FEM-BEM coupling analysis of vertically loaded rock-socketed pile in multilayered transversely isotropic saturated media

Computers and Geotechnics 120 (2020) 103437

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Research Paper

FEM-BEM coupling analysis of vertically loaded rock-socketed pile in multilayered transversely isotropic saturated media

T

⁎

Zhi Yong Ai , Yuan Feng Chen Department of Geotechnical Engineering, Key Laboratory of Geotechnical and Underground Engineering of Ministry of Education, College of Civil Engineering, Tongji University, Shanghai 200092, PR China

A R T I C LE I N FO

A B S T R A C T

Keywords: Finite element method Boundary element method Rock-socketed pile Transverse isotropy Multilayered saturated media

The rock-socketed pile is modelled by the finite element method (FEM), while the behavior of soil and rock mass is represented by applying the boundary element method (BEM). By using the consolidation solution of layered saturated foundation as the kernel function of the BEM, the flexibility matrix for soil and rock mass is deduced. With the aid of force equilibrium and displacement continuity condition at pile-soil-rock mass interface, the pilesoil-rock mass interaction problem is solved by compiling a MATLAB program. The solutions obtained by the present theory agree well with those from existing solutions and ABAQUS. Some examples are presented to reveal the effect of socketed length, the ratio of pile-rock mass modulus, length-diameter ratio and the stratification of the socketed rock mass on the load-deformation characteristic of rock-socketed pile in multilayered transversely isotropic saturated media.

1. Introduction In engineering practice, the construction of heavy structures, such as high-rise buildings, airports, bridges and high-speed railways, industrial facilities and offshore oil and gas platforms, requires stable foundations to ensure safe working condition. However, in many areas [1,2] the shallow soils do not have the ability to offer sufficient resistance to limit settlement to an acceptable level. For this reason, the rock-socketed pile is often utilized as an engineering solution to transfer the large vertical load imposed by these structures to the stronger bedrock, so that the settlement of upper structures could be well controlled. Therefore, it exerts great significance to study the load-deformation behavior of vertically loaded rock-socketed piles for engineering practice. So far, many analytical and numerical methods such as the finite element method [3–13], the finite difference method [14], the load transfer method [16–22] and other methods [23–24], have been proposed for the research of rock-socketed piles subjected to vertical loads. As a numerical method with wide adaptability, the FEM was adopted for the research of rock-socketed piles under vertical loads in early time. Pells and Turner [3] employed the FEM to deduce some elastic solutions for rock-socket analysis and design, where two different programs are utilized to provide a check on accuracy. Eid and Shehada [4,5] adopted the FEM to estimate the elastic settlement of the piles

⁎

which are entirely embedded in nonhomogeneous rock under vertical loads. The above studies mainly focus on the elastic load-deformation behavior of vertically loaded rock-socketed piles. Some research has also been conducted to investigate the elastic-plastic response of rocksocketed piles under vertical loads by the FEM. Donald et al. [6] studied the load-settlement behavior of rock-socketed piles by conducting an elasto-plastic finite element analysis. Rowe and Pells [7] studied the behavior of piles socketed in a homogeneous rock mass by the FEM, and by which the interface behavior is simulated with a pointwise strainsoftening model. Rowe and Armitage [8,9] further investigated the load-deflection response of drilled piers in rock with the FEM, and then proposed a procedure that explicitly considered the effect of slip for the design of rock-socketed piles. Leong and Randolph [10] presented a finite element model for rock-socketed piles subjected to vertical loads, in which the shaft responses with and without intimate base contact are particularly considered. Hassan and O'Neill [11] performed an elasticplastic finite element analysis for the research of drilled piles that socketed into soft argillaceous rock. Recently, Li et al. [12] studied the bearing capacity characteristics of rock socketed short piles in weathered rock site by the FEM. Rajan and Krishnamurthy [13] utilized the FEM to predict the termination criteria of vertically loaded rock socketed bored piles. As another alternative numerical method, the finite difference method (FDM) was also utilized for the research of rocksocketed piles under vertical loads. Considering the socket roughness,

Corresponding author. E-mail address: [email protected] (Z.Y. Ai).

https://doi.org/10.1016/j.compgeo.2019.103437 Received 9 July 2019; Received in revised form 31 December 2019; Accepted 31 December 2019 0266-352X/ © 2020 Elsevier Ltd. All rights reserved.

