Rationality of load transfer approach for pile analysis

Rationality of load transfer approachfor pile analysis

Wei Dong Guo*, Mark F. RandolphGeomechanics Group, The University of Western Australia, Perth, WA, Australia

Received 23 May 1997; received in revised form 4 June 1998; accepted 8 June 1998

Abstract

New closed form solutions accounting for non-homogeneous, elastic plastic or visco-elasticsoil properties have been established for a single pile subjected to vertical loading, and forvertically loaded pile groups. The solutions are based on load transfer approach, with the

pile±soil interaction being characterised by springs distributed down the pile shaft and at thepile base. As such, the accuracy of the solutions lie in the load transfer factors linking thegradient of the load transfer curves with elastic modulus values. This paper presents an

extensive investigation into the load transfer approach, by comparing the new closed-formsolutions for a pile embedded in an elastic medium with rigorous numerical analysis. By thismeans: 1. The rationality of the load transfer approach has been extensively explored; 2. Load

transfer factors have been calibrated against the numerical solutions to account for a numberof factors, such as the ratio of the depth to the underlying rigid layer and the pile length, thedegree of soil non-homogeneity, Poisson's ratio for the soil, and the pile slenderness ratio; 3.Expressions for estimating the load transfer factors have been provided. With the new load

transfer factors, the closed form solutions may be applied with high accuracy over a very widerange of pile geometries and soil conditions. # 1998 Elsevier Science Ltd. All rights reserved..

Nomenclature

The following symbols are used in this paper:Roman

A Coe�cient for estimating shaft load transfer factorAg Constant for soil shear modulus distributionAh Coe�cient for estimating `À'', accounting for the e�ect of H=L

Computers and Geotechnics 23 (1998) 85±112

0266-352X/98/$Ðsee front matter # 1998 Elsevier Science Ltd. All rights reserved.

PII: S0266-352X(98)00010-X

* Corresponding author at Department of Civil Engineering, National University of Singapore, 10

Kent Ridge Crescent, Singapore 119260, Singapore.

Aoh The value of Ah at a ratio of H=L � 4Ap Cross-sectional area of an equivalent solid cylinder pileB Coe�cient for estimating shaft load transfer factorCv z� � Function for assessing pile sti�ness at a depth of z, under vertical loadingCvo Limiting value of the function, Cv z� � as z approaches zeroCvb Limiting value of the function for the ratio of base and head loads as z

approaches zeroCl Coe�cient for estimating `À'', accounting for the e�ect of ld ro� � Diameter (radius) of a pileEp Young's modulus of an equivalent solid cylinder pileG Elastic shear modulusGb Shear modulus at just beneath pile base levelGL Shaft soil shear modulus at just above the pile base levelH The depth to the underlying rigid layerIm; Imÿ1 Modi®ed Bessel functions of the ®rst kind of non-integer order, m and

mÿ 1, respectivelyKm Modi®ed Bessel functions of the second kind of non-integer order, mKmÿ1 Modi®ed Bessel functions of the second kind of non-integer order, mÿ 1L Embedded pile lengthm 1/(2+n)n Power of the shear modulus distribution, non-homogeneity factorPb Load of pile baseP z� � Axial force of pile body at a depth of zPt Load acting on pile headw Local shaft deformationw z� � Deformation of pile body at a depth of zwb Settlement of pile basewt Pile-head settlementz Depthzt An in®nite small depth

Greek

� Shaft load transfer factorl Relative sti�ness ratio between pile Young's modulus and the initial soil

shear modulus at just above the base level, Ep=GL

�p Poisson's ratio of a pile�s Poisson's ratio of soil�b Pile base shear modulus non-homogeneous factor, GL=Gb

�g Ratio of the average soil shear modulus over the pile embedded depth tothe modulus at depth L

�o Shear stress on pile±soil interface�oave Average shear stress on a pile±soil interface over all the pile length�v Ratio of shaft and base sti�ness factors for vertical loading

86 W.D. Guo, M.F. Randolph/Computers and Geotechnics 23 (1998) 85±112

! Pile base shape and depth factor!h Coe�cient for estimating ``!'', accounting for the e�ect of H=L!oh The value of !h at a ratio of H=L � 4!� Coe�cient for estimating !, accounting for the e�ect of �s!o� The value of !� at a ratio of �s � 0:4

