Jardine Et All (1986)

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JARDINE, R. J., Porrs, D. M., FOURIE, A. B. & BURLAND, J. B. (1986). GCorechnique 36, No. 3, 377-396

Studies of the influence of non-linear stress-strain characteristics in soil-structure interaction

R. J. JARDINE,* D. M. POTTS,* A. B. FOURIEt and J. B. BURLAND*

Recent field and laboratory studies have shown that, even at very small strains, many soils exhibit non-linear stress-strain behaviour. Nevertheless, because of its convenience, linear elasticity will continue to play an important role in the analysis of such problems as settlement, deformation and soil-structure interaction. In this Paper the measured non-liqear stress-strain properties of a low plasticity clay are used in the finite element analysis of footings, piles, excavations and pressuremeter tests to assess the influence of small strain non-linearity in comparison with linear elastic behaviour. In all cases non-linear behaviour results in the concentration of strain and deformation towards the loading boundaries. This is shown to have important consequences for soil-structure interaction problems such as settlement profiles, pile group interaction and contact stress distributions. Small strain non- linearity also has a significant influence on the interpretation in terms of equivalent elastic moduli of in situ deformation tests (e.g. plates and pressuremeters) and of field measurements. It is concluded that, although linear elasticity remains a convenient tool for expressing measurements of soil stiffness, unless the non-linear nature of soils is taken into account, soil-structure interaction computations and the interpretation of field measurements can be misleading.

Des essais in situ et des Ctudes en laboratoire de date rtcente ont dtmontrb que beaucoup de sols ont un comportement contrainte_dCformation non-IinCaire, m&me g des dkformations extrtmement faibles. Cependant $ cause de sa commoditB l&lasticit& lintaire continuera B jouer un rble important dans Ianalyse des problbmes tels que le tassement, la dkformation et Iinteraction entre le sol et la construction. Dans cet article les prop- riCtts contrainte42formation non-liniaires dune argile de faible plasticit ont utilis&s dans Ianalyse B &men& finis des semelles, des pieux, des excavations et des essais pressiomttriques afin dkvaluer Iinfluence de la non-IinCaritk $ faibles contraintes en comparaison avec le comportement Clastique IinCaire. Dans tous les cas le comportement non-lintaire produit une concentration de contraintes et de d&formations vers les limites de chargement. On dkmontre que ceci a des cons&quences importantes pour les problimes dinteraction entre le sol et la construction, tels que les profils de tassement, Iinteraction entre des groupes de pieux et les distribu-

Discussion on this Paper closes on 1 January 1987. For further details see inside back cover. * Imperial College of Science and Technology. t University of Queensland; formerly Imperial College of Science and Technology.

tions de contraintes au contact. La non-1inCaritC B faibles contraintes influence aussi de faqon importante IinterprCtation des essais de d&formation en place, par exemple, plaques et pressiomitres, et les mesures in situ en fonction de modules klastiques kquivalents. Tandis que 16lasticitt lintaire reste encore une methode cor- recte pour exprimer des mesures de la rigidit du sol, on tire la conclusion que les calculs de Iinteraction entre le sol et la construction et Iinterpr&tation des mesures in situ peuvent induire en erreur, & moins quon ne tienne compte de la nature non-IinCaire des ~01s.

KEYWORDS: elasticity; excavation; field tests; piles; settlement; soil-structure interaction.

INTRODUCTION

Analyses of soil-structure interaction frequently involve the prediction of deformations and stresses, both in the surrounding soil mass and over areas of contact with the loading boundaries. In recent years it has become possible to compute solutions with increasingly complex descriptions of the soil properties. However, the use of non-linear calculations in engineering prac- tice is restricted by time and cost. Moreover high quality stress-strain data are difficult to obtain. There is therefore a need for sensitivity studies using advanced soil models to investigate the significance of various features of soil behaviour such as non-linearity at small strains and local failure.

The most common types of analysis continue to be based on the theories of linear elasticity. The underlying assumption is either that at working loads the soil mass is behaving in a linearly elastic manner or that the stress changes in the soil are close to those given by linear elasticity even though the soil itself may be non-linear. As pointed out by Eisenstein & Medeiros (1983), the work of Wroth (1971) and Burland (1975) has encouraged the former view for stiff clays and weak rocks. The finding that the vertical stresses beneath flexible loaded areas are relatively insensitive to the stress-strain law has greatly pro- moted the second assumption (Morgenstern & Phukan, 1968). The accuracy of predictions has thus been seen to hinge on the determination of appropriate in situ elastic moduli (E,, E, G, K etc.) and their variations with depth. However, it

377


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378 JARDINE, POTTS, FOURIE AND BURLAND

has proved to be difficult to measure these elastic parameters. In particular the results of conventional laboratory tests frequently give stiffnesses which are far lower than those back analysed from field measurements. This discrepancy has been explored for London Clay by St John (1975) and has resulted in a strong move towards in situ testing (Marsland, 1971; Windle & Wroth, 1977).

The combination of careful field measurements and linear elastic theory for back analysis and prediction has been relatively successful, but several limitations have become apparent. For example, it is well known that in many problems local enclaves of fully plastic behaviour can develop at working loads. Their existence leads to stresses and patterns of deformations which depart significantly from elastic behaviour. If the material is brittle such local failure zones may propagate rapidly leading to instability at relatively small displacements.

Field observations have identified other incon- sistencies. Burland & Hancock (1977) for example, drew attention to the fact that the profile of ground movements outside excavations cannot be accounted for by linear elasticity. Simpson, ORiordan & Croft (1979) showed that the use of a bilinear stress-strain relationship before failure with an initial very stiff portion accounted considerably for the observed behaviour.

Almost concurrently new laboratory techniques began to be developed for the accurate measurement of strains locally on soil samples (Daramola, 1978; Costa-Filho, 1980; Burland & Symes, 1982). The results obtained with these techniques throw into doubt the validity of the assumption of linear elastic behaviour under working conditions. The tests show that the initial stress-strain behaviour of many soils is much stiffer than indicated by conventional strain measurements, and that the undrained stress-strain characteristics of a wide range of soil types are markedly non-linear (Jardine, Symes & Burland, 1984). Careful analysis of specially instrumented field tests confirms that these characteristics are also representative of in situ behaviour (Jardine, Fourie, Maswoswe & Burland, 1985).

