the surface, using refraction surveys or Rayleigh wave techniques, or at depth, using the cross-hole or down- hole techniques (Yoshimi, Richart, Prakash, Barkan & Ilyichev, 1977). Once again, the strains involved in field dynamic tests are not constant, decreasing from a maximum close to the source. The shear strain amplitude can be obtained from the ratio of the particle velocity to the shear wave velocity, E = u&, so that estimates of the strain amplitude can be made in individual experiments

STIFFNESS OF FINE-GRAINED SOIL AT VERY SMALL STRAINS 251

by using calibrated geophones. Field dynamic tests generally develop strains in the field of 10-3-10-4% and less.

Direct comparison between the shear modulus obtained from dynamic tests and the very small strain shear modulus relevant to monotonic loading in triaxial compression or extension tests is difficult, as the rates of strain and the modes of shearing are very different in the two types of test. Recent laboratory test data are available comparing the stiffness of sands under dynamic and static conditions (Iwasaki, Tatsuoka & Takagi, 1978; Ni, 1987; Bolton & Wilson, 1989). The results of these tests indicate that the stiffness of sands at very small strain is independent of the rate of loading. Less experimental evidence is available for saturated clays, probably because of the difficulties connected with the correct definition of the drainage conditions and the measurement of pore pressures generated during fast dynamic loading. Nevertheless, the present evidence is that values of shear modulus obtained from dynamic and slow loading triaxial tests are approximately equal (Rampello & Pane, 1988; Georgiannou et al., 1991).

If the mechanical behaviour of soil is taken to be essentially frictional the mechanical properties, including both strength and stiffness, vary linearly with the mean stress. However, if the soil is ideal- ized as an assembly of elastic spheres in contact, the theories developed by Hertz as reported by Richart, Hall & Woods (1970) lead to the result that the shear modulus should depend on the mean stress raised to the power of l/3.

The observed behaviour of soil lies somewhere between these two limits. For sands Wroth & Houlsby (1985) proposed a general expression relating shear modulus to mean stress in the form

G t ” -_=A p

Pr 0 P,

where the dimensionless parameters A and n depend primarily on the nature of the soil and on the current strain. The reference pressure p, is included in equation (1) so that the parameters A and n are dimensionless but their numerical values will depend on the choice of reference pressure. Experimental results from both dynamic and static tests on sands indicate that the value of n varies significantly with strain from values near 0.5 at very small strain to 1.0 at large strain (Wroth, Randolph, Houlsby & Fahey, 1979).

Recent numerical studies of the mechanical behaviour of large random systems of elastic spheres of different sizes show that the overall deformation always includes both elastic deformations of the particles and slippage and rearrangement (Dobry, Ng & Petrakis, 1989),

with the contribution from inelastic slipping and rearrangement becoming larger with increasing strain and stress ratio. Thus the overall behaviour of soil should be expected to correspond to a value of n a little greater than l/3 at very small strains, because of the small contribution from slipping and rearrangement, with n increasing to a value very close to 1.0 at large strains as the contribution from the elastic deformations of the particles becomes relatively small.

Based on experimental observations on sands, Hardin & Black (1966) proposed that the shear modulus at very small strains could be related to the mean effective stress raised to a power of 0.5 and to the voids ratio and stress history. Hardin & Black (1968) assumed that the same relationship would hold also for normally consolidated clays. A more general expression was proposed by Hardin (1978) based on theoretical elastic stress-strain relationships by Rowe (1971) and empirical equations for initial tangent modulus by Janbu (1963) and Hardin & Black (1968). This can be written in the form

G, = Sf(u)OCRkp,‘-“p’” (2)

where S is a dimensionless coefficient which depends on the nature of the soil, f(u) is a function of the specific volume, p’ is the mean effective stress, p, is the atmospheric pressure and OCR is the overconsolidation ratio defined as the ratio of the maximum past stress to the current stress. Results of tests on soils in resonant column tests (Hardin & Drnevich, 1972) show that n is less than 1.0 and k increases from 0 to 0.5 as the plasticity index increases from 0 to 100.

In order to consider the effect of anisotropic stress states on the very small strain stiffness of soils, Ni (1987) proposed a more general form of equation (2)

G, = Sf(o)OCRkp,’ -“c:“‘cr;“‘c$’ (3)

where oe’ is the stress in the direction of wave propagation, or’ is the effective stresses in the direction of particle motion and cr8’ is the effective stress orthogonal to the plane of shear, and n = n, + n, + r18. From the results of resonant column tests on hollow cylindrical samples of dry washed mortar sand, both in biaxial loading conditions and in true triaxial conditions, Ni (1987) concluded that G, is affected equally by cre’ and err’ (n, w nr) while the influence of erg’ is practically negligible, so that nB z 0.