1. Introduction

Load transfer analysis is an uncoupled analysis, which treats the pile±soil inter-action along the shaft and at the base as independent springs [1]. The sti�ness of theelastic springs, expressed as the gradient of the local load transfer curves, may becorrelated to the soil shear modulus by load transfer factors [2]. Given suitable loadtransfer factors, the analysis provides a close prediction to a continuum basednumerical analysis as reported by Randolph and Wroth [2] for piles in an in®niteelastic half-space. The question is whether the load transfer factors are signi®cantlya�ected by a number of features: (a) non-homogeneous soil pro®le, (b) soil Poisson'sratio, (c) pile slenderness ratio, and (d) the relative ratio of the depth of any under-lying rigid layer to the pile length. How these features a�ect the ®nal pile predictionhas not yet been fully explored. In addition, the assumption of proportionality ofload transfer gradient to the soil shear modulus, uniformly down the pile, has notbeen rigorously justi®ed.Ideally, load transfer analysis should give identical results to that of a continuum

based numerical analysis. Therefore, continuum based analysis should be used tocalibrate the load transfer factors.The purpose of this paper is devoted to (1) investigating the adequacy of the load

transfer approach; (2) calibrating the load transfer factors by means of continuumbased analysis using the program FLAC [3] considering the above mentioned (a) to(d) conditions. The results are then expressed in formulae for the load transfer fac-tors, derived by means of least square ®ts between the two forms of analysis.The deduced load transfer factors are thereafter adopted in the new closed form

(CF) solutions to estimate pile-head sti�ness, and the ratio of pile base and headload over a wide range of slenderness ratios, ®nite layer ratios, soil Poisson's ratios,non-homogeneity and relative pile±soil sti�ness factor, with the purpose to re-examine the suitability and the accuracy of the load transfer approach throughcomparison with previous publications and the current FLAC analysis.

2. Rationality of load transfer approach

2.1. Closed form solutions

Closed form solutions for a pile in an elastic non-homogeneous soil have beengenerated by Guo [4], Guo and Randolph [5] based on the load transfer approach(see Fig. 1). In the solutions, the following three conditions are adopted:

W.D. Guo, M.F. Randolph/Computers and Geotechnics 23 (1998) 85±112 87

1. The shear modulus is assumed as a power of depth, z, according to

G � Agzn �1�

where n is the power for the pro®le, and Ag is a constant giving the magnitudeof the shear modulus. The base shear modulus jump is described by�b � GL=Gb (referred to as the end-bearing factor), where GL, Gb are the valuesof shear modulus of the soil just above the level of the pile tip, and beneath thepile tip.

2. The shaft displacement, w, is related to the local shaft stress, �o, and shearmodulus, G, by Randolph and Wroth [2]

w � �oroG� �2�

where ro is pile radius, and � is the shaft load transfer factor.

3. The base settlement is estimated through the solution for a rigid punch actingon an elastic half-space, as suggested by Randolph and Wroth [2]

wb � Pb 1ÿ �s� �!4roGb

�3�

where wb is the base settlement, Pb is the mobilised base load, ! is the base loadtransfer factor, and �s is Poisson's ratio for the soil.

Fig. 1. Pile and soil properties.


Following the above three conditions, taking the shaft load transfer factor, � asindependent of depth, the displacement, w, and load, P, of a pile at any depth, z, canbe expressed, respectively, as [4,5]

w z� � � wbz

L

� �1=2 C3 z� � � �vC4 z� �C3 L� �

� ��4�

P z� � � ksEpApwb

L

z

L

� � 1�n� �=2 C1 z� � � �vC2 z� �C3 L� �

� ��5�

where Ep, Ap are Young's modulus and cross-sectional area of an equivalent solidcylindrical pile, and L is the pile embedded length. The functions C1 to C4 are givenby

C1 z� � � ÿKmÿ1Imÿ1 y� � � Kmÿ1 y� �Imÿ1 �6�

C2 z� � � KmImÿ1 y� � � Kmÿ1 y� �Im

C3 z� � � Kmÿ1Im y� � � Km y� �Imÿ1C4 z� � � ÿKmIm y� � � Km y� �Im

with the modi®ed Bessel functions Im y� �, Imÿ1 y� �, Kmÿ1 y� �, and Km y� � being writtenas Im, Imÿ1, Kmÿ1, and Km at z � L, and m � 1= 2� n� �. The ratio �v in Eqs. (4) and(5) is given by