The purpose of this Paper is to examine the significance of the laboratory-observed stress- strain characteristics in a range of practical problems. A typical low plasticity clay is considered which exhibits both realistic non-linear behaviour before failure and plastic flow when its failure criterion is satisfied. Finite element analyses are presented which allow a detailed study of the undrained response of this one soil when loaded by footings, piles, cavity expansion and the con-

struction of a strutted excavation. Comparisons with linear elastic behaviour are then made to draw out the significance of the non-linear soil properties.

To facilitate the interpretation of the results, only the simplest form of failure criterion is used. This allows the problems to be treated in terms of total stresses but in each study the initial ground stresses were specified in relation to appropriate K,, effective stress conditions.

The objectives of the present study are

(a) to identify several important features of behaviour which stem from non-linear stress- strain characteristics

(b) to discuss the problems of selecting appropriate values of apparent elastic moduli for simple elastic calculations

(c) to draw attention to the difficulties of inter- preting in situ tests and field measurements using linear elasticity, particularly when the derived soil moduli are to be used in a different type of boundary value problem.

Before presenting the results of the various boundary value analyses, a brief resume is given of the characteristics observed in the recent laboratory experiments. Following from this, the details of the simple soil model developed for these studies are set out.

UNDRAINED STRESS-STRAIN CHARAC- TERISTICS OBSERVED IN SPECIALLY INSTRUMENTED LABORATORY TESTS

Recent research by Daramola (1978), Costa- Filho (1980) and Jardine et al. (1984) has shown that conventional methods of determining axial strains in triaxial experiments can lead to important errors. Even with the most careful sample preparation and calibration for the compliance of equipment, bedding effects at the specimen ends and rotation of the sample under load can cause the externally measured strains to exceed those measured locally on the sample. In the early stages of a test the external strains can be ten times larger than those measured on the sample. By means of displacement transducers mounted on the soil specimens, Costa-Filho & Vaughan (1980) found that the true secant stiffness of samples of London Clay, at around 0.1% strain, agreed well with average stiffness values obtained from the back analysis of field measurements (St John, 1975). It therefore appears that the previously reported differences between laboratory and field stiffnesses of London Clay result more from inadequacies in conventional laboratory strain measuring techniques than from sampling disturbance or time-dependent threshold effects.


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NON-LINEAR STRESS-STRAIN CHARACTERISTICS 379

1 2 3 4

Awal siran E %

(a)

Ax1.3 stram E: %

03

Fig. 1. Measured undrained stress-strain behaviour of a reconstituted K, consolidated low plasticity clay at various overconsolidation ratios as indicated: (a) strain on an arithmetic scale; (b) strain on a logarithmic scale

Using new local strain measuring techniques, Jardine (1985) carried out a comprehensive range of triaxial tests on a wide spectrum of soils. Experiments on intact and reconstituted low and medium plasticity clays, sand and intact chalk all exhibited high initial stiffness and non-linear stress-strain behaviour, both these factors depending on soil type, method of formation and stress history. To illustrate these features Fig. l(a) shows the undrained stress-strain behaviour of reconstituted samples of a low plas-

ticity clay which were consolidated from a slurry to various overconsolidation ratios. Fig. I(b) shows the same data with strain plotted on a logarithmic scale to show the detail of the initial stages of loading. In Fig. 2 the results of tests RI and R2 have been plotted as E,jC, against log(strain) where E, is the secant undrained Youngs modulus. The results of tests on two lightly overconsolidated intact samples of the same soil (I1 and 12) are also shown and give similar results. Sample I1 was tested unconsoli-


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Axial strain E. %

Fig. 2. Stiffness-&rain curves for a low plasticity clay: Rl and R2 resonstituted, sheared from K, conditions; I1 and 12 lightly overconsolidated intact clay from 672 m below the sea bed; I1 sheared from unconsolidated conditions, 12 reconsolidated to K. = @52 before undrained shearing

dated and sample 12 was anisotropically reconsolidated to its estimated in situ stress state. Further details of these tests are given by Jardine et al. (1984).

EMPIRICAL UNDRAINED STRESS-STRAIN RELATIONSHIP

To use the measured stress-strain relationships in the analysis of boundary value problems it is necessary to find a simple mathematical expression that fits the data reasonably well. In this section such an empirical expression is described. No doubt other expressions could be found which fit the data equally well.

The data presented in Fig. 2 suggest that the general form of the relationship between the secant Youngs modulus E, and the logarithm of axial strain before failure can be conveniently represented by a periodic logarithmic function

E -=4+Bcos{ol[log,,(~)~} (1) CU

as shown in Fig. 3. When considering more general effective stress models, equations of the same form may be used to describe the variations in shear and bulk modulus with their respective strain invariants. The empirical constants A, B, C, c( and y can be quickly determined from test data as described in Appendix 1. Equation (1) only holds for a specified range of strain values. For strains below a lower limit E,,,~ and above an upper limit E,,,, fixed tangent stiffnesses are assumed. Over this elastic range a Poissons ratio of 0.49 is specified and, if yield is to be modelled, a suitable criterion and flow rule must be included. Care is required to ensure compat- ibility between E,,, and the onset of plastic yield.

In Fig. 4 equation (1) is fitted to a selection of the stiffness test data given by the low plasticity clay. The values of the associated empirical constants are given in Table 1. It can be seen that the formulation is appropriate for soil behaviour before yield.

