a modified analysis method for the nonlinear load transfer behaviour of axially loaded piles

KSCE Journal of Civil Engineering (2012) 16(3):325-333DOI 10.1007/s12205-012-1341-y

− 325 −

www.springer.com/12205

Geotechnical Engineering

A Modified Analysis Method for the Nonlinear Load Transfer Behaviour ofAxially Loaded Piles

Jingpei Li*, Yongwei Tan**, and Fayun Liang***

Received October 9, 2010/Revised 1st: January 27, 2011, 2nd: April 18, 2011/Accepted May 25, 2011

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Abstract

A modified analytical solution is proposed to calculate the load-settlement curve of axially loaded piles, which may consider thedegradation of the skin friction of piles, distribution of shear strength along the pile shaft, and a combination of different stress statesof the pile-soil system. The parameters for the calculation model can be obtained by means of equivalent analysis of the in-situmeasured skin friction. The results of the presented approach are verified with data from the available literature. Compared with otherapproaches, this method can be applied to practical engineering more efficiently; therefore, it is feasible to perform nonlinear analysisfor load transfer of axially loaded piles. Furthermore, it can also be used for layered soils.Keywords: axially loaded piles, load transfer, nonlinear behaviour, skin friction, ultimate shear strength, stress stages

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1. Introduction

In recent decades, a number of studies on the load-settlementcurve for axially loaded piles have been reported. Typically,analytical methods based on elasticity theories such as Poulos(1968), Poulos and Davis (1980), and Randolph and Wroth(1978) are widely applied to engineering practices. However,elastic solutions are suitable merely for pile foundations in anelastic state under working load conditions. To clarify the non-linear and plastic behaviour of the pile-soil system, a “load trans-fer approach” was introduced to calculate the load-displacementrelationships along piles (Seed and Reese, 1957; Coyle andReese, 1966; Kraft et al., 1981; Chow, 1986). To simulate thenonlinear pile-soil interaction, Satoru (1965) proposed an analy-tical solution which is based on an elastic-perfectly plastic loadtransfer function for the soil surrounding the pile shaft. Ernesto(1994) obtained an approximate elastic-perfectly plastic solutionin a closed-form for an axially loaded pile based on a τ -s curve,which can consider the linear variation of shear strength alongthe pile shaft. Francesco and Michele (2002) also proposed anapproximate approach for the analysis of nonlinear responses ofvertically loaded piles, in which the nonlinear pile-soil-pileinteraction is modelled by a hyperbolic load-transfer function. Itshould be pointed out that the above mentioned solutions cannotdescribe the degradation characteristic of skin friction whenslippage occurs between the piles and the soil surrounding theshaft.

Based on a tri-linear softening model for pile–soil interactionalong the shaft and a tri-linear fully plastic model for the pile–soil interaction at the base, Liu et al. (2004) proposed a matrixsolution for the load–settlement curve of piles in layered soils,which may consider the softening characteristics and differentstress states of the pile-soil system. However, Liu et al. (2004)did not consider the variation of shear strength along the pileshaft. This is not easy to achieve for practical applications sincethere are many parameters which need to be determined for thecalculations. To overcome the limitations of the method of Liu etal. (2004), a modified analytical solution is proposed in thispaper to calculate the load-settlement curve, which can take intoconsideration simultaneously the degradation of skin friction, thedistribution of the shaft shear strength with depth and a com-bination of different stress states of the pile-soil system. Theresults of the proposed approach are verified with data from theavailable literature. Compared with other approaches, the presentsolution is computationally efficient and inexpensive. Thus, it isfeasible for performing a nonlinear analysis for load transfer ofaxially loaded piles.

2. Calculation Model

A number of research papers based on the load transfer law foraxially loaded piles have drawn the following conclusions. First,the ultimate skin shear strength surrounding the pile shaft has alinear variation with depth. Second, for friction piles or end-

*Professor, Dept. of Geotechnical Engineering, Tongji University, Shanghai 200092, China (E-mail: lijp2773@ tongji.edu.cn)**Graduate Student, Dept. of Geotechnical Engineering, Tongji University, Shanghai 200092, China (E-mail: [email protected])

***Associate Professor, Dept. of Geotechnical Engineering, Tongji University, Shanghai 200092, China (Corresponding Author, E-mail: [email protected])

Jingpei Li, Yongwei Tan, and Fayun Liang

− 326 − KSCE Journal of Civil Engineering

bearing friction piles, the skin frictional resistance from top to tipdevelops in advance, then the toe resistance, and the course ofthe stress state of the surrounding soil is from elastic to softening,then to plastic. Using a tri-linear load transfer function to dealwith the softening process, the variation of skin friction withdepth and the different stress states of the surrounding soil shouldbe taken into account.