�v � 2��2p

� 1ÿ vs� �!�b

��

l

r�7�

where l is the relative pile±soil sti�ness, Ep=GL. The sti�ness factor, ks is providedby

ks � L

ro

��2

l�

s�8�

The variable y is given by

y � 2mksz

L

� �1=2m�9�

The pile-head sti�ness may be estimated by

Pt

GLwtro� �

��2l�

sCvo �10�


where the coe�cient Cvo is the limiting value of Cv z� � as z and y approach zero (forthe depth at the ground surface). Therefore,

Cvo � Limz;y!0

Cv z� � � Limz;y!0

C1 z� � � �vC2 z� �C3 z� � � �vC4 z� �

z

L

� �n=2 �11�

The ratio of load at the pile base (Pb) to that at the pile head (Pt) is

Pb

Pt� 4

1ÿ �s� ��!1

ks�blCvb

L

ro�12a�

where Cvb is the limiting value of the following function as z and y approach zero:

Cvb � Limz;y!0

C1 z� � � �vC2 z� �C3 L� �

� �z

L

� � 1�n� �=2 �12b�

where Pt, Pb are the pile-head and base load, respectively.

2.2. Calibration procedures

The accuracy of the closed form (CF) solutions lies in the two critical load transferfactors, � and !, which are essentially dependant on each other. As shown later, theshaft load transfer factor can reasonably be assumed as a constant with depth. It isthus possible to back-analyse the two factors in the CF solutions through compar-isons with more rigorous numerical (FLAC) analyses. The base behaviour will becalibrated uniquely against Eq. (3), while the shaft load transfer factor can be cali-brated against two independent non-dimensional ratios.

(a) pile-head sti�ness, de®ned as Pt= GLwtro� �;(b) the ratio of base and head loads, Pb=Pt.

If the load transfer approach is accurate compared with the FLAC analysis, thenidentical values of � and ! should be obtained irrespective of which ratio is used forthe calibration.The calibration procedure can be detailed as:

1. FLAC analysis is performed for a given set of soil and pile parameters.2. The pile-head sti�ness, ratio of base to head loads, and base sti�ness (Pb=wb)

are obtained from the FLAC analysis. With these results, the value of ! maybe deduced from Eq. (3).

3. For the same problem, the pile-head sti�ness or the load ratio, Pb=Pt may beobtained from Eq. (10) or Eq. (12a) using an estimated value of �.

4. The estimated initial value of � is adjusted iteratively, so that the estimatedpile-head sti�ness or load ratio matches (within a desired accuracy) thatobtained from the FLAC analysis. Therefore, � is obtained.


This process has been ful®lled through a purpose written program in FORTRAN77. The calibration against pile-head sti�ness in step (4) will be referred to asmatching pile-head sti�ness. To examine the accuracy of the load transfer approach,calibration against the load ratio between pile base and head has been performed aswell, and which is later termed as matching ``load ratio''.

2.3. FLAC analysis

The continuum based ®nite di�erence (FLAC) analysis has been used to explorethe validity of the load transfer approach, and to assess optimum values of loadtransfer factors, � and !. Before undertaking parametric studies, an extensive sensi-tivity study was carried out in order to assess the required ®neness and lateral extentof the FLAC mesh. Full details of this study may be found in Guo (1997), and onlythe main ®ndings are summarised here:

. The width of the mesh was taken as the maximum of 2.5L or 75ro.

. The base boundary was fully ®xed, but the lateral boundary was constrainedonly in the radial direction, except when exploring very deep soil layers(H=L > 4), where it was found necessary either to ®x (fully) the lateralboundary, or extend the mesh much further laterally.

. The mesh adopted was 21� 50, as adopted by Guo and Randolph [5], exceptfor evaluating the base load transfer factor, !, where a ®ner mesh of 21� 100was adopted.

. It was found that a soil layer depth of H � 4L represented a limit beyondwhich the soil layer may be taken as in®nitely thick.

. Poisson's ratio of the pile has a very minor e�ect (see Table 1), and a value of�p � 0:2 has been adopted.

