Randolph, M. F. (2003). Geotechnique 53, No. 10, 847–875

847

Science and empiricism in pile foundation design

M. F. RANDOLPH

Scientific approaches to pile design have advanced enor-mously in recent decades and yet, still, the most funda-mental aspect of pile design—that of estimating the axialcapacity—relies heavily upon empirical correlations. Im-provements have been made in identifying the processesthat occur within the critical zone of soil immediatelysurrounding the pile, but quantification of the changes instress and fabric is not straightforward. This paperaddresses the degree of confidence we can now place (a)on the conceptual and analytical frameworks for estimat-ing pile capacity, and (b) on the quantitative parametersrequired to achieve a design. The discussion is restrictedto driven piles in clays and siliceous sands, with particu-lar attention given to extrapolating from design ap-proaches derived for closed-ended piles of relatively smalldiameter to the large-diameter open-ended piles that areused routinely in the offshore industry. From a practicalviewpoint, we need design approaches that minimisesensitivity to the estimated pile capacity. This may beachieved partly through a greater reliance on pile loadtesting, where significant advances have been made in thelast decade, but also by adopting design approaches thatare focused more on guarding against unacceptable de-formation of the complete foundation. Example applica-tions in the paper are drawn both from offshoreapplications, where current challenges include estimatingthe axial capacity of ultra-thin-walled, large-diametercaissons, and from onshore applications such as bridgepiers and piled raft foundations, where inelastic displace-ment of the piles is not only acceptable, but oftenessential for efficient design.

KEYWORDS: axial capacity; dynamic testing; pile driving; pilefoundations; pile groups; piled rafts

Les methodes scientifiques servant a la conception despiles ont fait d’enormes progres pendant les dernieresdecennies et pourtant l’aspect le plus fondamental de cetravail de conception–l’estimation de la capacite axiale despiles–s’appuie encore lourdement sur des correlationsempiriques. Des ameliorations pour identifier les proces-sus qui se produisent dans la zone critique de sol dans levoisinage immediat de la pile ont ete faites, mais laquantification des changements de contrainte et de struc-ture n’est pas simple. Cet expose s’interroge sur le degrede confiance que nous pouvons desormais accorder (a)aux cadres de travail analytiques et conceptuels pourl’estimation de la capacite des piles et (b) aux parametresquantitatifs requis pour leur conception. Cette etude selimite aux piles enfoncees dans des argiles et sablessiliceux, et extrapole a partir des methodes conceptuellesderivees pour des piles fermees de diametre relativementpetit et des grosses piles ouvertes qui sont utilisees regu-lierement dans l’industrie offshore. D’un point de vuepratique, nous avons besoin de methodes conceptuelles quiminimisent l’importance de la capacite estimee de la pile.On peut y arriver en partie en accordant une plus grandefiabilite aux essais de chargement de pile qui ont fait desprogres significatifs au cours des dix dernieres annees,mais aussi en adoptant des methodes conceptuelles quis’attachent davantage a empecher une deformation inac-ceptable de toute la fondation. Les exemples donnes danscet expose sont tires des applications offshore ou lesdifficultes actuelles sont d’estimer la capacite axiale descaissons de gros diametres aux parois ultra minces ; cesexemples sont egalement tires d’applications sur terrecomme les piles de ponts et les fondations radeaux a pilespour lesquelles un deplacement non elastique des piles estnon seulement acceptable mais egalement, souvent, essen-tiel a la reussite de la construction.

INTRODUCTIONThis paper provides an opportunity to reflect on the consid-erable advances that have been made over the last twodecades in the design of piles and pile groups, and toidentify those aspects of pile performance that may beestimated by sound conceptual models and analysis, andthose aspects where we still need to rely on empiricalcorrelations. In the latter case, if we are to extrapolate topile geometries or soil conditions outside the current data-base, we must take care to ensure that the correlations areconsistent with our understanding of mechanics and notdistorted by limitations in the database.

Much of the design of pile foundations is still dominated

by estimation of axial capacity, even in applications such aspile groups for buildings and bridge piers, where the criticalissue is more likely to be the magnitude of displacementsunder operating conditions. Indeed, one of the recommenda-tions proposed later for onshore applications is to endeavourto weight design criteria more towards limiting displace-ments, even for the ultimate limit state, by means of non-linear analysis of pile group response, rather than expressingthem solely in terms of the capacity of individual piles. Bycontrast, in the offshore field, particularly where individualpiles are used as anchors, axial capacity plays a necessarilydominant role in design, and here the main challenge isextrapolation to the extreme geometries now used, includingsuction-installed caissons with diameters over 5 m and wallthicknesses as low as 0.5% of the diameter.

In order to limit the scope to manageable proportions, thispaper is restricted to the following topics:

(a) axial capacity of displacement piles (driven or jacked)in clay and sand

(b) the role of pile testing, and in particular interpretationof dynamic pile tests

(c) performance of pile groups and piled rafts.

Manuscript received 19 March 2003; revised manuscript accepted 6October 2003.Discussion on this paper closes 1 June 2004, for further details seep. ii. Centre for Offshore Foundation Systems, The University ofWestern Australia, Crawley, Australia.The Centre for Offshore Foundation Systems is established andsupported under the Australian Research Council’s research centresprogramme.

This choice is consistent with my belief that we may neverbe able to estimate axial pile capacity in many soil typesmore accurately than about 30%. We therefore need to relyon pile tests conducted early during the construction phaseto refine the final design (generally in terms of varying theembedded pile length, but possibly also the diameter ornumber of piles). Hopefully, however, results from load testsmay allow adequate performance of the pile group to bedemonstrated, allowing for inelastic pile response, eventhough extreme loads on individual piles exceed their nom-inal design capacity.

In each of the three areas above, my aim will be to separatethe ‘scientific’ and ‘empirical’ components on which we relyfor design calculations, to identify any empirical correlationsthat appear inconsistent with theoretical reasoning, and tosuggest areas where improvements may be possible, either bynew analysis or by gathering more specific data to resolvecurrent uncertainties. Each of the areas is illustrated bypractical examples based on case histories.

AXIAL CAPACITY OF DRIVEN PILES IN CLAYOverview

Any scientific approach to predicting the limiting shaftfriction that may be mobilised along the shaft of a drivenpile must consider the changes that occur during installation,equilibration of excess pore pressures, and loading of thepile (Fig. 1). As the pile is driven, the soil immediatelyadjacent to the pile will undergo severe distortion andchanges to the fabric, with a degree of remoulding and thepotential formation of residual shear planes (Bond & Jar-dine, 1991). The soil outside the immediate vicinity of thepile will be displaced outwards, with a strain field thatresembles spherical cavity expansion ahead of the pile tip,merging to cylindrical cavity expansion along the pile shaft.In clay with moderate to low yield stress ratio, which is themain focus here, the mean effective stress in the soiladjacent to the pile will gradually reduce during the cyclicshearing action as the pile is driven, and the interfacefriction angle, , will reduce to a residual value consistentwith the high rates of shearing and relatively low level ofeffective stress (Lehane & Jardine, 1994), both of whichmoderate the degree of damage.

At the end of installation, an excess pore pressure fieldwill exist around the pile, arising partly from changes inmean effective stress due to shearing of the soil, butprimarily from increases in total stress as the soil is forcedoutwards to accommodate the volume of the pile. As posi-tive excess pore pressures dissipate, pore water will flowradially away from the pile, and soil immediately around the

pile will undergo consolidation, with decrease in watercontent and increase in mean effective stress. Outside thiszone, which may extend to a few times the diameter of thepile, the radial strains are tensile during equilibration(Randolph & Wroth, 1979). The timescale of equilibrationwill be proportional to the square of the pile diameter, d,and inversely proportional to a coefficient of consolidation,ch, that reflects (a) primarily horizontal drainage, and (b)partial consolidation and partial unloading of the soil domain(Fahey & Lee Goh, 1995).

The final phase comprises loading of the pile, resisted byshaft friction along the pile shaft, and end-bearing pressureat the pile tip. The limiting shaft friction, s, will bedetermined by the local radial effective stress at failure, 9rf ,and an interface friction angle, , according to

s ¼ 9rf tan (1)

The magnitudes of and, particularly, 9rf will depend onthe very complex processes that occur during pile installa-tion and subsequent consolidation of the soil close to thepile. Partial ‘healing’ of any residual shear surfaces gener-ated during pile installation may occur, although it is alsolikely that will reduce to a residual value quite rapidly asslip occurs between pile and soil.

The dependence of pile shaft capacity on conditions in avery narrow zone in the immediate vicinity of the pile nodoubt contributes to the scatter in results from pile loadtests. Even on a single site, it is common for values of shaftfriction, normalised by the average shear strength, su, orvertical effective stress, 9v0, to vary quite widely, emphasis-ing the sensitivity to details of the installation process. Asan extreme example, in the database of pile shaft frictionmeasured in nine separate tests at Pentre, Chow (1997)quotes values for s/su or s= 9v0 that range by more than35% from the average values, with no apparent trend withdepth or other soil characteristic.

The complexity of the changes in stress and fabric in thesoil immediately adjacent to a driven pile has limitedanalytical treatment of the processes involved, and mostpractical design still relies on correlations (O’Neill, 2001).It is now accepted that the simple correlation parameters Æ(s/su) and (s= 9v0) are complex functions of soil para-meters—in particular the yield stress ratio and, more deba-tably, plasticity index, sensitivity and so forth. As theundrained strength ratio, su= 9v0, is also a function of theyield stress ratio, correlations for shaft friction that arefunctions of both shear strength and vertical effective stresswere introduced. Originally this was in the form of thelambda coefficient (º ¼ s= 2su þ 9v0ð Þ; Vijayvergiya &Focht, 1972), and more recently the American PetroleumInstitute (API, 1993) guidelines, based on Randolph &Murphy (1985), have proposed estimating the shaft frictionas the larger from the following two expressions:

s ¼ 0:5ffiffiffiffiffiffiffiffiffiffiffisu 9v0

ps ¼ 0:5s0:75

u 90:25v0 (2)

In all these correlations, there appears to be an effect ofpile length, or embedment ratio, L/d, with the averagenormalised shaft friction decreasing with increasing embed-ment ratio. This has been addressed by incorporating correc-tions for values of L/d above a certain threshold (Semple &Rigden, 1984), or by using a power law correlation such asthat proposed by Kolk & van der Velde (1996):

s ¼ 0:55s0:7u 9

0:3v0

40

L=d

0:2

(3)

(a) (b) (c)

Fig. 1. Three main phases during history of driven pile: (a)installation; (b) equilibration; (c) loading

848 RANDOLPH

It is clear, however, that correlations of the type given inequations (2) and (3) are entirely empirical, and coefficientsof variation mostly exceed 25%.

Length effectThe apparent decrease in normalised shaft friction with

increasing embedment ratio has been attributed to two mainmechanisms, associated respectively with the installation andthe loading phases. The latter has been addressed byRandolph (1983), who showed that, where the load transferresponse along the shaft exhibits strain-softening, progressivefailure of a pile could lead to a significant reduction incapacity. Fig. 2 shows the displacement profile down atypical pile, and the relative states along load transfer curvesat positions A, B and C. The design chart presented byRandolph (1983) showed that, for piles where the end-bearing capacity was much less than the shaft capacity, thereduction factor, Rf , defined as

Rf ¼Qactual

Qrigid

(4)

where Qactual is the actual pile capacity and Qrigid is the idealcapacity of a rigid pile (calculated as the integrated peakshaft friction), could be expressed as a function of (a) thedegree of strain softening, ¼ residual/peak, and (b) therelative compressibility of the pile.

The pile compressibility may be expressed conveniently asthe ratio of the elastic shortening of the pile, treated as afree-standing column subjected to a load equivalent to theideal shaft capacity, dL(peak)average, to the local displace-ment, ˜wres, required for degradation from peak to residualshaft friction. Thus the compressibility factor, K, is definedas

K ¼ dL2peak=(EA)pile

˜wres

(5)

where (EA)pile is the cross-sectional rigidity of the pile. Thereduction factor, Rf , will also be affected to some degree bythe soil stiffness (or local displacement to peak shaft fric-tion) and the precise shape of the load transfer curves.Therefore the actual reduction should be evaluated for anygiven case, by means of numerical analysis. However, to afirst approximation for preliminary design calculations, thereduction factor may be expressed as

Rf 1 (1 ) 1 1

2ffiffiffiffiK

p 2

for K . 0:25 (6)

with Rf taken as (approximately) unity for smaller valuesof K.

The strain-softening load transfer response arises fromreduction of the radial effective stress, 9r, at the pile shaftand, more significantly, the reduction in interface frictionangle, , to a residual value. Ring shear tests suggest thatthe softening factor, , may lie in the range 0.5–0.8 (com-pared with a recommendation of 0.7 in the AmericanPetroleum Institute guidelines: API, 1993), with the lowerrange possible for high-plasticity clays at moderate to largeeffective stress levels. Ring shear tests show that moststrain-softening occurs within relatively small displacements(10–30 mm), although it is possible that ˜wres for full-scalepiles might be somewhat larger. For modern offshore pilegeometries, where the L/d ratio rarely exceeds 60, typical Kvalues would not exceed 5–10, giving rise to reductionfactors in the range 0.65–0.9. Progressive failure can there-fore still lead to a significant reduction in the ideal capacity.

The other source of length effect is that associated withstress changes during installation. This has been quantifiedby means of measurements from instrumented piles, particu-larly the extensive research programme undertaken at Imper-ial College (Jardine and co-workers Bond, Lehane andChow). A summary of radial stress changes measured at theend of jacked pile installation from three different clay siteshas been presented by Lehane & Jardine (1994), as shownin Fig. 3; each value of radial stress has been normalised bythe local cone resistance, qc. The three sites compriseheavily overconsolidated London clay, a stiff glacial till(Cowden), and a lightly overconsolidated silty clay or clayeysilt (Bothkennar).

The measured radial stresses have been fitted by powerlaw curves of the form

A

B

C

τ

τ

τ

w

w

w

Displacement profile

Fig. 2. Progressive failure of pile in strain-softening soil

0 0.2 0.4 0.6 0.8 1.00

5

10

15

20

25

30

Dis

tanc

e fr

om p

ile ti

p, h

/d

Normalised radial stress, σri /qc

n 0.6 0.35 0.2

Decaycurves(d/h)n

Bothkennar

Cowden

London

Profile fromstrain path method

(Whittle, 1992)

Fig. 3. Radial stress changes during jacked pile installation inclay (after Lehane & Jardine, 1994)

SCIENCE AND EMPIRICISM IN PILE FOUNDATION DESIGN 849

ri

qc

/ d

h

n

(7)

where h is the distance from the pile tip (equivalent toL z, where z is the depth and L is the embedded pilelength). Deduced values of n range from 0.2 to 0.6, althoughreasons for the higher values of n have been discussed byCoop & Wroth (1990). Also shown in Fig. 3 is a predictionfrom Whittle (1992), using the strain path method with soilparameters based on the Bothkennar site. By contrast withthe measured data, the analytical prediction shows a varia-tion in normalised radial stress only in the lower fewdiameters, where it reduces from a value close to unity downto a value of 0.5, after which it remains constant.