2.1 Basic AssumptionsThe main assumptions are made as follows:(1) The pile is a shaft with a uniform cross-section, made of an

elastic material.(2) A tri-linear load transfer function, shown in Fig. 1, is

introduced to describe the elastic, softening and plastic stressstates of the surrounding soil. Here, τu1 is the ultimate skinfrictional resistance in an elastic stage; Su1 is the ultimate relativedisplacement; τu2 is the residual frictional resistance after thedegradation of the surrounding soil’s strength; Su2 is the criticalrelative displacement just as the soil goes into a plastic state; λ1

and λ2 are coefficients of the elastic shear stiffness and soften-ing shear stiffness, respectively, and are constant along the pileshaft.

(3) A bilinear load transfer function is used to describe theelastic and hardening stress state of the tip soil (the soil at the tipof piles), as shown in Fig. 1. Here, k1 is the stiffness coefficient ofcompression resistance in an elastic stage; k2 is the stiffnesscoefficient of compression resistance in the hardening stage; τu2

is the residual frictional resistance after the degradation of thesurrounding soil’s strength; Sbu is the critical relative displace-ment of the tip soil when the hardening happens; and σbu is the

related normal stress of the pile tip. (4) The ultimate skin frictional resistance in an elastic stage has

a linear increase with depth, as shown in Fig. 1. Here, τu10 is theultimate skin frictional resistance at the top of the pile; α is thevariation coefficient of the ultimate skin frictional resistance withdepth; and η is the ratio of τu2 and τu1, which does not changewith depth.

2.2 Stress StagesAccording to the stress characteristic of the surrounding soil,

five stress stages can be put forward as follows:(1) Elastic shear stage. When the load on the top of the pile is

small, the ultimate skin frictional resistance τu1 cannot been fullyused, and the displacement of the pile head is smaller than theultimate relative displacement Su1. The soil surrounding thewhole pile shaft is in an elastic shear state.

(2) Softening-elastic shear stage. When the load on the top ofthe pile is relatively large, the ultimate skin frictional resistanceτu1 can be fully brought into play, and the displacement of thepile head is larger than the ultimate relative displacement Su1.The soil surrounding the pile shaft gradually changes into thesoftening shear state from the top downwards, and z1 is theboundary depth of the softening region.

(3) Plastic-softening-elastic shear stage. When the load onthe top of the pile continues to increase, the displacement of theupper pile shaft will be larger than the critical relative displace-ment Su2, and the skin frictional resistance will stay constant afterdegradation. The soil surrounding the whole pile shaft has threedifferent stress states (plastic, softening and elastic state) simul-taneously from top to tip, and z2 is the boundary depth of theplastic region.

(4) Plastic-softening shear stage. With the increase of theload on the pile head, the displacement of the soil surroundingthe pile tip will be larger than the ultimate relative displacementSu1. This means that there is no elastic region in the surroundingsoil and the whole lower part of the pile shaft is in a softeningstress state.

(5) Perfectly plastic shear stage. When the load on the pilehead rises to a certain limit, all the soil surrounding the pile willgo into a plastic state after degradation and maintain a constantresidual strength.

According to the stress characteristic of the tip soil, the stressstages can be divided into two kinds:

(1) Elastic compression stage. When the load transferredfrom top to tip is small, the displacement of the tip soil under thepile will not exceed the ultimate displacement Sbu. The soil underthe pile tip is in an elastic compression state.

(2) Hardening stage. When the load transferred to the tip islarge, the displacement of the tip soil will exceed the elasticultimate displacement Sbu. The soil under the pile tip is in ahardening state.

According to the different combinations, six different stressstages of the pile-soil system can be defined, as shown in Fig.2.

Fig. 1. Curves of Calculation Parameters: (a) Surrounding Soil, (b)Tip Soil, (c) Ultimate Shaft Shear Strength, (d) Shear Stiff-ness Coefficient

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A Modified Analysis Method for the Nonlinear Load Transfer Behaviour of Axially Loaded Piles

Vol. 16, No. 3 / March 2012 − 327 −

3. Equations for Load Transfer

3.1 Basic Governing EquationsBased on the stress characteristics of the pile-soil system in

each stage and the static equilibrium conditions for each smallsegment of pile shaft, a set of differential governing equationsare gained, as follows:

(1a)

(1b)

(1c)

where S is the displacement of the pile shaft at depth z; U is theperimeter of the cross section of the pile; E is the elastic modulusof the pile concrete; A is the cross section area of the pile body;and L is the length of the pile.