. Overall, FLAC tends to give a slightly higher pile-head sti�ness than theboundary element approach (which has been used most widely for pile analy-sis). Table 2 shows comparisons among boundary element analysis (BEM, [2]),variational method (VM, [6]) and the FLAC analysis for single piles inhomogeneous soil (n � 0, H=L � 4). The FLAC and BEM analyses were basedon a Poisson's ratio of 0.4, while in the VM analysis, the soil Poisson's ratiowas selected as 0.5. Since a higher Poisson's ratio generally leads to higher

Table 1

Comparison of the e�ect of Poisson's ratio of the pile (�p � 0 and 0.2, �s � 0:49, L=ro � 40, l � 1000)

n 0 0.25 0.5 0.75 1.0

Pt

GLwtro

59:08

59:04

51:93

51:91

46:64

46:63

42:62

42:61

39:56

39:55

wt

wb

1:64

1:64

1:60

1:60

1:58

1:58

1:55

1:55

1:53

1:53

Pb

Pt

7:08

7:09

8:75

8:76

10:5

10:51

12:07

12:08

13:68

13:69

Numerator and denominator for �p � 0:2 and 0, respectively.


sti�ness (as shown later), the sti�ness from FLAC analysis is slightly higherthan the predictions from the other approaches. Table 3 shows a further com-parison between FEM analysis [2], and the FLAC analysis for both homo-geneous (n � 0) and Gibson soil (n � 1). Note that the results shown in Table 3were based on a constant shear modulus below the pile base level.

2.4. Variation of shaft load transfer factor with depth

FLAC analysis is utilised to ®nd the base and shaft load transfer factors. The basefactor, ! is directly back-®gured by Eq. (3), in which the base load, Pb was estimatedthrough linear extrapolation from the stresses of the last two segments from the baseof the pile, and the base displacement,wb was taken as the node displacement of the pilebase. The shaft load transfer factor has been evaluated in terms of the shaft load transfermodel and the closed form solutions for single pile as described, respectively, below.

1. Variation of � with depth: with the shaft shear stress and displacement along apile obtained by FLAC analysis, the shaft load transfer factor has been back-®gured by using Eq. (2), as represented by ``FLAC'' in Fig. 2(a).

2. Average value of � over the pile embedded depth: taking � as a constant withdepth, the value of the factor, � has been back-®gured in light of the calibration

Table 2

Comparison of FLAC analysis with other approaches (n � 0)

Pt�GLwtro

FLAC

BEM

VM

69:7065:70

72:2 ��

64:3861:3

65:1

53:6052:00

54:9

36:5136:80

38:7

wt�wb

FLAC

BEM

VM

1:051:05

ÿÿ

1:181:12

1:19

1:551:49

1:59

2:922:66

3:25

Pt � �GLwtro

FLAC

BEM

109:0

102:2

85:0

85:2

61:6

61:6

36:2

38:0

wt � �wb

FLAC

BEM

1:18

1:16 1:542:96

2:68

7:99

6:75

l � Ep=GL

ÿ �10,000 3000 1000 300

*L=ro � 40, **L=ro � 80, ***rigid pile. VM analysis was based on �s � 0:5, while BEM and FLAC ana-

lyses were based on �s � 0:4. Also for FLAC analysis, H=L � 4.

Table 3

Comparison between FEM and FLAC analyses (n � 0; 1)

Pt

GLwtro

FLAC

FEM

43:95

41:5

56:84

53:6

63:89

65:30

Pt

GLwtro

FLAC

FEM

29:89

25:0

37:21

34:8

39:53

35:81.0

L=ro 20 40 80 n

Both FEM and FLAC analyses were based on �s � 0:4, l � 1000. However, H=L � 2 for FEM and

H=L � 2:5 for FLAC analyses.


Fig. 2. E�ect of di�erent back-estimation procedures for � on the pile response (L=ro � 40, �s � 0:4,l � 1000, H=L � 4).


procedures described previously in the section ``Calibration Procedures'', and hasbeen illustrated in Fig. 2 as matching ``load ratio'' and pile-head ``sti�ness''.

Fig. 2(a) shows that the variation of � with depth can be taken as approximatelyconstant. With average values of �, the predicted loads and displacements along thepile, using Eqs. (4) and (5), respectively, are very close to those from the FLACanalysis, as illustrated in Figs. 2(d) and (e), respectively. Therefore, the shaft factor,�, can generally be assumed as a constant with depth.

3. Expressions for load transfer factors

3.1. Base load transfer factor

Load distribution prediction is sensitive to the base load transfer factor. Thus, amore accurate value of the factor has been provided here as

! � !h

!oh

!�!o�

!o �13�

where !h, !� are the parameters that re¯ect the e�ect of H/L and soil Poisson'sratio; !oh is !h at H=L � 4, and !o� is !� at �s � 0:4.The inverse of the factor ``!'' re¯ects the base sti�ness (Pb 1ÿ �s� �= 4Gbrowb� �).

