Influences of soil consolidation and pile load on the development of negative skin friction of a pile

Computers and Geotechnics 36 (2009) 1265–1271

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Influences of soil consolidation and pile load on the development of negative skin friction of a pile R.P. Chen a,*, W.H. Zhou b, Y.M. Chen a a

Key Laboratory of Soft Soils and Geoenvironmental Engineering of Ministry of Education, Department of Civil Engineering, Zhejiang University, 388 Yuhangtang Road, Hangzhou 310058, China b Department of Civil and Structural Engineering, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, China

a r t i c l e

i n f o

Article history: Received 23 September 2008 Received in revised form 11 March 2009 Accepted 14 May 2009 Available online 10 June 2009 Keywords: Pile Negative skin friction Soil consolidation Hyperbolic interface model Pile–soil interface

a b s t r a c t Negative skin friction (NSF) along a pile caused by soil consolidation is of great concern to engineers. The development of NSF is time-dependent because soil consolidation is also time-dependent. In this paper, a numerical solution is provided for the development of negative skin friction of a pile in nonlinear consolidated soil under different loads on a pile top. A hyperbolic interface model is also developed. This model considers the development of shear strength during soil consolidation and loading–unloading scenarios at the pile–soil interface. One-dimensional nonlinear consolidation theory is invoked to estimate the soil settlement and shear strength. The distributions of NSF and the axial force along the pile are obtained using the differential quadrature method (DQM). The influences of soil consolidation and different pile loads on the negative skin friction of a pile are discussed. Ó 2009 Elsevier Ltd. All rights reserved.

1. Introduction A thorough understanding on the development of negative skin friction (NSF) of piles is essential for the analysis and design of pile foundations. Generally, NSF has two effects on pile foundations: the development of additional compressive axial force (dragload) in piles and excessive pile settlement (downdrag). Due to the development of soil deformation and the shear strength of the pile–soil interface, the distribution of NSF along the pile is timedependent. As the excess pore pressure dissipates with time, there is a corresponding increase in soil settlement and effective stress. This, in turn, causes an increase in shear strength at the pile–soil interface and a corresponding increase in the dragload and downdrag of the pile. This process continues until the soil becomes fully consolidated. Commonly used interface models include the linear elastic model, the linear elastic plastic model, and the hyperbolic model. For NSF piles, some key factors should be considered in modeling the interface behavior, such as a nonlinear interface stress–strain relationship, the irrecoverable relative displacement between pile and soil under unloading situation, and shear strength development during soil consolidation. Current methods for the predictions of NSF on pile foundations fall into the following four broad categories: (I) the empirical * Corresponding author. Tel.: +86 571 88208769; fax: +86 571 88208685. E-mail addresses: [email protected] (R.P. Chen), [email protected] (W.H. Zhou), [email protected] (Y.M. Chen). 0266-352X/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.compgeo.2009.05.011

method [6,25], (II) the load transfer method [1,24], (III) the continuum method [9,10,15,18–20], and (IV) the finite element method [14,16]. Although much work has been done to investigate NSF, most past research has focused on the distribution of NSF when the soil consolidation is completed. The differential quadrature method (DQM) was introduced by Bellman and Casti [2] and Bellman et al. [3] as a rapid means for finding the solution of ordinary and partial differential equations. The fundamental idea of the DQM is to approximate various derivatives of a function by means of a polynomial expressed as a weighted linear sum of function values at all grid points. It has been claimed in numerous studies that DQM is capable of yielding highly accurate solutions for both initial- and boundary-value problems with minimal computational effort. Malik and Civan [17] have written a comprehensive comparison for linear and nonlinear convection–diffusion–reaction problems. They reported that DQM stood out in numerical accuracy as well as computational efficiency over the finite difference model (FDM) and finite element model (FEM). One can find detailed discussions and pertinent references in the review paper [4]. The application of DQM in geotechnical engineering is still limited. Chen et al. [8] introduced DQM to the analysis of linear and nonlinear one-dimensional consolidation problems. In this paper, the load transfer method is adopted to analyze the NSF problem of piles. An improved hyperbolic interface model is developed to describe the pile–soil interface behavior while considering the shear strength development and the irrecoverable