The divergence between the ‘science’ of the strain pathmethod prediction and the ‘empiricism’ of the fit to fielddata suggests that further study of the processes involved isrequired, and we must explore what facet of soil behaviour,or of experimental technique, may have led to this differ-ence. For practical application, it will also be necessary todecide how to extrapolate from the field measurements,which are on full-displacement, closed-ended piles, to allowestimation of stress changes around partial-displacement,open-ended pipe piles. These aspects may be explored con-veniently through the analogy of cavity expansion.

Cavity expansion analogy for excess pore pressures andequilibration times

The analogy of cylindrical cavity expansion to model theinstallation of displacement piles formed the basis of earlyattempts to quantify stress changes due to pile installation(Kirby & Esrig, 1979; Randolph et al., 1979). Subsequently,the strain path method, pioneered by Baligh at MIT (Baligh,1985, 1986), provided more realistic and detailed predictionsfor the strains and stress changes in the immediate vicinityof the pile, particularly in respect of the zone of very highstress gradients ahead of, and behind, the pile tip and thetransition to quasi steady-state conditions (in terms of nor-malised stresses) along the pile shaft. Comparison of the twoapproaches shows that, ignoring the few diameters close tothe pile tip, the radial displacement fields are extremelysimilar apart from immediately adjacent to the pile shaft(within a zone of thickness about 10% of the pile radius, fora full-displacement pile).

Assuming that pile installation occurs under undrainedconditions, the radial displacement, r, for soil at finalradius, r, may be deduced as

r

req

¼ r

req

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

r

req

2

1

s(8)

where req is the pile radius for a closed-ended pile, and foran open-ended pile is the radius of an equivalent solid pilethat gives the same volume of displaced soil. For thin-walledpiles of wall thickness t the equivalent pile radius anddiameter are

req ffiffiffiffiffidt

p; deq 2

ffiffiffiffiffidt

p(9)

where it is assumed implicitly that the pile is installed in anunplugged manner, with the top of the internal soil plugremaining (approximately) level with the external soilsurface.

The relationship in equation (8) is shown in Fig. 4 for aclosed-ended pile (req ¼ rpile), and also an open-ended pilefor a d/t ratio of 40, which is a typical value for steel pipepiles. It is shown dashed in the region close to the (solid)pile, where the cavity expansion solution is no longerdeemed accurate. The location of the open-ended pile is

indicated, and the thicker line for r/req greater than 3.2 isapplicable to the open-ended pile. The right-hand axis givesthe radial displacement for the open-ended pile, normalisedby the actual pile radius, rpile, rather than the equivalentradius, req. Note that, for d/t of 40, the ‘area ratio’ (of pilewall to the gross cross-sectional area of the pile) is r ¼ 0.1( 4t/d), and the equivalent radius is 0.32 times the actualradius.

The assumed radial expansion for the open-ended pileshown schematically in Fig. 4 is such as to accommodatethe full wall thickness, essentially modelling the pile as aperfect sampling tube. Support for this assumption comes,experimentally, from the observation that, under the dynamicconditions of pile driving, the soil plug does indeed appearto progress up the pile, with only small variations in theposition of the top of the soil plug relative to the originalground surface.

The excess pore pressures generated by pile installationarise from two sources: changes in mean effective stressduring shearing and partial remoulding of the soil (whichwill give rise to positive excess pore pressures for lightlyoverconsolidated clay, and negative pore pressures for poten-tially dilatant, heavily overconsolidated clay), and increasesin mean total stress due to outward ‘expansion’ of the soilto accommodate the pile volume.

Simple cavity expansion theory, applied to an elastic,perfectly plastic soil with shear modulus G and undrainedshear strength su would give rise to an excess pore pressuredistribution of (Gibson & Anderson, 1961)

˜u

su

¼ lnrG

su

2 ln

r

rpile

> 0 (10)

Although this expression does not account for changes inmean effective stress as the soil is sheared and remoulded,these may be accounted for approximately for lightly over-consolidated clays by adjustment of the rigidity index, Ir ¼G/su. The main features of a logarithmic decay with radius,and typical values of maximum pore pressure adjacent tofull displacement piles of 4su to 6su (in lightly overconsoli-dated soils) agree well with results from the strain pathmethod (Baligh, 1986). An important feature of equation(10) is the term accounting for the area ratio, r, for open-ended piles, where the reduction in excess pore pressurecompared with a full-displacement (solid or closed-ended)pile is su ln(r).

The excess pore pressure fields around closed-ended andopen-ended piles (with d/t ¼ 40) based on cylindrical cavityexpansion, taking G/su ¼ 100, are shown in Fig. 5, togetherwith isochrones during dissipation. Baligh (1986) has com-

Open-endedClosed-ended

Open-ended pile (d/t 40)

Normalised final radius, r/req

321 5 1070

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

δr/r

eq

δr/r

pile

0

0.05

0.10

0.15

0.20

0.25

0.30

Fig. 4. Radial displacement field for closed- and open-endedpiles

850 RANDOLPH

mented that the excess pore pressures predicted by cavityexpansion may be overestimated, as a result of not followingthe correct strain path, but for lightly overconsolidated soilsthis will be offset by ignoring the excess pore pressures dueto shearing of the clay (with corresponding reduction inmean effective stress). The isochrones of excess pore pres-sure shown in Fig. 5 have been derived using the radialconsolidation solution of Randolph & Wroth (1979), withthe non-dimensional times expressed as

T ¼ ch t

d2; Teq ¼ ch t

d2eq

(11)

where t is the time and ch is an appropriate coefficient ofconsolidation for horizontal drainage. During the consolida-tion process, the outer soil (beyond 3–5 pile radii) under-goes swelling, while the inner soil consolidates: hence thecoefficient of consolidation must reflect this fact (Fahey &Lee Goh, 1995), and is most easily assessed through piezo-cone dissipation tests.

An interesting (and somewhat surprising) feature of Fig. 5is that isochrones for equal proportions of excess porepressure, ˜u/˜umax, occur at very similar non-dimensionaltimes, T and Teq for the two pile types, as remarked on byWhittle (1992). This is illustrated in Fig. 6, where thenormalised excess pore pressure is plotted against the twoalternative time factors, for closed and open-ended piles ofdifferent wall thickness ratios. For comparison, a dissipationcurve based on the strain path method, as presented byWhittle (1992), is also shown. Whittle’s original curve waspresented using a time factor expressed in terms of verticalpre-consolidation pressure, 9p, and horizontal permeability,kh, and the results in Fig. 6 have been scaled by assuming

ch 15 9p kh=ªw, where ªw is the unit weight of water. Thefigure shows that the dissipation curves for all the piles ofdifferent wall thickness fall in a narrow band, when ex-pressed in terms of Teq, based on the equivalent pilediameter, rather than the true diameter. Note also that thetimescale of consolidation is affected by the original magni-tude of the excess pore pressure ratio, ˜umax/su (and hencethe lateral extent of the pore pressure field), and the resultsshown in Fig. 6 are for an initial excess pore pressure ratioof 4.6, corresponding to G/su ¼ 100 for the cavity expansionanalogy.

In passing, it may be noted that, in their analysis ofdissipation around a piezocone, Teh & Houlsby (1991)proposed a ‘generalised’ time factor, T, given by

T ¼ 4ch t

d2ffiffiffiffiI r

p (12)

in order to bring together dissipation curves for different soilrigidity indices. This contrasts with the normalisation usingTeq in Fig. 6(b), where normalisation using deq can be shownto be equivalent to taking Teq as inversely proportional to Ir

(rather than to the square root of Ir). In fact, the optimalnormalisation depends on (a) the range of Ir values that needto be considered, and (b) whether the focus is on the earlydissipation response (up to T50) or the later response (timesgreater than T50). For the interpretation of piezocone tests,with a likely range for Ir between 50 and 500 and with thefocus on the early dissipation response, the normalisationproposed by Teh & Houlsby (1991) may be optimal. How-ever, dissipation around open-ended piles, where the focus ismore on the times for 50–90% dissipation, and the range ofrIr (in the light of equation (10)) that need to be consideredis very broad, the normalisation shown in Fig. 6(b) using Teq

appears more useful.Despite the approximations involved in the cylindrical

cavity analogue for pile installation, it appears thatthe general pattern of excess pore pressure, and the

T 0

T 0.01

T 0.8

T 2

T 10

T 0.1

T 0

Teq 0.03

Teq 0.2

Teq 0.8

Teq 2

Teq 10

Pile wall

1 3 5 7 90

1

2

3

4

5

∆u/

s u

1 3 5 7 90

1

2

3

4

5

∆u/

s u

0.9

0.7

0.5

0.3

∆u/∆umax

0.9

0.7

0.5

0.3

0.1

d/t 40, rpile 3·2req

∆u/∆umax 0.1

Normalised radius, r/req

Normalised radius, r/req

Fig. 5. Excess pore pressures generated by pile installation: (a)closed-ended pile; (b) open-ended pile

G/su 100

G/su 100

Closed

d/t 20

d/t 40

d/t 80

d/t 160

Closed

d/t 20

d/t 40

d/t 80

d/t 160

SPM: Whittle

T cht/d2

1010.10.010.0010

0·1

0·2

0·3

0·4

0·5

0·6

0·7

0·8

0·9

1·0

∆umax

∆u

∆umax

∆u

Teq cht/deq2

0

0·1

0·2

0·3

0·4

0·5

0·6

0·7

0·8

0·9

1·0

1010.10.010.001

Fig. 6. Dissipation of excess pore pressures at pile shaft

SCIENCE AND EMPIRICISM IN PILE FOUNDATION DESIGN 851

consolidation response, can be predicted reasonably for bothpiezocones and driven piles. Two examples illustrating thisare shown in Fig. 7. The first example is for a closed-endeddriven pile, which was subjected to dynamic tests at differ-ent times after driving (data kindly supplied by Mr AntonioAlvez, PhD student at COPPE, Federal University of Rio deJaneiro). Dissipation from a piezocone test, from whichappropriate values of Ir and ch were deduced, is comparedwith theoretical dissipation curves and also the measuredincrease of shaft resistance, Qs (where Qs,1 and Qs0 repre-sent long-term and initial shaft resistance respectively). Twodifferent dynamic pile–soil interaction models were used inanalysing the pile tests, a continuum model and the Smith(1960) model, both of which are described later. Althoughthe increase in shaft friction is not precisely proportional tothe decrease in excess pore pressure, because of stressrelaxation effects (discussed later) and changes in radialeffective stress during loading, it seems that consolidationtheory gives a sufficiently accurate estimate of the timescaleof increase in shaft resistance.

The second example, in Fig. 7(b), is from centrifugemodel tests on very thin-walled suction caissons, reported byCao et al. (2002). The prototype dimensions of the caissonare shown, with a d/t ratio of 80. However, as the caissonwas installed using suction, the outward soil movement maybe less than for a driven pile, as more soil is drawn insidethe caisson (Andersen & Jostad, 2002). Hence the operatived/t ratio may be nearer 160 than 80 in terms of outwardmovement of soil.

In both of these examples, the experimental data arematched reasonably well by the theoretical dissipationcurves. The first example, in Fig. 7 (a), also demonstrates

that simple scaling of piezocone dissipation times, by thesquare of the diameter ratio (equivalent pile diameter di-vided by piezocone diameter), should give a reasonableestimate of the consolidation times for a pile.

Referring to Fig. 6, two important observations may bemade. The first is that dissipation times for typical open-ended piles (d/t 40) and suction caissons (d/t 200) willbe respectively one and two orders of magnitude shorterthan for a closed-ended pile of the same diameter. Thesecond observation is that significant dissipation, with 20%reduction in pore pressure, occurs for Teq 0.1. For typicalvalues of consolidation coefficient in the range 3–30 m2/yr,this value of Teq corresponds to 0.5–5 days for an open-ended offshore pile 2 m in diameter, or 0.3–3 days for aclosed-ended onshore pile 0.5 m in diameter. These timesare longer than most installation times (except in the case ofequipment breakdown). However, for the 0.1 m diameterinstrumented pile used to obtain the data in Fig. 3, 20%pore pressure dissipation would occur in 0.3–3 h, comparedwith total jacking periods of 1–5 h (Lehane & Jardine,1994).

It appears, therefore, that partial pore pressure dissipationduring installation may account, at least in part, for the h/deffect deduced from radial stress measurements, and thedivergence between the trends in the data and theoreticalpredictions from the strain path method. Rapid initial porepressure dissipation may also account for the low valuesreported by Karlsrud (1999) in low-plasticity clays. Suchclays, with high silt content, are likely to show shorterconsolidation times, comparable with pile installation times,leading to greater damage to the soil (lower residual inter-face friction angles, because of the higher effective stresslevels during installation), less ‘set-up’ following installation,and thus lower shaft friction values than for higher plasticityclays.

Radial stress changes during installation, equalisation andloading

The pile shaft friction depends on the radial effectivestress acting around the shaft, according to equation (1), andthis may be estimated by considering the sequential changesduring pile installation, consolidation and loading. Measure-ments of radial total stress, ri (less the in situ pore pressure,u0) immediately after installation, and radial effective stress, 9rc, at the end of consolidation, both normalised by the insitu vertical effective stress, 9v0, are shown in Fig. 8 forvalues of h/d . 10. The data were assembled by Lehane(1992), and Fig. 8(a) shows his proposed trend lines (seealso Lehane et al., 1994).

During installation, the trend of radial total stress ratio, ri u0ð Þ= 9v0, increases in proportion to the yield stress

ratio to the power of about 0.4, from a value of 2 fornormally consolidated soil, to just under 10 at very highyield stress ratio. As Lehane (1992) observed, the gradient isapproximately parallel to the correlation of K0 with over-consolidation ratio proposed by Mayne & Kulhawy (1982),with a radial total stress ratio of 3–3.5 times K0.

Although the trend in the data on Fig. 8(a) is evident, thelogarithmic scales can lead to quite significant deviationfrom the mean line. At present, the only viable analyticalapproach for quantifying detailed stress changes during pileinstallation and consolidation appears to be the strain pathmethod, but simpler quasi-analytical approaches are neededfor routine design. Potential approaches, admittedly some-what speculative, are discussed here.