Equations (2) and (3) contain the specific volume u, the current stess cr’ or p’ and the overconsolidation ratio. If the overconsolidation ratio is redefined in terms of the stress at the intersection of a swelling and reconsolidation line with the intrinsic normal consolidation line for recon-


Intrinsic normal

;

5 g ” _ _________-------

0 E P lz

(7)

I P’

In p’ P’P

Fig. 2. Relationship between specific volume, stress and overconsolidation ratio

stituted samples as in Fig. 2, the state at A is defined by any two of the parametrics v, p’ and R, It is usual to choose the current stress as one of the parameters to define the current state of soil, but the choice of the other parameter v or R, is a matter of convenience. For coarse-grained soils, for which it is often difficult to locate the intrinsic normal consolidation line, the most con- venient parameter is the specific volume; for fine- grained soils the overconsolidation ratio is usually preferred.

The work described in this Paper was experimental and consisted largely of tests on reconstituted samples in a hydraulic triaxial cell in which values of G, were measured using bender elements. The principal purpose of this work was to examine the variation of G, with current stress and overconsolidation ratio and to evaluate the parameters for some typical fine-grained soils. Additional work was carried out to examine the variation of the shear modulus with stress and overconsolidation at larger strains and to determine values of G, for London clay in laboratory tests on undisturbed samples and in situ.

At large strains the states of normally consolidated and lightly overconsolidated soils are on the state boundary surface and the stiffness at a particular strain increases linearly with the mean stress p’ (Wroth, 1971; Wroth et al., 1979) so that n = 1 in equation (1). From the results of undrained triaxial tests on normally consolidated and overconsolidated samples of a glacial till soil, Atkinson & Little (1988) found that the value of G/p’ at a particular strain increased linearly with the logarithm of the overconsolidation ratio, which is consistent with the behaviour derived by Wroth (1971) from a reinterpretation of tests on undisturbed London clay by Webb (1967). These results can be expressed in the form

G G -= P' 0 --$ / + c 108 Ro)

or

G/G”, = 1 + c log R, (5)

where G,, is the stiffness of a normally consolidated sample at the same strain and the same mean effective stress and c is a constant. Houlsby & Wroth (1991) expressed the variation of stiffness with overconsolidation ratio using a power function of the type

G G -= PI 7 nc 0 R;;

LABORATORY TEST EQUIPMENT AND PROCEDURES

The laboratory tests were carried out on 38 mm dia. samples in a computer-controlled hydraulic triaxial cell of the type described by Atkinson, Evans & Scott (1985). The apparatus is shown in Fig. 3. The very small strain shear modulus G, was measured using bender elements of the type developed at the Norwegian Geotech- nical Institute by Dyvik & Madshus (1985), and the stiffness at larger strains was measured using both an external displacement transducer and a pair of Hall-effect local axial gauges (Clayton & Khatrush, 1986). For tests on reconstituted samples which were reconsolidated in the hydraulic triaxial cell the measurements of axial strain were approximately the same for both sets of instruments, provided allowance was made for the compliance of the apparatus in the readings obtained from the external displacement transducer. Using either an external transducer or

I Slgnal generator

Local axial . . ,. gauges

Bender jijjjj j.: ‘:jj::i:iji:

elements jj:jj:ji/: ,I.j::,:.: ::::: . . . ..A .:: .:.:.::.:

~~~ 1

.‘,.,. :. ..:..: . . . / I’ ::::. ::::: ::::. . . ..A.. .. :. .:.:.: : :.::.,.:. :;‘;i.:::+-::. .::.

. . L.. : :

Lziijiiqei Fig. 3. Laboratory test apparatus


local axial gauges the smallest reliable measurement of strain was about 0.002% and the first reliable determination of stiffness was at a strain of about 0.004%.

Reconstituted samples were prepared by one- dimensional consolidation of a slurry in a tall floating ring oedometer until the samples were sufficiently strong to handle. They were then transferred to the hydraulic triaxial cell and consolidated, usually isotropically, to the required initial state. The samples were then loaded and unloaded as required by the particular test and the very small strain shear modulus G, measured at various states during the test using the bender elements. At the same time the overall strains in the sample were measured using the external displacement transducer or the local axial gauges and a conventional Imperial College type volume gauge. Additional tests were carried out on a 38 mm dia. undisturbed sample of London clay prepared by hand-trimming a 100 mm dia. thin wall tube sample.

The test results were interpreted in terms of the deviatoric stress parameter q’ = (oa’ - or’) and the mean stress parameter p’ = l/3(0,’ + 2~,‘), where era’ and or’ are the axial and radial effective stresses respectively. The corresponding strain parameters were the shear strain E, = 2/3(~, - E,) and the volumetric strain a, = a, + 2~,, where E, and E, are the axial and radial strains. The state of soil is defined by the current values of q’, p’ and the specific volume u.

Bender elements were used to determine the shear modulus at very small strains G, by measuring the velocity of shear waves through the sample. Bender elements are piezoelectric electro- mechanical transducers that bend as an applied voltage is changed or, conversely, mechanical bending of the elements produces a change of voltage. A transmitter element and receiver element are fixed into the top and bottom platens of the triaxial cell so as to protrude about 3 mm into the sample. A change of voltage applied to the transmitter causes it to bend and generate a shear wave that propagates through the sample. The arrival of the shear wave is recorded as a change of voltage by the receiver.

The electronics required to operate the bender elements are shown in Fig. 3. A Farnell FGl function generator was used to supply the transmitter with the driving voltage. This normally consisted of a square wave with a frequency of 50 Hz and an amplitude of 10 V (20 V peak to peak). The frequency of the square wave was always sufficiently low that the subsequent step of the wave did not interfere with the received wave generated by the previous step. The amplitude of the wave was limited by the necessity of avoiding depolar- ization of the bender elements. The signal used to

drive the transmitter element and the output signal from the receiver were displayed on a Tek- tronix 2211 50 MHz digital storage oscilloscope.

The shear modulus G, was calculated from

G, = pV,’ = pL2/t2 (8)

where p is the density of the soil and V, is the velocity of the shear wave which is determined from the effective length L through which the shear wave travels and the travel time t. The values of p for a cylindrical sample can be determined with high precision, but there are uncertainties about the value for L (which could vary between the overall length of the sample and the distance between the tips of the bender elements) and the precise instant of arrival of the shear wave at the receiver.

These uncertainties were examined in detail by Viggiani (1992) and Viggiani & Atkinson (1995). By means of direct calibration they found that the effective length of travel should be taken as the distance between the tips of the elements. They also found that the arrival of the shear wave cor- responds closely to the first major reversal of polarity of the received signal rather than to the point of first deflection which probably corre- sponds to the arrival of the near field components of the shear wave (Salinero, Roesset & Stokoe, 1986), travelling at the velocity of compression waves. Both findings are in good agreement with previous experimental observations and numerical studies (Brignoli & Gotti, 1992; Dyvik & Madshus, 1985; Mancuso, Simonelli & Vigale, 1989).

For the tests described in this Paper the results were always interpreted by taking the effective length as the distance between the tips of the elements and the arrival of the shear wave as the first significant reversal of polarity of the received signal. From the work described by Viggiani & Atkinson (1995), the calculated values of G, could have overestimated the true values by up to 14%. However, because the choice of the arrival time was consistent throughout the analysis of the data, this error does not apply to comparisons between shear moduli and affects only their absolute values when compared with other methods of measurement.

At strains sufhciently large to be measured using an axial displacement transducer or local strain gauges the tangent shear modulus G was calculated from the test results from the gradient of the deviatoric stress-shear strain curve. In comparing the values of shear modulus determined from bender element tests and from direct measurements of deviatoric stress and strain on the same sample it has been assumed that the results are comparable, despite differences that


may arise due to different modes of shearing, stress path rotations and rate effects.

VARIATION OF G, WITH ISOTROPIC STATE Tests were carried out on reconstituted samples

of speswhite kaolin clay, powdered slate dust, London clay and North Field clay. North Field clay is a glacial till from a test bed site adjacent to the Building Research Establishment at Watford (Abbiss, 1981). These tests examined the variation of the very small strain shear modulus G, with isotropic effective stress p’ and overconsolidation ratio R, given by

R, = P~'IP‘ (9)