The boundary conditions at the pile head and pile tip could beexpressed as follows:

(2a)

(2b)

(2c)

The continuous conditions for the juncture of the differentstress regions could be expressed as follows:

(3a)

(3b)

(3c)

(3d)

3.2 Equations for Stress Stage I of the Pile-Soil SystemIn this stage, the soil surrounding the pile shaft is in an elastic

shear state, and the soil under the tip of the pile is in an elasticcompression state. The related main governing equations includeEqs. (1c), (2a) and (2b), where z1 = z2 = 0, and the whole region 0≤ z ≤ L is in an elastic state.

By solving the governing equations, the analytical solution tothe load-settlement curve of the pile head is obtained as follows.The derivation procedures are listed in the Appendix of thispaper.

(4)

where ϕ = (β + γ thβ)/(γβ + β 2 th β), α1 = , γ = k1L/E, β = α1L.

3.3 Equations for Stress Stage II of the Pile-Soil SystemIn this stage, the soil surrounding the pile shaft is in a soften-

ing-elastic shear state, and the soil under the tip of the pile is stillin an elastic compression state. The related main governingequations include Eqs. (1b), (1c), (2a), (2b), (3a) and (3b), wherez2 = 0 , the region 0 ≤ z ≤ z1 is in a softening state, and the regionz1 ≤ z ≤ L is in an elastic state.

By solving the governing equations, the analytical solution tothe load-settlement curve of the pile head is obtained as follows:

d2Sdz2-------- U

EA-------η τu10 αz+( )= 0 z z2≤ ≤( )

d2Sdz2-------- U

EA------- Su1λ1 λ2 S Su1–( )–[ ]= z2 z z1≤ ≤( )

d2Sdz2-------- U

EA-------λ1S= z1 z L≤ ≤( )

P0 EAdSdz------

z 0=–=

Pb EAdSdz------

z L=– k1ASb k1AS z L== = = Sb Sbu≤( )

Pb EAdSdz------

z L=

– k2 Sb Sbu–( )A k1SbuA+ k2 S z L= Sbu–( )A k1SbuA+= = =

Sb Sbu≥( )

τ z z1= τu1 z z1=τu10 αz1+= =

S z z1= Su1 z z1=τu1 z z1=

λ1⁄= =

τ z z2= τu2 z z2=ητu1 z z2=

η τu10 αz2+( ) ηλ1Su1 z z2== = = =

S z z2= Su2 z z2=1 η–( ) λ2⁄ 1 λ1⁄+[ ] τu10 αz2+( )= =

S0 P0Lϕ AE⁄=

Uλ1 AE( )⁄Fig. 2. Pile-Soil System Model at Different Stress Stages



(5a)

(5b)

where, β1 = (1/λ1 + 1/λ 2)α , β2 = (1/λ1 + 1/λ2)τu10, β ' = α1L', ϕ '= (β ' + γ ' thβ ')/(γ 'β ' + β '2 thβ '), γ ' = k1L'/E, L' = L − z1, α2 =

.

3.4 Equations for Stress Stage III of the Pile-Soil SystemIn this stage, the soil surrounding the pile shaft is in a plastic-

softening-elastic shear state, and the soil under the tip of the pileis still in an elastic compression state. The related main govern-ing equations include Eqs. (1a), (1b), (1c), (2a), (2b), (3a), (3b),(3c) and (3d), where the region 0 ≤ z ≤ z2 is in a plastic state; theregion z2 ≤ z ≤ z1 is in a softening state; and the region z1 ≤ z ≤ L isin an elastic state.


(6a)

(6b)

3.5 Equations for Stress Stage IV of the Pile-Soil SystemIn this stage, the soil surrounding the pile shaft is in a plastic-

softening shear state, and the soil under the tip of the pile is stillin an elastic compression state. The related main governingequations include Eqs. (1a), (1b), (2a), (2b), (3c) and (3d), wherez1 = L, the region 0 ≤ z ≤ z2 is in a plastic state; and the region z2 ≤z ≤ L is in a softening state.


(7a)

(7b)

where K1 = k1(β1L + β2)/E+β1.

