An Empirical Method For Analysis Of Load Transfer And .
Geotech Geol Eng (2010) 28:483–501DOI 10.1007/s10706-010-9307-7ORIGINAL PAPERAn Empirical Method for Analysis of Load Transferand Settlement of Single PilesJ. R. Omer R. Delpak R. B. RobinsonReceived: 16 August 2006 / Accepted: 1 February 2010 / Published online: 24 February 2010Ó Springer Science Business Media B.V. 2010Abstract An empirical method is developed forestimating the load transfer and deformation ofdrilled, in situ formed piles subjected to axial loading.Firstly, governing equations for soil–pile interactionare developed theoretically, taking into accountspatial variations in: (a) shaft resistance distributionand (b) ratio of load sharing between the shaft andbase. Then generic load transfer models are formulated based on examination of data from 10 instrumented test piles found in the literature. Thegoverning equations and load transfer models arethen combined and appropriate boundary conditionsdefined. Using an incremental-iterative algorithmwhereby all the boundary conditions are satisfiedsimultaneously, a numerical scheme for solving thecombined set of equations is developed. The algorithm is then developed into an interactive computerprogram, which can be used to predict the loadsettlement and axial force distribution in piles. Todemonstrate its validity, the program is used toanalyse four published case records of test piles,which other researchers had analysed using thefollowing three computationally demanding tools:(a) load transfer (t–z), (b) finite difference and (c)finite element methods. It is shown that the proposedmethod which is much less resource-intensive, predicts both the load-settlement variation and axialforce distribution more accurately than methods:(a–c) above.J. R. Omer (&)Faculty of Engineering, Kingston University,London KT1 2EE, UKe-mail: j.r.omer@kingston.ac.ukLLcR. Delpak R. B. RobinsonFaculty of Advanced Technology,University of Glamorgan,Wales CF37 1DL, UKKeywords Piles Settlement Load-transfer Cohesive soils Weathered rocksList of Symbols½fs ðzÞ z¼LA½fs ðzÞ z¼0BCDbDsEpEiEuILs0.001%LCompound parameter (see Eq. 6b)Diameter of pile baseDiameter of pile shaftElastic modulus of pile materialInitial tangent modulus of soilUndrained modulus of deferomationof soilIntercept on the Ps/HDs axis of thebest line of fit of Ps/HDs versus HDsTotal pile lengthLength of permanent casing for pile(or length of upper pile segment notembedded in soil or passing throughweak soil)Length of pile shaft transferring loadto soil by means of shaft resistance123
s(z)fubfusj(k1k2limsnr2w(z)z123Geotech Geol Eng (2010) 28:483–501Bearing capacity factorSPT blow count for the ith stratumalong pile shaftAverage value of NSPT along pileshaft, after weighting with respect tostrata thicknessNumber of blow count in standardpenetration testMobilised base resistance of pileApplied pile head loadMobilised shaft resistance of pileUltimate pile base resistanceMaximum pile head capacityMaximum shaft resistance of pileAxial force in pile at depth z belowpile head levelTangent slope, at the origin, of thecurve of unit base resistance versusbase movement of pileUndrained shear strength of soil atdepth zCompression of pileUnit base resistance of pile mobilisedat a particular loading stageMaximum unit shaft resistance at acertain depth along pile (Kim et al1999 and Balakrishnan et al 1999)Average unit shaft resistance of pilemobilised at a particular loading stageLocal unit shaft resistance mobilisedat depth z along pileMaximum unit base resistance of pileMaximum unit shaft resistance of pileNumber of Db increments applied inSEM analysis programEmpirical constants in the expressionfor n (Eq. 15a)Thickness of ith stratum along pileshaftGradient of the regression trend lineof Ps/HDs versus HDsCritical shaft settlement divided bypile shaft diameter (i.e., Dsc/Ds)Correlation coefficient in linearregressionPile movement at depth z below pilehead levelDepth below pile head levelaa 1, a 2, a 3, a 