R.P. Chen et al. / Computers and Geotechnics 36 (2009) 1265–1271

relative displacement at the pile–soil interface during soil consolidation. The numerical solution of the model is obtained by using DQM. The influences of nonlinear soil consolidation and pile loads on the development of NSF are discussed. 2. Mathematical model of a pile Generally, the equilibrium of an elastic pile element along its axis (Fig. 1) can be written as [23]: 2

d wp ðzÞ dz

2

¼

2pr 0 s0 ðzÞ Ep Ap

ð1Þ

where wp(z) is the vertical displacement of the pile shaft, s0 is the shear stress at the pile–soil interface, Ep is the Young’s modulus, and Ap and r0 are the cross-sectional area and the radius of the pile, respectively. For piles with other than circular cross-section geometry, an equivalent circular pile can be used [27]. The boundary conditions at the pile top and the pile tip are:

dwp Q ¼ E p Ap dz dwp Pb ¼ dz Ep Ap

z¼0: z¼L:

ð3Þ

wp  ws  Dsr 1 sf =wf

þ si 

C

0.8 0.6

ks

0.4 0.2

B Residual relative 0 displacement ks

0.0 -0.2 -0.4 -0.6

ksr=ks

-0.8

A

-1.0 -15

-10

-5

0

5

10

15

20

25

30

Relative displacement at pile-soil interface Fig. 2. Improved hyperbolic model.

ð2Þ

where Pb is the mobilized base load, Q is the load on pile top, and L is the pile length (Fig. 1). During the process of soil consolidation, the strength and settlement of soils increase. Therefore, the load transfer model at pile– soil interface should consider the increase of the limiting shear stress and the loading–unloading conditions. An improved hyperbolic model, as shown in Fig. 2, assumes that the unloading stiffness factor ksr is equal to the initial stiffness factor ks. The mathematical formula for the soil model can be written as follows:

s0 ¼

1.0

Shear stress/Limiting shear stress

1266

ð4Þ

Rfs ðwp ws Dsr Þ

sf

where ws is the soil displacement as a result of soil consolidation and sf is the limiting shear stress along the pile shaft. The latter can be written as sf ¼ K tan /0 ðc0 z þ Dr0 Þ, where K is the lateral earth pressure coefficient, u0 and c0 are the effective friction angle and submerged unit weight, and Dr0 is the increased effective vertical stress due to soil consolidation [7]. Rfs is a hyperbolic constant, which is assumed to be unity in this study. Dsr is the residual relative displacement at the pile–soil interface due to unloading, as shown in Fig. 2, where Dsr = OB. The variable si represents different

loading conditions. Under the unloading condition AB in Fig. 2, si = 0, and Eq. (4) can be reduced to a linear unloading formula:

s0 ¼

sf wf

 ðwp  ws  Dsr Þ

whereas in other cases, such as OA and BC in Fig. 2, si is the sign of (wp – ws – Dsr) and is equal to 1 or 1. wf is the necessary shear displacement for reaching half of the limiting shear stress sf. Generally, only extremely small movements (2–5 mm) are necessary to generate shear stress or to reverse the direction of shear at the pile–soil interface [5,12,26]. Alonso et al. [1] reported laboratory tests on the pile–soil interface behavior, and found that the shear displacement necessary to induce peak strength of the pile–soil interface (wu) is almost invariable in the range of 1–3 mm, which seems independent of the soil type and the confining stress. They proposed a simplified elastic–plastic load transfer model for analyzing NSF of piles, where a constant relative displacement of 2 mm was assumed for the wu value. In the present improved hyperbolic model, the value of wf is assumed to be 1=4 of the wu value. Thus, the shear stress will reach 80% of the limit shear stress, sf, at the shear displacement, wu. The pile base settlement can be estimated through the solution for rigid punch acting on an elastic half-space, as suggested by Randolph and Wroth [23]:

wp ðLÞ  ws ðLÞ ¼

0

r

z ws τ0

Ep

τ0

Pb ð1  v r Þx 4r 0 Gb

ð6Þ

where x is the pile base shape and depth factor, which is generally chosen as unity since it can only lead to a rather small (normally <6% [13]) difference in predicting pile-tip stiffness; Gb is the shear modulus of the soil below the pile tip level; and vr is the Poisson’s ratio of the bearing layer. Using a hyperbolic model, the base load–displacement relationship can be given by:

Q

q

ð5Þ

E s, vs

L

Pb ¼

wp ðLÞ  ws ðLÞ 1 kb

þ Rfb

ð7Þ

wp ðLÞws ðLÞ Pfb

2r0

Table 1 Drainage conditions of the upper boundary and lower boundary.