Therefore, all the ®gures will be illustrated in the form of ``1=!'' to be consistentwith that for pile-head sti�ness. The following conclusions have been observed:

1. The ratio of ``1=!'' generally increases slightly with the pile slenderness ratio,when the ratio of L=ro is higher than 20, as shown in Fig. 3. As the non-homogeneity factor, n, increases from 0 to 1, the factor ``1=!'' increases byabout 0.15. Therefore, it can be approximated by

1=!0 � 0:67ÿ 0:00029L=ro � 0:15n �L=ro < 20� �14�1=!o � 0:6� 0:0006L=ro � 0:15n �L=ro � 20�

2. As Poisson's ratio increases, ``1/!'' decreases slightly. However, once �sexceeds 0.4, it increases as shown in Fig. 4. Thereby

1=!� � 1=!o � 0:3 0:4ÿ �s� � �s � 0:4� � �15�1=!� � 1=!o � 1:2 �s ÿ 0:4� � ��s > 0:4�

3. The ``1=!'' calibrated is sensitive to the grid used for the case of di�erent valuesof H/L, as described previously. Following careful exploration, it maybe con-cluded that ``1=!'' can be predicted by the following equation

1=!h � 0:1483n� 0:6081� � 1ÿ e 1ÿHL� �

� �0:1008nÿ0:2406�16�

Eq. (16) compares well with the results from FLAC analysis, as shown in Fig. 5.


4. Guo (1997) has shown that increase in the pile±soil relative sti�ness can lead toa slight increase in the value of ``1=!'', particularly at high pile-soil relativesti�ness ratios. However, it is generally su�ciently accurate to ignore the e�ectof pile±soil relative sti�ness, over a practical range for the sti�ness ratio, lbetween 300 and 3000.

In summary, it may be seen that typical values of the base load transfer parameter,!, lie in the range 1.1 to 1.7 (1=! in the range 0.6 to 0.9), compared with the com-monly adopted value of unity [2]. However, it may also be seen from Fig. 2 thatvalues of the shaft load transfer parameter, �, reduce near the bottom of the pile,implying higher load transfer in proportion to the amount of displacement. Thus,although the detailed expressions given previously for ! allow accurate assessmentof the amount of load transferred at the pile base, for practical load transfer analysis,where � is assumed constant with depth, it is often su�cient to assume a value of 1 for!. As discussed later, the e�ect of ! on the overall pile-head sti�ness is extremely small.

3.2. Shaft load transfer factor

Back-®gured shaft load transfer factors are slightly di�erent, as noted before,depending on the back-estimation procedures of either matching the pile-head sti�ness

Fig. 3. Base load transfer factor vs slenderness ratio relationship (H=L � 4, �s � 0:4, l � 1000).


or load ratio. In this section, expressions for estimating the values of � will beobtained through curve ®tting those values back-®gured from the process ofmatching ``pile-head sti�ness'', although the corresponding values back-®gured frommatching load ratio will be attached for comparison as well.

Fig. 4. Base load transfer factor vs Poisson's ratio relationship (L=ro � 40, H=L � 4).

Fig. 5. E�ect of soil layer thickness on base load transfer factor (L=ro � 40, �s � 0:4, ls � 1000).


The shaft load transfer factor is mainly a�ected by the combination of pile slen-derness ratio L=ro, the soil non-homogeneity factor n, and the soil Poisson's ratio �s.The shaft load transfer factor, �, can be approximated by the following expression [4,5]

� � ln A1ÿ �s1� n

L

ro

� �� B

� ��17�

The parameters `À'' and ``B'' have been estimated through ®tting Eq. (17) to thevalues of � obtained by the approach of matching pile-head sti�ness. Fig. 6 showsthe variation of � with pile slenderness ratio and soil non-homogeneous pro®ledescribed by Eq. (1), for �s � 0:4, H=L � 4 and L=ro � 40. The curve ®tting to thisvariation results in B � 1:0 and

A � 1

1� n

2

1ÿ 0:3n

� ��18�

The prediction from Eq. (17), with `À'' from Eq. (18) and B � 1, has been shown inthe ®gure and termed as ``Current equation''. Eq. (18) is limited to Poisson's ratio,�s � 0:4. Generally the factor, � varies with Poisson's ratio in a way as shown inFig. 7, which may be simulated by a modi®cation of Eq. (18), so that `À'' may berewritten as