4azDbDbcDhDsDscs1, s2, s3sbssxwAdhesion factorCompound coefficients in themodelled functions: fs(z) and P(z)fs(z)/cu(z)Pile base displacementCritical pile base displacement(Db corresponding to Pub)Pile head settlementAverage shaft settlementCritical shaft settlement (Dscorresponding to Pus)Shear strength of 1st, 2nd, 3rd soillayer respectively along pile shaftShear strength of soil at pile baselevelAverage shear strength of soil aroundpile shaft, weighted with respect tostrata thicknessValue of z/L at the point of maximumfs(z)Mobilised shaft resistance divided bypile head load1 IntroductionOne of the most widely used methods of predictingthe load–displacement behaviour and axial forcedistribution in vertically loaded piles is the loadtransfer (t–z) analysis (Coyle and Reese 1966). Themethod has three main advantages in that it can beapplied to:1.2.3.multi-layered strata having different load transfercharacteristicspiles having variable cross sectional area withdepthpile materials having non-linear stress/strainrelationships.The major disadvantages are that:(a)measurement of t–z relationships for eachstratum penetrated by a pile is not only difficultbut also expensive(b) t–z relations are highly site-specific and therefore extrapolation of results between differentsites is seldom successful(c) Although the modified t–z approach suggestedby Kim et al. (1999) accounts for effects of
Geotech Geol Eng (2010) 28:483–501485displacement of a given soil layer due to shearon the other layers, it assumes that soil is anelastic and homogeneous material.(a)PhInitial pile head leveland soil surface(b)ΔhTherefore, in order to contribute a simpler and morepractical solution, an opportunity has been taken todevelop an alternative method that can be solvedusing a spreadsheet and requires basic soil parametersfrom a standard site investigation report. By linkingbasic concepts in soil–pile interaction theory withempirical models derived from actual pile tests, theproposed method aims to represent pile behaviourrealistically. At present, the formulation is appropriate to bored piles formed in cohesive soils andweathered cohesive rocks, although there is scope toextend the method to accommodate other pile typesinstalled in different ground profiles. The validity ofthe proposed method is tested against published loadtest data from instrumented piles.Figure 1a is a schematic illustration of an unlined pileof length L and diameter Ds supporting a vertical headload Ph, through mobilised shaft and base resistancesPs and Pb, respectively. If the vertical displacementsof the pile head and base are denoted Dh and Db,respectively and the compression of the pile is ep, thenignoring the weight of the pile, it is obvious that:ð1aÞandð1bÞDh ¼ ep þ DbConsider an element of length dz, located at depthz below the pile head level as illustrated in Fig. 1b.The shaft resistance dPs mobilised, along the segmentdz, equals the change in axial force dP(z). The unitshaft resistance fs(z), mobilised along dz, is related toP(z) through the equation:dPðzÞ¼ pDs fs ðzÞdzzLPs(z)Ps(L-ep)P(z)δzδPsð1cÞwhere the negative sign indicates that P(z) decreasesas z increases.If the elastic modulus of the pile is denoted Ep thenignoring any vertical soil movements, the displacement w(z) of the pile element at depth z can beexpressed as follows:P(z) δP(z)ΔbPile diameter DsElastic modulus E pPbPbFig. 1 Forces and displacements in a loaded pile-soil system:a schematic representation, b forces on a pile element in‘‘equilibrium’’dwðzÞ 4PðzÞ¼dzpD2s Ep2 Basic Equations for Pile–Soil InteractionPh ¼ Ps þ Pb ;Phð2ÞEquation (2) is differentiated with respect to z anddP(z)/dz is substituted for, using Eq. (1), to yield thefollowing:d2 wðzÞ4¼fs ðzÞ2dzD s Epð3ÞThe above equations, which are based on forceequilibrium and displacement compatibility considerations, are valid irrespective of the pile type andsoil classification.The variation of fs(z) with depth is influenced bynumerous factors, most of which are not yet wellunderstood. The factors include not only the pile andsoil properties but also:(1) pile–soil interface geometryand slip characteristics, (2) technique used for pileinstallation, (3) stresses acting on the pile–soil interface and (4) method and rate of pile loading. Therefore,it is helpful to develop a more realistic fs(z) relationon the basis of observed performance of loaded piles.This is carried out in the following section.3 Equations for Idealised Loaded PileFigure 2a depicts normalised load transfer curves inthe form of shear strength reduction factor, az,123