Bearing layer E , v Fig. 1. Mathematical model for negative skin friction problem.

Boundary drainage

a1

b1

a2

b2

Free draining Impervious

0 1

1 0

0 1

1 0

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R.P. Chen et al. / Computers and Geotechnics 36 (2009) 1265–1271 Table 2 Parameter values in three case studies. Case

1 2 3

Pile

Soil around the pile shaft 3

Bearing layer 2

Loading

r0 (m)

L (m)

Ep (kPa)

c (kN/m )

Ktan/

mv0 (m /kN)

L (m)

wu (mm)

wf (mm)

Er (kPa)

vr (–)

Pfb (kN)

q (kPa)

Q (kN)

0.25 0.25 0.25

10 10 10

2  107 2  107 2  107

10 10 10

0.3 0.3 0.3

2  104 1  104 1  104

10 10 10

2 2 8

0.5 0.5 2

5  105 5  105 1.5  105

0.3 0.3 0.3

500 900 600

75 150 150

0 0, 200, 400 200

0

0

where kb = 4Gbr0/(1  vs)x, Pfb is the limiting base load, and Rfb is a hyperbolic constant. 3. Numerical modeling of soil consolidation The governing differential equation for one-dimensional (1-D) nonlinear soil consolidation is described as follows [8]:

"  2 # @u @2u 1 @u þ RðtÞ þ ¼ cv @t @t 2 q þ r00  u @z

ð8Þ

permeability is proportional to the decrease in soil compressibility. Thus the coefficient of consolidation is constant during soil consolidation. According to Eq. (8), the discrepancy between linear and nonlinear consolidation increases with an increase in load level. The parameter Nr is defined as the ratio of final effective stress to the initial effective stress, N r ¼ r0f =r00 . When Nr = 1, i.e., q approaches to zero, Eq. (8) degenerates to the case of linear consolidation. The boundary conditions are:

@u 1 ð0; tÞ  b uð0; tÞ ¼ 0 @z @u 2 a2 ðH; tÞ  b uðH; tÞ ¼ 0 @z a1

where u is the excess pore water pressure; q(t) is the uniformly distributed load applied on the top surface of the soil; R(t) = dq(t)/dt is the rate of loading; t and z are the variables of time and space, respectively; and cv is the coefficient of consolidation. Cv can be written as cv = kv0/(mv0cw), in which kv0 is the vertical initial coefficient of permeability, cw is the unit weight of water, and 0:434C c is the initial coefficient of compressibility with Cc mv 0 ¼ ð1þe 0 0 Þr0 denoting the compression index of the soil and e0 the initial void ratio of soil corresponding to the initial effective stress r00 . The nonlinear soil consolidation formula in Eq. (8) is based on Davis and Raymands’ theory [11], i.e., it assumes that the decrease in soil

(a)

-60

-40

-20

Positive skin friction 0

20

40

Positive skin friction

Negative skin friction

-80 -70 -60 -50 -40 -30 -20 -10

0

10

20

30

40

50

60

70

80

0

60

U=5% U=10%

0

Terzaghi Consolidation Nonlinear Consolidation Nσ= 3.5

2

ð10Þ

where H is the total thickness and a1, b1, a2, and b2 are coefficients that depend on the specific boundary conditions. Table 1 presents the particular values of these coefficients for free draining and impervious boundaries. The initial excess pore water pressure was u0(z).

(a) Negative skin friction

ð9Þ

Limiting Shear stress

2

U=25%

Nonlinear Consolidation Nσ= 11

4

4

U=50%

Nonlinear Consolidation Nσ= 21

6

U=75%

6

U=100%

8

8

10

10 -60

-40

-20

0

20

40

60

-80 -70 -60 -50 -40 -30 -20 -10

(b)

0

Nonlinear Consolidation Nσ= 6

2

Nonlinear Consolidation Nσ= 11

3

Depth z (m)

Depth (m)

20

30

40

50

60

70

80

1

Nonlinear Consolidation Nσ= 21

4

10

0

Terzaghi Consolidation Nonlinear Consolidation Nσ= 3.5

2

0

Shear stress τ (kPa)

Shear stress τ (kPa)

(b)

Depth z (m)

Depth (m)

Nonlinear Consolidation Nσ= 6

6

4 5 6 7

8

U = 25%

U = 5%

U = 50%

8

U = 10%

U = 75% 100%

9 10

10 0

100

200

300

400

500

Axial force Ν (kN) Fig. 3. Influences of nonlinear soil consolidation on (a) shear stress distribution and (b) axial force distribution along the pile.