Table I. Non-linear soil parameters from three tests

Test A B c: % a B e,(min): % EJmax): %

RI 850 1000 OGO8 2.023 0.5943 0@05 0.20 R2 3100 3200 0.0007 1,349 0.6385 0,003 0.20 12 1420 1380 0.009 2.098 0.5050 0.0045 1.5


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3

Y u=

/- / /

E = c-

0 10~3 r-

NON-LINEAR STRESS-STRAIN CHARACTERISTICS

,Observed, or protected. maximum

/

E,/C, = A + E cos [a (log,, dC)yI

Projected minimum

I Fig. 3. Curve fitting to stiffnessstrain data

The majority of numerical procedures make use of the tangent modulus E,, rather than the secant modulus. Differentiating and rearranging equation (1) gives

E,, Buyl- c = A + B cos (d) - x sin (czP) (2)

where I = log,, (E,/C). Equation (2) can be gen-

381

eralized by substituting the deviatoric strain invariant

E = 0

; 12[(~1 - Q) + (et - E#

+ (E2 - &3)2]12 (3)

for 312&, and this allows equation (2) to be incor- porated into non-linear elastic finite element

- Test data Equatfon (1) (Rl, R2)

; Equation (1) (12)

Fig. 4. Curve fitting for tests Rl, R2 and I2


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-0C = 110 kN/m

Experiment R2 - - o - - Finite element slmulatfon

0. 103 10~2 10-i 100

eA %

Fig. 5. Comparison between experimental and fitted curves for test R2

computer programs. For the present analyses the empirical formulation has been combined with a perfectly plastic (non-hardening) Tresca failure criterion and plastic potential. The undrained stress-strain relationship chosen for this study is referred to as LPC2 and corresponds to the relationship obtained for test R2 given in Fig. 1 (see Table 1 for the appropriate empirical constants).

Figure 5 shows a comparison of the source data from test R2 and the finite element simula- tion using material LPC2. The agreement is excellent over almost four logarithmic cycles of strain. Test R2 was chosen for the studies described in this Paper, as its yielding behaviour most closely approximated to the simple Tresca criterion.

Model LPC2 represents a stiff low plasticity clay with an overconsolidation ratio of 2. For the range of data presented by Jardine et ul. (1984) the model shows a higher normalized stiffness than average but is not particularly non-linear.

SIMPLE PRACTICAL ILLUSTRATIVE PROBLEM

In this section a simple problem is analysed to illustrate some of the practical conclusions that arise from the effects of non-linear stress-strain behaviour. An approximate method is employed for this preliminary study but in the remainder of the Paper a finite element analysis is used. The problem considered is that of a rigid circular load which might, for example, represent a foundation for which the underlying settlements are required or an instrumented loading test from which deformation properties are to be deduced.

Figure 6(a) shows a rigid smooth circular footing of diameter D resting on a layer of uniform clay of thickness 5D underlain by a rigid

layer. The footing is loaded undrained to a mean bearing pressure of 3C, where C, is the undrained strength of the clay. Hence the load factor on undrained bearing failure is approx- imately 0.5. Throughout the clay layer it is assumed that the soil has the same initial stress history and stress-strain properties as given by the low plasticity clay in test R2 (see Fig. 5).

For this illustrative problem it is assumed that the changes in the total stresses crv and CT~ beneath the centre of the footing can be obtained by means of linear elasticity. At any depth Z/D the value of (a, - a,)/2 can be calculated and the corresponding value of vertical strain obtained from Fig. 5. Fig. 6(b) shows the distribution of vertical strain beneath the centre of the footing obtained using this procedure. In Fig. 6(c) the variation in normalized settlement 6/S, is plotted against depth and compared with the distribution for a homogeneous linear elastic material. Fig. 6(d) shows the variation in secant modulus E,/C, with depth corresponding to the strain distribution given in Fig. 6(b).

The following important practical conclusions can be drawn from this simple illustration. (a) From Fig. 6(b) it can be seen that the axial

strains beneath the centre are always less than O-l%. Therefore, even though the load factor is as high as 0.5, the ground response is domi- nated by its small strain properties. Thus laboratory testing to evaluate the stiffness properties of the ground will require precise measurement of strains to at least an accuracy of 0.01%.

(b) From Fig. 6(c) it is evident that the settlement for LPC2 reduces much more rapidly with depth than for a homogeneous linear elastic material.


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Y CD_*

E: %

0 0.1

T-r :2-0

(c) The results given in Fig. 6(d) show that, had field measurements of settlement at various

(4

Apparent E,IC,

5000

E,/C, = 1000 + 1250UD (apparent linear varlallon Of Stiffness with depth)

id)

Fig. 6. Approximate calculations of strain, settlement and apparent stiffness beneath the centre of a rigid footing: (a) circular footing on a uniform layer of clay; (b) vertical strains deduced under the centre line for an elastic stress distribution; (c) variations in settlement with depth for LPC2 and a homogeneous linear elastic soil; (d) variation in apparent linear modulus with depth for LPC2 using elastic stresses and strains from (b)

depths been made down to Z/D = 3.5, a linear elastic interpretation would have pointed to a linearly increasing stiffness with depth. This conclusion has been reached from many back-analysed field measurements. An erroneous conclusion that a given soil profile has increasing stiffness with depth, rather than non-linear small strain stiffness properties, can have significant and unfortunate practical consequences. Of course in many practical situations both effects will be present.

The remainder of this Paper is devoted to exploring some of the practical implications of small strain non-linearity (and failure) and the limitations of linear elastic predictions when such non-linearity is present.

ANALYSIS OF SOME BOUNDARY VALUE PROBLEMS

As mentioned previously, the purpose of this study is to analyse a range of undrained boundary value problems using non-linear elasto-plastic stress-strain characteristics in such a way as to identify any major differences between the results obtained and those predicted from linear elastic

,Applled vertical displacements

1 st element yields

100 m-

0 02 0.4 0.6 0.8 10 Normalired settlement of footing 8/D %

Fig. 7. Pressuresettlement curve for a rigid footing resting on soil type LPC2


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theory. The boundary value problems considered fall in four groups which will be discussed separately

(a) footings (6) cavity expansion problems (c) axially loaded piles (d) strutted excavations.

The analyses were carried out by means of the finite element method using eight-noded iso- parametric elements and reduced integration.