486Geotech Geol Eng (2010) 28:483–50100.250.50.751Ph(b)Shear strength reduction factor α z (dimensionless)(a)0Relative depth Lr (dimensionless)(c)fs (z 0)1.25Ph-0.1-0.2ωL-0.3-0.4L-0.5PsPs /Ph ψfs (max)Pb (Ph-Ps ) Ph(1-ψ)-0.6-0.7-0.8-0.9Pile S1Pile MT1Pile S2Pile S3Pile PR2Pile PR3Pile MTOfs( z L ) Afs( z 0 )-1fs (z L)PsPbPbFig. 2 Load transfer characteristics of bored cast in situ piles: a observed: relative depth versus shear strength reduction factor (afterReese 1978), b modelled shaft resistance distribution, c modelled axial force distributionversus relative depth, Lr, for seven instrumentedtest piles installed in clay, at the stage of mobilisation of maximum shaft resistance. The datawere reported by Reese et al. (1976), Reese (1978)and Wright and Reese (1979). The definitions of azand Lr are as follows: (1) az fs(z)/cu(z), wherecu(z) is the undrained cohesion at level depth z and(2) Lr z/L, where z depth below ground surface and L pile length or, for piles havingenlarged bases, L length of pile minus basethickness.Bowles (1996), Wright and Reese (1979) andSchmidt and Rumpelt (1993) suggested that loadtransfer curves such as those shown in Fig. 2a can bemodelled using parabolic functions. Based on thissuggestion, fs(z) can be a represented as follows:fs ðzÞ ¼ a1 z2 þ a2 z þ a31.2.3.4.ð4Þin which a1, a2 and a3 are constants for a given Phvalue. Combining Eqs. (1c) and (4), the followingensues: a a21 3PðzÞ ¼ pDsz þ z 2 þ a 3 z þ a4ð5aÞ32where a4 is also constant for a given Ph value.Typical curves modelling fs(z) and P(z) variations areillustrated in Fig. 2b and c, respectively. Let x bethe value of z/L corresponding to the location of fs123(max) as shown in Fig. 2b. Denoting Ps/Ph by w,expressions for a1 to a4 can be derived using thefollowing boundary conditions:5.dfs(z)/dz 0 when z xL, representing thelocation of fs (max). The parameter x is allowedto vary with the ratio, Ph/Puh, of the applied loadto maximum head loadP(z) Ph when z 0, describing force equilibrium at the pile head levelP(z) Pb (1 - w)Ph when z L, describingforce equilibrium at the pile toe levelThe shapes of the fs(z) and P(z) curves areadditionally controlled by the ratio [fs(z)]z L/[fs(z)]z 0, where [fs(z)]z L and [fs(z)]z 0 are thevalues of fs at the bottom and top of pile shaft,respectively. At present, the equations are notvalid if [fs(z)]z 0 \ 0, i.e. the case of negativeskin friction along the upper segment of the pile.Infact the method is confined to positive shaftfriction cases only.The ratio [fs(z)]z L/[fs(z)]z 0 is allowed to varywith Ph/Puh so that positive values of fs(z) andP(z) are guaranteed for all values of z as Ph/Puhincreases from zero to unity.Letting A [fs(z)]z L/[fs(z)]z 0, Omer (1998)determined based on parametric studies of fs(z) andP(z) for different piles, that generally:
Geotech Geol Eng (2010) 28:483–501Ph 0:4PuhPhfor 0:4\ \0:6PuhPhfor 0:6Puhforð5bÞHowever, in the present work, more precise valuesof A, for given Ph/Puh, are determined iterativelywithin the same algorithm used to satisfy boundaryconditions (1–4).From the above boundary conditions, the following solutions are explicitly obtained:8Ph a4 ¼ pD s wP h a1 ¼pDs LCð6aÞ a2 ¼ 2xLa1 2 : a3 ¼ wPh L a1 La22pDs L 3in which 2 822 L C ¼ ð1 AÞþ L3 2xL xL2 ð1 AÞ if[fs ðzÞ z¼0 ¼ 0; or C ¼ ðB2 A L2 Þ þ L2 2xLðBA LÞ xL2 3ð1 AÞð1 AÞ :if[fs ðzÞ z¼0 [ 0Plots of fs versus depth for pile S1, as reported byReese (1978), are shown in Fig. 