3.6 Equations for Stress Stage V of the Pile-Soil SystemIn this stage, the soil surrounding the pile shaft is still in a

plastic-softening shear state, but the soil under the tip of the pileis in a hardening state. The related main governing equationsinclude Eqs. (1a), (1b), (2a), (2c), (3c) and (3d), where z1 = L, the

region 0 ≤ z ≤ z2 is in a plastic state; and the region z2 ≤ z ≤ L is ina softening state.


(8a)

(8b)

where, K2 = k2(β1L + β2)/E + β1 + (k1 − k2)Sbu/E.

3.7 Equations for Stress Stage VI of the Pile-Soil SystemIn this stage, the soil surrounding the pile shaft is in a perfectly

plastic shear state, and the soil under the tip of the pile is in thehardening state. The related main governing equations includeEqs. (1a), (2a) and (2c), where z1 = z2 = L, and the region 0 ≤ z ≤L is in a plastic state.


(9)

3.8 Critical Load

3.8.1 The Critical Load between Stage I and Stage IIThe critical condition is that the surrounding soil at the pile

head is just entering into the softening state. This means that thedisplacement of the surrounding soil at the pile head is equal tothe ultimate relative displacement Su1, and the skin frictionalresistance reaches the peak quantity τu1 in the elastic stage. If weset z1 = 0, the corresponding critical load at this condition can beobtained as follows:

(10)

3.8.2 The Critical Load between Stage II and Stage IIIThe critical condition is that the surrounding soil at the pile

head is just entering into the plastic state. This means that theskin frictional resistance of the surrounding soil at the pile headis equal to the residual stress τu2 after degradation; and thedisplacement is equal to the critical relative displacement Su2. Forz2 = 0, the corresponding softening depth z1 can be obtained bysolving the governing equations as follows:

(11)

P0 EA τu10 αz1+( ) L'ϕ 'λ1( )⁄ β1+[ ]cosα2z1 EAβ 1–= EA τ u10 αz1+( )α2 λ2⁄ sinα2z1⋅+

S0 τu10 αz1+( ) L'ϕ 'λ1( )⁄ β1+[ ]/α2 sinα2z1⋅=τ u10 αz1+( ) λ2 cosα2z1⋅⁄ β2+–

Uλ2 AE( )⁄

P0 EA τu10 αz1+( ) L'ϕ 'λ1( )⁄ β1+[ ]cosα2 z1 z2–( )=EA τ u10 αz1+( )α2 λ2⁄ sinα2 z1 z2–( )⋅ EAβ 1 Uη τu10 αz2 2⁄+( )z2+–+

S0 τu10 αz1+( ) L'ϕ 'λ1( )⁄ β1+[ ]cosα2 z1 z2–( ) z2 β 2+⋅= τu10 αz1+( )α2sinα2 z1 z2–( ) z2⋅ λ2⁄+ Uη αz2 3⁄ τu10 2⁄+( )z2

2 AE⁄ η τu10 αz2+( ) λ2⁄–+

P0

η τu10 αz2+( )λ2

----------------------------- λ2sinα2 L z2–( )k1

E----cosα2 L z2–( )–⎝ ⎠

⎛ ⎞ K1+

cosα2 L z2–( )k1

Eα2---------sinα2 L z2–( )+

------------------------------------------------------------------------------------------------------------------------ β1– EA=

Uηαz22 2⁄ Uητu10+ z2+

S0

η τu10 αz2+( )λ2

----------------------------- α2sinα2 L z2–( )k1

E----cosα2 L z2–( )–⎝ ⎠

⎛ ⎞ K1+

cosα2 L z2–( )k1

Eα2---------sinα2 L z2–( )+

------------------------------------------------------------------------------------------------------------------------- β1– z2=

β1z2 β2 Uη αz2 3⁄ τu10 2⁄+( )z22 AE⁄ η τu10 αz2+( ) λ2⁄–+ + +

P0

η τu10 αz2+( )λ2

----------------------------- α2sinα2 L z2–( )k2

E----cosα2 L z2–( )–⎝ ⎠

⎛ ⎞ K2+

cosα2 L z2–( )k2

Eα2---------sinα2 L z2–( )+

------------------------------------------------------------------------------------------------------------------------- β1– EA=


S0

η τu10 αz2+( )λ2

----------------------------- α2sinα2 L z2–( )k1

E----cosα2 L z2–( )–⎝ ⎠

⎛ ⎞ K1+

cosα2 L z2–( )k1

Eα2---------sinα2 L z2–( )+

------------------------------------------------------------------------------------------------------------------------- β1– z2=