where pP’ is the stress at the intersection of a swelling line with the isotropic normal consolidation line (see Fig. 2). As soils may creep at constant mean effective stress, the value of pp’ may

2.5 , , , , , , , , I, I I,, I I

* Dynamic reading of Go

- p’ = 50 kPa 100 kPa 200 kPa 400 kPa

Nominal values

Mean effective stress p’: kPa

(b)

Fig. 4. (a) Isotropic stress states at which bender element tests were carried out; (b) variation of G, with mean stress p’ and nominal overconsolidation ratio

not correspond to the maximum past stress p,‘. Also, this definition of R, is different from the usual definition of overconsolidation ratio in which the current vertical effective stress is related to the maximum past vertical effective stress. With R, defined as in equation (9), the current specific volume is defined by the current values of p’ and R, and so is not independent.

Each sample was isotropically compressed and swelled following the path in Fig. 4(a). Typically the samples were brought to states with p’ = 50- 400 kPa and R, = l-8. Due to the hysteresis in the unloading-reloading loops the values of R,, defined by equation (9), at a particular stress on a particular loop are different and so the values of R, indicated in Fig. 4(a) are nominal values.

The samples were generally compressed and swelled by continuous drained loading and unloading, but in a few stages the load was applied as a single increment followed by isotropic consolidation. The solid line in Fig. 4(a) is the state path calculated on the assumption that any excess pore pressures could be neglected. The small vertical sections at the end of some stages represent additional small volume changes due to dissipation of small excess pore pressures developed during continuous loading or unloading or to creep. The line for R, = 1 in Fig. 4(a) through the equilibrium states represents the isotropic normal compression line.

Figure 4(b) shows the values of Go measured at the states indicated in Fig. 4(a). The data show that the value of Go increases with mean effective stress p‘, although the variation is non-linear, and that the value of G, at a particular stress increases with overconsolidation ratio. At a particular nominal value of R, and at a given stress, values of G, may be slightly different. This is

I

I 105 -

d ‘0 0

Nominal values 1

104’ .-lo

I

102

P’!Pr

J 103

Fig. 5. Variation of G, with stress and overconsolidation for reconstituted samples of speswhite kaolin


because, due to hysteresis in the unloading- reloading loop in the An p’ plane, the specific volumes and hence the actual values of R, are different.

Figure 5 shows a typical set of values of G, measured on samples of kaolin clay at the states indicated in Fig. 4, together with additional values obtained at intermediate states on the normal compression line. This shows values of G,/p, against p’/p,, both to a logarithmic scale, where p, is a reference pressure which, for convenience, has been taken as 1 kPa. The data points for the normally consolidated samples fall close to a single straight line given by

or

(11)

where A and n are non-dimensional soil parameters. For the test data for speswhite kaolin shown in Fig. 5 the values are A = 2088 and n = 0.653, with coefficient of correlation r2 = 0.996 and standard deviation a = 0.009. The values of A and n obtained by fitting a straight line through all the available data for normally consolidated speswhite kaolin were A = 1964 and n = 0.653. The general form of equation (ll), which relates the shear stiffness of kaolin clay at very small strain G, to the current stress, is essentially the same as equation (l), which was proposed by Wroth & Houlsby (1985) for the shear modulus of sands. The values of A and II depend on the value taken for the reference pressure.

tional tests on reconstituted speswhite kaolin, the values of n in equation (11) found by fitting straight lines through all the available data for a particular overconsolidation ratio indicate that n is independent of R, , at least for R, less than 4.

Figure 6 shows the same data as those in Fig. 5, together with data from additional tests on reconstituted kaolin clay. The data are plotted as (GdGe,,) where GOnf is the value of G, corresponding to normally consolidated samples at the same mean effective stress. The data points fall reasonably close to a straight line which, making use of equation (1 l), is given by

%! = A p’ ‘&” P, 0 P,

(12)

In’Fig. 5 the data points for overconsolidated where m can be regarded as another soil para- samples fall above the line for normally consoli- meter. For the data shown in Fig. 6, m = 0.196 dated samples and, for each value of R, , they fall with coefficient of correlation r2 = 0.830 and close to lines parallel to the line for normally con- standard deviation e = 0.021. The increase of G, solidated samples. Considering the data shown in with log R, given by equation (12) is similar to Fig. 5, together with results of a number of addi- that reported by Houlsby & Wroth (1991).

g , , , , , ,,(

1 2 3 4 5 6 769

Ro

3

Fig. 6. Variation of G,, with stress aad overconsolidation for reconstituted samples of speswhite kaolin

Plasticity index

0 Speswhite kaolin 0 London clay

Plastiaty index

A North Feld clay 0 Slate dust

Plasticity index

l Various clays (Weller, 1966) l Fucino clay (Pane & Butghignoli, 1966)

Fig. 7. Variation of stitTness parameters for G, with plasticity index


500,, , , , , , , , , , , , , , , , , , , , , , , I -

‘1’ = 0.75 043 0.30 0.00 -

4OO-

a',: kPa

Fig. 8. Anisotropic stress states at which bender element tests were carried out on reconstituted samples of speswhite kaolin

Results obtained from tests on the other soils all conformed to the general relationships found for speswhite kaolin clay shown in Fig. 6, but with different values of the parameters A, n and m. Values for these parameters are shown in Fig. 7 plotted against plasticity index. The broken lines in Fig. 7 are not best-fit lines and indicate