A � 1

1� n

0:4ÿ �sn� 0:4

� 2

1ÿ 0:3n

� �� Cl �s ÿ 0:4� � �19�

where Cl � 0, 0.5 and 1.0 for l � 300, 1000 and 10000. Eq. (17) with `À'' from Eq.(19) o�ers a reasonably good ®t as illustrated in the ®gure.The shaft load transfer factor decreases as the ®nite layer ratio decreases. This

e�ect can be accounted for by simply decreasing the parameter, A. To accommodatethis adjustment, the parameter, A, may be rewritten in the following format

A � Ah

Aoh

1

1� n

0:4ÿ �sn� 0:4

� 2

1ÿ 0:3n

� �� Cl �s ÿ 0:4� �

� ��20�

where Aoh is Ah at a ratio of H=L � 4, Ah is given by the following equation

Ah � 0:124e2:23�g 1ÿ e1ÿH

L

� �� 1:01e0:11n �21�

where �g � 1= 1� n� �. Estimation of Eq. (21) is simpler than it looks. It has nophysical implication but compares well with that back-®gured from FLAC analysis,as illustrated in Fig. 8. The main modi®er of 1ÿ exp 1ÿH=L� � is similar to thatrecommended by Lee [7].The shaft factor, �, is only slightly a�ected by the pile±soil relative sti�ness factor,

l, as shown in Fig. 9 [and expressed by the factor of Cl in Eq. (20)], and thereforemay be approximately taken as independent of l.


Fig. 6. Load transfer factor vs slenderness ratio (H=L � 4, �s � 0:4).


Fig. 7. Shear in¯uence zone vs Poisson's ratio relationship (H=L � 4, L=ro � 40).


Fig. 8. Load transfer factor vs H=L ratio (L=ro � 40, �s � 0:4).


3.3. Accuracy of load transfer approach

The back estimation of `À'' has been based on matching either load ratio or head-sti�ness. There are some di�erences in the values of the back-®gured `À'' from thetwo approaches, particularly for the following listed cases: (1) homogeneous soilpro®le, and (2) cases of higher slenderness ratio but lower sti�ness, e.g. l � 300(Figs. 6 and 9). In these cases, as shown previously, the accuracy of load transferapproach might be less than that for other cases. However, generally, the values of`À'' from the two methods are consistent with each other.The expressions in the previous section are believed to allow accurate estimate of

the load transfer parameter, �. However, in many cases, simpler expressions for theload transfer parameters may prove su�cient, at the expense of some loss in accu-racy. The two parameters that most a�ect � are (a) the soil layer depth ratio, H=L,and (b) the non-homogeneity factor, n. The shaft load transfer parameter, �, may beestimated from Eq. (17), taking B � 1 and A given approximately by:

A � 1� 1:1eÿ��np

1ÿ e1ÿHL

� ��22�

For deep layers, previous recommendations have been to take B � 0, and A � 2:5[2,8], compared with taking B � 1 and a limiting value of A � 2:1 from Eq. (22) (forhomogeneous soil, with n � 0). The small discrepancy appears due to the fact thatFLAC analysis gives a slightly higher pile-head sti�ness for a pile in an in®nite layerthan the boundary element approach, hence the lower value of A.It must be remembered that generally even a 30% di�erence in choice of `À''

value, leads to less than 10% di�erence in the prediction of head sti�ness from Eq.(10). However, the accuracy of `À'' becomes important when estimating pile±pileinteraction factors, as noted by Guo and Randolph [8] and further explored by Guo[4]. Fig. 10 shows a comparison of Eq. (22) with results from the FLAC analyses.The base contribution to the pile-head sti�ness is generally less than 10%. There-

fore, taking ! as unity will result in less than 6% di�erence in the predicted pile-headsti�ness. Fig. 11 shows a comparison of the back-®gured values of � using the simple! � 1 and the more precise values of ! for two extreme cases of higher (L=ro � 80)and lower (L=ro � 10) slenderness ratio, together with the prediction by the currentequation, Eq. (17).