β1z2 β2 Uη αz2 3⁄ τu10 2⁄+( )z22 AE⁄ η τu10 αz2+( ) λ2⁄–+ + +

S016---Uη

AE-------- α 3EL2

k2------------ L3+⎝ ⎠⎛ ⎞ 3τu10

2ELk2

---------- L2+⎝ ⎠⎛ ⎞+–=

1k2A-------- L

EA-------+⎝ ⎠

⎛ ⎞P0k1 k2–

k2--------------Sbu–+

P1 EAτu10 Lϕλ1( )⁄=

τu10 αz1+( ) L'ϕ'λ1( )⁄ β1+[ ]λ2sinα2z1 α2⁄ ητu10+τu10 αz1+( )cosα2z1– 0=


Vol. 16, No. 3 / March 2012 − 329 −

Substitution of z1 and z2 = 0 into Eq. (6a) gives the correspondingcritical load as follows:

(12)

3.8.3 The Critical Load between Stage III and Stage IVThe critical condition is that the surrounding soil at the pile tip

is just entering into the softening state from the elastic state. Thismeans that there is no longer an elastic region along the pileshaft; the skin frictional resistance of the surrounding soil at thepile tip is equal to τu1 and the displacement is equal to theultimate relative displacement Su1. For z1 = L, the correspondingplastic depth z2 can be obtained by solving the governing equa-tions as follows:

(13)

Substitution of z2 into Eq. (7a) gives the corresponding criticalload as follows:

(14)

3.8.4 The Critical Load between Stage IV and Stage VThe critical condition is that the soil under the pile tip is just

entering into the hardening state, and the surrounding soil alongthe pile shaft is still in the plastic-softening shear state simul-taneously. This means that the displacement of the soil under thepile tip is equal to the critical relative displacement Sbu. If z = L,the corresponding plastic depth z2 can be obtained by solving thegoverning equations as follows:

(15)

Substitution of z2 into Eq. (8a) gives the corresponding criticalload as follows:

(16)

3.8.5 The Critical Load between Stage V and Stage VIThe critical condition is that the soil surrounding the pile tip is

just entering into the plastic state after degradation. This meansthat the soil surrounding the whole pile shaft is in a plastic state,and the skin frictional resistance of the surrounding soil at thepile tip is equal to the residual stress τu2. For z2 = L, the corres-ponding critical load can be obtained as follows:

(17)

4. Case Histories and Verifications

4.1 Case History ITo validate the presented solution, a case history of the load

test in situ at Jia-He Central in Ningbo, China (Qin, 2005) isintroduced here.

A cross-section of the pile-soil system and the measured datafrom the load test are shown in Fig. 3 and Tables 1-4. The lengthof the pile L is 55 m, the pile diameter D is 1 m, and the elasticmodulus of the pile E is 30000 MPa.

Using the data in Table 3 and Table 4, Fig. 4(a) shows therelative curve of the skin friction and displacement for eachtested pile section under different load levels. It is quite obviousthat there is a phenomenon of degradation of the skin frictionwith increasing pile depth.

In order to obtain the value of the model parameters forcalculation, the equivalent method is shown as follows:

(i = A, B, C ··· I) (18)

P2 EA τu10 αz1+( ) L'ϕ'λ1( )⁄ β1+[ ]cosα2z1 EAβ1–= EAsinα2z1 τ u10 αz1+( )α2 λ2⁄⋅+

λ2

α2-----

η τu10 αz2+( )λ2

----------------------------- α2sinα2 L z2–( )k1

E----cosα2 L z2–( )–⎝ ⎠

⎛ ⎞ K1+

cosα2 L z2–( )k1

Eα2---------sinα2 L z2–( )+

--------------------------------------------------------------------------------------------------------------------------sinα2 L z2–( )

η τu10 αz2+( )cosα2 L z2–( )+ τu10 αL+=

P3

η τu10 αz2+( )λ2

----------------------------- α2sinα2 L z2–( )k1

E----cosα2 L z2–( )–⎝ ⎠

⎛ ⎞ K1+

cosα2 L z2–( )k1

Eα2---------sinα2 L z2–( )+

-------------------------------------------------------------------------------------------------------------------------- β1– EA=


Sbu1α2-----

η τu10 αz2+( )λ2

----------------------------- α2sinα2 L z2–( )k1

E----cosα2 L z2–( )–⎝ ⎠

⎛ ⎞ K1+

cosα2 L z2–( )k1

Eα2---------sinα2 L z2–( )+

-------------------------------------------------------------------------------------------------------------------------sinα2 L z2–( )–=