only the general trends of the data. Also shown in Fig. 7 are points representing tests on various other fine-grained soils reported by Pane & Burghignoli (1988) and Weiler (1988). They generally fall near the trends for the data obtained as part of the present work. These data illustrate the significant influence of plasticity index on small strain stiffness through the parameter A, and the relatively small importance of overconsolidation ratio through the parameter m.

VARIATION OF G, WITH ANISOTROPIC

STRESS STATE

For anisotropic states for which ua’ # 6,’ the value of G, might vary primarily with the mean stress p’ as in equation (1 l), or primarily with the stress in the direction of travel of the shear waves, or with one of the stresses orthogonal to the direction of travel. Following Ni (1987), equation (11) can be extended for anisotropic states of stress in a similar way to that in which equation (2) was extended to equation (3). In a bender element test the direction of wave propagation is always the direction of the axial stress and oe’ = 0,‘; the direction of particle motion is orthogonal to the surface of the bender element and the direction normal to the plane of vibration is

o u'r= lOOkPa q a',= 2OOkPa A dr = 400kPa

o u',= 50kPa q o',=lOOkPa A 0'8 = POOkPa

Fig. 9. Variation of G,, with axial and radial stress for normally consolidated speswhite kaolin: (a) compression, a.’ > a,‘; (b) extension, u,’ < u,’


parallel to the bender element. In a triaxial sample these are equal and err’ = CT*’ = Q,‘, where ur’ is the radial stress.

As the influence of the stress orthogonal to the plane of vibration has a negligible influence on G, (because n, x 0 in equation (3)), equation (11) can be written as

where B,’ and cr’ are the axial and radial stresses in a triaxial sample, A, is equivalent to A in equation (11) and, in general, the parameters n, and n, are not equal. The data shown in Figs 4-l were obtained from tests on isotropically compressed samples for which p’ = oq’ = or’ and so A, = A and n = n, + n,. To examme the variation of G, with axial and radial stress for anisotropic states, tests were carried out on reconstituted samples of kaolin clay consolidated to the isotropic and anisotropic stress states following the stress paths shown in Fig. 8, where the stress ratio 9’ is q’/p’. For the paths in Fig. 8 the value of the mean effective stress was increased and so the samples can be considered to be normally consolidated throughout all stages of each test.

The test results are shown in Figs 9 and 10. In Fig. 9 the observed value of G, increases with both axial stress CT=’ and radial stress or’. In Fig. 9(a), for compression, the gradients of the lines and hence the values of n, and n, are approximately equal, the exponent n, being only slightly larger than the exponent n,. This indicates that, in triaxial compression, G, depends about equally on the stress in the direction of wave propagation and on the lateral stress, which is consistent with the observations by Ni (1987). In Fig. 9(b), for extension, the gradients of the lines and hence the values of n, and n, are significantly different from each other, the exponent n, being significantly larger than the exponent n, although n, + n, is approximately equal to n in both triaxial

extension and compression. Ni (1987) did not report the results of triaxial extension tests. In Fig. 10 there is a unique relationship between G, and p’ for the normally consolidated samples which is independent of the value of the stress ratio g’ and is the same for compression and extension. The line in Fig. 10 is the same as that obtained for isotropically compressed samples shown in Fig. 5. These data indicate that the value of G, in reconstituted normally consolidated kaolin clay depends primarily on the value of mean stress p’ rather than on the value of the stress either in the direction of travel of the shear waves or orthogonal to the direction of travel, at least for stress ratios in the range -0.60 < q’ < 0.75 and which are not close to failure.

106 _ I

Compression Extension

0 ?J’ = 0.00 . ‘I’ = 0.00 0 ‘I’ = 0.43 . ‘I’ = -0.36 A ?f = 0.75 . ‘I’ = -0.60 * rf = 0.30 l Tf - -0.27

,drlos -

s :

I

102

P’lPr

Fig. 10. Variation of G, with mean stress and stress ratio for normally consolidated samples of speswhite kaolin

VARIATION OF SHEAR MODULUS DURING TRIAXIAL COMPRESSION

A constant p’ drained triaxial compression test was carried out on a reconstituted sample of kaolin clay and values of G, were measured using the bender elements at various points on the stress path as indicated in Fig. 11(a). The corresponding stress-strain curve is shown in Fig. 11(b). The overall stress-strain behaviour is represented by the smooth non-linear curve drawn through the many data points; the stars indicate bender element tests.

The test results are shown in Fig. 12. The tri- angles represent values of G, determined from shear wave velocities using the bender elements plotted at values of overall strain measured using the external axial displacement transducer. The circles represent values of the tangent shear modulus calculated from deviator stresses and shear strains. Values for the shear strains were calculated from axial strains measured using an external linear variable differential transformer (LVDT) and from volumetric strains measured using an external volume gauge. The samples were reconstituted and reconsolidated in the hydraulic triaxial cell, and so bedding and seating errors were very small and the axial strains measured using the external LVDT were not significantly different from those measured using the local axial gauges. The lower limit of strain for which the values of tangent stiffness become unre- liable was estimated to be about 0404%. For strains of up to 0.02% the tangent stiffnesses were determined from a fifth-order polynomial fitted to the data points. For strains greater than 0.02% the tangent stiffnesses were calculated using the methods described by Atkinson, Richardson & Woods (1986).