After installation, the radial total stress (less the in situpore pressure) may be expressed as

∆u∆umax

1·0

0·8

0·6

0·4

0·2

00·001 0·01 0·1 1 10 100

1·0

0·8

0·6

0·4

0·2

0

Theory: G/su 100Theory: G/su 50

Cone: Mid-faceCone: ShoulderPile: Continuum

Pile: Smith model

Theory: d/t 160

Theory: d/t 80

Test SAT06

Test SAT08

Normalised time, T cht/d2

65 mm24 m

5·2 m

1·0

0·8

0·6

0·4

0·2

00·001 0·01 0·1 1

Normalised time after installation , T cht/d20·0001

(a)

(b)

∆u∆umax

Qs Qs0

Qs Qs0

Fig. 7. Measured dissipation around closed and open-endedpiles: (a) piezocone dissipation and pile shaft resistance inhigh-plasticity clay (data provided by Mr Antonio Alvez); (b)pore pressure dissipation around thin-walled caisson (data fromCao et al., 2002)

852 RANDOLPH

ri u0 ¼ 9ri þ ˜umax ¼ ( 9ri p9i) þ p90 þ ˜ p (13)

where ˜umax is the maximum excess pore pressure and p90and p9i are respectively the original in situ mean effectivestress and the value just after pile installation (adjacent tothe pile shaft). The bracketed term has a relatively narrowrange [negative, owing to the slight unloading strains next tothe pile according to the strain path method (Baligh, 1986;Whittle, 1992), but limited in magnitude to the currentundrained shear strength, allowing for any remoulding thatmay have occurred as the pile is installed]. The in situ meaneffective stress, p90, may be estimated through K0, and theincrease in mean total stress, ˜p, required to accommodatethe pile should prove amenable to estimation through numer-ical analysis (strain path or cavity expansion methods).Estimating these quantities with any accuracy at present isnot straightforward, but the approach represents a possiblescientific way forward.

During equilibration, the excess pore pressure reduces tozero and the radial effective stress increases to a final valuedenoted by 9rc. The data for the final radial effective stressratio, 9rc= 9v0, in Fig. 8(a) have been correlated with linesthat lie nearly parallel to the trend of the installationstresses, but are offset by varying amounts, depending onthe sensitivity of the clay (Lehane, 1992; Jardine & Chow,1996). During consolidation there is some relaxation in totalstress (so that the final radial effective stress is less than theinitial radial total stress), and the data suggest that thedegree of relaxation is high for low yield stress ratios (alsohigh sensitivity) and reduces as the yield stress ratio in-creases (and sensitivity reduces).

Such a trend is consistent with radial consolidation mod-els, which show the outer soil (beyond 3–5 times the pileradius) swelling, while the inner core consolidates (Fahey &Lee Goh, 1995). It is the difference in stiffness of these twozones that gives rise to the relaxation in total radial stress(with no relaxation in classical solutions where the soil isassumed elastic and homogeneous). The relaxation gradientat any stage during consolidation is d 9r=du, and it may beargued that this quantity will become progressively less thanunity the softer the inner soil is relative to the outer(swelling material), and will therefore be a function of therelative magnitude of the current radial effective stress, 9r,and the preconsolidation or yield stress, 9vc. This effect maybe captured by a function such as

d 9rdu

¼ ºe( 9r 9ri)= 9vc (14)

where º and are adjustable parameters.Integrating this expression over the change in excess pore

pressure from ˜umax down to zero, the final radial effectivestress is given by

9rc 9v0

¼ 9ri 9v0

þ R

ln 1 þ º

R

˜umax

9v0

(15)

where R is the yield stress ratio, 9vc= 9v0. This expression isplotted in Fig. 8(b), adjusting º to unity and to a value of5, in order to give a reasonable fit to the data (identical tothe data in Fig. 8(a)). Although this approach is speculative,and significant further work is needed before it might beuseful in design, the concept of a relaxation gradient thatvaries during consolidation is consistent with physical argu-ments of the conditions around the pile, and also with fieldmeasurements by Lehane (1992), which indicate a gradualreduction of jd 9r=duj during consolidation. One importantconsequence is that the net relaxation ratio, ( ri u0)= 9rc,will be higher (for a given soil) for an open-ended pile thanfor a closed-ended pile.

The final phase of the pile’s history to consider is theloading phase. By the end of consolidation, the radial effec-tive stress will have become the largest of the three normalstresses (vertical, radial and circumferential) close to thepile. During loading of the pile, a reduction in the radialstress is therefore expected. Lehane (1992) and the designapproach proposed by Jardine & Chow (1996) suggest thatthe reduction may be taken as about 20%, independent ofthe yield stress ratio, so that 9rf in equation (1) is then0:8 9rc.

Example design calculations: new horizonsThe offshore industry continues to face new challenges as

it moves into deeper waters and new regions of the world.Currently, one of the most active offshore areas is off thewest coast of Africa, where very high-plasticity clays havebeen encountered in water depths of 1000 m. Typical ‘gener-ic’ soil properties, based on data from a number of sites, aresummarised in Table 1.

The combination of high plasticity index with high fric-tion angles measured in triaxial compression and simpleshear is unusual and, in a similar fashion to Mexico Cityclay, lies well outside common correlations of friction anglewith PI (Mesri et al., 1975). Unlike Mexico City clay,however, interface friction angles are significantly lower,particularly at residual. This characteristic poses a particularchallenge in estimating the shaft capacity of driven pipepiles and thin-walled suction caissons in these clays, astraditional approaches based on correlations with su and 9v0

will diverge from more fundamental approaches based onequation (1).

10After installation

(σir u0)/σv′0

Rad

ial s

tres

s co

effic

ient

s(σ

ir

u0)

/σ′ v0

and

σ′ rc/

σ′ v0

1

0·1

Increasingsensitivity

After consolidationσr′c/σv′0

1 10 100

Yield stress ratio, R

After installation(σir uo)/σv′0

After consolidationσr′c/σv′0

Rad

ial s

tres

s co

effic

ient

s(σ

ir

u0)

/σ′ v0

and

σ′ rc/

σ′ v0

10

1

0·11 10 100

Yield stress ratio, R

Equation (15)

(a)

(b)

Fig. 8. Radial stress coefficients after installation and consolida-tion (data from Chow, 1997): (a) relaxation ratio as function ofsoil sensitivity; (b) relaxation ratio derived from function ofcurrent yield stress ratio

SCIENCE AND EMPIRICISM IN PILE FOUNDATION DESIGN 853

To illustrate the ideas discussed earlier, two differentgeometries of offshore piles will be considered:

(a) a conventional pipe pile, 2 m in diameter and with50 mm wall thickness (d/t ¼ 40, r ¼ 0.1) embedded100 m (L/d ¼ 50)

(b) a suction caisson, 6 m in diameter with wall thicknessof 30 mm (d/t ¼ 200, effective area ratio, allowingfor suction installation, of r ¼ 0.01) embedded 20 m(L/d ¼ 3.3).

The shaft capacity of these piles will be estimated using theapproach described here (equations (10), (13) and (15)) andalso the method of Jardine & Chow (1996) (referred to hereas the MTD method, as it is known in the offshore industry).The MTD method for piles in clay is based on the empiricalcorrelations of Lehane (1992) and Lehane et al. (1994). Incontrast to the MTD method, the effect of h/d is ignored inthe present approach apart from for h/t , 10, where thenormalised radial total stress is assumed to increase gradu-ally by a maximum factor of 2 at the pile tip (as suggestedby the strain path method results shown in Fig. 3). Simplis-tically, the radial effective stress just after installation ( 9ri)has been estimated using equation (1), assuming that theshaft friction during installation is equal to the remouldedshear strength, and the maximum excess pore pressure gen-erated by a solid pile has been taken as 4.6su.

The profiles of peak shaft friction obtained from thesetwo approaches are compared with that estimated using theAPI guidelines (API, 1993) in Fig. 9 for each pile geometry.In Fig. 9(a), the strong h/d effect from the MTD method isevident, with lower shaft friction over most of the pile shaft,apart from close to the tip. For the particular combination ofsoil properties, it turns out that the approach described heregives a shaft friction profile that is remarkably similar tothat obtained from API (1993), although this is something ofa coincidence and will be affected by the interface frictionangle, . Average values of shaft friction from the differentmethods are quite close, with the MTD method about 10%lower than the other two methods.

For the caisson (Fig. 9(b)), there is a much greaterdivergence of shaft friction profiles. The API (1993) profileis identical to that for the driven pile, whereas the approachsuggested here gives lower shaft friction, largely because ofthe low area ratio of the caisson and hence lower excesspore pressures generated during installation and lower finalradial effective stresses. The MTD method was not intendedto apply to piles with such low L/d or high d/t ratios, whichfall well outside the database used to calibrate the method.It is an instructive comparison, however, reminding us thatextrapolation of any design method must be carried out withcare, particularly where the method is based on empiricalcorrelations.

Whereas the suction caisson may be considered as effec-tively rigid, in terms of strain-softening effects during axialloading, the driven pile is relatively flexible. The calculatedload–displacement responses, assuming strain softening by40% ( decreasing from 208 to 128) over relative pile–soil

slip of 50 mm, are shown in Fig. 10. The reduction factordue to strain-softening is around 10%, which is somewhatless than the value of 15% estimated from equations (5) and(6), mainly because of the triangular distribution of shaftfriction arising from the linearly increasing shear strengthwith depth.

SummaryThe ‘science’ in estimating driven pile capacity in clay

provides the framework within which the different phases ofthe installation, consolidation and loading history of the pileare considered. It also extends to different analytical ap-proaches, such as the strain path method, and cavity expan-sion, which allow quantification of certain aspects of eachprocess. Magnitudes of total stress increase, quantification ofthe differences between full and partial displacement piles,and estimation of the timescale for consolidation may all betreated analytically. However, design calculations still rely on

Table 1. Clay properties offshore West Africa

Parameter Typical values

Shear strength, su: kPa 1.5z (with z the depth in m)Effective unit weight, ª9: kN/m3 3.5Yield stress ratio, R 1.8Sensitivity, St 4Plasticity index, PI: % 100Friction angle, 9: degrees 35Interface friction angle, : degrees 20 (residual value 12)

Shaft friction: kPa

160120804000

20

40

60

80

100

120(a)

Shear strength profile

MTDapproach

Presentapproach API (1993)

Dep

th: m

Dep

th: m

(b)

Shaft friction: kPa

3025201510500

5

10

15

20

25

Presentapproach

Shear strength

MTDapproach

API (1993)

Fig. 9. Profiles of peak shaft friction for offshore piles: (a)driven pipe pile (L/d 50, rr 0.1); (b) suction caisson (L/d3.3, rr 0.01)

854 RANDOLPH

empirical correlations in order to quantify those aspects thatare dominated by the complexities of soil response, such asreduction in effective stresses and degree of remouldingduring pile installation, relaxation of radial total stressduring consolidation, and reduction in radial effective stressduring loading.

There appears to be divergence between analysis and fieldmeasurements in respect of the h/d effect during pile instal-lation, although partial consolidation appears partly respon-sible. Resolution of this is important, and requires carefulreview of what fundamental mechanisms might lead to anh/d effect. An improved model to quantify stress relaxationduring consolidation is also needed, perhaps through numer-ical parametric studies, as this is an area where considerablescatter in the database exists. The concept of a relaxationgradient that changes as consolidation proceeds, as therelative stiffness of the inner and outer soil zones evolves,has been proposed as a possible way forward.

The challenge of providing anchors in deepwater, usingsuction-installed caissons, requires extension of our currentdesign approaches to low aspect ratio (L/d , 6) and lowarea ratio (r 0.01) pile geometries. A rational scientificbasis is essential for this.

AXIAL CAPACITY OF DRIVEN PILES IN SANDOverview

Over the last decade there have been two major advancesin design approaches for driven piles in sand. The first ofthese is the capturing, through instrumented pile tests, of thegradual degradation of shaft friction at any given depth asthe pile is driven progressively deeper (Lehane et al., 1993),and the second is the linking of key parameters such as baseresistance and maximum shaft friction to the cone resistance,qc, which has evolved from the early correlations ofBustamante & Gianeselli (1982). Both of these advances areempirical in nature, but they embody principles that could,in due course, be quantified more scientifically.

Historically, pile design in sand has been based on simplelinear relationships for both shaft friction and base resis-tance, but with limiting values at some ‘critical depth’expressed either in absolute terms or normalised by the pilediameter (Vesic, 1967, 1970; Coyle & Castello, 1981). Therationale behind this approach has been challenged(Kulhawy, 1984), and alternative explanations offered for theexperimental finding that increasing lengths of piles driveninto sand do not yield proportional increases in capacity.

For base resistance, the influence of decreasing frictionangle with increasing stress level, and the non-linear rela-

tionship between stiffness and stress, both contribute to adecreasing gradient of base resistance with depth (Randolphet al., 1994). For shaft friction, although equation (1) stillprovides the physical basis, the normal effective stress, 9rf ,at any given depth has been found to degrade as the pile isinstalled, owing to gradual densification of the surroundingmaterial. These components of the axial capacity of drivenpiles in sand, and the necessary adjustments for open-endedpiles, are explored here in the context of recent designrecommendations (Jardine & Chow, 1996).

Base resistanceAlthough it is natural to correlate the end-bearing resis-

tance of a pile with the cone resistance, consideration mustbe given to the displacement needed to mobilise a givenproportion of cone resistance. Fleming (1992) proposed ahyperbolic relationship for bored piles, relating the end-bearing pressure, qb, and the base displacement, wb, giving anormalised end-bearing resistance, qb/qc, expressed as

qb

qc

wb=d

wb=d þ 0:5qc=Eb

(16)

where Eb is the Young’s modulus of the soil below the pilebase.

For a bored pile, with initially zero base pressure at zerodisplacement, this relationship will lead to end-bearing pres-sures mobilised at a base displacement of 0.1d of around15–20% of qc (Lee & Salgado, 1999). However, for drivenand jacked piles, significant residual pressures are locked inat the pile base during installation (equilibrated by negativeshear stresses along the pile shaft, as if the pile were loadedin tension). This will lead to a stiffer overall pile response incompression, and significantly higher end-bearing stressesmobilised at small displacements.

The magnitude of residual base stress will depend on therelative magnitudes of shaft and base capacity, as well as onthe method of installation. For jacked piles the residual basestress can be as high as 70–80% of the lesser of shaft or(ultimate) base capacity (Poulos, 1987). For driven closed-ended piles the residual stress will be lower, but may still beas high as 75% of the base capacity (Maiorano et al., 1996).The lowest residual base stress is likely to be for open-endedpiles, unless they become fully plugged during driving.

Equation (16) can be generalised to allow for a residualpressure, qb0, locked in below the pile base at the start ofloading, to give

qb

qc

wb=d þ 0:5qb0=Eb

wb=d þ 0:5qc=Eb

(17)

The resulting end-bearing responses are illustrated in Fig. 11for Eb/qc ¼ 1.25 (lower set of curves for each value ofqbo/qc) and Eb/qc ¼ 5 (upper set of curves for qb0/qc ¼ 0.3and 0.7). This range of Eb/qc reflects conservative valuessuggested for bored and driven piles (Poulos, 1989; Fleming,1992).