Circular rigidfooting The problem considered is that of a vertically

loaded rigid smooth circular footing of diameter D resting on the surface of a uniform layer of clay. The depth assumed for the clay layer was 5D, as this case approaches the asymptote for infinite depth with a linear elastic material (Poulos & Davis, 1974).

Two soil types were considered in the study

(a) the undrained non-linear soil model LPC2 described previously, with K,, = 0.72 and C, = 220 kPa (Fig. 5)

(6) undrained linear elastic soil with E, = 1056 MN/m2 (i.e. the initial value for LPC2) and Poissons ratio 0.49.

The finite element mesh used for the analysis is shown by the inset in Fig. 7 and loading was carried out by applying increments of uniform vertical displacements to the footing surface. For the elastic case very close agreement was obtained with the exact solution given by Poulos & Davis (1974).

The load-settlement relationship predicted with LPC2 is plotted in Fig. 7. The analysis was continued until a settlement ratio 6/D of 0.02 had been reached, by which point the mean bearing pressure was within 2% of the solution qua z (3 + n)C, given by Vesic (1975) and others.

The analysis gives first yield under the centre line at a load factor L, = 0.58 and plastic flow first develops at the edge of the footing at a lower factor of 0.38. The latter ratio is dependent on the finite element mesh geometry, and if an infinitely fine mesh were to be used for the analysis yield would occur under the perimeter at even the lightest loads. The approximation allows the effects of non-linearity and local failure to be separately assessed.

Figure 8 gives plots of normalized settlement against depth and these show settlement reducing far more rapidly with depth than elastic theory predicts. These features develop long before first yield occurs and the profile at L, = 0.52 shows that the trend is accelerated by the onset of local failure.

6/6, 0 0.2 0.4 0.6 0.8 1 .o

A

SOL Fig. 8. Profiles of normalized settlement with depth for a rigid footing (see Fig. 7)

Figure 9 shows the predicted ground surface profiles at load factors of 0.3 and 0.52 (i.e. before and after first yield) for the LPC2 model compared with linear elasticity. It can be seen that the influence of the stiffness variation before yield, compared with linear elasticity, is to concentrate, the settlements strongly around the loaded area. The onset of local yield further accentuates this behaviour. It is important to note that a linear elastic material with stiffness increasing with depth (a Gibson soil) also exhibits a concentration of settlement towards the loaded area (Gibson, 1967; Burland, Sills & Gibson, 1973). These steep local gradients of displacement have important implications for soil-structure interaction.

Figure 10 presents contours of deviatoric strain at a load factor of 0.52. The small region of local failure is shown shaded. It can be seen that at a normal working load most of the soil is expe- riencing smaller shear strains than those developed in a triaxial test at 0.1% axial strain.

In Fig. 11 the elastic stress changes beneath the centre of the loaded area are compared with the LPC2 model for load factors of 0.3 and 0.52. It can be seen that the influence of the non-linearity of the stress-strain characteristics is to increase the vertical and horizontal stresses. The effects are most pronounced for the radial stresses at a depth beneath the footing and are less marked for the vertical stresses. It is of considerable interest to note that the changes in deviator stress are much less sensitive to non-linearity. Thus the approximation employed earlier in the simple illustrative problem would not have given rise to


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0

1 ,o 2-O 3.0 4.0 50 6.0

L, = O-52 >__--

__ ______-_--------- _______-----

-

_----- --

Fig. 9. Profiles of surface settlement adjacent to a rigid footing (see Fig_ 7)

\ \

0

C \

e contour 1 E: % +%EiT ; I ;:y D 0.05 E 0.1 F 0.35 G 1 0.5 I I \

Fig. 10. Contours of deviatoric strain beneath a rigid footing when L, = 05 (see Fig. 7)

large errors in the calculation of undrained centre line settlement.

Figure 12 shows the vertical base contact stress distributions. It can be seen that the influence of non-linearity and local yield is to distribute the

stresses more uniformly by shedding load towards the centre and to decrease the stress concentrations considerably at the edges (where the stresses are infinite in the linear elastic cases).

The assumption that, for a known surface stress distribution, soil stresses can be calculated from linear elasticity is central to routine foundation engineering. To investigate this aspect calculations were carried out for a circular flexible load with Z/D = 5.0. The plots of centre line stresses for load factors of 0.3 and 0.5 are compared with elastic profiles in Fig. 13. In this case the vertical stresses are insensitive to the constitutive law, but elasticity underestimates the radial stresses and overpredicts the deviator stress profile. This result provides further evidence of the validity of employing Boussinesq theory for the vertical stresses in settlement calculations.

In summary, with footings, the non-linear stress-strain characteristics have a dominant influence on the form and scale of the displace-

(4 (b) (0

Fig. 11. Distributions of stress increment beneath the centre of a rigid footing (see Fig. 7): (a) radial stress changes; (b) vertical stress changes; (c) deviator stress changes


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---'Elastic' ----_L, = 03

D *- -L, = 0.52

2 9 average footing stress

4

PA

: 2z

iO8-

iz v) -__-_---

m _e -_----

k 5 04

I I

0 0.2 0.4 0.6 O-8 10

Fiad~al distance from centre of footing r/r0

Fig. 12. Vertical contact stresses beneath a rigid footing (see Fig. 7)

ment distributions and a less marked, but none the less significant, influence on the stress distribution. It is of interest to note that a Gibson soil (linearly elastic with stiffness increasing hn- early with depth) gives rise to very similar effects, i.e. a concentration of vertical displacements beneath and around the loaded area, a reduction in variations in base contact stresses beneath rigid footings and relatively small deviations of vertical stresses from homogeneous elasticity beneath uniformly loaded areas.

Figure 14(a) shows the predicted relationships between P/P,,, and A V/( V, + A V) where P is the increase in cavity pressure from the initial radial stress, V, is the initial volume of the cavity and V is the change in the volume of the cavity. The ratio P/P,,, is a load factor L, and it can be seen that yield occurs at a relatively early stage.