η τu10 αz2+( )λ2

-----------------------------cosα2 L z2–( )– β1L β2+ +

P4

η τu10 αz2+( )λ2

----------------------------- α2sinα2 L z2–( )k2

E----cosα2 L z2–( )–⎝ ⎠

⎛ ⎞ K2+

cosα2 L z2–( )k2

Eα2---------sinα2 L z2–( )+

-------------------------------------------------------------------------------------------------------------------------- β1– EA=


P5 k2A τ u10 αL+( ) 1 η–( ) C2⁄ 1 C1⁄+[ ] k1 k2–( )ASbu+= UηαL2 2⁄ Uητu10+ L+

τjτij hi⋅

L------------∑=

Fig. 3. Cross-Section of Pile-Soil System (Qin, 2005)



(i = A, B, C ··· I) (19)

where τ j is the equivalent skin friction under some load level j, Sj

is the relative displacement, and hi is the length of each pilesegment.

Figure 4(b) shows the relative curve of the equivalent averageskin friction and displacement under different load levels. It isobvious that there is a peak point when the load reaches level 3,and there is a phenomenon of degradation when the load reaches

level 4, after which the curve changes slightly. Thus, the equival-ent average ultimate τuu = τ3 = 26 kPa, and the residual skinfriction τuur = (τ4 + τ9)/2 = 23.9 kPa. Here, the residual strengthratio η = τuur/τuu = 0.92, and softening shear stiffness coefficientλ 2 = (1 − η)τuu/(S4 − S3) = 0.11 kPa/mm.

The elastic shear stiffness coefficient λ1 was defined as theratio of the equivalent ultimate skin friction τuu to the relativedisplacement Suu by Qin (2005). This definition may have arelatively large discreteness. Here λ1 can be redefined as the

SjSij hi⋅

L------------∑=

Table 1. Parameters for the Surrounding SoilNo. Soil Name w (%) γ (kN/m3) c (kPa) ϕ (o) Es (MPa) fk (kPa) qsu (kPa)1 fill 2.36

3-1 organic silty clay 45.6 17.4 9.8 4.2 9.20 80 53-2 silty sand 27.5 19.1 7.5 26.8 2.62 140 164 clay 44.0 17.4 15.8 5.7 6.32 100 135 silty clay 28.3 19.5 56.8 7.3 9.65 210 25

6-1 sandy silt 28.9 19.4 18.2 23.6 8.79 200 206-2 sandy silt 30.0 19.2 13.4 23.6 6.50 220 227-1 silty clay 33.0 18.9 26.7 6.9 7.45 180 167-2 silty clay 24.5 19.8 60.8 8.1 12.3 235 278 silty sand 19.6 20.2 9.4 30.3 11.3 260 309 silty clay 27.9 19.2 63.5 11.1 8.39 220 2810 clayey silt 29.8 19.2 36.4 15.0 7.03 250 26

11-1 silty clay 32.6 19.0 48.2 7.1 6.90 190 2511-2 silty sand 25.0 19.8 12.5 29.3 8.39 190 2812 silty clay 26.5 19.5 53.9 12.4 7.27 220 2713 gravelly sand with clayey soil 24.1 19.3 8.0 24.5 8.05 300 38

Table 2. Axial Force of Piles Under Different Load Levels (P/kN)

No. Depth (m)Load (kN)

0 2400 3600 4800 6000 7200 8000 8800 9600 10400A 0~2.5 0 2359 3555 4753 5958 7160 7960 8760 9560 10361B 2.5~15.0 0 2051 3233 4443 5660 6872 7680 8483 9286 10090C 15.0~23.0 0 1794 2923 4143 5391 6613 7428 8235 9040 9845D 23.0~32.0 0 1174 2083 3193 4487 5723 6547 7364 8172 8980E 32.0~40.0 0 664 1383 2372 3704 4950 5783 6605 7417 8227F 40.0~43.0 0 474 1103 2047 3408 4662 5501 6328 7144 7956G 43.0~45.0 0 369 917 1812 3203 4467 5315 6141 6960 7774H 45.0~50.0 0 0.00 394 1052 2483 3759 4613 5449 6271 7088I 50.0~55.0 0 0.00 0.00 303 1747 3036 3894 4739 5565 6385

Table 3. Skin Friction of Piles Under Different Load Levels (τ /kPa)