0

100

200

300

400

500

600

700

800

Axial force N (kN) Fig. 4. Distributions of (a) shear stress and (b) axial force along the pile at different soil consolidation degrees under zero pile load.

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R.P. Chen et al. / Computers and Geotechnics 36 (2009) 1265–1271

The DQM solution to nonlinear soil consolidation is discussed in detail in Chen et al. [8]. Thus, the soil settlement is calculated as follows:

ws ¼

Z

Z

H

em dz ¼ 0

¼ 2:3mm0 r00

0

Z

H

0

H

e0  e dz ¼ 1 þ e0  0 r dz log 0

Z

H

0

Cc log 1 þ e0

 0 r dz 0

r

ð11Þ

r0

4. Numerical solution to NSF of piles The differential quadrature method (DQM) is a numerical solution technique for initial- and/or boundary-value problems proposed by Bellman and Casti [2] and Bellman et al. [3]. In DQM, the derivative of a function with respect to a variable can be approximated by a weighted linear combination of function values at the grid points in the domain of that variable. Consider a function y = y(x) on the domain 0  x  a. Let the function values be known at a set of preselected sampling points xa (a = 1, 2, . . . , N). Then, the DQM formula for an rth-order derivative of the function y(x) at a point x = xa can be approximated by the weighted linear sum of function values as:

 N X @ r y ðrÞ ffi Dab yðxb Þ;  r @x x¼xa b¼1

a ¼ 1; 2; . . . ; N

ð12Þ

In this study, Quan and Chang’s method [22] is adopted to obtain the weighting coefficients and equally spaced points are selected for computation. With the assumption that soil consolidation is not affected by the existence of the pile, the DQM solution of soil displacement during consolidation obtained by Chen et al. [8] is adopted to solve soil consolidation settlement ws in this study. In that solution, both traditional Terzaghi consolidation theory and nonlinear consolidation theory based on the work of Davis and Raymond [11] are embedded.  s ¼ Rfs Dsr =wf ,  p ¼ Rfs wp =wf , w  s ¼ Rfs ws =wf , D  ¼ Rfs s0 =sf , w Let s and P b ¼ Rfs P b =kb wf , z ¼ z=L. Substituting Eq. (4) into Eq. (1), one obtains the governing equation of the negative skin friction of single piles. The dimensionless form of governing equation and boundary conditions, Eqs. (2) and (3), can be written as follows: 2  sr s  D p p  w d w w ¼ C1   sr Þ p  w s  D dz2 1 þ si  ðw   p dw  þ C2 ¼ 0 dz z¼0   p jz¼1  w  p  s jz¼1 w dw  þ C3  ¼0  p jz¼1  w  s jz¼1 Þ dz z¼1 1 þ a1  ðw

ð13Þ ð14Þ ð15Þ

where C1 = 2pr0L2sf/EpApwf, C2 = RfsQL/EpApwf, C3 = kbL/EpAp, and a1 = kbRfbwf/PfbRfs. The pile and the surrounding soil are discretized into N sampling points according to the equally spaced rule. Eqs. (13)–(15) can be reduced to the following DQM formulation:

ðrÞ

where Dab are the weighting coefficients of the rth derivative, r 6 N  1.

(a)

(a)

Positive skin friction

Negative skin friction

-80 -70 -60 -50 -40 -30 -20 -10

0

0

10

20

30

40

50

60

70

80

Negative skin friction

Positive skin friction

-80 -70 -60 -50 -40 -30 -20 -10

0 0

U=5%

U=5% U=10%

10 20 30 40 50 60 70 80

U=10%

Limiting shear stress

2

Limiting shear stress

2

U=25%

U=25% 4

U=75% 6

U=75% U=100%

6

U=100%

8

8

10 -80 -70 -60 -50 -40 -30 -20 -10 0

10

20

30

40

50

60

70

80

10 -80 -70 -60 -50 -40 -30 -20 -10 0

0

100

200

300

400

500

600

700

800

900

1000

0

(b)

U = 5%

2

10 20 30 40 50 60 70 80

Shear stress τ (kPa)

Shear stress τ (kPa)

(b)

4

U=50%

U=50%

0

100

200

300

400

500

600

700

800

900 1000 1100

0

2

4

Depth z (m)

Depth z (m)

U = 5% U = 10% U = 25%

6

U = 50% U = 75%

8

4

U = 10%

6

U = 25% U = 50%

8

100%

75% 100%

10

10 0

100

200

300

400

500

600

700

800

900

1000

Axial force N (kN) Fig. 5. Distribution of (a) shear stress and (b) axial force along the pile at different soil consolidation degrees under a 200 kN pile load.