Expansion of long cylindrical cavity Cylindrical cavity expansion analysis is of con-

siderable practical interest, as it is used to interpret pressuremeter data, and has some common

The curve predicted using LPC2 may be treated as being equivalent to field data obtained from an ideal undrained pressuremeter test, and it is of interest to plot the curve of normalized Youngs modulus EJC, against P/P_, and P/C,, as shown in Fig. 14(b). In most pressuremeter testing a single shear stiffness G = EJ3 is evaluated from an unload-reload loop, the magnitude of which can vary between 0.X and 3.5C,, depending on the judgement of the operator. A linear shear modulus is then determined from a mean line drawn through the loop. It is clear from Fig. 14(b) that the deduced modulus is likely to be highly sensitive to the size of the imposed pressure cycle for soils such as LPC2.

features with the horizontal loading of piles. The problem geometry was idealized by considering a disc-like assembly of finite elements, with the mesh extending to 100 times the internal diameter of the cavity. Plane strain in the vertical direction was imposed, and increments of radial displacement were applied at the inner radius of the cavity. (Solutions to this problem can be derived without resort to finite element methods although it is frequently just as convenient to make use of them.) The analyses were again performed using LPC2 and a linear elastic soil with E, = 1056 MN/m and v, = 0.49. The analysis was continued until the cavity had been expanded to 1.5 times its initial diameter, when LPC2 gave a total pressure of 10,9C,.

0 02 04 06 02 04 06 08 10

(a) ib) (Cl

Fig. 13. Distributions of stress increment beneath the centre of a flexible circular footing with the geometry shown in Fig. 7: (a) radial stress changes; (b) vertical stress changes; (c) deviator stress changes


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4 a 12 16 20

AVl(V, + AV). %

(a)

P/C

2 4 6 8 10 I I I

I ( I I I I

\ 1 Range for magnitude

I

o-2 0.4 0.6 0.8 1 .o Load factor P/P,,,

ib)

Fig. 14. (a) Pressure-volume change curves for cylindrical cavity expansion and (h) the variation in apparent secant modulus with load factor for cylindrical cavity expansion in soil type LPC2


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erkal displacements applied to top 01 pile

de7 A+7 + 50 m +

Fig. 15. Finite element mesh for a pile 30 m long

The analysis of a full pressuremeter test was not attempted because of the assumptions which would have to be made concerning the stress- strain characteristics associated with loading reversals. However, the curves obtained from high quality tests in clays often show a non-linear response similar to that given in Fig. 14(a) (Windle & Wroth, 1977; Powell & Uglow, 1985).

Axial loading of pile 30 m long The third class of problem to be investigated

was that of a compressible pile. A solid pile, 0.75 m in diameter and 30 m long, was selected with a modulus of 30 x 10 MN/m. Such a stiffness is appropriate to either a steel pipe pile or a reinforced concrete pile, and was 28 times the maximum soil stiffness. The finite element mesh for the study is shown in Fig. 15, and no account was taken of any effects of installation on soil properties or initial conditions. Loading was simulated by applying increments of vertical displacement to the top of the pile. Three cases were considered

(a) PI: the soil was everywhere represented by the non-linear model LPC2 including the soil immediately adjacent to the pile shaft (i.e. Lx = 1)

(b) P2: the soil was linear elastic with E, taken as

the maximum value given by LPC2 (as described later, additional elastic runs were carried out with a range of E, values to aid in the interpretation of equivalent soil stiffness from load-settlement data)

(c) P3: as for Pl with the pile modulus increased by a factor of 103.

The relationships between load and settlement of the pile head are given in Fig. 16. In both of the elasto-plastic cases local plastic failure (indicated by arrows) was reached at settlements of less than 2 mm and the stiffness of the pile can be seen to have an important influence on the behaviour. (The finite element mesh designed for this study was not chosen to give particularly thin elements close to the pile. With the shaft shear stresses being projected from points around r,J5 from the interface, there was a tendency for the ultimate capacity to be overpredicted by as much as 7%. In analyses where the calculation of ultimate loads is of greater importance, the accuracy can be improved by using a finer mesh.)

Figure 17 shows the radial profiles of relative surface settlement (6,/d,), where 6, is the settlement at radius r from the pile centre and 6, is the settlement of the pile. The profiles for Pl and P3 correspond to a load factor of 0.5. It is apparent that linear elastic theory gives a poor estimate of the surface settlement profile around a typical pile


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LPC2(cr=1)(P1)Ep=30X103MN/m2

- - - ~ Linear e&AC (I??) E, = 30 X 1 O3 MN/m* -----xx LPC2(a = l)(P3)Ep = 30X106 MN/m

(Arrows lndlcate points of first yield)

0 4 8 12 16 20

Settlement: mm

Fig. 16. Load-settlement curves for a pile 30 m long

installed in a soil with the stress-strain characteristics of LPC2. However, it is of interest to note that the elastic profile can be almost recovered if the pile-soil stiffness ratio is increased by 103, i.e. the pile becomes very stiff.

The location and extent of local failure at working load is shown in Fig. 18 in which contours of a deviatoric strain E are plotted for case Pl with a load factor of 0.5. It can be seen that yielding has occurred near the ground surface and that, except for a very narrow zone immediately adjacent to the pile, the bulk of the soil

experiences only very small strains, with E


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Zone of plastic behaviour close to pile I

Fig. 18. Contours of deviatoric strain around a pile 30 m long when L,O= 05 (see Fig. 16)

overall behaviour is sensitive to the combination of large strain properties close to the pile and the small strain characteristics of the surrounding soil. These features are significant in design and are likely to affect the capacities and working settlements of both single piles and pile groups. It is apparent that linear elastic theory will tend to

Normalized shear stress: TJC,

overpredict group settlement ratios and to exag- gerate the non-uniformity of loads within rigidly capped pile groups. In assessing the interaction of pile groups it is thus necessary to consider the initial response of the soil to shearing with the full accuracy afforded by the new laboratory techniques, and this is particularly so when considering the response of large offshore piled structures (Jardine, 1985).