The data in Fig. 12 show that the value of G, remains nearly constant, irrespective of the deviator stress or the shear strain. The slight reduction in G, by about 5% can be attributed to small excess pore pressures in the sample which would have the effect of reducing the current value of mean effective stress p’. The tangent shear modulus decreases smoothly from a value that approaches the value of G,, corresponding to very small strains, towards a value of G = 0 at large strains. At very small strain, G, varies with p’” as shown in equation (11). At strains greater than about 0.1% the shear modulus G has been found to vary linearly with p’ (Atkinson & Little, 1988). A question then arises as to the nature of the relationship between stiffness and mean stress in the intermediate range of small strain.

Figure 13 shows stiffness-strain curves for a set of constant p’ drained triaxial compression tests carried out on normally consolidated samples of

l_l. 0 20 40 60 80 100 120 p’: kPa

(a)

kaolin clay. The data points start at strains of about 0X)04%, which was the limit of reliable measurements of stiffness. Each stiffness-strain curve has the characteristic shape, with G

* decreasing with increasing strain. Also shown in Fig. 13 are the values of G, obtained from bender element measurements in tests at different values of p’.

Figure 14 shows values of stiffness at a particular strain extracted from the data in Fig. 13 plotted against mean stress, both normalized with respect to the reference pressure pr as in Fig. 5. The uppermost line represents results obtained from the bender elements corresponding to Go. For each strain the variation of stiffness with mean stress can be represented by equation (1 l), with the values of A and n both varying with increasing strain. Figure 15 shows the values of A and n obtained from the data in Fig. 14 plotted against strain on a logarithmic scale. The values of A decrease by a factor of about three over the strain range OXrO5%-0.05%, which is a consequence of the highly non-linear behaviour of soil over this range of strain. In Fig. 15(b) the parameter n increases smoothly in the strain range 0.005-0.05% from n = 0.12, which is just above the value for Go obtained using the bender elements, towards n = 1, corresponding to large strain when G is proportional to p’.

G,’ IN UNDISTURBED LONDON CLAY A further test was carried out on an undis-

turbed sample of London clay. The sample was trimmed from a 100 mm dia. tube sample taken from a depth of about 6 m at a site at Chattenden in Kent about 45 km east of central London

I””

Dynamic reading of Go

0 2 4 6 8 10

Shear strain: %

(b)

Fig. 11. Stress path and stressstrain curve for special triaxial test with additional bender element tests

50

____~---*----b---d-*___

40

m

P’S0

0:

20

10

0 10-d 10-S 10-z lo-’ 1 10

Shear Wan: %

Fig. 12. Variation of G and G, with strain for the special triaxial test shown in Fig. 11

STIFFNESS OF FINE-GRAINED soIL AT VERY SMALL STRAINS 259

q p’ = 400 kPa A D’ = 200 kPa

p’ = 200 kPa Go = 62 MPa

p’ = 100 kPa Go = 39 MPa

0 p’ = 50 kPa 60 m

$

ij 40

p’ = 50 kPa Go = 25 MPa

Shear strain: %

Fig. 13. Variation of G with strain for a set of constant p’ trinxial compression teats oa reconstituted kaolin clay

(Abbiss & Viggiani, 1994). The sample was compressed and swelled in the hydraulic triaxial cell to a number of different isotropic and anisotropic states which encompassed the range of states estimated for the sample in situ, and at each state the value of G, was measured using bender elements.

Figure 16(a) shows the variation of Go for the undisturbed sample of London clay with p’, both normalized with respect to a reference pressure p, = 1 kPa as before. The undisturbed sample was heavily overconsolidated in situ and remained overconsolidated even at the maximum stress achieved in the triaxial cell, so that its overconsolidation ratio varied with reconsolidation pressure as indicated in Fig. 16(a). The values for the overconsolidation ratio R, given in Fig. 16(a) were calculated from equation (9) taking the iso-

‘““5

P’IP.

10-Z lo-’ 1 10

Shear strain: %

(b)

Fig. 14. Variation of G with mean stress and strain Fig. 15. Variation of the parameters A aad II extracted extracted from the data in Fig. 13 from the data in Fig. 14

2000

t

+ From bender elements A = 1964

Shear strain: %

(a)

1.1 :

1.0 At larger strain - ~ : rl=l.)

t t From bender elements

0.6 ” = 0.653


tropic normal compression line for reconstituted samples. Also shown in Fig. 16(a) is the relationship between G, and p’ obtained from tests on normally consolidated reconstituted samples.

In Fig. 16(a) the data points for the undisturbed samples represent states with different overconsolidation ratios because the maximum stress applied in the triaxial cell was always considerably smaller than the estimated maximum past stress in the ground. Consequently the data points for the undisturbed sample fall above those for the reconstituted samples, and the best- fit lines through the two sets of data are not parallel.

The data for the intact sample fall close to a line given by

Go P' ’ -_=c( -

PI 0 P, (14)

where a = 3000 and p = 0.50. From equations (9) and (12)

so a = A(JQ’/~,)~ and p = n - m. (Test results for reconstituted London clay gave n = 0.76, m = 0.25 and n - m = O-51, which is very close to the value obtained from Fig. 16(a).)

Equation (12) can be rewritten as

Go ” P -=,‘I - 0 P~Ro" P, (16)

Fig. 16(b) shows values of Go’/p,Rom plotted against p’/p,, both to a logarithmic scale; the value of m was taken as 0.25 corresponding to the value obtained from the tests on reconstituted samples of London clay already described. The data points corresponding to the results obtained from the undisturbed samples now fall very close to the line obtained from the tests on the reconstituted samples. This result demonstrates that, for the London clay examined, the value of Go depends on the current state (determined by both the current stress and the overconsolidation ratio) and is unaffected by structure and fabric. This apparently surprising result can be explained if, at the very small strains involved in dynamic tests, most of the deformation is connected to elastic deformations and local slipping at the points of contact between soil particles rather than to major slippage and rearrangement of particles.

VARIATION OF G, WITH OVERCONSOLIDATION

The general variation of shear modulus at very small strain G, with overconsolidation can be