0 2400 3600 4800 6000 7200 8000 8800 9600 10400A 0~2.5 0 5.22 5.73 5.99 5.35 5.19 5.05 5.00 4.98 4.95B 2.5~15.0 0 7.85 8.19 7.89 7.60 7.30 7.15 7.07 6.99 6.91C 15.0~23.0 0 10.20 12.34 11.95 10.70 10.30 10.03 9.86 9.80 9.73D 23.0~32.0 0 21.91 29.73 33.60 31.98 31.50 31.13 30.83 30.68 30.59E 32.0~40.0 0 20.31 27.84 32.66 31.13 30.79 30.43 30.21 30.05 29.97F 40.0~43.0 0 20.16 29.73 34.45 31.46 30.48 29.87 29.39 29.06 28.82G 43.0~45.0 0 16.78 29.62 37.48 32.57 31.04 30.08 29.65 29.33 29.04H 45.0~50.0 0 23.49 33.29 48.36 45.84 45.06 44.53 44.10 43.82 43.59I 50.0~55.0 0 0.00 25.08 47.71 46.87 46.05 45.61 45.17 44.94 44.74


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weighted average ratio with depth of each layer’s τ1i, asfollows:

(i = A, B, C ··· I) (20)

From our calculation, λ 1 is 7.3 kPa/mm.Figure 5 shows the relative curve of the axial force and the

displacement of the pile tip under different load levels. Clearly,the third data point is a turning point. Therefore, the elasticultimate displacement at the pile tip Sbu = 20 mm.

The stiffness coefficient of compression resistance in theelastic stage k1 can be obtained by the following equation:

(21)

where k11 and k12 are the starting slopes of the first two datapoints, and S11 and S12 are the corresponding relative displace-ments. From our calculation, k1 = 114.6 kPa/mm.

Regarding the stiffness coefficient of compression resistance inthe hardening stage k2, the line segment between the third andfourth measured points is selected in view of its large span andgood agreement with the bilinear assumption for the tip soil.From our calculation, k2 = 45.8 kPa/mm.

The variation coefficient of ultimate skin frictional resistancewith depth α could also be obtained by the weighted averagewith depth for each segment’s variation coefficient α i, asfollows:

(i = AB, BC, CD ··· HI) (22)

where α i is the variation coefficient of ultimate skin frictionalresistance with depth for each segment; hi is the segment length,and H is the total length.

By calculation, α = 0.79 kPa/m. The equivalent average ultimate skin friction τuu is equal to the

ultimate skin friction τu1 at depth L/2, so the ultimate skin frictionτu1 at the top of the pile can be deduced from the variationcoefficient α. From our calculation, τu10 = τuu − α · L/2, τu10 =4.17 kPa.

Figure 6 shows the distribution curve of the skin friction under

λ1λ1i hi⋅

L-------------∑=

k1k11 S11⋅ k12 S12⋅+

S11 S12+--------------------------------------=

α αi hi⋅H

------------∑=

Table 4. Displacement between Piles and Soil under Different Load Levels (s/mm)


0 2400 3600 4800 6000 7200 8000 8800 9600 10400A 0~2.5 0 0.63 0.84 6.63 24.54 60.83 74.53 93.55 107.47 118.79B 2.5~15.0 0 0.20 0.31 5.88 23.69 59.95 73.53 92.29 106.0 116.92C 15.0~23.0 0 0.00 0.02 4.98 22.55 58.75 72.18 90.59 104.01 114.38D 23.0~32.0 0 0.00 0.00 4.16 21.69 57.82 71.13 89.25 102.45 112.38E 32.0~40.0 0 0.00 0.00 3.57 20.93 56.99 70.17 88.02 100.99 110.5F 40.0~43.0 0 0.00 0.00 3.25 20.51 56.51 69.61 87.29 100.12 109.37G 43.0~45.0 0 0.00 0.00 3.14 20.34 56.31 69.37 86.97 99.74 108.88H 45.0~50.0 0 0.00 0.00 3.02 20.14 56.06 69.07 86.57 99.25 108.23I 50.0~55.0 0 0.00 0.00 2.91 19.90 55.74 68.68 86.04 98.60 107.37

Fig. 4. Relationships between Skin Friction and Displacement: (a)Measured Data Under Different Load Levels, (b) EquivalentAverage

Fig. 5. Curves of Axial Force and Pile Tip Displacement under Dif-ferent Load Levels



different load levels. By calculation, the correlation coefficientbetween the equivalent data and measured data reaches 0.96,which indicates a good agreement.

