PAPER

Se;:eiIIenI;an( ~eave ooverconso ic a~:ecc BIfs-

Eu

Cu

1500

1000'00

Note

Curves NLI

illustrative c

London Clay, unweather

a simplified non-linear methodof calculation

OL0.001 0.005 0.01 0.05 0.1

Vertical strain (%)

05 1.0

ByAS O'rien and P Sharp, Mott MacDonald0

0

Normahsed undrained settlement Sz/50

0.5 1.0

IntroductionSettlement at working load is usually the critical design considerationfor vertically loaded foundations on stiff overconsolidated clays.Foundation settlement is usually calculated by assuming that theground behaves as a linear elastic material. The reliability of thiscalculation is mainly dependent on an appropriate choice of elasticmodulus.

However, modern laboratory testing techniques have identified thehighly non-linear stress strain characteristics of overconsolidatedclays, for example, refer to Jardine et al 1984'nd Figure 1'. Thisexplains the wide range of values for "linear elastic" moduli, whichhave been reported in the technical literature; for example E/Cu ratiosof between 150 and 1500 have been quoted in CIRIA SP 27'. As a resultthe selection of an appropriate value for a linear elastic modulusbecomes a matter of considerable engineering judgement. The recentGround Engineering article'xemplifies some of the difficulties thatengineers face in selecting appropriate deformation moduli.

An additional problem is that linear elasticity incorrectly predictsthe pattern of settlement adjacent to and beneath a loaded area. Forexample, if total settlement of a structure is correctly predicted, set-tlement at depth or remote from the structure will be incorrectly pre-dicted. In view of these problems, the applicability of linear elastic cal-culations for overconsolidated clays is limited. However, the alterna-tives to conventional linear elastic calculations, such as non-linearfinite element techniques, can be complex, expensive and time con-suming, requiring high level expertise and considerable engineeringinterpretation.

Hence, there is a need for a simplified method which enables theengineer to gain an understanding of foundation deformation behav-iour under loads of varying intensity. This paper describes a methodwhich enables the non-linear stress strain behaviour of overconsoli-dated clays to be modelled in a manner which is relatively simple andis appropriate for routine design calculations. The calculations aremost conveniently undertaken by computer. however, the method isreadily amenable to hand calculation. Undrained and total settlement(or heave) can be calculated under foundation loading of any shape andof varying intensity using a varying ground stiffness depth profile.

Background to proposed methodField observations of ground movementDuring the 1970s, the BRE performed several large scale plate loadingtests. Figure 2 shows settlement recorded beneath a 0.9m diameter platetest on London Clay. The data is shown in dimensionless form(subsurface settlement divided by plate settlement at founding level) andcan be compared with the prediction of settlement within ahomogeneous linear elastic medium. The observed subsurfacesettlements are quite localised, most settlement occurs within a depth ofabout 0.6times the plate diameter.

In contrast the linear elastic calculation predicts a significantlygreater depth of influence. Figure 2 also shows normalised subsurfacesettlement recorded beneath a large building founded on glacial till oflow plasticity'. The pattern of settlement with depth is similar to that

0.5 >-

ccCl'aoi

ioEo

1.0,

1.5'ey:

ao (kN/m ) Sofmml zfm) Foundation Formation Reference';geometry

250 18 3.3 agua gie,Baca iiiB=m L=BB',

Il egal g

137 7 3 3 Ae ac e Ai ac e

220 4 17 5 'o cacao ua va ca'aco" .. 2

ea e'actce ieeic e iceomc oi .".."'are a*eeoc Aoa eo ace ai:

for the plate test on London Clay, with most settlement occurring with-in a depth of about 0.6 times the building width. An interesting featureof this field data is that the building settlement increases by more thana factor of 2.5, when bearing pressure increases by a factor of onlyabout 1.8. Also as bearing pressure is increased, normalised sub sur-face settlement becomes concentrated closer to the building founda-tion. Linear elastic calculations would not predict this pattern ofbehaviour.

Figure 3 summarises some settlement and heave observations forrelatively large rigid structures founded on overconsolidated clays.Reviews of published case histories of settlement of buildings con-structed on overconsolidated clays by, for example, Simons and Som1970', and Morton and Au 1975'ndicate that settlement at the end ofconstruction is about 60'o of the total settlement. Good case historiesof time dependent heave are, by comparison, relatively rare.