‘OS5 J

_a-- *9-

_Q_-9=- _--*

g 104. Jl,r;

’ : 0 Ro= 62 _____ Undisturbed - a&=43 -m-- Reconstituted (R, = 1)

OR,=38 . R. = 21

- AR,=18 AI&=13

- OR,=12

103 I 10 102 103

P’IP,

6)

lo55 L -I

@ 910’:

2 -

o Undisturbed l Reconstituted

lo’l 10 102 103

P’lP,

(W

Fig. 16. Variation of GO with mean stress and overconsolidation ratio for undisturbed and reconstituted samples of London clay

represented by equation (12); for the shear modulus at larger strains the corresponding relationship could be either equation (4) or equation (6), which is similar to equation (12). However, for the data shown in Fig. 6 from which equation (12) was developed, the overconsolidation ratios were limited to about 7 due to the maximum pressures available in the hydraulic triaxial cell; for tests on intact samples the overconsolidation ratios were larger since the maximum stresses in the ground were generally greater than those available in the triaxial cell.

Figure 17(a) shows values of G,/G,,, obtained from bender element tests on undisturbed and reconstituted samples of London clay plotted against R,, both to a logarithmic scale corresponding to equation (6). The data points obtained from the tests on the undisturbed sample fall very close to the best-fit straight line through the results obtained from the tests on reconstituted

STlFFNESS OF FINE-GRAINED

A Undisturbed q Reconstituted

Best-fit line from reconstiiuted samples

-F

L

100

A Undisturbed q Reconstituted

est-fit line from reconstituted samples

0.5P 1 10 100

z

Fig. 17. Variation of G, with overcoasolidatioo ratio for undiiturbed and reconstituted samples of London clay

samples. These results again indicate that the stiffness of soil at very small strain is basically unaffected by whether the sample is undisturbed or reconstituted, provided that undisturbed and reconstituted samples are brought to the same mean effective stress and overconsolidation ratio. The unique relationship between log (Go/Go,,)

SOIL AT VERY SMALL STRAINS 261

and log R, in Fig. 17(a) agrees with the relationship proposed by Houlsby & Wroth (1991) and test results given by Hardin & Drnevich (1972).

Figure 17(b) shows the same data as those in Fig. 17(a), but with G,/G,,, plotted to a natural scale corresponding to equation (4). In this case the data points for the undisturbed sample at values of R, greater than about 10 depart from the best-fit line through the data for reconstituted samples. On the basis of these results G, can be related to stress and overconsolidation by an expression of the form of equation (6) or equation (12) over a wide range of overconsolidation ratios.

MEASUREMENTS OF G, IN SITU AND IN LABORATORY TESTS

Measurements of Go in situ can be made by observing the velocities of either shear waves or Rayleigh waves in the ground; a number of different techniques are available (Yoshimi et al., 1977; Atkinson & Sallfors, 1991). Measurements were made using Rayleigh waves at the site on London clay at Chattenden from which the undisturbed sample was obtained. Rayleigh waves, also known as surface waves, travel largely in a layer that is about one wavelength deep, and their velocity varies with the shear stiffness of the ground. The Rayleigh wave velocity V, is related to the shear wave velocity V, through Poisson’s ratio v’, but the relationship is not sensitive to the value of v’. For most practical purposes it is suffi- cient to take V, as 0.9V, and the error will be limited to less than 5% for a very wide range of values of v’ (Richart et al., 1970).

In the most common form of the Rayleigh wave method (Richart et al., 1970) continuous seismic waves generated at the surface by a vibrator are detected by two geophone receivers placed at the surface at a distance from the vibrator and at a known distance apart as in Fig. 18. For a half space in which the stiffness increases with depth the velocity of the Rayleigh waves between the two receivers varies with the wavelength, and hence with the frequency, of the excitation

Vibrator

Oscillator and amphfier

00 -

Fig. 18 Apparatus for in situ measurements of shear modulus using Rayleigh waves


because different wavelengths sample material with different stiffnesses.

The travel time between the two receivers is given by

T(f) = ?!a L 2% f

(17)

where f is the excitation frequency and @ is the phase shift between the two received signals. The Rayleigh wave velocity and wavelength corresponding to the frequencyfare given by

v,(f) = wxf) (18)

b(f) = UfM- (19)

where d is the distance between the receivers. The Rayleigh wave velocity determined by this method may be thought of as representative of the properties of the ground in a characteristic layer. Following Vettros (1990) and Gazetas (1982), this was taken as one third of the wavelength determined from equation (19). By repeat- ing the experiment with different excitation frequencies, a profile of Rayleigh wave velocity with depth can be obtained. Hence profiles of shear wave velocity and shear modulus G, are obtained.

Details of the equipment and experimental methods used for the Rayleigh wave tests are

- yGo Field observations -

0 Summer 1989 _ D Summer 1990

.-- Based on laboratory results

Fig. 19. Variation of G, with depth in the ground in London clay at Chattemien

given by Abbiss & Viggiani (1994). These are similar to those described by Nazarian & Stokoe (1986a, 1986b), except that they generated Ray- leigh waves as single pulses rather than as continuous vibrations at fixed frequencies. The general configuration of the equipment used in the present work is shown in Fig. 18. Rayleigh waves were generated using a Ling Dynamics 400 elec- tromagnetic vibrator attached to a plate, about 0.2 m in diameter, resting on the ground and tests were carried out at frequencies in the range 8-200 Hz. The receivers were Sensor SM6 vertically pol- arized geophones. The signals were recorded on a Hewlett Packard dual channel spectrum analyser and the phase shift of the main Fourier components was determined.