Figures 7(a) and 7(b) show the load-settlement curve of thepile head and the distribution curve of the axial force with depthderived using different methods. It can be seen from the figuresthat there is a good agreement between the computed results andthe measured data of the load test on the in situ piles, and thecomputed results by analytical solution proposed in this paperare more reasonable and accurate than Chen et al.’s solutions(2000), in which only the variation in the ultimate skin frictionalresistance with depth is considered but the degradation of skinfriction is ignored.

Figures 8(a) and 8(b) show the load-settlement curve of thepile head and the distribution curve of the axial force with depthby Satoru’s solution (1965). By comparison, it is quite obviousthat the computed results in this paper are also more reasonableand accurate than Satoru’s solution which doesn’t consider thevariation of the ultimate skin frictional resistance with depth.

Indeed, when η = 1.0 and λ 2 = 0, the results obtained from theretrogression of the analytical solutions proposed in this paperagree with Chen et al.’s solution (2000). Otherwise, when η =1.0, λ2 = 0 and α = 0, Satoru’s solution will also be obtainedfrom the degenerate solution proposed in this paper.

4.2 Case History IIFor further validation, an example of the load test on in situ

manually dug piles in Zhuzhou, China from Liu’s paper (2004) ischosen here. Similar to Case History I, the parameters calculatedby the equivalent method are shown as follows:

L = 13.35 m, D = 1 m, E = 32000 MPa, λ1 = 5.7 kPa/mm, λ 2 =0.1 kPa/mm, τu10 = 5.53 kPa, k1 = 108 kPa/mm, k2 = 6.17 kPa/mm, Sbu = 18 mm, α = 3.47 kPa/m, η = 0.9.

Figure 9 shows the calculation result of the load-settlement

curve of the pile head. It can be seen from the figure that there isquite a good agreement between the calculated results and themeasured in situ data.

5. Conclusions

By considering the degradation of the skin friction of piles andthe distribution of shear strength along the pile shaft, a modifiedanalytical solution is proposed to calculate the nonlinear loadtransfer behaviour of axially loaded piles. Compared to theavailable literature, the presented approach is verified with quitegood agreement. Hence, it is feasible to perform the nonlinearanalysis for load transfer of axially loaded piles. Furthermore,

Fig. 6. Distribution Curves of Skin Friction under Different LoadLevels

Fig. 7. Comparison between Different Methods and Measured-Data: (a) Load-Settlement Curve, (b) Axial Force Distribu-tion


Vol. 16, No. 3 / March 2012 − 333 −

more accurate results for layered soil can also be taken intoconsideration. The solution for layered soil is given in theAppendix A-4.7.

Acknowledgements

This work was supported by the National Natural ScienceFoundation of China (Grant No.: 41172246), the Science andTechnology Commission of Shanghai Municipality (Grant No.:09231200900), and the Shanghai Rising-Star Program of theScience and Technology Commission of Shanghai (Grant No.:10QA1407000). The authors wish to express their gratitude for

the above financial support. The anonymous reviewers’ com-ments have improved the quality of this paper and are alsogreatly acknowledged.

References

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Liu, J., Xiao, H. B., Tang, J., and Li, Q. S. (2004). “Analysis of load-transfer of single pile in layered soil.” Computers and Geotechnics,Vol. 31, No. 2, pp. 127-135.

Poulos, H. G. (1968). “Analysis of the settlements of pile groups.”Géotechnique, Vol. 18, No. 3, pp. 449-471.

Poulos, H. G. and Davis, E. H. (1980). “Pile foundation analysis anddesign.” John Wiley & Sons, New York.

Qin, Y. X. (2005). Calculation and analysis of the pile's settlementunder pile-soil softening, Master's Thesis, Zhejiang University,Hangzhou (in Chinese).

Randolph, M. F. and Wroth, C. P. (1978). “Analysis of deformation ofvertically loaded piles.” Journal of Geotechnical Engineering, Vol.104, No. 12, pp. 1465-1488.

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Seed, H. B. and Reese, L. C. (1957). “The action of soft clay alongfriction piles.” Transactions, ASCE, Vol. 122, pp. 731-754.

Fig. 8. Comparison between Different Methods and MeasuredData: (a) Load-Settlement Curve, (b) Axial Force Distribu-tion

Fig. 9. Curves of Load-Settlement of Pile by Different Methods andMeasured Data

a modified analysis method for the nonlinear load transfer behaviour of axially loaded piles

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