The exact form of the end-bearing response is of coursedebatable. However, the main principles illustrated in Fig. 11are as follows:

(a) Steady-state conditions are reached after large displace-ment (4–10 diameters for zero residual stress),with theend-bearing resistance of a pile approaching the coneresistance, after appropriate averaging of the latterquantity to reflect the larger size of the pile.

(b) At limited displacements, such as 10% of the pilediameter as is often taken as the practical definition of‘ultimate’, the end-bearing resistance will be signifi-cantly lower than the cone resistance, and will also

0·100·080·060·040·0200

5000

10000

15000

20000

25000

30000

35000

40000P

ile h

ead

load

: kN

MTD method

Pile head displacement: m

Presentapproach

Idealcapacities

Fig. 10. Load–displacement response of driven pipe pile

SCIENCE AND EMPIRICISM IN PILE FOUNDATION DESIGN 855

depend strongly on any residual stresses locked in atthe pile base at zero displacement.

The large displacements necessary to mobilise a true ‘ulti-mate’ end-bearing capacity lead to a form of scale effect incomparing pile end-bearing and cone resistance, as the(much greater diameter) pile will react more slowly tochanges in stratigraphy than the cone. To overcome this, it isessential to average the cone resistance over a number ofdiameters above and below the pile base level. Distancesrecommended range up to 8 diameters above the pile base(to allow for the gradual development of residual stresses),and 2 diameters below the pile base, with a weightingtowards the minimum envelope of the cone profile (Fleming& Thorburn, 1983). In practice, averaging over a shorterlength, between 1 and 2 diameters above and below the pilebase, is often acceptable, provided there are no strongstratigraphic changes within the wider range.

Chow (1997) assembled a database of high-quality pileload tests, and her data for the end-bearing resistance ofclosed-ended piles driven into sand are shown in Fig. 12.The values of cone resistance have been obtained by aver-aging over 1.5d relative to the pile base, and the ultimateend-bearing resistance, qbu, is that mobilised at a pile basedisplacement of 0.1d. The design curve proposed by Jardine& Chow (1996) is indicated, and is expressed as

qbu

qc

¼ 1 0:5 logd

dcone

> 0:13 (18)

At first glance, the design curve appears a reasonable fit tothe data, in spite of some scatter. However, the data forsmall pile diameters are dominated by jacked piles, wherethe full cone resistance (appropriately averaged according tothe pile diameter) would be mobilised at each stroke, andhigh residual stresses (or at least a high reloading stiffness)will be retained. An annotated version of the database isshown in Fig. 13(a), with jacked piles indicated and also avibro-driven pile, where the normalised end-bearing capacityfalls below the other data. The driven pile result from theAkasaka (AK: BCP Committee, 1971: see legend in Fig. 12)pile tests plots above the jacked pile data, but the reportedload–displacement plot (see their figure 9) is anomalous,with a base resistance that suddenly falls after a displace-ment of one pile diameter, with a corresponding jump in theshaft friction (the total load remaining largely unchanged).Correction for that anomaly would result in a normalisedend-bearing (qbu/qc) of 0.4 for the driven pile.

Load cells or strain gauges in instrumented driven pilestend to undergo zero shifts during installation, due tochanges in fabrication strains within the pile caused by thehigh dynamic stresses. It is therefore usually necessary tozero strain gauges prior to testing the pile statically, and torely on alternative means to estimate any residual base loads.A common approach is to assume equal shaft capacity intension and compression, although this will tend to over-estimate residual base loads (see later). Correction forresidual loads varies among the pile tests, but examples ofhow this may change the deduced end-bearing capacity areshown in Fig. 13(a) for the Baghdad (BG: Altaee et al.,1992, 1993) and Hunters Point (HP: Briaud et al., 1989) piletests (the arrows linking uncorrected and corrected data).From the current database, however, and notwithstandingsome inconsistency in respect of allowing for residual baseloads, a design end-bearing capacity of around 0.4qc, inde-pendent of diameter, appears reasonable. This may turn outto be conservative in cases where high residual stresses canbe justified, for example for jacked piles or where thetransient base pressures mobilised during pile driving can beshown to be a high proportion of qc.

The vagaries of data interpretation, and the inevitablesubjectivity involved, are well illustrated in Fig. 13(b), whichshows a recent reinterpretation of the same database byWhite (2003). The reinterpretation includes:

q b/q

c

0 1 2 3 40

0.2

0.4

0.6

0.8

1·0

Normalised displacement, wb/d

0.9

0.6

0.3

qbo/qc 0.7

qbo/qc 0.3

00 0.1 0.2

Eb/qc 5

Eb/qc 1.25

Fig. 11. Development of end-bearing resistance

0 0.2 0.4 0.6 0.8 1.0

Pile diameter: m

Key to individual pile testsfrom Chow (1997)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

q bu/

q c

Design curve fromJardine & Chow (1986)

KA (Franki)

KA (Cone)

D (C,d)

DK (C,j)

HP (C,d)

AK (C,j)

S (C,d)

E (O,d)

A (C,d)

G (C,d)

LB (C,j)

HT (C,d)

AK (C,d)

BG (C,d)

Fig. 12. Normalised end-bearing capacities for closed-ended piles from Chow (1997)

856 RANDOLPH

(a) adjustment of design cone resistance values to allowfor partial penetration into a dense sand layer [Kallo(KA: de Beer et al., 1979); Lower Arrow Lake(E: McCammon & Golder, 1970)]

(b) correction to include residual base load [Drammen (D:Gregersen et al., 1973)]

(c) reservations on the quality of the data, such as

estimation of qc from SPT data, and insufficient basedisplacements to estimate qbu.

In the light of these caveats and adjustments, evidence for asignificant diameter effect is unconvincing, provided appro-priate averaging of the cone resistance is undertaken and‘ultimate’ base capacity is assessed in terms of relativedisplacement (proportion of pile diameter) not absolute dis-placement. Particular care should be taken in strongly strati-fied soils, for example where piles are driven through weakmaterial and penetrate only 1 or 2 diameters into a densesand layer. For such cases the design cone resistance needsto be weighted to reflect the overlying weaker material(Meyerhof, 1976; Meyerhof & Valsangkar, 1977).

Corresponding end-bearing data for open-ended piles areshown in Fig. 14. There are many fewer data points, andthey are very sparse for diameters in excess of 1 m, which isthe main area of interest for offshore applications. Againthere appears to be a decreasing trend of normalised end-bearing resistance with increasing pile diameter. However,scrutiny of the data reveals that:

(a) the data for piles of diameter 1 m (HO: Kusakabe etal., 1989) and 1.2 m (K: Ishihara et al., 1977) areprojected from tests where the base movement was only0.5% of the pile diameter

(b) the data point for the pile of 2 m diameter (T: Shioi etal., 1992) has been normalised using a cone resistanceof 35 MPa, whereas the pile tip was very close to thetop of a much softer stratum (see Fig. 15).

Certainly for design a much more conservative value of qc

would be adopted in this case, possibly as low as 10 MPa.In order to arrive at an acceptable design approach for

large diameter open-ended piles, it is necessary to considerthe mechanics of the soil plug (Fig. 16). If the soil plugstarts to slip relative to the pile, then the shear stressesaround the plug, which are themselves a function of theaverage vertical effective stress in the plug, will lead to anexponential growth in the vertical stress within the soil plug.It may be shown (Randolph et al., 1991) that the availableend-bearing resistance may be expressed as

qbplug

9vbase

e4hp=di (19)

where hp is the height of the soil plug, di is the internal pilediameter, and 9vbase is the ambient vertical effective stress atthe base of the plug (taken as ª9hp). As for the externalshaft friction, the ratio ¼ s= 9v may be expressed asK tan , where is the interface friction angle. Although the

qbu

/qc

0 0.2 0.4 0.6 0.8 1.0Pile diameter: m

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

qbu

/qc

0 0.2 0.4 0.6 0.8 1.0Pile diameter: m

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.9

1.0

Jacked piles

Vibro-driven

Suggested design value(diameter independent)

Residual loadcorrections

Driven pile (suspect data point)

qc reassessed (shallowpenetration of sand layer)

Possible claylayer at base

qc estimated from SPT

Zero residualload observed

(a)

(b)

0.8

Fig. 13. Commentary on database of pile end-bearing capacity:(a) annotated database from Chow (1997); (b) alternativeinterpretation of data by White (2003)

0 0.5 1.0 1.5 2.0Pile diameter: m

0

0.1

0.2

0.3

0.4

0.5

0.6

q bu/

q c

Limited basemovement (0.5 % of d)

Overestimated qc

DK (O,d)

DK (O.d)

CH (O,d)

T (O,d)

G (O,d)

HO (O,d)

K (O,d)

CR (O,d)

Key to individual pile testsfrom Chow (1997)

Fig. 14. Normalised end-bearing capacities for open-ended piles from Chow (1997)

SCIENCE AND EMPIRICISM IN PILE FOUNDATION DESIGN 857

value of K may be as high as unity close to the pile tip(Paik & Lee, 1993; De Nicola & Randolph, 1997; Lehane &Gavin, 2001), it has been found to decay rapidly along thelength of the soil plug. Minimum values of may bededuced from Mohr’s circle considerations, and lie in therange 0.15–0.2 for typical soil friction angles (Randolph etal., 1991).

From equation (19), the available end-bearing pressurerises rapidly with the plug length, so that lengths of only afew diameters can provide sufficient internal resistance toensure ‘plugged’ failure mode under static loading, regard-less of the pile diameter. This contrasts with recommenda-tions of Hight et al. (1996) and Jardine & Chow (1996),where driven piles of diameter greater than2(Dr 0.3) metres (with Dr the relative density, expressedas a fraction) are assumed not to plug.

Part of this divergence of opinion revolves around thesemantics of ‘plugged’ or ‘unplugged’. Here, the term‘plugged’ is restricted to the situation where, during staticloading, the top of the soil plug does not slip relative to the

pile wall. Near the tip, however, relative slip will occurduring loading, owing to compression of the soil plug.

Lehane & Randolph (2002) have considered the separateresponse of the soil plug, soil below the pile base, and pile–soil interaction around the annular tip, in order to establishminimum values for the end-bearing capacity of open-endedpiles in sand. Their recommendations are shown in Fig. 17,based on conservative assumptions that ignore any increasein the stress ratio, K, near the pile tip or densification of thesoil within the plug or beneath the pile base (together withany associated residual stress systems). Even moderate re-laxation of these assumptions suggests that a design end-bearing capacity of qb/qc 0.2 is reasonable, and such avalue is broadly consistent with the database in Fig. 17,taking account of limitations in the data plotted for the pilesof diameter greater than 1 m. Results of centrifuge modeltests in dense sand also support this as a lower bound designvalue (Bruno, 1999; De Nicola & Randolph, 1999).

Shaft frictionSince the pioneering work of Vesic in the 1960s (Vesic,

1967, 1970), it has been realised that in sand and other soilsof high permeability the magnitude of shaft friction at agiven depth can reduce as the pile is driven further, with thenet effect that the average friction along the pile shaft canreach a limit and even reduce as the pile embedment in-creases. However, this effect has only recently been quanti-fied, through the carefully instrumented pile tests undertakenby the research group at Imperial College. The phenomenonof ‘friction degradation’ is illustrated in Fig. 18 (Lehaneet al., 1993), with profiles of shaft friction measured in thethree instrument clusters at different distances (h) from thetip of a pile 6 m long and 0.1 m in diameter, as it is jackedinto the ground. For comparison, the cone profile is plottedon the same scale, but with qc factored down by 100. Theshaft friction measured at h/d ¼ 4, in particular, follows theshape of the qc profile closely, allowing for differences incone and pile diameter. Comparison of the profiles from theinstrument clusters at h/d ¼ 4 and h/d ¼ 25 shows that thefriction measured at the latter position is generally less than50% of that measured close to the pile tip.

The physical basis for friction degradation is the gradualdensification of soil adjacent to the pile shaft under thecyclic shearing action of installation. This process is en-hanced by the presence of crushed particles from the pas-sage of the pile tip, which gradually migrate through thematrix of uncrushed material (White & Bolton, 2002). Thefar-field soil acts as a spring, with stiffness proportional toG/d (where G is the soil shear modulus), so that any

0 10 20 30 40 50 60TP

24.4m

TP55.0m

Pile tip

35 MPa assumedbut could be 10 MPa

40

30

20

10

Dep

th b

elow

sea

bed:

m

Fig. 15 Cone resistance profile for Tokyo Bay pile test (afterShioi et al., 1992)

Soil plugPile wall

τ σv′

σv′

σv′ dσv′

γ′

Fig. 16. Equilibrium of soil element within soil plug

Driven piles

wb/d 0·2

wb/d 0·15

wb/d 0·1

0·4

0·3

0·2

0·1

0

q bu/

q c

Bored pile (Lee & Salgado, 1999)

0 0·2 0·4 0·6 0·8 1·0

Relative density, Dr

Fig. 17. Normalised end-bearing capacity for open-ended piles(after Lehane & Randolph, 2002)

858 RANDOLPH

densification close to the pile results in reduced radial effec-tive stress. The operative value of G will be high, as the soilis heavily overconsolidated having moved through the zoneof high stress close to the pile tip during installation and isbeing unloaded.

The incremental volume change, and hence reduction inradial effective stress, is likely to depend on the currentstress level, with greater changes at higher stress levels. Thissuggests an exponential variation of radial stress along thepile shaft of the form (Randolph et al., 1994)

K ¼ 9r 9v0

¼ s

9v0 tan cv

¼ Kmin þ (Kmax Kmin)eh=d

(20)

where Kmax may be taken as a proportion of the normalisedcone resistance, typically 1–2% of qc= 9v0, and Kmin lies inthe range 0.2–0.4, giving a minimum friction ratio, s= 9v0,of 0.1–0.25 (Toolan et al., 1990). The coefficient may betaken in the region of 0.05 for typical pile diameters,although there are some indications that the value decreasesas the pile diameter increases and vice versa. Indeed, muchhigher values of are needed to match data from centrifugemodel tests (Bruno, 1999), although scaling problems relatedto the spring stiffness of the surrounding soil may occur forcentrifuge modelling of piles in sand (Fioravante et al.,1999; Fioravante, 2002).

Other key variables that affect the rate of degradationinclude

(a) the unloading modulus of the soil (probably close tothe small strain value, G0), with higher unloadingmodulus leading to more rapid degradation

(b) the number of ‘shearing cycles’ per diameter ofadvance (or blow count for driven piles).

Assuming that cyclic stress reversal is the major trigger forcompressive volumetric strain in the shearing zone, the rateof degradation should be very low for continuous jacking(De Jong & Frost, 2002), and maximum for driven pileswith high blow count. For intermittently jacked piles such asthe Imperial College instrumented pile tests, the rate ofdegradation would be intermediate, lower than most drivenpiles, although this may be compensated for by scale effectsassociated with the small diameter of the pile. As noted byFioravante (2002), the unloading stiffness of the surrounding

soil annulus is proportional to G/d, and hence the reductionin radial stress resulting from any contraction of soil in theinterface layer will be higher for smaller-diameter piles.