Figure 19 shows the variation of shear modulus with depth obtained from Rayleigh wave measurements at the site on London clay at Chatten- den. The values of Go measured in situ are generally in the range lo-30 MPa, increasing slightly with depth and with a few higher values recorded in the top metre in the crust. (The differ- ence in the values measured one year apart has not yet been explained satisfactorily.)

Values for G, in the ground can be calculated from the results of laboratory tests together with estimates of the current state of stress and overconsolidation in the ground. The relationship between G, , stress and overconsolidation ratio is given by equation (12). Values for the parameters A, n and m were determined from bender element tests on both undisturbed and reconstituted samples of London clay from the site at Chatten- den; from these tests A = 400, n = 0.76 and m = 0.25. The in situ stresses were calculated from the measured unit weights of undisturbed samples and from the position of the water table, with values of K, varying with overconsolidation ratio as proposed by Wroth (1975). The reduction in vertical effective stress due to erosion of 1500 kPa was estimated using data given by Skempton (1961) for a site where the geology resembles that at Chattenden. The values of shear modulus G, calculated from the laboratory tests are shown in Fig. 19; they fall within the scatter of the results obtained from the in situ measurements.

SUMMARY AND CONCLUSIONS The work investigated the variation of the

shear modulus of fine-grained soils with state (i.e. current stress and overconsolidation ratio) and strain. From the results of laboratory tests on reconstituted samples of a number of different fine-grained soils using bender elements, it was found that the shear modulus at very small strains G, could conveniently be related to the current state through an expression of the form of


equation (12), where p, is a reference pressure included so that the parameters A, n and m are dimensionless, although their numerical values depend on the choice of the reference pressure.

Values for the parameters A and n depend on the plasticity index of the soil. The value of n is in the region of 0.6-03, depending on the plasticity index as shown in Fig. 7(b). The value of A decreases with plasticity index as shown in Fig. 7(a), and this shows the very significant influence of plasticity on soil stiffness. The relationship between G, and current state given by equation (12) was found to hold for both isotropic and anisotropic stresses that were not close to failure.

A limited investigation of undisturbed London clay showed that the stiffness at very small strain G, was the same as for reconstituted samples at the same state. Furthermore, values of G, measured in situ using Rayleigh waves compared well with the values obtained in the laboratory tests normalized to the in situ stress and overconsolidation ratio.

At strains larger than those corresponding to the region of very small strain in Fig. 1, the dependence of shear stiffness on state can still be expressed using equation (12), but with values of the parameters A, n and m that depend addi- tionally on the magnitude of the strains. In particular, for a given soil, the value of A was found to decrease with increasing strain as shown in Fig. 15(a); this is a consequence of the highly non-linear stress-strain behaviour of soil. The value of the exponent n gradually increases with strain as shown in Fig. 15(b) and, at large strains, n x 1.0. The change of n can be attributed partly to a change from essentially elastic behaviour at very small strains to essentially plastic behaviour at large strains.

The value of shear modulus at very small strain G, is a useful parameter for characterizing the highly non-linear stress-strain behaviour of soil for monotonic loading. It is also required for analyses of dynamic and small strain cyclic loading of soils. The present work has shown that the variation of G, with stress and overconsolidation is similar to that for the shear modulus at larger strains. Values for G, can be obtained relatively easily from measurements of shear wave velocities in field or in laboratory tests using a number of techniques. In laboratory tests shear wave velocities can be measured using bender elements which can be incorporated into conventional soil testing apparatus.

ACKNOWLEDGEMENTS Dr Viggiani was supported by an SERC Case

award in collaboration with the Building Research Establishment. The field tests at Chat-

tenden were carried out in collaboration with Dr C. P. Abbiss.

NOTATION A

d

f G

G “E

GO k

L

LR m

n, nar ne, n,, ns> 4 OCR

PI P,

PP

P,

2 S

T

v, v, a

nondimensional factor relating G, to pl distance between receivers in Ray- leigh wave test frequency shear modulus shear modulus for normally consolidated soil shear modulus at very small strain exponent of overconsolidation ratio effective length of travel of shear waves in bender element tests wavelength of Rayleigh wave exponent for R, exponents relating G, to p’” overconsolidation ratio defined as l&‘/a’ mean effective stress, 1/3((r,’ + 20,‘) maximum past mean effective stress effective stress at the intersection of a swelling line with the normal compression line reference pressure (taken as 1 kPa) deviator stress, ur’ - 6,’ overconsolidation ratio defined as Pp’lP’ non-dimensional factor in equation (2) time shift between received signals in Rayleigh wave test travel time of shear waves in bender element tests specific volume Rayleigh wave velocity shear wave velocity non-dimensional factor relating G, to p’ for intact samples shear strain, 2/3(s, - E,) volumetric strain, 6. + 2.5, exponents for mean pressure for intact samples soil density effective stress in the direction of wave propagation effective stress in the direction of particle motion effective stress out of the plane of wave motion phase shift in Rayleigh wave tests

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viggiani, g., and atkinson, j. h._1995_geotechnique

Documents

dynamic tests

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current strain

tests simple expressions

shear stiffness of soil

small strain g

shear strain amplitude