Computers and Geotechnics 120 (2020) 103437

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Fig. 1. Vertically loaded rock-socketed pile in layered foundation.

isotropic saturated rock mass. Compared with the finite element software such as the ABAQUS, the proposed coupling model of FEM-BEM has the advantage of smaller number of elements and fewer unknown equations to be solved, which has higher computational efficiency for the interaction problem of pile and stratified foundation. In this study, the pile is treated as a 1D structure and modelled as 3-node element with the FEM. Meanwhile, by using the consolidation solution of layered saturated media [32] as the kernel function of the BEM, the boundary integral equations of layered foundation can be established. Then, the flexibility matrix of foundation can be obtained by solving the integral equations with numerical integration methods. Through considering the force equilibrium and displacement continuity condition between pile and soil-rock mass, the equations for pile-soil-rock mass interaction system can be established, and the solutions can be obtained by solving these equations. Finally, the validity and precision of the present theory are verified by comparing it with the solution in existing literature. The influences of socketed length, ratio of pile-rock mass modulus, length-diameter ratio and the stratification of socketed rock mass are then discussed.

Kong et al. [14] analyzed the elastic behavior of piles in mudstone by utilizing the FDM and the Rocket program [15]. Different from the FEM and FDM that are classified as the continuum methods, the load transfer method assumes the soils as springs attached to the pile shaft and has been widely used for the investigation of vertically loaded rocksocketed piles. Kim et al. [16] investigated the load-deformation behavior of drilled shafts under vertical loads by a load-transfer approach, in which the emphasis was on quantifying the load-transfer mechanism at the interface between shafts and surrounding highly weathered rocks. Basarkar and Dewaikar [17] analyzed the load-deformation behavior of piles socketed in weathered rocks, where the rock mass was typically found in Mumbai region. Jeong et al. [18] revealed the load transfer characteristics by proposing a shear load transfer function, for the research of vertically loaded drilled shafts socketed into rocks. Kulkarni and Dewaikar [19] studied the load-deformation characteristic of piles socketed into weathered stratum in Mumbai region by incorporating rock mechanics principles. Kodikara and Johnston [20] proposed analytical solutions to predict the load-settlement characteristic of a compressible rock socketed pile under vertical loads. By using the load transfer method, Carrubba [21] evaluated the ultimate skin friction among the pile-rock interface. Seol et al. [22] evaluated the load-deformation characteristic of vertically loaded rock-socketed piles, where the effect of coupled soil resistance was analyzed by the 2D elasto-plastic FEM. In addition, the machine learning algorithms [23] and empirical methods [24] can also be utilized to predict the ultimate load of rock-socketed piles. As is known to all, due to the preferred orientation in the sedimentation process, soils usually exert transversely isotropic characteristics [25,26]. Meanwhile, in virtue of the differences in sedimentary history and the directional alignment of the rock particles, as well as the influence of structural planes such as bedding, joints and schistosity in the rock mass [27], the mechanical characteristics of the vertical and horizontal directions of rock mass are generally different, which can also be considered as multilayered transversely isotropic material. However, it should be pointed out that most of the solutions mentioned above consider the soil and rock mass as multilayered isotropic media in the research of vertically loaded rock-socketed piles. In addition, few solutions have been put forward to consider the influence of saturated porous media on the rock-socketed piles. On the other hand, the BEM is usually considered as the most efficient and practical method for infinite-domain problems due to its high efficiency in calculating the fundamental solutions for the stratified media, and has been adopted by many scholars [28–31] to study the behavior of vertically loaded piles. Considering the advantages of the FEM and the BEM, we utilize a FEM-BEM coupling approach to study the problem of vertically loaded piles socketed in the transversely

2. Modelling of vertically loaded rock-socketed piles in layered media As illustrated in Fig. 1, an elastic rock-socketed pile with length L and diameter D is installed in the layered saturated porous media. A vertical uniformly distributed load V acts on the pile top. The rock mass and soils are both modelled as transversely isotropic saturated porous media. The corresponding parameters of the jth layer are: shear modulus Gvj , the horizontal and vertical Young's moduli Ehj and Evj , the Poisson's ratios vhj and vvhj , the permeability coefficients k vj and khj . The thickness of the jth layer is Δhj . Actually, the pile-soil-rock mass interface is quite complicated. To simplify the analysis, it is assumed that the deformation behavior of a vertically loaded rock-socketed pile is consistent with one-dimensional compression theory, and the pile is modelled with 3-node bar element to improve the calculation accuracy. The contact force along the interface of pile and foundation is considered as a vertical uniformly distributed load. The coupling points between foundation and pile are set on the central axis of the pile before the external load is applied. Therefore, the pile-soil-rock mass interaction system can be decomposed as shown in Fig. 2. In the following, the pile is discretized with FEM and the contact force qiz is acting on the ith node of pile element, while the foundation solution is deduced with BEM and the contact force pz(i) denotes the boundary force acting on the boundary of ith element of foundation. 2

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Fig. 2. The interaction system of pile-soil-rock mass (a) the contact forces acting on the pile (b) the boundary forces acting on the soil-rock mass media.