The MTD method, derived from the Imperial College fieldstudies and database of high-quality pile tests, expresses theshaft friction for driven piles in sand as

s ¼qc

45

9v0

pa

0:13d

h

0:38

þ ˜ 9rd

" #tan cv (21)

where pa is atmospheric pressure (100 kPa) and ˜ 9rd is a(relatively small) stress increase due to dilation during load-ing (Jardine & Chow, 1996). The minimum h/d is takenconservatively as 4 (in the absence of data at lower h/dratios), and for open-ended piles the diameter, d, is replacedby the equivalent diameter, deq. The method adopts a powerlaw degradation, rather than an exponential decay, but thisleads to similar shapes of shaft friction profiles.

A comparison between the MTD and the present method,using equation (20) with Kmax ¼ 0:01qc= 9v0, is provided inFig. 19, for a 1 m diameter open-ended pile driven 40 m intosand. The main difference is close to the pile tip, where theMTD method yields identical values of shaft friction foropen- and closed-ended piles (for h/deq , 4). The presentmethod gives different maximum values of shaft friction,dictated by Kmax, and it is suggested that Kmax is increasedto 0:015qc= 9v0 for closed-ended piles in view of the highernormalised end-bearing resistance. The shaft friction ratiobetween open and closed-ended piles implied by the twomethods is quite similar, with an average ratio of around0.7, although the MTD method gives a ratio that decreasesfrom unity at the pile tip down to (deq/d)0:38 (typically0.65) whereas the present method gives an increasing ratioas K approaches Kmin for both geometries. The average ratioof 0.7 may be compared with the API (1993) designrecommendation of 0.8, but also with recent experimentalstudies that show a much lower ratio of just under 0.5 (Paiket al., 2003).

Shaft friction in tension and compressionThe tensile capacity of piles in sand has been found to be

less than the shaft capacity measured in compression, andmost design guidelines include a reduction of 10–30% toallow for this (API, 1993). Two factors were identified byDe Nicola & Randolph (1993) that contributed to lowertensile shaft friction: the first was a reduction in effectivestress levels adjacent to the pile compared with loading incompression (even for a rigid pile), and the second was thePoisson’s ratio reduction in diameter (and consequential

Local shear stress: kPa

0 10 20 30 40 50 600

1

2

3

4

5

6

Dep

th o

f ins

trum

ent:

m

Cone resistanceqc/100

h/d 25

h/d 14

h/d 4

Fig. 18. Measured profiles of shaft friction (Lehane et al., 1993)

Shaft friction: kPa

0 100 200 300 4000

5

10

15

20

25

30

35

40

45

Dep

th: m

MTD method

Present method:µ 0·03, 0·05 and 0·07

Open-ended pile: L 40 m, d 1 m, ρ 0.1

qc 5 1z MPa, γ′ 11 kN/m3, δcv 28°

Kmax 0·01qc/σ′v0, Kmin 0·3

Fig. 19. Example profiles of shaft friction for driven pile in sand

SCIENCE AND EMPIRICISM IN PILE FOUNDATION DESIGN 859

reduction in radial effective stress). These two effects werequantified for piles fully embedded in sand, by the expres-sion

(Qs)tens

(Qs)comp

1 0:2 log10

100

L=d

(1 8þ 252) (22)

where Qs is the shaft capacity and ¼ p(L/d)(Gave/Ep)tan ,with Gave, Ep and p being respectively the average soilshear modulus, Young’s modulus of an equivalent solid pileand Poisson’s ratio for the pile.

The two factors that contribute to reduced tensile capacitytend to compensate as the pile aspect ratio increases, withthe average change in effective stress level decreasing andthe effect of Poisson’s ratio contraction increasing. This isshown in Fig. 20 where, for a typical modulus ratio ofEp/Gave ¼ 400, the shaft capacity ratio is 0.8 for a rangeof L/d. Even for quite wide extremes of Ep/Gave, the shaftcapacity ratio remains within 0.7–0.85.

Although other effects, such as local stress changes due todilation, will influence the shaft capacity ratio, the expres-sion in equation (22) provides a reasonable design basis forassessing the reduced shaft capacity for loading in tension,compared with that for loading in compression.

Euripides pile testA major field investigation of driven pile capacity in

dense sand, the Euripides pile test, was undertaken in the1990s, funded by a number of companies operating in theoffshore area and managed by Fugro BV. The data are nowin the public domain and are in the process of beingpublished: a brief comparison of measured and predictedpile resistance is presented here. The pile test comprised aheavily instrumented pile, 0.76 m in diameter and 35 mmwall thickness, driven open-ended into very dense sand.Cone resistance profiles are shown in Fig. 21, and it may beseen that, in spite of some variability in the cone resistancebelow 30 m, the average qc rises to between 60 and 70 MPa.The cone resistance over the upper 22 m is very low, andmost of the test pile capacity was generated below thatlevel.

The pile resistance mobilised in the dense sand has beenestimated using the MTD method, and also that presentedhere, adopting the design qc profile shown in Fig. 21. TheMTD method gives an end-bearing ratio of qb/qc ¼ 0.17,whereas a ratio of 0.2 has been adopted for the alternativemethod. Values of Kmax, Kmin and have been taken as0:01qc= 9v0, 0.3 and 0.05 respectively, whereas the value ofcv measured in ring shear tests was 308.

The resulting profiles of pile resistance are shown in Fig.22, compared with the measured loads at a pile displacementof 0.1d. The test pile was initially driven to depths of30.5 m, 38.7 m and 47.0 m, with compression and tensiontests being conducted at each level. The pile was thenextracted and driven without pause to 46.7 m penetration,after repairing some of the damaged instrumentation. Over-all, the agreement between either prediction method and thetest data appears reasonable over the range where theinstrumentation survived the driving process. However,the measurements show very low friction mobilised over thedepth range 22–30 m, and a need is indicated for somerefinement of the average cone resistance (possibly averagingover a greater distance above pile base level). At greaterdepths, the MTD and exponential decay methods give shaftfriction values (or gradients of axial load) that lie respec-tively slightly above and below the measurements. Predic-tions of end-bearing resistance appear reasonable, with closeagreement between the MTD method and measured baseload at the intermediate depth.

The average shaft friction ratio between tensile and com-pressive load tests ranged between 0.6 and 0.9, but with noclear pattern among the four separate load test depths.Further assessment of any residual loads (or load cell zeros)may lead to some revision of these estimates, but the rangespans that shown in Fig. 20 based on the approach of DeNicola & Randolph (1993).

0·9

0·8

0·7

0·6

0·50 10 20 30 40 50 60

Pile aspect ratio, L/d

Sha

ft ca

paci

ty r

atio

400

200

Ep/Gave 800

tanδ = 0·5

νp = 0·3

Fig. 20. Ratio of shaft capacity in tension and compression (DeNicola & Randolph, 1993)

55

50

45

40

35

30

25

200 20 40 60 80 100

Dep

th: m

Simplified design profile

Cone resistance, qc: MPa

Fig. 21. Cone resistance profiles from Euripides site

50

40

30

20

10

00 5 10 15 20

Field data: location 1

Field data: location 2

MTD method

Exponential decay: µ 0.05

Pile load: compression: MN

Dep

th: m

Fig. 22. Measured and calculated load distributions for Eur-ipides pile tests

860 RANDOLPH

SummaryWe now have a much clearer picture of changes that

occur in the soil around piles driven into sand, even thoughour design calculations rely heavily on empirical correla-tions. Use of the cone resistance provides a better quantita-tive basis for these correlations, but we need to review howbest to average qc, taking account of the pile diameter, thedisplacement needed to reach steady-state conditions, andthe soil stratigraphy. The effect of pile diameter on designend-bearing resistance, or on the plugging of pipe piles, isan area of apparent divergence between science and empiri-cism, which needs to be resolved. Analysis of the soil plugresponse suggests that the (static) end-bearing resistance ofpipe piles that do not plug during driving may be takenconservatively as about half that for a comparable closed-ended pile, but that design values of qbu/qc may be stronglyaffected by the magnitude of residual base pressure, qb0, orthe extent to which the base response has been pre-stiffenedduring the installation process.

Modern design methods must take account of frictiondegradation, but further work is needed in order to explorehow the rate of degradation is affected by pile diameter,method of installation (particularly blow counts during driv-ing), and soil modulus.

The design approaches considered here are conservative intwo respects. The true ultimate base resistance will exceedthe design value based on a limited displacement, with theaverage end-bearing resistance ultimately approaching thecone resistance. Secondly, recent studies have suggested thatthe shaft friction of piles in sand shows significant increasewith time (Chow et al., 1998), with gains of 50–100%possible. Although the resulting shaft friction may provesomewhat brittle, and so should not be considered in con-junction with large displacements to mobilise the baseresistance, further understanding and quantification of thisphenomenon would be valuable.

DYNAMIC PILE TESTINGOverview

Load tests to verify capacity are an essential part of mostpiling contracts, reflecting the relatively high level of uncer-tainty in predictive methods. Traditional static load tests,using kentledge or reaction systems, undoubtedly provide themost precise method of evaluating the load–displacementresponse, with minor limitations in terms of interaction(mainly affecting the pile stiffness) and loading rate effects.However, static pile tests are relatively expensive, and alsogive limited information on the distribution of resistancealong the pile unless it is instrumented. For large diametercast-in situ piles, external reaction systems become prohibi-tively expensive, and alternative devices such as Osterbergcells (Osterberg, 1989), which use part of the pile itself toprovide reaction, offer a more effective means of measuringcapacity. An example application of this technique is pre-sented later.

Dynamic pile tests, using high-energy piling hammers,provide an alternative to static loading tests, at a cost that istypically two orders of magnitude lower, but require sophis-ticated numerical simulation of the measured stress waves inorder to interpret the test. Alternative methods of testing, atintermediate loading rates (and costs) have appeared overthe last decade, in particular the Statnamic test (Berming-ham & Janes, 1989), in which a fast-burning fuel is used toaccelerate a mass away from the pile, thereby loading it inreaction. Reviews of various alternative methods for pileload tests have been given by England & Fleming (1994),who focus on different procedures for static load tests, and

Poulos (1998), who compares the relative merits of static,Statnamic and dynamic tests

Further discussion here is confined to the interpretation ofdynamic pile tests, which have the potential to be extremelycost-effective, but where commercial practice has fallenbehind significant advances in modelling the dynamic inter-action between pile and soil. This has contributed to agrowing scepticism regarding the confidence that may beplaced in such tests. The discussion will focus on modellingthe inertial (or radiation) damping from the soil surroundingthe pile, and on treatment of open-ended piles, but a briefdescription of the main principles is included first.

A schematic diagram of the main features of dynamic piletesting is shown in Fig. 23. The test comprises monitoringof strain and acceleration by means of instruments attachedto the pile shaft above ground, and preferably at least twodiameters below the pile head, during blows applied bystandard piledriving hammers or large drop-weights. Nor-mally pairs of instruments are located at opposite sides ofthe pile in order to minimise effects of bending, and the rawdata are converted to force, F (by multiplying the measuredstrain by the cross-sectional rigidity, EA, of the pile) andvelocity, v (by integrating the accelerometer data).

Initially, the stress wave travels down the pile unimpeded,during which time the measured force and velocity aredirectly proportional to each other: that is, F ¼ Zv, where Zis the pile impedance, EA/c, with c being the wave speed inthe pile. Any resistance to movement of the pile, forexample due to shaft friction, or change in pile section(more precisely, impedance) including at the pile tip, willcause an upward-travelling wave to be propagated back uptowards the pile head. Monitoring two properties (force andvelocity) allows separation of downward (subscript, d) andupward (subscript, u) components of the stress wave, accord-ing to

Fd ¼ 0:5(F þ Z)

Fu ¼ 0:5(F Z) (23)

It may be shown that, for sufficient magnitude of dynamicforce, the total soil resistance encountered by the pile duringthe passage of the stress wave down and up the pile is equalto the algebraic sum of the initial downward-travelling(maximum) force plus the upward-travelling force that ar-rives back at the instrumentation at a time 2L/c after theinitial maximum (Rausche et al., 1985; Randolph, 1990).

Simplified methods have been proposed for estimating thestatic component of the total soil resistance mobilised duringa dynamic test. However, the most reliable approach toanalyse dynamic tests is by computer simulation of the

Instrumentation:accelerometers

strain cells0 2h/c 2L/c

Time

Reflectionsfrom shaftresistance

Wavespeed,

c

1

DownwardtravellingDepth

Pileimpedance:Z EA/c

Shockwave

L

h

v0

Upwardtravelling(reflectionfrom base)

Fig. 23. Schematic of stress wave travel down pile

SCIENCE AND EMPIRICISM IN PILE FOUNDATION DESIGN 861

pile–soil interaction, endeavouring to match computed andmeasured stress-wave signals. The computer model isthe dynamic equivalent of the load transfer approach forstatic (non-linear) analysis, with the pile modelled as a one-dimensional elastic column and static load transfer springsreplaced by an appropriate dynamic model.

Dynamic pile–soil interaction modelsWorldwide, the majority of dynamic pile tests are ana-

lysed using pile–soil interaction models based on Smith(1960), with the soil resistance expressed as

¼ Min 1,wp

Q

(1 þ J)s (24)

where wp is the local pile displacement, and s are thedynamic and static shaft friction values (or equivalent inend-bearing pressure), Q is the displacement, or ‘quake’, tomobilise the limiting static resistance, and J is a dampingparameter multiplying the local pile velocity, v.

In the Smith model, separate viscous and inertial effectsare lumped into a single parameter, and the model does notdifferentiate properly between conditions prior to pile–soilslip and those after slip. A more rational model is shown inFig. 24, as proposed by various researchers in the 1980s(Simons & Randolph, 1985; Lee et al., 1988) and subse-quently implemented in many research-oriented, but fewcommercial, codes. In what will be referred to here as thecontinuum model, the far-field soil response may be mod-elled accurately using an elastic spring in parallel with adashpot representing inertial damping (Novak, 1977), withthe shaft response expressed as

Gw

dþ

s

< lim (25)

where vs is the shear wave velocity in the soil. Here, w andv are the displacement and velocity of the soil immediatelyadjacent to the pile–soil interface, rather than the pile,which is an important distinction.

The interface itself is modelled using a limiting shaftfriction that is velocity dependent, such as

lim ¼ s 1 þ m˜v

0

n" #

(26)

where m and n are viscous parameters and ˜v is the relative(or slip) velocity between pile and soil, normalised by v0

(taken for convenience as 1 m/s). Studies by Litkouhi &Poskitt (1980) suggest that the exponent n typically lies in

the range 0.2–0.5, and m is 0.3–0.5 for sand, and as high as2 or 3 for clays.