0

100

200

300

400

500

600

700

800

900 1000 1100

Axial force N (kN) Fig. 6. Distributions of (a) shear stress and (b) axial force along the pile at different soil consolidation degrees under a 400 kN pile load.

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R.P. Chen et al. / Computers and Geotechnics 36 (2009) 1265–1271

(a)

(b)

Q = 400kN q=150kPa

1.0 0.8

1

0.6

2

A 2 .5 0 m

3

Depth (m)

Shear stress τ / σz'

0

4 5 6

B

7

C

6 .2 5 m

0.0 0.2 0.4 0.8 1.0

9

-10

10

1.0

(d)

0.8

10

20

30

1.0 0.8

B Shear stress τ / σz'

0.2 0.0 0.2 0.4

0.4 0.2 0.0 0.2 0.4

0.6

0.6

0.8

0.8

1.0 0

5

10

15

C

0.6

0.4

-5

0

Relative displacement ws-wp (mm)

0.6

Shear stress τ / σz'

0.2

0.6 7 .5 0 m

8

(c)

A

0.4

20

Relative displacement ws-wp (mm)

1.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0

Relative displacement ws-wp (mm)

Fig. 7. Stress histories of A, B, and C points along the pile shaft under a 400 kN pile load.

N X

 pa ¼ C 1a  D2ab w

b¼1

 sra  sa  D  pa  w w  sra Þ ;  pa  w  sa  D 1 þ sia  ðw

a ¼ 2; 3; . . . ; N  1 ð16Þ

N X

 b þ C2 ¼ 0 D11b w

ð17Þ

b¼1 N X b¼1

 b þ C3  D1Nb w

 pN  w  sN w ¼0   sN Þ 1 þ a1  ðwpN  w

ð18Þ

where D1ab and D2ab are the weighting coefficient matrices [8,22] of the first-order and second-order derivative, respectively, and  sra are the dimensionless displacement values at the  sa and D  pa ; w w ath discretized point of the pile and soil. The set of algebraic equations can be solved by the medium-scale algorithm and using the trust-region dogleg method [21].

5. Results and discussion In this section, the present method is applied to analyze the pile NSF problems. The distributions of NSF and axial force along the pile are presented and discussed. Three influencing factors are considered in three cases, respectively. The factors include nonlinear soil consolidation, pile load magnitude, and start time of pile loading. In the case studies, a circular pile is considered. The length of the pile is 10 m, and the soil surrounding pile shaft is one uniform soil layer. The pile base is embedded in a stiff bearing layer, as shown in Fig. 1. The values of parameters used in calculation are summarized in Table 2.

5.1. Case 1: influence of nonlinear soil consolidation It is clearly shown in Eqs. (13)–(15) that the soil settlement is the main trigger in the development of NSF of piles. In previous studies [1,27], the soil settlements were usually obtained by Terzaghi consolidation theory, due to its simple and direct formulation. However, the influence of nonlinear soil consolidation on the pile NSF distribution has seldom been discussed. This factor is taken into account in this section. As discussed earlier, the nonlinear soil consolidation formula in Eq. (8) assumes that the decrease in soil permeability is proportional to the decrease of soil compressibility. That is to say, under the same surcharge q, the Terzaghi consolidation theory overestimates the soil settlement because it fails to consider the decrease of soil permeability and compressibility during the consolidation. In this case study, the surcharge q is 75 kPa, as indicated in Table 2; the larger Nr value will result in smaller final settlement due to nonlinear soil consolidation. Fig. 3 shows the distributions of shear stress and axial force along the pile at different Nr values at the end of consolidation.