2.1. FEM equation for rock-socketed pile

functions which can be written as [35]

In order to improve the calculation accuracy, based on the one-dimensional compression bar theory, the pile is modelled by FEM and is discretized using a 3-node bar element, where the 3-node element has been applied for the dynamic analysis of piles and pile groups [33,34]. As shown in Fig. 3, there are three degrees of freedom defined on it: vertical displacements on each node uk , ul and um . The vertical displacement u along the element is approximated by a set of shape

u = ϕ1 uk + ϕ2 ul + ϕ3 um

(1)

where

ϕ1 =

1 ζ (ζ − 1), 2

ϕ2 = 1 − ζ 2,

ϕ3 =

1 ζ (ζ + 1), 2

(2)

in which ζ is the elemental dimensionless coordinate and ζ = 2(z − z2) L , (−1 ⩽ ζ ⩽ 1) ; L is the pile element length and L = z 3 − z1, z2 = (z1 + z 3) 2 . By using the principle of virtual displacements and the shape functions defined above, the stiffness matrix of pile element i can be deduced as follow [35]:

Kip =

EA ⎡ 7 − 8 1 ⎤ − 8 16 − 8 3L ⎢ 1 − 8 7 ⎥ ⎦ ⎣

(3)

where E denotes the Young's modulus, and A denotes the cross section area of the pile element. In terms of the finite element integration theory, we can assemble the global stiffness matrix of a rock-socketed pile, which is listed as follow: Fig. 3. Finite element definition of 3-node element. 3

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excess pore pressure under the slowly changing load can also be expressed as ∼ σ͠ (α, β, s ) = E (α, β , s )∼ p (α, s ) (10) ∼ ∼ ∼ where E (α, β, s ) = s·Φ(α, β, s ) , in which the expression Φ(α, β, t ) stands for the excess pore pressure of point β (rβ, φβ , z β ) arising from the unit vertical load that acts on point α (rα , φα , z α ) at the initial time. Actually, the pile-foundation boundary surface is a spatial surface. For the convenience of calculation, we convert the point load into ring load and let the sum of the ring load be the unit; moreover, we also let the radius of the ring force be the same as the radius of the pile section. Due to the above processing method, the integration on the x − y plane could be avoided, and the flexibility coefficients can be deduced by integration in the depth direction. Through the integration, the boundary integral equations of vertical displacement and excess pore pressure that arising from the loads in the depth direction within the distance L can be expressed as

(4) The global force balance equation of pile can be expressed as

KpUp (t ) = Fp (t ) − Q (t )

(5)

where Up (t ) , Fp (t ) and Q(t ) are the nodal displacement vector, external load vector and equivalent contact force vector at time t , respectively. 2.2. BEM equation of layered foundation For the saturated layered foundation, the flexibility coefficients of foundation under a constant load have been derived by Ai and Cheng [32] by applying the precise integration method [36–38], and the derivation process is illustrated as follows: Firstly, on the basis of the fundamental governing equations of axisymmetric consolidation of transversely isotropic saturated media, the ordinary differential matrix equation in the transformed domain is deduced with the aid of a Laplace-Hankel transform. Then, an extended precise integration method for internal loading situations is used to solve the ordinary differential matrix equation in the transformed domain. Finally, the actual solution for the saturated layered foundation can be recovered with the aid of a numerical inverse transformation. With the inherent advantages of the precise integration method, the present method can avoid exponential overflow in numerical calculation and its efficiency and precision can be guaranteed thoroughly. Moreover, for a pile embedded in saturated foundation, the side resistance along the pile depth changes with time due to the existence of the pore water. Therefore, for a quasi-static case, the displacementtime solution of foundation under a slowly changing load should be deduced as follow [31]:

uz (α, β, t ) = Ψ(α, β, t ) p (α, 0) +

∫0

t

Ψ(α, β, t − τ )

∂p (α, τ ) dτ ∂τ

σ͠ (L, β , s ) =

f1 (t ) ∗ f2 (t ) =

∫0

f1 (τ ) f2 (t − τ )dτ

p (α, t ) = ΦP (e) (t )

(13)

∼(e) u͠ z (L(e) , β , s ) = P (s ) (6)

∼(e) σ͠ (L(e) , β , s ) = P (s )