However, comparing the settlement case histories with those forheave, it is apparent that for comparable foundation type, ground con-ditions and net change in foundation pressure, time dependent heave

28 GRoLINI) ENGINI3ERING ocTOBER 2001

PAPER

due to unloading is greater than time dependent settlement due to load-ing. The fundamental difference in behaviour between the develop-ment of settlement and heave is highlighted by comparing the ratio ofthe end of construction movement to the time dependent movement,R = 6u ih,d. The settlement records typically indicate R to vary between1.4 and

PAPER

Dividing the compressible strata into n layers, then for layer i:

5u, [A(r,. - v(kirh + Airh )]H,1u

From Equation 8:

eu = ui = [Aire - V (3(rh + Arrh )]5u 1

uut ij 2iH, Eu,

(8)

(9)

Normalised subsurface settlement az/rro

0.2 0.4 0.6 0.8 1.0

1.0

2.0

Normaiised distance X/R

1.0 2.0 3.0 4.0 5.00

0.2

0.4

0.6

0.8

The undrained secant Young's modulus, Eu., is assumed to be a func-1

tion, only, of the vertical strain which layer i experiences. Mean effectivestress during undrained loading (or unloading) is assumed to be con-stant. Equation 9 has to be iterated until the vertical strain calculated forthe layer is compatible with the strain assumed for estimating Eu .

iFollowing a successful iteration, the final calculated value of Eu. is then

Ithe mobilised undrained secant Young's modulus for layer i, E.

"i(mob)'ollowinga successful iteration eu, is determined and the cumula-UVitive undrained vertical displacement is calculated for the compressiblestrata:

i=n

3.0

4.0

5.0a. Profiles of normalised subsurfacesettlement for a rigid foobng

1.0b. Profiles of normalised surface settlementadiacent to a ngid footing

Linear elasticNon-linear, Fs = 3.3Non-linear, Fs = 2.0

8u = X (euu H,)i=i

(10)

Mobilised ground stiffnessThe key element of the proposed method is the facility to allow for thedependence of mobilised secant Young's modulus on the magnitude ofstrain (and additionally, in the case of drained modulus, on themagnitude of mean effective stress).

For the proposed method, the characterisation of ground stiffnesscomprises two main elements:(a) definition of the variation of secant Young's modulus (at a particularstrain magnitude) with depth;(b) definition of the change in secant Young's modulus with changes inthe magnitude of vertical strain.

Typically in the absence of advanced insitu or laboratory test data,the engineer can define the change in secant Young's modulus withchanges in strain magnitude by utilising published literature. Suchdata is frequently normalised to allow data obtained for different sites,depths and types of test to be compared in a rational way. This alsoallows data for different soil types to be compared. The most commonnormalising parameter is undrained shear strength, that is, it is oftenassumed that undrained, or drained, secant Young's modulus (at a par-ticular strain magnitude) is proportional to Cu. Therefore, the engineermay utilise a profile of Cu with depth in order to derive a correspond-ing profile of secant Young's modulus (at a particular strain level).However, it should be noted that stiffness profiles obtained by using Cuas a normalising parameter may be less reliable than those which usemean effective stress or the product of specific volume and mean effec-tive stress as the normalising parameter. These normalising parame-ters have been used by Jardinet and O'rien et al, 1992'.

4.0,

3.0 l

2.0 L

1.0

Qi0.002 0.005 0.01 0.02 0.05 0.1 0.2 0.5

Vertical strain, r,(N) Logarithmic scale

d NL2

d NLD2

shold

1.0 2.0

Modification of drained secant Young's modulus for averagemean effective stress during loadingThe drained secant Young's modulus is dependent upon the averagemean effective stress during the load increment (or decrement).Therefore, the variation of the initial drained secant Young's modulus,E ', with depth (for a particular strain magnitude) needs to be modified totake account of the increase (or decrease) in mean effective stress (withdepth) which occurs during drained loading (or unloading). The averagemean effective stress during the load increment (or decrement) isdefined as follows:

(14)

Hence the mean effective stress during a load increment (or decrement)for layer i may be expressed in terms of the change in vertical effectivestress for layer i as follows:

Rearranging Equation 11:

pa = po+ pr2

Pa;=Po,+ ~p;2

From equation 14, substituting for Ap'n Equation 15:

(15)

For isotropic elastic materials under one-dimensional loading, thechange in horizontal effective stress is related to the change in verticaleffective stress by the Poisson's ratio:

v(1 v')

(12)

Ap: 1 (3(r + Ao ht + 3(rh2)3

(13)

Then from Equation 12, substituting for Arr'h, and r1(r', in Equation 13:

The increment or decrement of mean effective stress, Ap', can be relat-ed to the change in vertical and horizontal effective stress as follows:

(18)8(1- v)

Hence, p',. can be calculated from a knowledge of only p', and 2(rr', Inorder to modify the initial drained secant Young's modulus to allow forthe increase (or decrease) in mean effective stress, the initial drainedsecant Young's modulus can be pro-rated as follows:

Ec,=Eo iP oi

(17)

From Equation 17, the variation E'c. of with depth under the specifiedloading can be defined. It should be noted that the magnitude of F.' isnot usually unduly sensitive to the magnitude of p', assumed.

30 (iROI/NI) RN(;INKRR(N('(rTOI(KR 2001

PAPER

Calculate undrainedsettlement/heave

0 Input profile of Euo with depth,at normalising strain magnitude

0 Input vanation of Eu with strain0 Define loading intensity and

foundation geometry, calculateSay aah aah

with depth beneath point of interest

04-Calculate strain within each

tof n layers

0 0M

PAPER

Conf. on Settlement of Structures, Cambridge, pp183 to 203. Pentech Press. London.9. NV Mettvear (1984). The short term behaviour of a deep cutting in London clay, '.4IScdissertation. Imperial College.10. ND Pierpoint (1996). The prediction and back analysis of excavat>on behaviour inOxfordClay'hDThesis. Sheffield University.11.D Burford(1992) Private communication, kIarch 1992.12. D Burford (1988). Heave of tunnels beneath the Shell Centre. London. 1939-1986.Technical Note, Geb(ec)u>i que Vol 38, No.l. pp.136 to 137.13. RJ Jardine, DM Potts. AB Fourie & JB Burland (1986).Studies of the influence of non-linear stress-strain characteristics in soil-structure interaction. Gdu(echnique, Vol.36,No.3, pp377-396.

14. Henkel (1971).The relevance of laboratory measured parameters in tield studies. Proc.Roscoe Nlemomal Symposium. pp669 676 ( ambrions for soil anI

K

Po

P,,

P(

Ap'

P.

CCn>

uV

a,bc(

e

7Su

Su.Su)

F>T

SsF>o

Sx

DZ

X

RLSPT N

the normalising level of strain)adjusted value of undrained secant Young's modulus forlayer i (for a calculated level of shear strain)final undrained secant Young's modulus for layer i (forthe solution strain level)secant Young's modulus at a strain 6secant Young's modulus at 0.1 'v straindrained secant Young's modulus for vertical loadingdrained secant Young's modulus for layer irate of increase of E with depthinitial drained secant Young's modulus for layer i (for thenormalising level of strain)corrected drained secant Young's modulus for layer i (forthe norma)ising level of strain and the average value ofmean effective stress for the loading underconsideration)adjusted value of drained secant Young's modulus forlayer i (for a calculated level of shear strain)final drained secant Young's modulus for layer i (for thesolution strain level)drained secant Young's modulus immediately beneathfoundation (at load / unload boundary)drained secant Young's modulus for horizontal loading

shear modulus in vertical planecoefficient of volume compressibility for verticaldirection for layer iconstant which depends on the degree of stiffnessanisotropy. For isotropic materials (with v'=0) k., = 1,typically for over consolidated clays k, = 0.9vertical strain for layer i (undrained loading/unloading)total vertical strain for layer i (drained loading/unloading)shear strainundrained settlement or heaveundrained settlement or heave, for layer itime dependent settlement or heavetotal settlement or heavetotal settlement or heave, for layer itotal settlement at depth ztotal settlement at depth z = 0total settlement at depth z=0, at distance x from edge offoundationdepthof excavationvertical depth below loaded/unloaded boundary(underside of foundation)horizontal distance from edge of foundationwidth of loaded arearadius of circular loaded arealength of loaded areauncorrected blow count for standard penetration test

32 GR(iUNI) ENGINI')I RIN(1 ocTon(IR 2001