LPC2,Ei,=30X103MN/m(P1)

--x ~ LPC2 Ep = 30 X 1 O6 MN/m2 (P3)

Fig. 19. Variation in shear stress with depth down a pile 30 m long for two pile stiffnesses when L, = 05 (see Fig. 16)

Strutted excavation The prediction of structural forces and ground

movements around deep excavations has important implications for construction in built-up areas. These problems have prompted much research, and some of the difficulties of assuming linear elastic soil behaviour have first become apparent from the monitoring of such structures. It was therefore considered important to include an analysis of a hypothetical strutted excavation in the present series of studies.

The excavation considered was infinitely long, 40 m wide, 15.26 m deep and was supported by a diaphragm wall 20 m deep, propped at the surface by means of rigid struts before excavation. The finite element mesh is shown in the inset to Fig. 20 and excavation was simulated by sequen- tially removing layers of elements from within the excavation. The final depth of 15.26 m was chosen to give a factor of safety F, of 1.5 in terms of undrained shear strength. The wall adhesion factor was taken as 0.X,. The material compos- ing the diaphragm wall was specified as being linear elastic with E, = 28 x lo3 MN/m and


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employlng-

analws

3.0 2.0 1 ,o Dlsplacemenl towards excavatfon: cm

Fig. 20. Variation in maximum horizontal wall displacement with depth of excavation for a propped diaphragm retaining wall

v, = 0.15. Two cases were considered with the soil modelled as

(a) LPC2 (b) linearly elastic with E, = 1056 MN/m and

v, = 0.49.

For both cases the initial stresses were assumed to increase linearly with depth Z such that crV = 202 kN/m and un = 0,7a,. The undrained shear strength was taken everywhere as 110 kN/m.

Figure 20 shows the relationship between maximum inward displacement of the wall, wmaxr and the depth of excavation. Local failure is initi- ated before the load factor L, = l/F, reaches 0.2. The load-displacement curve computed with the LPC2 model is clearly non-linear with wall deflexions accelerating as excavation takes place. Fig. 21 shows the normalized horizontal deflected profiles of the wall, w/wtoer at the final depth of excavation found with the two soil models. It can be seen that the shapes of the wall profiles do not differ appreciably.

Figure 22 shows the profiles of surface settlement with distance from the wall, expressed as a proportion of w,,,. Unlike the displacement profile for the wall, the surface settlement profile

i Excavation depth 15-26 m

7 1 ,o 0.6 0.2

6

0

Normalzed horizontal displacement w/wloe

Fig. 21. Horizontal deflected profiles of a diaphragm wall for a depth of excavation of 1526 m (see Fig. 20)

is very sensitive to the soil model. The incorpo- ration of non-linearity and plasticity into the model gives rise to a pronounced settlement trough close to the excavation which cannot be matched using linear elasticity. As mentioned in the introduction, this phenomenon was noted in the field by Burland & Hancock (1977) and was later reproduced analytically by Simpson et al. (1979) using a bilinear elastic perfectly plastic soil model.

The non-linear and elastic solutions also give rise to different stress distributions over the retaining wall, and hence different bending moments and prop forces. Thus, for a depth of excavation of 15.26 m the LPC2 model predicts a prop force of 387 kN/m while linear elasticity gives 316 kN/m.

Contours of deviatoric strain are given in Fig. 23 with the excavation at 15.26 m. A zone of contained plastic behaviour is found with the non-linear soil, and the contours show that the deviatoric strains in the surrounding ground fall between 0.3% and 0.05%.

These findings have been further reinforced by recent field and analytical studies of the strutted excavations for the Bell Common cut and cover tunnel (Fourie, 1984; Tedd, Chard, Charles &


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Linear elastic analysis

Fig. 22. Vertical displacement profiles adjacent to a strutted excavation (see Fig. 20)

Symons, 1984; Hubbard, Potts, Miller & Burland, 1984). Fourie carried out finite element analyses of the excavations using an elasto-plastic effective stress model for London Clay, in which the pre-yield behaviour was described in a similar way to the LPC2 model. The stiffness parameters were derived from the instrumented laboratory tests described by Jardine et al. (1985) and it is encouraging that excellent agreement was found between the measured and predicted behaviour.

INTERPRETATION OF LOAD-DISPLACEMENT BEHAVIOUR USING LINEAR ELASTICITY

The finite element studies presented here provide insight into the effects of non-linear soil properties in soillstructure interaction. In this section the results of the various studies are used to investigate the choice of equivalent elastic design parameters and to draw attention to the difficulties of linear elastic interpretations of in situ tests and full-scale field monitoring.

It is common to interpret field loaddisplacement behaviour in terms of linear elasticity. An apparent Youngs modulus EUA, is

RIgId

Fig. 23. Contours of deviatoric strain around a strutted excavation for a factor of safety on strength F. = 1.5 (see Fig. 20)

calculated by relating a characteristic displacement to a known loading condition, such as centre line settlement to mean bearing pressure. The same method has been applied to the loaddisplacement data calculated using model LPCZ. Thus the computed loaddisplacement curves are treated as if they were experimental data gathered in the field. The variations in apparent modulus with load factor L,, produced by the different boundary value problems, can then be compared.

Figure 24 shows the variation in E,*/C, and L, for the rigid footing on a deep clay layer. It can be seen that even for load factors as low as one- third the value of E,* reduces from its initial value by about 40%. The broken line in Fig. 24 represents the variation in secant E,* with L, for a triaxial test with the soil model LPC2 (i.e. test R2 in Figs l(b) and 5). In this case L, = (q - qO)/(qr - qo) where q is the deviator stress and q. and qf are the initial and failure values respectively. It can be seen that the two curves are almost identical up to a load factor of about 0.5, i.e. for most practical ranges of working load. However, as L, increases above 0.5 and the zones of local failure spread, the apparent moduli derived from the displacements of the footing fall below the values from the triaxial test.