Table 3 Summary of results in Case 3. Start time of pile loading

U = 0%

U = 50%

U = 95%

U = 100%

Position of neutral point (m) Axial force (kN) Settlement at pile top (mm) Settlement at pile base (mm) Pile compression (mm) Resistant force at pile base (kN)

7.49 459.2 12.78 11.90 0.89 372.23

7.81 439.8 11.26 10.40 0.86 352.97

8.275 402.28 9.0365 8.2012 0.835 317.85

8.55 341.22 7.60 6.90 0.704 291.89

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R.P. Chen et al. / Computers and Geotechnics 36 (2009) 1265–1271

It can be seen that the nonlinearity of the soil consolidation has some effect on the shear stresses and axial forces along the pile. The position of the neutral point is higher for the case of nonlinear soil consolidation with larger Nr values, while the axial force is overestimated by about 20% using the Terzaghi consolidation theory when the nonlinearity of soil consolidation is significant (Nr = 21). 5.2. Case 2: influence of different pile load magnitudes The distributions of the shear stress and axial force along the pile at different degrees of consolidation for an end bearing pile are shown in Figs. 4–6. The consolidation is induced by an external uniform load, q = 150 kPa, which is calculated using the Terzaghi linear consolidation solution. The results show that the neutral point varies during the consolidation. The final neutral point is higher when the pile load magnitude is larger. Fig. 7 presents the stress under the loading and unloading situations at the pile–soil interface at different locations (A–C) along the pile during soil consolidation, as shown in Fig. 7, which results in the variation of the neutral point with time. 5.3. Case 3: influence of start time of pile loading Since the improved hyperbolic interface model reflects the influence of stress histories on the interface shear stress, the effect of the time of pile loading can be studied. Consider a pile with length 10 m (values of parameters are shown in Table 2) installed

(a) -30

-20

-10

0

10

20

30

40

0.0

Depth z (m)

U=10% U=30% U=50% U=50% Pile loading U=80% U=90% U=100%

0.2

0.4

0.6

0.8

-20

-10

0

10

20

30

40

Shear stress τ (kPa)

(b)

0

50

100

150

200

250

300

350

0.0

450

500

U=10% U=30% U=50% U=50% Pile loading U=80% U=90% U=100%

0.2

Depth z (m)

400

0.4

0.6

0.8

1.0 0

50

100

150

200

250

300

350

400

450

6. Conclusions In this paper, an improved hyperbolic model is developed to describe the load transfer behavior of pile–soil interface, including the development of shear strength during soil consolidation and the loading–unloading conditions. The differential quadrature method is employed to solve the initial boundary-value problem. Based on the present method, three influencing factors of pile NSF problems are considered in the parametric studies. The factors include nonlinear soil consolidation, pile load magnitude, and start time of pile loading. It is found that under the same surcharge, the final soil settlement is smaller when considering the soil nonlinear consolidation and the Terzaghi linear consolidation theory overestimates the maximum pile axial force. The behavior of NSF piles under different pile load magnitudes and start time of pile loading have been discussed. It is also revealed that the direction of shear stresses at some locations along the pile shaft reverses during soil consolidation. Parametric studies show that the stress history is one of the factors that affects the NSF distribution, as the position of the neutral point varies during soil consolidation. Future studies should focus on the application of the method in practice. Acknowledgements

1.0 -30

in the soil layer, while the soil is consolidated under a 150 kPa uniform surcharge. Pile loading of 200 kN is applied at different soil consolidation degrees (U = 0%, 50%, 95% and 100%). The soil consolidation is calculated by using the Terzaghi linear consolidation solution. Table 3 summarizes the position of the neutral point, maximum axial force, settlements at pile top and pile base, pile compression, and resistant force at the pile base at the end of soil consolidation for each start loading time condition. It can be seen that the pile loading time has some effect on the final pile axial force and settlement. If the pile loading is applied at a larger consolidation degree, the final position of neutral point is higher, while the final pile axial load and settlement is smaller. Fig. 8 shows the distributions of shear stress and axial force at different degrees of consolidation for the case of pile loading starting at 50% consolidation degree. It can be seen that the position of neutral point initially descends with increases in the soil consolidation degree. At 50% consolidation, the neutral point jumps to a higher position as a result of the pile load application. It then descends again as soil consolidation progresses.

500

Axial force N (kN) Fig. 8. Influences of pile loading time on (a) shear stress distribution along the pile and (b) axial force distribution along the pile.

The authors wish to thank the National Natural Science Foundation of China for financial support (Research Grant: 50878193) and the Program for New Century Excellent Talents in University (NCET).

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