3. The undrainedshear strength varies approximately linearly withdepth as shown. Tracing the locus of maxima of thecurves in Fig. 3, values of x corresponding to eachPh value are estimated. For the pile S1, Wright andReese (1979) reported Pus to be 859kN, although Puband Puh were not mobilised. Therefore, the authorshave extrapolated Puh and Pub to be 1720kN and643 kN respectively using the method proposed byChin (1972). As recommended by Darrag (1987), areduction factor of 0.9 has been applied to the resultsfrom Chin’s (1972) method.In Fig. 4, plots of x versus Ph/Puh, Ps/Pus and Pb/Pub for pile S1 are presented. Although the data islimited to one pile and also low values of Ps and Pbare not available, it appears that there is an approximately linear relationship between x and Ph/Puh(and also between x and Ps/Pus) giving r2 0.97.However, there is no discernible relation between xand Pb/Pub. So the following expression can bemodelled from Fig. 4:x ¼ b1ð6bÞwhere B WL and W a constant, the effect of whichis discussed later.For most piles, the top of the shaft segment is atground level, where the horizontal effective stress iszero therefore so that [fs(z)]z 0 0 and therefore thefirst option in Eq. (6b) is used. On the other handsome piles might be permanently sleeved over partof their length from ground level to a certain depth,thereby preventing contact with soil. In such a casethe top of the embedded segment (the lowersegment) is considered as depth z 0 and has[fs(z)]z 0 [ 0 because the horizontal effective stressacting there is greater than zero. In this case thesecond option in Eq. (6b) is used, however anappropriate value of B must be judged, throughselectingW. Fortunately, as discussed later, the shapeof the shaft friction distribution curve has almostnegligible effect on the calculated load-settlementcurve so that the parameters {A,x,W} are relativelyless important.Phþ b2Puhð7Þwhere b1 0.2902 and b2 0.3375. While it isaccepted that much more data is needed to verify thetrend in Fig. 4, Eq. (7) is consistent with theDeveloped shear stress (tsf)00.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 11.1 1.2 1.3 1.4 1.50-5-10Depth (ft)8 0:7 A ffi 2:0 : 3:5487Head load10 T-1520 T50 T70 T140 T-20120 T[1 ft 0.305 m; 1 tsf 95.8 kN/m2]Shear strengthVs depthLocus ofmaxima-25Fig. 3 Distribution of mobilised shaft resistance for pile S1Reese et al. (1976) [1 ft 0.305 m; 1 tsf 95.8 kN/m2]123
488Geotech Geol Eng (2010) 28:483–50111Ds ¼LDepth ratio, ω (non-dimensional)Pile S1 in clay0.8ω versus Pb/Pubω 0 .2902 4 a1 4 a2 3 a3 2z þ z þ z þ a4 z þ C1 dzDs Ep 12620Ph 0 .3375Puhð10Þ0.6ThusDs ¼0.4ω versus Ps/Pus2r 0.970.2000.20.40.60.81Load ratios, P h/Puh, Ps/Pus and Pb/Pub (non-dimensional)Fig. 4 Modelled trend for x versus Ph/Puh (pile S1)observations by Schmidt and Rumpelt (1993) andZhu and Chang (2002) that the centre of frictiontransfer generally moves downwards with increasingpile head load.Using Eq. (5a) to substitute for P(z) in Eq. (2)results in: dwðzÞ4 a1 3 a2 2¼z þ z þ a3 z þ a 4dzD s Ep 32It should be borne in mind that Eqs. (4)–(11) areonly defined when, for a given Ph value, w is knownfrom which a1, a2 and a3 can be calculated. This canbe achieved by developing an iterative scheme thatsatisfies Eqs. (1–11) simultaneously thereby enablingw to be determined. These equations describe theequilibrium and deformation of a pile assumedisolated from the soil and acted upon by skinresistance forces and discreet forces at the extremities. The next section focuses on the idealised soilcomponent and considers the state of shear resistancemobilisation due to progressive deformation.4 Behavioural Models for Soilð8Þwhere C1 is a constant, which can be determined fromthe boundary condition w(z 0) Dh (Db ? ep), inwhich ep is the compression of the pile and Db is thebase movement (see Eq. 1b). Thus, C1 is found to be:ð9bÞInvoking the final boundary condition, w(z L) Dbin Eq. (9a) results in 4Ds Ep Db a1 4 a2 3 a3 2ep ¼ L L L a4 LDs Ep41262 Dbð9cÞFrom