Similar relationships between E,* and L, have been derived for the other boundary value problems and are plotted in Fig. 25, together with the result from the triaxial compression test. This latter curve may be conveniently used as a basis for comparison. The relationship for the 30 m long pile needs special mention since it is compli- cated by the effect of pile compressibility. The relationship between E,* and 1oad:settlement ratio was obtained by carrying out eight linear elastic analyses in which E,* was varied but the pile stiffness was fixed. Hence, for any given point


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- Rigld fooilng Z/D = 5.0

-- - - Tnaxml test R2

0 02 04 O-6 08 10

Load factor L,

Fig. 24. Variations in apparent secant modulus with load factor for a rigid footing and an undrained triaxial test for soil type LPC2

5OOOr

on curve Pl in Fig. 16, it was possible to obtain the apparent modulus E,* and thereby to derive the relationship between E,*/C, and L, given in Fig. 25.

For all the cases referred to in Fig. 25, with the exception of the strutted excavation, the apparent modulus E,* relates to the deflexion of the point of application of the load (or stress), and the load factor is clearly defined. For the strutted excavation the value of E,* relates to the maximum horizontal deflexion of the wall and the load factor is the inverse of the factor of safety on undrained strength.

Bearing in mind that Fig. 25 relates to a spe- cific non-linear soil model applied to particular boundary value problems the following observations can be made.

(a) In each case the apparent modulus reduces continuously as the load factor increases, and in no case can the continuum behaviour be properly described as linear elastic at working loads.

(b) For any given load factor there is a considerable range of values of E,*/C,. For example, at L, = 0.5 the rigid pile gives the stiffest response (E,*/C, = 3600) and the cavity

RIgId 30 m long, 0.75 dia. ptle

0 0.2 0.4 0.6 0.6 1 .o

LOad factor L/L,,,, PIP,,, and 1 IF s

Fig. 25. Summary diagram showing variations in apparent modulus with load factor (l/F, for the excavation)


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expansion curve gives the softest response (E,*/C, = 300) and these two curves fall to either side of the triaxial characteristic. (Intuitively, similar results might be expected for the horizontal loading of the pile.) Thus field moduli deduced from experiments in the same soil, but with different types of boundary conditions, can be radically different even for an isotropic material such as LPC2.

(c) First plastic yield occurred over a wide range of load factors. Hence the values of E,*/C, at the onset of yield vary from 3700 for the strutted excavation (L, = 0.14) to 800 for the triaxial test (L, = 1.0). If an infinitely fine mesh had been employed for the footing analysis, first yield would have occurred with L, = 0.0 when E,*/C, = 4800.

The two extreme cases in Fig. 25 are of interest (i.e. the expanding cavity and the rigid pile) since both form the basis of in situ tests. It is clear that great care is needed in evaluating the stiffness of the ground from such data. The value of these tests would be increased if the full characteristic of apparent modulus with load factor were reported, rather than a single arbitrary stiffness value. It may be feasible to use such a characteristic to estimate the non-linear stress-strain characteristics of individual soil elements.

It is of considerable practical interest to note the reasonably good agreement between the results derived from the settlement of footings and the triaxial element test. Good agreement is also found between the 30 m pile and the triaxial test when a realistic pile stiffness is assumed. However, a rigid pile gives a far stiffer characteristic than the triaxial curve and explains why the modulus back calculated from pile tests is often so much higher than that obtained by other methods.

DISCUSSION AND CONCLUSIONS The series of analyses are intended to give a

preliminary appraisal of the effects of the non- linear soil behaviour observed in recent laboratory tests. To restrict the number of variables, only the undrained behaviour of a homogeneous layer of an isotropic material under monotonic loading is considered.

The simple empirical stress-strain expression used for the calculations provides a good fit to the undrained behaviour of a lightly overconsolidated low plasticity clay in triaxial compression. Non-homogeneity can be considered without undue difftculty, but if drained conditions (or cycles of loading) were to be considered a more complex model is required. The material considered, LPC2, is probably stiffer than most soils but is not unusually non-linear. Recognizing

the limitations of the study, the following preliminary conclusions can be made.

There can be considerable dihiculties in applying linear elastic theory to the prediction of ground movements and soil stresses induced by different types of structure. In all the practical cases studied, the modelling of realistic small strain non-linearity and the consideration of local failure have important implications in considering soil-structure interactions at working loads.

In footings and excavations the small strain characteristics appear to have the greatest influence on the deflexion profiles around the loaded boundary. With piled foundations the onset of local failure appears to be at least as significant, and the combination of these two kinds of non- linearity appears to control pile group interaction and progressive failure.

For all the cases studied, the large mass of the soil influenced by the boundary loading was strained to less than 0.1% deviator strain and frequently to less than 0.05%. If representative soil parameters are to be determined experimen- tally, highly accurate measurements are required. Thus the precision offered by the new laboratory techniques is of considerable practical value.

In problems such as footings it may be reason- able to combine stresses predicted from linear elastic theory and measured non-linear stress- strain characteristics to carry out approximate evaluations of centre line settlements. In other cases, such as the estimation of group displacements for large piles, or the calculation of wall bending moments in deep excavations, linear elastic theory is less satisfactory.

The studies show that the back analysis of full- scale performance or in situ tests is likely to lead to a wide range of possible values for deformation moduli, even for a uniform isotropic material.

Recent research has confirmed the expected sensitivity of stiffness to boundary conditions and the loading level in field and laboratory tests on London Clay (Jardine et al., 198.5). The initial stiffnesses observed in triaxial tests can exceed overall values deduced from either high quality in situ tests or the back analysis of full-scale performance.

Although linear elasticity remains a convenient tool for expressing measurements of soil stiffness, its limitations must be recognized. In particular the importance of load factor and the tendency towards concentrations of strain close to loading boundaries must be taken into account. If the non-linear nature of soils is not acknowledged, comparisons of field and laboratory measurements can be confusing.

The use of instrumentation systems in field monitoring which allow the direct determination


On: Fri, 26 Aug 2011 17:33:26


of strains (and their comparison with stresses) is likely to be of great value in understanding soil behaviour and assessing the applications of the laboratory techniques.

Although the apparent agreement between laboratory non-linear characteristics and measured field behaviour is most encouraging, further studies are required to investigate the effects of different stress paths, drainage conditions and strain rates.