4. Validation of load transfer approach

The current solutions of pile-head sti�ness and the load ratio are in the form ofmodi®ed Bessel functions as illustrated previously [4,5]. The numerical evaluation ofthe solutions have been performed through a spreadsheet program, which operatesthrough a macro sheet in Microsoft EXCEL, with the shaft load transfer factorgiven by Eq. (20) and the base load transfer factor generally given by Eq. (13), exceptfor comparison with the FLAC analyses, where a value of unity for the base loadtransfer factor has been used. All the following CF solutions result from this program.


Fig. 9. Load transfer factor vs relative sti�ness (�s � 0:4, H=L � 4).


Fig. 10. E�ect of soil layer thickness on load transfer parameters, A and � (L=ro � 40, �s � 0:4, l � 1000).


Fig. 11. Variation of the load transfer factor due to using unity and the realistic value for the base factor o.


4.1. Comparison with existing solutions

Table 4 shows that the pile-head sti�ness predicted by Eq. (10) and the ratio ofpile base and head load by Eq. (12a) are consistent with but slightly higher thanthose reported by Rajapakse [6]. However, the FLAC analyses give higher headsti�nesses and load ratios than those obtained from the VM analysis [6], particularly,at higher relative pile-soil sti�ness (e.g. l>10000). The current CF approaches gen-erally yield a prediction which lies in between that from the FLAC and VM analyses.

4.1.1. Slenderness ratio e�ectFig. 12(a) shows the variation of pile-head sti�ness with slenderness ratio obtained

for a pile in a homogeneous, in®nite half space by Butter®eld and Banerjee [9], andChin et al. [10]. As expected, the present analysis (termed as ``CF'') yields slightlyhigher head sti�ness than those by other approaches. Analysis using a value ofA � 2:5 has been performed as well and shown in the ®gure as ``CF (A � 2:5)''. Thepredicted results compares well with those from the more rigorous numericalapproaches shown in the ®gure. Fig. 12(b) provides a further comparison of the headsti�ness for a pile in a Gibson soil (n � 1) obtained by Chow [11], Banerjee andDavies [12] and the present closed form solutions. Increase in slenderness ratio asshown in Fig. 12 can lead to an increase in pile-head sti�ness, but this tendency islimited to a certain value, beyond which any increase in the slenderness ratio willlead to a negligible di�erence in the pile-head sti�ness.

4.1.2. Soil Poisson's ratio e�ectPoisson's ratio re¯ects the compressibility of a soil; the more incompressible

(higher Poisson's ratio) the soil is, the higher is the pile-head sti�ness, as shown inFig. 13. The di�erence in the sti�ness due to variation of Poisson's ratio between 0and 0.5 can be as high as 25%.

Table 4

Comparison of FLAC analysis with the VM approach

Pt�GLwtro

FLAC

VM

CF

74:8272:2 ��71:68

68:4165:1

66:75

56:7354:9

56:59

38:0938:7

38:68

Pb�Pt

(%)FLAC

VM

CF

8:475:4

6:63

8:015:2

6:38

6:94:6

5:69

4:473:1

4:0

Pt � �GLwtro

FLAC

VM

CF

52:4444:46

49:79

48:0840:38

45:93

39:13ÿ

38:13

24:722:2

24:3

l � Ep=GL

ÿ �10 000 3000 1000 300

*n � 0, **n � 1:0, ***rigid pile. VM and CF analyses were based on �s � 0:5, while FLAC analysis was

based on �s � 0:495, and H=L � 4.


Fig. 12. Comparison between pile-head sti�ness vs slenderness ratio relationship.


Fig. 13. Pile-head sti�ness vs Poisson's ratio relationship (L=ro � 40, H=L � 4).


Fig. 14. Pile-head sti�ness vs the ratio of H=L relationship (L=ro � 40, �s � 0:4).


Fig. 15. Comparison between current closed form analysis (dashed line) and the numerical result (solid

line) by Butter®eld and Douglas [15].


4.1.3. Finite layer e�ectThe ®nite layer ratio, H=L can in¯uence the pile response, if it is less than a

limiting value (approximately H=L � 4 when L=ro � 40). The limiting value of H/L is a�ected by the critical pile slenderness ratio, beyond which any increase inthe pile slenderness ratio results in negligible increase in pile-head sti�ness. Fig.14 shows that as the ratio of H=L increases from 1.25 to 4, the head sti�nessincurs about 15% reduction, but only a slight decrease in base load is noted(not shown). At this particular slenderness ratio (L=ro � 40), the percentage inreduction of sti�ness with H=L is consistent with those reported by Poulos [13]and Valliappan et al. [14] (not shown). The e�ect of the ratio of H=L can bewell represented for di�erent slenderness ratios by the current load transfer factors,as illustrated in Fig. 15, together with the results from Butter®eld and Douglas [15].The overall comparison demonstrates that Eq. (13) for ! and Eq. (17) for � are

su�ciently accurate for load transfer analysis.