Eq. (9a), the average shaft displacement Dscan be interpreted as the mean value of w(z) over theinterval from z 0 to z L and is therefore given by123ð11Þ4.1 Shaft Load TransferEquation (8) is integrated with respect to z to yield: 4 a1 4 a2 3 a3 2z þ z þ z þ a4 z þ C1wðzÞ ¼ð9aÞDs Ep 1262 D s Ep ep þ Db4 4 a1 4 a2 3 a3 2 a4L þ L þ L þ L þ C1Ds Ep 6024622r 0.96ω versus Ph/PuhC1 ¼ZLColumns 1–5 and 9 of Table 1 present data frominstrumented bored piles reported by Wright andReese (1979) (clay) and by Omer(1998) (weatheredmudstone). Graphs of shaft resistance Ps versusaverage shaft displacement Ds are shown in Fig. 5aand b for the piles in clay and mudstone, respectively.Plotting Ps/HDs versus HDs as shown in Fig. 6a andb, it is found that the trends of variation arereasonably linear, and can therefore be representedas follows:pffiffiffiffiffiPspffiffiffiffiffi ¼ ms Ds þ Ið12aÞDswhere ms slope and I vertical axis intercept.On examining plots Ps versus Ds curves in Fig. 5aand b plus curves (not shown due to space limits)from the other piles in the present database, approximate formulae for expressing maximum shaft resistance Pus and critical shaft settlement Dsc (i.e., thevalue of Ds corresponding to Pus) in terms of ms and Iwere found as follows: Pus & [-I2/(4 ms)] andDsc & [I/(2 ms)]2. Using these relationships, the
Geotech Geol Eng (2010) 28:483–501489Table 1 Back-analysed Pus and Dsc values for bored piles in clay and in mudstoneReferencePileno.Site locationPiledimensions(m)Shaft: ss (kN/ Pus -I2/m2)(4 ms)(MN)Dsc [I/(2 ms)]2(mm)Wright andReese(1979)SASan Antonio,Texas, USALc* 3)20.210.97(20.96)13.8736.200.89(15.00) (40.50)S1Houston, Texas,USAL@s 8.174D#s 0.762Lc 0.000Ls 7.040Correlationcoeff. r2Soil typeClayDs 0.762S4T1 Houston, Texas,USALc 0.000Ls 13.725Ds 0.762Lc 0.000MT1 Austin, Texas,USALs 7.259TP3Lc 20.230912bLs 8.780Ds 0.900(NSPT 152)Lc 18.640486bLs 12.000(NSPT 81)Ds 0.762Omer(1998)TP5Cardiff, UKCardiff, UK8.55(8.77)32.250.99(35.66)Ds 0.900TP6Cardiff, UKLc 20.380714bLs 11.130(NSPT 119)WeatheredMerciamudstones12.2331.270.99(13.00) (28.00)Ds 0.900#Lc* length of casing, L@s pile length embedded in soil, Ds Pile shaft diameteranot directly availablebValues correlated from mean NSPT along shaft using ss 6NSPTEmboldened bracketed numbers are observed values of Pus and Dsccalculated values of Pus and Dsc are shown inColumns 6 and 7, respectively of Table 1. Thecorresponding correlation coefficients, r2, given incolumn 8, lie in the range 0.89–0.99, which isencouraging. Also, it is seen that the calculated Pusand Dsc are close to the measured ones (emboldenedand bracketed values in Table 1). Therefore, forDs B Dsc, the relationship between Ps and Ds and canbe approximated as follows:Ps Pus pffiffiffiffiffi 2Puspffiffiffiffiffi ¼Ds þ pffiffiffiffiffiffiffið12bÞDscDsDscUsing Eq. (12b), the calculated curves of Ps versusDs are included in Fig. 5a and b for comparison withmeasured pile data. As can be seen, there is goodagreement between the measured and modelledvariations.Expressing Dsc in general terms as Dsc nDs,where n depends on the properties of the soil, pile andsoil–pile interface, Eq. (12b) can be re-written �ffiffiffi!DsDsPs ¼ Pus2 ð13ÞnDsnDsDespite the simplicity of the foregoing equation, asearch of literature shows that so far there is noreliable formula for determining the value of n for agiven pile situation. This is because n is influenced byseveral factors linked to (1) pile dimensions andstiffness relative to soil, (2) pile–soil interfaceproperties, (3) the strength and stiffness of the soilalong the shaft and beneath the pile base, (4) the rateof application of pile head load, (5) cyclic loading.123 pa
transfer (t–z) analysis (Coyle and Reese 1966). The method has three main advantages in that it can be applied to: 1. multi-layered strata having different load transfer characteristics 2. piles having variable cross sectional area with depth 3. pile materials having non-linear stres
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