A typical response of the model described by equations(25) and (26) is shown in Fig. 25(a), compared with acorresponding result for the Smith (1960) model in (b) –note that both results are from the example applicationdescribed later. Various features are evident from this com-parison:

(a) The initial response of the continuum model isextremely stiff, and dominated by the inertial dashpotterm in equation (25). As noted by Randolph & Deeks(1992), for typical acceleration levels the dashpot termwill be at least a factor of 10 larger than the springterm during the initial response, and here slip starts tooccur after a pile displacement of 0.3 mm, when thedashpot accounts for 97% of the resistance.

(b) By contrast, the Smith model with a standard ‘quake’of 2.5 mm and damping value, J, of 0.2 s/m shows amuch more gradual development of resistance: theimmediate consequence of this is that the profile ofshaft friction deduced using the continuum model willbe offset to significantly greater depths, compared withthe Smith model (as the pile will take 1–3 ms to move2.5 mm, during which time the stress wave will havetravelled 5–15 m).

(c) The dynamic amplification of the static shaft friction isaffected by the velocity exponent, n (here taken as 0.2in the continuum model, and unity in the Smith model);a non-linear version of the Smith model is straightfor-

Pile–soil interface

Far-fieldresponse

Plasticslider

Elasticspring

Pile node

Viscousdashpot

Inertialdashpot

Fig. 24. Model for dynamic pile–soil interaction along shaft

She

ar s

tres

s, τ

/τs

Total

Static limit

Inertial dashpot

Elastic spring

Viscous dashpot

0 5 10 15 20 25

Pile displacement: mm

(a)

Total

Static response

Viscouscomponent

0 5 10 15 20 25

Pile displacement: mm

(b)

1.5

1.0

0.5

0.5

1.0

1.5

1.5

1.0

0.5

0

0.5

1.0

1.5

2.5

2.0

0

She

ar s

tres

s, τ

/τs

Fig. 25. Example responses of (a) continuum and (b) Smithshaft models

862 RANDOLPH

ward to implement, although it is rarely adopted. Evenso, the continuum model tends to give a more sustainedmaximum resistance, as the spring component continuesto rise as the inertial component decreases, whereas theformulation of the Smith model automatically gives adecline in resistance as the pile velocity decreases.

A similar ‘continuum’ model exists for the dynamic inter-action at the base of the pile (Deeks & Randolph, 1995),and again the inertial component dominates the initialresponse, although not to quite to the same extent as for theshaft (Randolph, 2000).

Open-ended pilesThe driving response of open-ended piles is usually

evaluated by lumping internal and external friction together(as external friction) and considering the end-bearing resis-tance on the annular base. However, the response of the soilplug is very different from that of the external soil, asenergy cannot be propagated to the far-field. A way ofmodelling the soil plug, as a column of nodal massesinterconnected by damped springs, and its interaction withthe pile was first proposed by Heerema & de Jong (1979)and extended using continuum-style soil models byRandolph (1987), as shown in Fig. 26. Recent studies byLiyanapathirana et al. (2001) have shown that this approachworks well for conditions where the incremental filling ratio,defined as the ratio of incremental plug movement withinthe pile, to the penetration of the pile (Brucy et al., 1991)exceeds 50–70%.

Example applicationExamples of dynamic testing of steel pipe piles are

presented here, drawn from a recent project in Perth, Wes-tern Australia, involving duplication of the main freeway

bridge across the Swan River. Fig. 27 shows a typicalgeometry of the steel pipe piles, with diameter of 0.61 mand wall thickness of 12.5 mm. In most piles, an annulardiaphragm was incorporated a few metres back from the piletip, in order to encourage the development of a fullyplugged condition. Without the diaphragm, it was found thatthe piles could be driven to large penetrations with relativelylow energy, and although the pile would undoubtedly haveresponded in a plugged manner to static loading, it was notpossible to verify the plugged capacity dynamically. The soilstratigraphy for the pile groups supporting the approachpiers and abutments is indicated in Fig. 27: it comprisedsand fill overlying alluvial soft clays and sand lenses abovea dense sand base in which the piles were to be founded.

Matching of computed and measured stress waves can beachieved by using one of the signals (such as the force) asinput, and matching the other signal (velocity). However, abetter approach is to combine the signals together in themanner of equation (23), using the downward-travellingwave (the externally applied force) as input, and matchingthe upward-travelling waves (due to soil resistance and tipreflection).

Figure 28 shows examples of matching the upward-travel-ling force wave using three approaches:

(a) explicit modelling of the soil plug using the continuumsoil model

(b) internal and external shaft friction lumped together,using the continuum soil model

(c) internal and external shaft friction lumped together,using the Smith (1960) model.

All results are for identical distribution of friction down theinside and outside of the pile, and identical base resistanceon the pile annulus. It is clear that lumping the internal shaftfriction with the external friction has had a significant effecton the shape of the computed stress wave, even using anidentical continuum model. Also, the response computedusing the Smith (1960) model is very different from thatusing the continuum model. In fact, for this particular open-ended pile test, no good match could be obtained withoutexplicit modelling of the soil plug, even by varying thedistribution of soil resistance.

The best (although not particularly close) fit to the

Shaft

Soil responsemodel

Soil plug

Base

Base responsemodel

Pile wall

Externalsoil

Fig. 26. Modelling dynamic response of soil plug

0.61 m

Sandfill

Softclay

Densesand

6m

33 m

Steelpipepile

Annular steeldiaphragm

Soilplug

Fig. 27. Partially plugged pipe pile

SCIENCE AND EMPIRICISM IN PILE FOUNDATION DESIGN 863

measured upward-travelling wave in Fig. 28 was obtainedwith the capacities summarised in Table 2 (Pile 1). Thequality of fit was relatively sensitive to the internal shaftfriction, and reasonable fits were obtained for mobilisedresistances in the range 6.5–7.5 MN. The static response ofthe pile was evaluated by lumping the internal shaft capacityinto the base resistance, and assuming that the resulting baseresistance of 4.91 MN would be mobilised at a displacementof 10% of the pile diameter (0.06 m). The distribution ofexternal shaft friction (and shear modulus) deduced from thedynamic test was incorporated unaltered into the staticanalysis. The resulting pile response is compared with theresults of a static load test on the pile in Fig. 29 and showssurprisingly good agreement considering the rather arbitrarymanner of incorporating the dynamic soil plug resistance inwith the base response.

One further example of predicting the static pile responsefrom a dynamic test is shown in Fig. 30, for a tension testcarried out as part of the same project. In this case theopen-ended pile was driven (with no internal diaphragm)through the upper sand fill to tip into the underlying clay,and a plug length of 10 m (compared with total pile penetra-tion of 16.5 m) was measured. The stress-wave matching hasbeen discussed by Randolph (2000), and the resulting capa-cities are summarised in Table 2.

The computed response under (static) tensile loading,taking just the external shaft resistance, is compared withthe measured response in Fig. 30. The measured capacity intension is about 15% lower than the external shaft capacitydeduced from the dynamic test, which is consistent with theratio of tensile and compressive shaft capacities from Fig.20. However, it should also be pointed out that the level ofconfidence in the back-analysed external shaft capacity ispoor, with a possible range of 1–1.5 MN (mainly by redis-tributing internal friction to external friction).

SummaryDynamic pile tests are arguably the most cost-effective of

all pile-testing methods, although they rely on relativelysophisticated numerical modelling for back-analysis. Theor-etical advances in modelling the dynamic pile–soil inter-action have been available since the mid-1980s, but havebeen slow to be implemented by commercial codes, most ofwhich still use the empirical parameters of the Smith (1960)model. In order to allow an appropriate level of confidencein the interpretation of dynamic pile tests, and estimation ofthe static response, it is high time that appropriate scientificmodels were used for pile–soil interaction, including explicitmodelling of the soil plug for open-ended piles.

There will still be a need for some empiricism, mainly inadjusting the static shaft friction to allow for the very highshearing rates during a dynamic test. Modelling viscous and

ComputedSmith modelclosed-ended

Computedcontinuum model

closed-ended

Computedopen-ended

Measured

Time: ms

10 20 30 400

1.0

0.5

0

0.5

1.0

1.5

2.0

2.5

3.0

For

ce: M

N

Fig. 28. Matching of measured and computed upward-travelling waves

Table 2. Summary of mobilised soil resistances from dynamic tests

External shaftresistance:

MN

Internal shaftresistance:

MN

Annular baseresistance:

MN

Total pileresistance:

MN

Pile 1 (compression) 2.43 3.97 0.94 7.34Pile 2 (tension) 1.02 0.98 0.06 2.06

Computed

Range of reasonablefits to dynamic test

Measured

0 20 40 60 80 100 120Displacement: mm

0

1

2

3

4

5

6

7

8F

orce

: MN

Fig. 29. Comparison of measured and computed pile response

864 RANDOLPH

other effects needs to be placed on a more rational basis, asapproaches such as equation (26) are far from ideal. Finally,it should be realised that the deduced soil resistance para-meters will not be unique. It is therefore important whenback-analysing dynamic pile tests to adopt appropriatelyconservative values (or distributions) of soil resistance thatstill give a reasonable fit to the stress wave data, rather thannecessarily adopting the ‘best fit’ set of parameters. Giventhe empirical nature of the soil models, the concept of ‘bestfit’ is of limited value, and a philosophy of ‘conservative,while still consistent’ is a more prudent approach.

DESIGN OF PILE GROUPSDesign philosophy

The main objective of pile group design is to ensure thatthe foundation does not undergo excessive displacementsduring its design life. This principle is represented by theserviceability limit state in design codes, but in additionthere is normally a requirement to ensure that there is anadequate material factor against ‘collapse’, or ultimate limitstate. Where piles exhibit strain-hardening, the ultimate limitstate for an individual pile is customarily (but somewhatarbitrarily) taken as the load to cause either an absolutedisplacement (such as 50 or 100 mm), or a relative displace-ment (such as 10% or 15% of the pile diameter).

In practice, however, for most applications the real ulti-mate limit state should be determined by considering theinteraction of the foundation with the superstructure, andmust consider the complete foundation system (pile group,with or without ground-contacting cap) rather than indivi-dual piles. As for serviceability limit states, the ultimatelimit state then also reverts to a displacement criterion, butone that is determined largely by structural considerations.

The advantage of moving more towards displacementcriteria, reducing the emphasis on the capacity of individualpiles, is illustrated in Fig. 31: axial pile capacity dependscritically on effective stress and fabric conditions at thepile–soil interface, which are difficult to determine accu-rately, whereas the deformation response is influenced pri-marily by soil conditions away from the pile, which are onlyslightly affected by the installation process. Simple, butrobust, analytical approaches for estimating the verticalstiffness of piles and pile groups are readily available, asoutlined below. Hence, provided the in situ stiffness of thesoil strata can be determined, the pile group load–displace-ment response up to working load can be estimated withgreater accuracy than the ultimate capacity.

The discussion here will be restricted to vertically loaded

pile groups, and the trend towards piled rafts where raftfoundations are supported by relatively low numbers of piles,placed strategically in order to minimise differential settle-ments and bending moments in the raft. However, a casestudy involving foundations subjected to general vertical,horizontal and moment loading will also be presented,illustrating the importance of considering the overall founda-tion response, rather than ‘safety factors’ on individual piles.

Analysis of vertically loaded pile groupsMany different approaches may be used to analyse the

response of single piles and pile groups subjected to verticalloading. The most widespread, and most rigorous for homo-geneous soil conditions, is the boundary element method(Poulos, 1968; Banerjee & Butterfield, 1981; Basile, 1999),which has the advantage that only the pile–soil interfacesneed to discretised, rather than the full continuum. A muchsimpler approach, leading to analytical solutions, is to adopta Winkler approximation, where values of shear stress downthe pile shaft, and end-bearing pressure at the base, aretaken as proportional to the local pile displacement. Thisapproach has been shown to lead to good agreement withresults from more rigorous numerical analysis (Randolph &Wroth, 1978), and recent analytical studies have evaluatedthe relationship between Winkler spring constant and elasticshear modulus in closed form (Mylonakis, 2001).

The original solution of Randolph & Wroth (1978)adopted the idealised assumption of shear stresses aroundthe pile that decayed inversely with radius, leading to alogarithmic decay in vertical displacements, together with amaximum radius of influence, rm. The shear stress at thepile wall is then expressed in terms of the local displace-ment, w, as

¼ 2Gw

d(27)

where

¼ ln2rm

d

The Winkler spring constant, k, relating the force per unitlength transferred from pile to soil to the local displacement,is then given by

k ¼ d

w¼ 2G

¼ G (28)

The value of is typically around 1.5 for floating piles, butincreases for ‘end-bearing’ piles (with wb ¼ 0) where it

0 10 20 30 40 50

Measured

Computed

Range of reasonablefit for external shaft

friction fromdynamic test

16·5 m

0·61 m

10 m

Soilplug

For

ce: M

N

0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

Steelpipepile

Displacement: mm

Fig. 30. Comparison of measured and computed pile responsein tension

(a) (b)

Fig. 31 Pile group capacity and stiffness: (a) capacity dependenton conditions at pile–soil interface; (b) stiffness determinedprimarily by far-field conditions

SCIENCE AND EMPIRICISM IN PILE FOUNDATION DESIGN 865

varies with aspect ratio, L/d, and stiffness ratio, Ep/G,according to (Mylonakis, 2001)

¼ k

G 1:3

2(1 þ )G

Ep

0:025

1 þ 7L

d

0:6" #

(29)

The resulting pile head stiffness, K ¼ P/wt, where P and wt

are the pile-head load and ground-level displacement respec-tively, may then be expressed as

K ¼ Ep Apºþ tanh(ºL)

1 þ tanh(ºL)(30)

The parameters and ºL represent non-dimensional basestiffness and slenderness ratio for the pile, expressed as

¼ Pb

wb Ep Apºand ºL ¼

ffiffiffiffiffiffiffiffiffiffiffik

Ep Ap

sL (31)

where Pb and wb are respectively the load and displacementat the pile base, and EpAp is the cross-sectional rigidity ofthe pile.

The above expression may be used to calculate thestiffness of single piles in homogeneous soil, or in multi-layered soil (essentially using equation (30) and the asso-ciated value of as a transfer function; Mylonakis &Gazetas, 1998). For soils where the stiffness varies withdepth, alternative solutions may be developed, involvingBessel functions of fractional order (Guo & Randolph,1997). Such solutions are readily evaluated using modernmathematical packages.