Finally the pre-yield non-linear formulation described in this Paper can be used more gener- ally and has been included in analyses where more sophisticated, effective stress, elasto-plastic models have been required; see Fourie (1984) and Jardine (1985).

REFERENCES Burland, J. B. (1975). Some examples of the influence of

field measurements on foundation design and construction. Proc. 6th Regional Conf for Africa Soil Mech. Fdn Engng, Durban 2.

Burland, J. B. & Hancock, R. J. R. (1977). Underground car park at the House of Commons: geotechnical aspects. Struct. Engr 55, 877100.

Burland, J. B., Sills, G. C. & Gibson, R. E. (1973). A field and theoretical study of the influence of non- homogeneity on settlement. Proc. 8th Inc. Con/ Soil Mech. Fdn Engng, Moscow 1.3, 31-46.

Burland, J. B. & Symes, M. (1982). A simple axial displacement gauge for use in the triaxial apparatus. Geotechnique 32, No. 1,62X5.

Costa-Filho, L. M. (1980). A laboratory investigation of the small strain behaviour of London clay. PhD thesis, University of London.

Costa-Filho, L. M. & Vaughan P. R. (1980). Discussion on A computer model for the analysis of ground movements in London clay. Geotechnique 30, No. 3, 336339.

Daramola, 0. (1978). The influence of stress history on the deformation of sand. PhD thesis, University of London,

Eisenstein, Z. & Medeiros, L. V. (1983). A deep retaining structure in till and sand: part II, performance and analysis. Can. Geotech. J. 20, No. 1, 131-141.

Fourie, A. B. (1984). The behaviour of retaining walls in sttrclays. PhD thesis, University of London.

Gibson, R. E. (1967). Some results concerning displacements and stresses in a non-homogeneous elastic half-space. Geotechnique 17, No. 1, 58-67.

Hubbard, H. W., Potts, D. M., Miller, D. & Burland, J. B. (1984). Design of the retaining walls for the M25 cut and cover tunnel at Bell Common. Geotechnique 34, No. 4, 495-512.

Jardine, R. J. (1985). Investigations ofpile-soil behaviour with special reference to the foundations of offshore structures. PhD thesis, University of London.

Jardine, R. J., Fourie, A., Maswoswe, J. & Burland, J. B. (1985). Field and laboratory measurements of soil stiffness. Proc. 11th Inc. Conf Soil Mech. Fdn Engng, San Francisco 2,51 I-514.

Jardine, R. J., Symes, M. J. & Burland, J. B. (1984). The

measurement of soil stiffness in the triaxial apparatus. Gtotechnique 34, No. 3, 323-340.

Marsland, A. (1971). Laboratory and insitu measurements of the deformation moduli of London clay. Proc. Symp. Interaction of Structure and Foundation, July. Midland Soil Mechanics and Foundation Engineering Society. (Also Building Research Station Current Paper CP 24/73.)

Morgenstern, N. R. & Phukan, A. L. T. (1968). Stresses and displacements in a homogeneous non-linear foundation. Proc. Int. Symp. Rock Mech., Madrid, pp. 3 13-320.

Poulos, H. G. & Davis, E. H. (1974). Elastic solutions fin soil and rock mechanics. New York: Wiley.

Powell, J. J. M. & Uglow, I. M. (1985). A comparison of Menard, self-boring and push-in pressuremeter tests in a stiff clay till. Adv. Underwat. Technol. Offshore Engng 3,201-219.

Simpson, B., ORiordan, N. J. & Croft, D. D. (1979). A computer model for the analysis of ground movements in London Clay. Geotechnique 29, No. 2, 149- 175.

St John, H. D. (1975). Field and theoretical studies of the behaviour of ground around deep excavations in London clay. PhD thesis, University of Cambridge.

Tedd, P., Chard, B. M., Charles, J. A. & Symons, I. F. (1984). Behaviour of a propped embedded retaining wall in stiff clay at Bell Common Tunnel. Gtotech- nique 34, No. 4, 513-532.

Vesic, A. S. (1975). Bearing capacity of shallow foundations (eds H. F. Winterkorn and H. Y. Fang), Chap. 3, Foundation Engineering Handbook. New York: Van Nostrand Reinhold.

Windle, D. 8~ Wroth, C. P. (1977). lnsitu measurements of the properties of stiff clays. Proc. 9th Inc. Conf Soil Mech. Fdn Engng, Tokyo 1, 347-352.

Wroth, C. P. (1971). Some aspects of the elastic behaviour of overconsolidated clay. Proc. Roscoe Memo- rial Symp., pp. 347-361. London: Foulis.

APPENDIX 1 Calculation of non-linear parameters from test data. Referring to Fig. 3, first locate the observed, or projected, maximum stiffness point. Now maxima for equation (1) occur when

cos {fIog,, (;)J} ?= 0 i.e. when

flog,, @)y = 2nn assuming n = 0 for the observed maximum gives log,, (EJC) = 0 and C = A.

Next the crossing point where the angular part of equation (1) must equal n/2 is located, so that

(5)

then the minimum point where the angular part must equal rr is located, so that


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396 JARDINE, POTTS, AND BURLAND

The two parameters A B can taken directlv the stiffness corresnondine. to strainsc, D E.

I I

Equation (1) should then be evaluated for a number of points to find the degree of departure from the test data near the upper and lower limits of strain. The

(7 limits should be selected to prevent negative tangent stiffnesses from being predicted, and the lower limit should not usually be less than O+JOl %I.

From the result of equation (7) a can be obtained The maximum, minimum and crossing points can be

from equation (5) and reselected if the degree of fit is unsatisfactory. If it is required to evaluate the expressions for A

42

a = Clog,, wc)Iy

-c C, problems will arise when raising the logarithmic terms to a fractional power. In this case pre- and post- multiplication by - 1 will be required.

The most common types of analysis continue to be based on the theories of linear elasticity. The underlying assumption is either that at working loads the soil mass is behaving in a linearly elastic manner or that the stress changes in the soil are

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