5. E�ect of soil pro®le below pile base

The above analysis is generally based on the shear modulus described by Eq. (1)through the entire soil layer of depth, H, and is referred to as Case I. In some cases,it is more appropriate to assume a constant value of shear modulus below the piletip level, and this is referred to as Case II.Guo [4] has presented modi®ed expressions for the parameter A in Eq. (17) for

Case II, to allow for the slightly softer pile response compared with Case I. Thedi�erence between the two cases is more pronounced for shorter piles (L=ro < 30)than for longer piles, particularly in soil where the strength increase with depth issigni®cant (high n values). In such cases, the di�erence in pile head sti�ness mayamount to 10%.

6. Conclusions

In this paper, extensive numerical analysis has been undertaken using the FLACprogram. With these numerical results, the suitability and rationality of load trans-fer analysis has been explored widely. The core of the load transfer approach lies inthe two factors, !, relating the base spring sti�ness to the Boussinesq solution for arigid punch, and �, relating the gradient of the shaft load transfer spring to the shearmodulus of the soil.A preliminary numerical check showed that a grid of 21� 100 was necessary to

obtain accurate estimation of the base load transfer parameter, !, that the lateralboundary should extend to the maximum of 75ro or 2.5L, and that a soil layerthickness of H=L � 4 may be taken as e�ectively an in®nitely deep layer.The numerical analysis shows that the e�ect of varying the soil Poisson's ratio can

be equally as important as the ratio of H=L and should be taken into consideration.The ®nite layer ratio of H=L can only lead to about 15% increase in head sti�ness


when H=L decreases from 4 to 1.25, but an increase in soil Poisson's ratio from 0 to0.499 (keeping the shear modulus constant) can result in about a 25% increase inpile-head sti�ness.The calibration using load transfer model shows that, generally, the shaft load

transfer factor can be taken as constant with depth. With average values of the shaftload transfer factor, the load transfer approach yielded close predictions of overallpile response compared with those obtained by FLAC analysis.The calibration using the closed form solutions demonstrates that the shaft load

transfer factor, �, (a) increases with increase in pile slenderness ratio; (b) decreaseswith increase in Poisson's ratio; (c) increases slightly with increase in the ratio ofH=L (H=L � 4), but (d) is nearly independent of the pile±soil relative sti�ness.The di�erence in the values of shaft load transfer factors, calibrated against pile-

head sti�ness and ratio of base and head load, implies that the load transferapproach is less accurate in the cases of (a) homogeneous soil pro®le; and (b) higherpile slenderness ratio but lower pile±soil relative sti�ness. However, an appreciable(e.g. 30%) di�erence in selection of the value of `À'' for Eq. (17) generally leads to aslight (e.g. about 10%) di�erence in the predicted pile-head sti�ness of a single pile.Therefore, generally load transfer analysis is su�ciently accurate for practical analysis.The back-®gured load transfer factors have been expressed in the form of simple

formulas and also implemented in a spreadsheet program. In comparison with thecurrent FLAC analysis and relevant rigorous numerical approaches, the simple for-mulas can well account for the e�ects of various relative thickness ratio of H=L(� 4), Poisson's ratio and pile slenderness ratio. Overall, the values of the parameter`À'' derived here are somewhat lower than values suggested previously; e.g. forhomogeneous soil, a limiting value of A � 2:1 (for n � 0, H=L > 4) emerges fromthe FLAC analysis, compared with 2.5 generally adopted. The di�erence is partlydue to the tendency for FLAC to overestimate the pile-head sti�ness. Allowing forthis, the limiting value of A � 2:5 appears adequate for estimating the pile-headsti�ness for piles in deep homogeneous soil layers.

Acknowledgements

The work reported here was undertaken during doctoral studies by the ®rst authorat the University of Western Australia. During the period, the ®rst author was sup-ported by an Australian Overseas Postgraduate Research Scholarship and by scho-larships from the University of Western Australia. This ®nancial assistance isgratefully acknowledged.

References

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Rationality of load transfer approach for pile analysis