For pile groups, the stiffness of each pile is reducedbecause of interaction effects, as indicated in Fig. 32.Mylonakis & Gazetas (1998) demonstrated that the inter-action factor, Æ (as defined by Poulos, 1968), must reflectnot only the (assumed) logarithmic decay in displacements,but also the reinforcing effect of the neighbouring pile. Thisleads to a reduction in the interaction factor, as indicated inFig. 33. For piles of the same length and diameter, theinteraction factor for a given spacing, s, may then be ex-pressed as the product of two terms representing the loga-rithmic decay and a ‘diffraction factor’, (Mylonakis &Gazetas, 1998), giving

Æ ¼ ln(rm=s)

ln(2rm=d)

(32)

where the diffraction factor, , is a function of and ºL,according to

¼

2ºL þ sinh(2ºL)

þ 2[sinh(2ºL) 2ºL] þ 2[cosh(2ºL) 1]

2 sinh(2ºL) þ 22 sinh(2ºL) þ 4 cosh(2ºL)(33)

This elegant capturing of the reinforcing effect of each pilein a group is a seminal advance in quantifying interactionbetween piles. The form of equation (33) is illustrated inFig. 34, where it may be seen that converges to 0.5 forlong (or compressible) piles, is less than 0.5 for end-bearingpiles (high ), and is between 0.5 and 1 for most floatingpiles (low ).

An equivalent expression to equation (33) may be derivedfor piles of the same length, but different diameters d1 andd2 (and corresponding values of , and ºL), leading to aninteraction factor, Æ21 (proportional settlement of Pile 2compared with Pile 1), given by

Æ21 ¼ ł21º1º2

º22 º2

1

º2

º1

sh1 þ

2º2

º1

ch1

sh2 þ2ch2

24

(1 þ tanh º1 L) 2sh2 þ ch2

2º2

º1

sh1 ch1

(1 þ1 tanh º1 L)(sh2 þ2ch2)

3775

(34)

P

w1t αw1t

1 2

S

d

L

Fig. 32. Interaction between two piles for axial loading

0 0·2 0·4 0·6 0·8 1·0

α 0.376 α′ 0.578

Pile 2adjusted

Pile 2logarithmic

decay

Pile 1

L/d 20Ep/G 500

ν 0.3s/d 3

0

0.2

0.4

0.6

0.8

1.0

Normalised displacement, w/w1o

Dep

th, z

/L

Fig. 33. Displacement profiles down loaded and adjacent piles

0 0.5 1.0 1.5 2.0 2.5

Ω 0

Ω 0.05

Ω 0.2

Ω 1

0.1

1.0

0.8

0.6

0.4

0.2

0

Dimensionless pile length, λL

End-bearing pile

Diff

ract

ion

fact

or,

Fig. 34. Diffraction factor, (Mylonakis & Gazetas, 1998)

866 RANDOLPH

where

ł21 ¼ ln(rm=s)

1

; shi ¼ sinh ºi L and chi ¼ cosh ºi L (35)

It may be shown that the reciprocal theorem is satisfied bythe above expression, as Æ21/K1 is identical to Æ12/K2, andthat the above expression reduces to that from Mylonakis &Gazetas (1998) as the pile diameters converge.

The effect of non-homogeneous soil conditions may beincorporated in an approximate manner, following Randolph& Wroth (1978), whereby the load transfer parameter, , isreduced by ln(r) (where r ¼ Gaverage/GL, the ratio ofaverage shear modulus to the value at pile tip level, rangingbetween 0.5 and 1), and also the tanh(ºL) term in thenumerator of equation (30) is multiplied by r.

The above approach has been used to evaluate the stiff-ness of square groups of piles, from 2 3 2 up to 30 3 30,for L/d ¼ 25, Ep/GL ¼ 1000, r ¼ 0.75 and ¼ 0.3. Theresults are shown in Fig. 35(a), where the pile groupstiffness, Kg (ratio of total applied load to average settle-ment), has been normalised by GLB, where B is the width ofthe pile group. Plotting the normalised stiffness against thenormalised width, B/L, leads to an envelope of curves thattends to the stiffness of a surface raft as B/L becomes large.The stiffness envelope may also be matched closely by usingan equivalent pier approximation of the pile group (Poulos& Davis, 1980; Randolph, 1994), demonstrating the robust-ness of calculations of pile group stiffness even with quiteapproximate models (Fig. 35(b)).

From a practical viewpoint, it is also useful to link the

average settlement of a pile group to the dimension, B, ofthe foundation and the factor of safety against ultimatecapacity. This has been explored by Cooke (1986) and morerecently by Mandolini (2003), who suggest that for fine-grained soils the average settlement is typically around 0.3–0.6% of the foundation width, B, for a factor of safety of 3.

Piled raftsAs may be seen from Fig. 35, for pile groups where

B/L , 1 the pile group stiffness is significantly greater thanthe stiffness of a raft foundation. Therefore, even if the pilecap rests directly on competent ground, it will contributelittle to the response of the overall foundation. Viggiani(2001) has referred to such foundations as ‘small’ pilegroups, where piles are needed to ensure adequate bearingcapacity, and the pile cap (or raft) can easily be madesufficiently stiff to eliminate differential settlements. Bycontrast, for ‘large’ pile groups, with B/L . 1, the pile capwill often provide sufficient margin against bearing failure,and will contribute significantly in terms of transferring loaddirectly to the ground. The design of such foundationshinges more on limiting the average and differential settle-ments to a acceptable level. As, for large rafts, the flexuralstiffness will be low, the location and length of any pilesupport should be chosen in order to minimise differentialsettlements. Design strategies for ‘small’ and ‘large’ pilegroups have been discussed by Viggiani (2001) andMandolini (2003).

The concept of using a limited number of piles, poten-tially loaded to near their ultimate capacity, beneath a raftfoundation in order to reduce differential settlements wasmooted by Burland et al. (1977), and also explored furtherby Randolph (1994) and Horikoshi & Randolph (1998). Theprinciple behind the design approach is illustrated in Fig. 36.For large rafts (B/L . 1) the stiffness of a pile groupoccupying the full area of the raft will be quite similar tothat of the raft alone, and a more effective approach toreducing differential settlements is to place a few piles overthe central region of the raft. Where a layer of soft soilexists beneath the raft, then it may be necessary to installshort piles extending through that layer over the full raftarea, but then to use longer piles in the central 25–40% ofthe raft area.

Piled rafts of this nature require careful analysis, but thereare a variety of analytical approaches of varying complexitythat are now available (Franke et al., 1994; Poulos, 1994,2001; Clancy & Randolph, 1996; Russo, 1998; Katzenbachet al., 2000). Studies on optimising pile geometry havemainly been restricted to uniform loading of the raft, and

L/d 25Ep/GL 1000ρ 0·75ν 0.3

s/d 2

s/d 3

s/d 5

s/d 10

Raft stiffness

Gro

up s

tiffn

ess,

Kg/

GLB

0 1 10

Normalised width of pile group, B/L

0

2

4

6

8

10

12

14

16

18

20

Gro

up s

tiffn

ess,

Kg/

GLB

0·1 1 10

Normalised width of pile group, B/L

0

2

4

6

8

10

12

14

16

18

20

Stiffness ofincompressible pier

80% of stiffness ofincompressible pier

Equivalent pier(same area and

length as pile group)

(a)

(b)

Fig. 35. Comparison of pile group and equivalent pier stiff-nesses: (a) normalised stiffness of pile groups; (b) comparison ofequivalent pier stiffness

Piled foundationRaft foundation Piled raft foundation

Fig. 36 Transition from raft to pile group to piled raftfoundation

SCIENCE AND EMPIRICISM IN PILE FOUNDATION DESIGN 867

these have suggested that piles of length greater than about70% of the width of the raft are required, situated over thecentral 25–40% of the raft area (Horikoshi & Randolph,1998; Prakoso & Kulhawy, 2001; Viggiani, 2001).

In practice, buildings often concentrate a significant pro-portion of the total load towards the outer edges of the raft,giving rise to concern over potential hogging of the raft. Inorder to address that concern, a recent study by Reul &Randolph (2004) has also considered the case where half ofthe applied load is distributed uniformly over the central25% of the raft, and the other half is transmitted around theraft edge. Fig. 37 shows the raft and pile geometry analysed,together with the two alternative load distributions.Horikoshi & Randolph (1997) proposed a raft–soil stiffnessratio defined as

Krs ¼ 5:57Er

2G

1

1 2r

ffiffiffiffiffiBr

Lr

rtr

Lr

3

(36)

where the subscript r denotes the raft properties of Young’smodulus, E, Poisson’s ratio, , breadth, B, length, L, andthickness, t. The two different raft thicknesses shown in Fig.37, of 1 m and 3 m, lead to raft–soil stiffness ratios of 0.06and 1.5, representing practical limits of flexible and stiffrafts.

The resulting settlement patterns for piled and unpiledrafts are presented in Fig. 38(a) for the flexible raft (wherethe bending moments are negligible), whereas Fig. 38(b)shows bending moment profiles for the stiff raft (where thedifferential settlements are negligible, less than 1% of theaverage settlement). It may be seen from this figure thatthe central pile support reduces differential settlements forthe flexible raft case by a factor of between 2.7 (uniformloading) and 3.5 (core-edge loading), with correspondingratios of differential settlement to average settlement for thepiled rafts of 0.15 and 0.24. For the stiff raft, maximum(absolute) bending moments are reduced by a factor of about2.4, at the expense of introducing greater hogging moments.Thus, even for situations where a significant fraction of theapplied load is concentrated towards the edges of thefoundation, central pile support appears a beneficial designapproach.

For any given design, the precise layout and geometry ofthe pile support would need to be fine-tuned by numericalanalysis. However, the principle is to optimise the design bylocating the pile support in such a way as to minimisedifferential settlements and bending moments. In this way,the foundation costs may be minimised without compromis-ing performance. Although few, if any, piled rafts have beenoptimised in this way, several studies have been publishedshowing how conventionally designed piled foundationswould have performed adequately, and with smaller differen-tial settlements, with as little as 30–50% of the actual

cumulative length (number times average length) of pilesused, but with the piles concentrated in the central part ofthe raft (Horikoshi & Randolph, 1998; Prakoso & Kulhawy,2001; Viggiani, 2001; Mandolini, 2003).

General loading: a case studyThe final topic covered in this paper concerns the founda-

tions for a major bridge in Vietnam, the construction ofwhich was funded jointly by the Vietnamese and Australiangovernments. The My Thuan bridge crosses one of the twobranches of the Mekong delta, and is a cable-stayed bridgewith a central span of 350 m and clearance of 37.5 m (Fig.39). Maunsell Australia Pty Ltd were the consulting engi-neers for the project, and the author was a member of thetechnical advisory group for the Australian Agency forInternational Development (AusAID). The case study isinstructive in two ways, first for its use of sophisticated piletests in order to finalise the design of the main pile supportfor the bridge towers (Chandler, 1998), and second as anillustration of the effects of redistributing load away fromheavily loaded piles, effectively allowing the ultimate limitstate to be considered in terms of allowable deformations.

The main towers of the bridge are supported on founda-

30 m

38 m

38 m

1 m

6 m

Core:edge (50:50)

Uniform

1 or 3 m

Fig. 37. Example piled raft showing two alternative loaddistributions

0.5 0.3 0.1 0.1 0.3 0.5

Krs 0.06

Piled raftuniform loading

Piled raftcore-edge loading

Unpiled raftuniform loading

Unpiled raftcore-edge loading

(a)

0.50

0.45

0.40

0.35

0.30

0.25

0.20

0.15

0.10

0

0.05

Nor

mal

ised

set

tlem

ent,

wG

B/P

tota

l

0.5 0.3 0.1 0.1 0.3 0.50.010

0.005

0

0.005

0.010

0.015

0.020

0.025Krs 1.5

Piled raftuniform loading

Piled raftcore-edge loading

Unpiled raftuniform loading

Unpiled raftcore-edge loading

Position across centreline of raft, x/B

Position across centreline of raft, x/B

Nor

mal

ised

set

tlem

ent,

wG

B/P

tota

l

(b)

Fig. 38. Settlement and moment profiles across raft centreline:(a) settlement profiles; (b) profiles of bending moment per unitlength

868 RANDOLPH

tions comprising 16 bored piles, arranged as two groups ofeight beneath each leg of the tower, and linked by a 6 mthick pile cap as shown in Fig. 40. The piles are 2.4 m(nominal) diameter, around 95 m long, and were constructedunder bentonite; base-grouting to 5 MPa was also performed.Under normal conditions the water depth is around 23 m,but a deep scour hole exists upstream of the bridge wheretwo branches of the river converge. The ultimate designconditions allow for migration of the scour hole, resulting inscouring to a depth of 47 m at the tower locations. The mostcritical design loads correspond to ship impact on one of thetower foundations, either parallel to the river (08) or at 458to the river, with design loads as given in Fig. 40. Thedesign vertical load of 315 MN corresponds to an averageload of just under 20 MN per pile.

A simplified soil stratigraphy is shown in Fig. 41 for thesouth tower and also for the first piers on the southern bank.Only limited strength data were obtained from the siteinvestigation, and instead the final pile length was based onthe results of two load tests on piles for the south bank pier,which were deliberately extended to reach the founding

stratum of dense sand deemed necessary for the tower piles.Two pile load tests were conducted on the south bank pier,each with a pair of Osterberg cells, as indicated. Furtherload tests were conducted on the south and north towerfoundations, using a single Osterberg cell.

The use of Osterberg cells at two levels within a test pileallows measurement of base capacity, shaft capacity over thepile section between the two cells, and shaft capacity of theupper pile section, by means of three test phases conductedin sequence (Osterberg, 1989). The results of the two southbank load tests are presented in Fig. 42. The measured baseresponse was consistent in both tests and showed no evi-dence of any accumulation of soft sediments from theconstruction process. The base response has been modelledusing a hyperbolic response with an ultimate end-bearingpressure of 4.5 MPa, and an initial stiffness that correspondsto a shear modulus of about 450 MPa.

The values of shaft friction measured in the first load testwere below expectations, and before the second pile wasconstructed, modifications were made to the constructionprocedures. These included steps to reduce the delay be-tween excavation and concreting, a reduction in the head ofbentonite above river level, and mechanical scarifying of the

Fig. 39. My Thuan bridge in Vietnam

~ 60 m

2.4 m

x

5.5 m

V 315 MN

M ≈ 320 MNm

H ≈ 20 MN

95 m

Fig. 40. Pile layout for tower foundations of My Thuan bridge

Silty clay(su ~ 200 kPa)

Water

Clayey sand(Φ′ ≈ 38°)

Silty clay(su ≈ 300 kPa)

Dense sand(Φ′ ≈ 40°)

South bank testpiles (86·4 m)

South pier pile (96 m)

93 m

75 m

56 m

42 m

23 m

0 m

Osterberg cells

83 m

68 m

51 m

40 m

0 m

Fig. 41. Soil stratigraphy and location of Osterberg cells

Test 2

20 40 60 80 100

Test 1 Displacement: mm

Simulation (RATZ)

Base

Middle section(between cells)

Test 1

Upper section(above top cell)

Test 2

100

75

50

25

0

2·5

5

Ave

rage

sha

ft fr

ictio

n: k

Pa

End

-bea

ring

pres

sure

: MP

a

Shaft

Fig. 42. Response measured from load tests on south bank piles

SCIENCE AND EMPIRICISM IN PILE FOUNDATION DESIGN 869

borehole shaft prior to casting the pile. The last step wasachieved by welding short strands of wire to the calliperingtool (Fig. 43), which was then raised and lowered within theborehole, rotating it as required in order to cover the fullcircumference of the shaft.

As may be seen from Fig. 42, these measures led to animprovement of about 30% in the shaft friction measured inthe second load test, with design values of 55 kPa and90 kPa adopted for the upper and middle zones respectively.Making allowance for slight differences in stratigraphy be-tween the south bank and tower locations, and also for thedesign scour depth of 47 m, the ultimate capacity of the95 m long tower piles was determined as 42.1 MN. Adoptinga material factor of 0.72 (Australian Standards, 1995), andallowing for the buoyant pile weight of 5.4 MN, a designgeotechnical ultimate capacity of 24.9 MN was arrived at forthe 95 m long piles. Subsequent load tests on the towerpiles, using a single Osterberg cell, resulted in a measuredultimate capacity of the upper section of pile of 26–27 MN,and confirmed an ultimate capacity well in excess of30 MN.

Analysis of the pile group under the design load condi-tions, using the software PIGLET (Randolph, 2003), leads toa distribution of loads among the piles as shown in Fig. 44.Assuming elastic response of the piles, it is found that threepiles under load case 1, and six piles under load case 2,exceed the design capacity of 24.9 MN. However, non-linearanalysis, with the axial load limited to a notional designcapacity, allows the loads to be redistributed among the pilesat a cost of increased deformations. The load distributionsfor the case where the axial loads have been limited to22 MN are shown in Fig. 44. For each load case, only fourpiles remain with axial loads less than the imposed limit of22 MN, suggesting that the pile group is close to failure inthe sense of no longer being able to find a load distributionin equilibrium with the applied loading.

Figure 45 shows the development of vertical displacement

at the pile group centroid for the non-linear analyses. Thelower two curves are conventional load–displacement re-sponse for the two load cases, as the applied loads arefactored up proportionally to their full values and with animposed axial load limit of 22 MN on individual piles. Finaldisplacements are 280 mm (load case 1) and 250 mm (loadcase 2), compared with the elastic value of 74 mm for bothcases.

From a design perspective, more useful information isgiven by the upper two curves in Fig. 45, which give thefinal vertical displacements for different magnitudes of thelimiting axial load. This shows the more critical nature ofload case 2, where displacements start to increase signifi-cantly once the axial load limit is reduced below 28 MN.For the actual design limit of 24.9 MN the vertical displace-ment is 140 mm for this load case (and 76 mm for load case1: little more than the elastic value).

An appropriate design strategy for bridge foundationsinvolves an iterative interaction with the structural designengineers, in order to determine what displacement of thefoundation constitutes an ultimate limit state. In the presentcase this allows confirmation that the geotechnical designlimit of 24.9 MN (and the resulting displacement of140 mm) is indeed adequate, and essentially weights thedefinition of ultimate limit state more towards a deformationcriterion, rather than the geotechnical capacity of individualpiles in the group.

Fig. 43. Tool to scarify borehole shaft prior to concreting

Load case 1:ship impact at 0°

40

30

20

10

0

10

20

30

4030 20 10 0 10 20 30

Load case 2:ship impact at 45°

Non-linear analysisaxial limit: 22 MN

Distance from pile group centroid: m

Axi

al lo

ad: M

N

Upload

Elasticanalysis

Download

Fig. 44. Computed axial loads for critical load cases

Loadcase 2

Final displacement asfunction of axial

capacity

Load-displacementresponse with axialcapacity of 22 MN

Loadcase 1

Loadcase 2

Elastic

0·300·250·200·150·100·0500

5

10

15

20

25

30

35

Vertical displacement of pile group centriod: m

Axi

al c

apac

ity:

MN

Ave

rage

axi

allo

ad: M

N

Fig. 45. Non-linear response of pile group under design loads

870 RANDOLPH

SummaryIn contrast to other areas of pile design, the science of

predicting the deformation response of pile groups is welladvanced, with a variety of methods available, differing incomplexity and degree of rigour. Under vertical loading,quantification of the reinforcing effect of piles in moderatinginteraction factors is a major improvement, and the approachleads to estimates of pile group stiffness that are consistentwith simple analogues based on equivalent pier or raftapproximations.

For piled rafts, the benefits of central pile support for(primarily) raft foundations have been documented widelyboth for uniformly loaded pile groups and for the moredemanding case where a significant proportion of the loadingis concentrated at the raft edges. As illustrated here, even inthat case, substantial reduction in differential settlements (forflexible rafts) and absolute magnitude of bending moments(in stiffer rafts) is possible using central pile support.

The trend towards design based on allowable deformationsmay also be appropriate for ‘small’ pile groups, such as forsupporting bridge piers, which are subjected to combinedvertical and horizontal loading. Non-linear analysis allowsconsideration of the consequences of limiting axial (orlateral) loads on individual piles to a notional design value,and hence, in conjunction with structural considerations,permits even the ultimate limit state to be weighted moretowards allowable deformations.

The key soil parameter required for predicting the defor-mation response of pile groups is the shear modulus, andback-analysis of the performance of large pile groups sug-gests that the relevant value is close to the small-strainmodulus, G0 (Mandolini & Viggiani, 1997). This may bemeasured relatively easily using modern methods such asseismic cone tests or the small strain response of high-quality samples. For piled rafts, with fewer piles and higheraverage shear stresses in the soil, some factoring of Go willbe necessary, and further analysis and field evidence isneeded in order to quantify the reduction. The use of amodified hyperbolic response to bridge between the initialstiffness (derived using G0) and the ultimate capacity ap-pears a promising approach (Mayne & Poulos, 2001; Mayne,2003). Pile load tests carried out near the start of the pilingcontract provide a useful means of verifying, or fine-tuning,the design, not just in terms of pile capacity, but also inrespect of shear modulus values deduced from the load–displacement response at different load levels.

CONCLUDING REMARKSEinstein once commented along the lines that:

‘‘Experimental data are believed by everyone, except theperson who did the experiment; while theory is believed bynobody, except the person who developed it.’’

This comment is particular pertinent to pile design, wherewe need to be vigilant over the quality of data relied uponin empirical approaches and to continue to improve ouranalytical methods. With this in mind, the aim of this paperhas been to explore advances in scientific approaches to piledesign, and the extent to which we still need to rely onempirical correlations.

The scope has been deliberately restricted, partly becauseof space limitations, but also reflecting the author’s interestsand experience. Summary comments have been provided atthe end of each topic and will not be repeated here.

In estimating the axial capacity of piles, areas have beenidentified where ‘empirical’ trends from measured perform-ance do not appear to be supported by ‘science’ in the formof conceptual or analytical models. These areas therefore

deserve closer scrutiny, particularly where design requiresextrapolation outside the existing database.

Overall, significant uncertainty exists in our ability toestimate the axial capacity of individual piles, and hence weneed to adapt our design and contracting strategies to en-courage the testing of piles in order to fine-tune designs,allowing more optimistic material factors as more preciseestimates of pile capacity become possible. Dynamic piletesting has an important role in this respect, but inertiawithin the industry has restricted the adoption of morescientific models of pile–soil interaction.

We should capitalise on the ductile response of most piledfoundation systems, weighting design criteria more towardsallowable deformations, with reduced dependence on thecapacity of individual piles. Efficient design of piled raftsmay allow the piles to operate close to their geotechnicaldesign capacity under working conditions, with their mainpurpose being to minimise differential settlements and bend-ing moments in what is primarily a raft foundation.

Despite adverse comments by some of the pioneers of soilmechanics, there is a significant role for scientific methodsin pile design. If we are to continue to attract the best ofeach new generation of engineers, then we must incorporatesuch science in our teaching and our practice, using empiri-cal approaches to validate and calibrate, but not replace,scientific theory.

ACKNOWLEDGEMENTSI would like to gratefully acknowledge support and con-

tributions from a wide range of people during preparation ofthis paper: first and foremost my wife, Cherry, and sons,Nick and Tom; mentors at key stages of my career, includ-ing John Burland, Peter Wroth, Andrew Schofield and JohnBooker; colleagues Martin Fahey and Barry Lehane fromUWA, and Carl Erbrich from Advanced Geomechanics;collaborators who have supplied data and taken time tocorrespond on some of the issues, particularly David Whitebut also Antonio Alvez, Fiona Chow, George Mylonakis andOliver Reul; and finally past and present staff and studentsin the Geomechanics Group at UWA who have made it sucha fun place to work. I am also grateful to AusAID forpermission to use data from the My Thuan bridge project.

NOTATIONA area (Ap for cross-sectional area of pile)

Br width of raft foundationc wave speed in pile

ch coefficient of consolidation for horizontal flowDr relative densityd diameter of pile (subscripts eq for equivalent, i for

internal)E Young’s modulus (subscripts b for soil at pile base, p for

pile, r for raft)F dynamic axial force in pileG shear modulus of soilh distance measured upwards from pile tip (subscript p

for soil plug)Ir rigidity index, G/su

J Smith damping factorK compressibility factor, horizontal stress ratio (subscript

0 for in situ value), pile head stiffnessKrs non-dimensional raft-soil stiffness ratio

k load transfer factor, permeability (with subscript h forhorizontal flow)

L pile length, (subscript r for raft length)m non-dimensional rate factorn exponentP load at pile head

PI plasticity index

SCIENCE AND EMPIRICISM IN PILE FOUNDATION DESIGN 871

p mean stress (subscripts i for initial, o for in situ, a foratmospheric pressure)

Q Smith quake, capacity (subscript s for shaft)q pressure (subscripts b for pile base, c for cone

resistance)R yield stress ratio

Rf pile capacity reduction factorr radial coordinate (subscript m for maximum radius of

influence)req equivalent pile radiusSt soil sensitivitysu undrained shear strength

T, T non-dimensional time factors (subscript eq forequivalent)

t timetr thickness of raftu pore pressure (subscripts o for in situ, max for

maximum value)v velocity (subscript s for shear wave velocity in soil)w settlement (subscript t for pile head)z depth co-ordinateÆ shaft friction correlation factor, interaction factor

between piles shaft friction correlation factorª unit weight (9 for effective, subscript w for water)˜ indicating change interface friction angle, load transfer ratio k/G, also

indicating increment load transfer parameter non-dimensional pile compressibility parameterº shaft compressibility inverse length, parameter in radial

consolidation model parameter in radial consolidation model and radial

stress degradation Poisson’s ratio (subscripts p for pile, r for raft) ratio of residual to peak shaft friction, diffraction

parameter mathematical constantr area ratio stress (subscripts p for vertical yield stress, rc for radial

after consolidation, rf for radial at failure, ri for initialradial, v0 for in situ vertical)

ı shear stress (subscripts peak for peak, res for residual, sfor shaft friction)

9 soil friction angleł geometry factor for pile interaction non-dimensional base stiffness parameter

REFERENCESAltaee, A., Fellenius, B. H. & Evgin, E. (1992). Axial load transfer

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VOTE OF THANKSHUGH ST JOHN, Director, Geotechnical Consulting Group

When I contacted Mark to ask him to write his ownintroduction (which at the time I thought I was going todo), he said that he would prefer me not to be sycophanticbut to refer to his return from exile after 16 years. I endedup taking responsibility for the vote of thanks, which givesme a lot more license, because he wouldn’t write thateither.

I would like to start off with a complaint about hisAustralian accent. This was supposed to be an overseaslecture, Mark. You made no effort to make it sound likeone.

I first met Mark around 30 years ago when he turned upat BRE, the bright young graduate fresh from Oxford whoseemed a little bewildered about what he was supposed tobe doing. I had preceded him by a few years but wasimmediately aware of the presence of a superior intellect. Igave up competing after we both decided to go in for theCooling Prize. At least I am a quick learner in somerespects. I think that these early years were very formativefor him. He found himself in a situation which suited him; agroup of people looking for theory to fit their excellent fielddata, and a leadership which encouraged self development.Mark soon found his problems to solve and after a shortwhile a subject that he could develop as his own, thebehaviour of piles. Tonight we have seen how this hasblossomed, initially under the mentorship of John Burlandand the late Peter Wroth, of course, two former RankineLecturers. I think that both of them instilled in him what hehas so ably demonstrated tonight, the importance of the‘why?’ when deciding on the ‘how?, but also the art ofdistilling what is a very complex problem through a seriesof logical steps, to something that can be understood.

Not once this evening have we seen an equation that ismore than half a line long. Not once have we seen a formulawith a multitude of variables. Mark’s hallmark is that hebreaks things down into a series of logical steps and thenuses the building blocks he creates to examine the problemin its entirety. Although this is an academic approach, heuses such models to solve practical problems, and applieshis engineering judgement and observation to challenge thelogic of both other peoples’ theories and the way in whichdesign is carried out. This is a very powerful combinationwhich he has obviously used to good effect in advising awide range of clients.

Mark expressed some concern to me that in part of hislecture he is challenging some of the assumptions made byothers in the derivation of design methods, and that theymay take umbrage as they are so close to home. I assured

874 RANDOLPH

him, probably unwisely that, as true scientists, trying torefine their own understanding of their data, they wouldwelcome such a side swipe. But maybe, as a non-academic Ihaven’t understood how things really work. There are manymore PhDs to be had from the subject, and I’m sure thateven those closest to the subject have seen something herethis evening which may stimulate a further thought.

I think that Einstein was right though about psyche of theScientist. I hadn’t realised before that even Einstein, likeeveryone else, only presented his best data. . .otherwise hewouldn’t be sceptical about it.

I tried to find a quote to match Mark’s, and decided that Ishould look for something from another adopted son ofAustralia, the late Spike Milligan, who had a theory aboutthe origin of rain. He wrote. . ..

There are holes in the sky where the rain gets in, but theyare ever so small. That’s why the rain is thin.

This is an acute observation, but not backed up by thescientific theory. However, it is an empirical relationship

between the sky and the size of the rain drops which alwaysworks.

Tonight Mark has more than achieved what we have cometo expect from a Rankine lecture. He has shown us the fruitsof his lifetimes work, challenged us to think further aboutthe assumptions that we make when we select a particulardesign method and shown us ways of doing better. He hasalso demonstrated through some fascinating examples howan understanding of the science and thus an appreciation ofthe mechanisms controlling soil–structure interaction canresult in finding better solutions to geotechnical problems,backed up of course by a means of verifying the result. Hehas been and will continue to be an inspiration to many bothwithin his own highly successful teams and to the geotechni-cal profession worldwide.

I would like you now to join me in giving our heartfeltthanks to Mark for all the blood, sweat and tears (probablylargely sweat) that he has put in to preparing and deliveringthe 43rd Rankine Lecture.

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