∫L U∼ (α, β, s)ΦdL

∫L E∼ (α, β, s)ΦdL

(14) (15)

Further, the expression of the two above variables caused by the contact force of the entire pile shaft can be gotten as m

u͠ z (Lp , β , s ) =

∼(e)

∑P

(s )

e=1 m

σ͠ (Lp , β , s ) =

∼(e)

∑P e=1

(s )

∫L U∼ (α, β, s)ΦdL

∫L E∼ (α, β, s)ΦdL

(16)

(17)

For each node of boundary surface, the displacement and excess pore pressure can be expressed as the above equations. Therefore, they can be written in matrix form as following: ∼ ∼∼ Uz (s ) = HP (s ) (18)

(7)

∼∼ σ͠ (s ) = YP (s ) (19) ∼ ∼ where Uz (s ) , σ͠ (s ) and P(s ) are the vertical displacement vector, excess pore pressure vector and boundary force vector of all the pile nodes, ∼ ∼ respectively; H and Y are the flexibility matrices calculated from Eqs. (16) and (17), respectively. In the coupling analysis of FEM-BEM, the key is to establish the relationship between the boundary force vectors in Eqs. (18) and (19) and the equivalent nodal force vector in Eq. (5) through a transformation matrix [40]. From the work of Ref. [41], the relationship between the equivalent nodal force vector and the boundary force vector can be expressed as

(8)

where L [f (t )] stands for the Laplace integral transformation of f (t ) . Then, applying the Laplace transformation to Eq. (6) and then combining with Eq. (8), we have

∼ u͠ z (α, β, s ) = U (α, β , s )∼ p (α, s )

(12)

where Φ = [ ϕ1 ϕ2 ϕ3 ], P (e) (t ) is the boundary force vector. After taking the Laplace transformation to Eq. (13) and combining with Eqs. (11) and (12), the formula of displacement and pressure, which are caused by the contact force within the depth of element e at arbitrary point β (rβ, φβ , z β ) , can be expressed as

where f1 (t ) and f2 (t ) represent two functions related to time t . For the convolution formula, the following relationship can be further obtained,

L [f1 (t ) ∗ f2 (t )] = L [f1 (t )]·L [f2 (t )]

∫L E∼ (α, β, s)∼p (α, s)dL

(11)

where u͠ z (L, β, s ) and σ͠ (L, β, s ) stand for the vertical displacement and excess pore pressure of point β (rβ, φβ , z β ) in the transformed domain, respectively. According to the discrete condition of the pile element, the interface of pile and foundation is also correspondingly discrete, as shown in Fig. 2. Therefore, the force in the unit can be represented by the nodal force through the interpolation function, which can be written as [31]

where uz (α, β, t ) stands for the vertical displacement of point β (rβ, φβ , z β ) arising from the slowly changing load that acts on point α (rα , φα , z α ) at the initial time; p (α, 0) is the vertical load that acts on point α (rα , φα , z α ) at the initial time; Ψ(α, β, t − τ ) is the flexibility coefficient, which represents the vertical displacement of point β (rβ, φβ , z β ) at time t arising from the unit vertical load that acts on point α (rα , φα , z α ) at time τ through t − τ period of time. Therefore, in order to solve the equation mentioned above, the following convolution formula is introduced [39]: t

∫L U∼ (α, β, s)∼p (α, s)dL

u͠ z (L, β , s ) =

(9)

where s is the Laplace transform parameter corresponding to t ; p (α, s ) are the vertical displacement and vertical load in u͠ z (α, β, s ) and ∼ ∼ ∼ the transformed domain, respectively; U (α, β , s ) = s·Ψ(α, β , s ) , which can be obtained from the solutions mentioned in Ref. [32]. Through the similar derivation, the transformed expression of 4

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Z.Y. Ai and Y.F. Chen

Fig. 4. Comparison of the settlement influence factor with Donald et al. [6].

Q(e) (t ) = T (e) P (e) (t )

T (e)

influences of socketed length, ratio of pile-rock mass modulus, lengthdiameter ratio and the stratification of the socketed rock mass are conducted in the following sub-sections. For the convenience, we define the following dimensionless variables: the dimensionless time E k τ = v1 v21 t , the dimensionless vertical displacement of pile top

(20)

T (e)

ΦT ΦdL(e)

= ∫L(e) is the transformation matrix, and . where The assembly method of global transformation matrix is similar to that of pile stiffness matrix, so we have: ∼ ∼∼ Q (s ) = TP (s ) (21) ∼ ∼ where Q(s ) and T denote the global equivalent nodal force vector and transformation matrix, respectively. Eqs. (18) and (21) can be combined to get the following expression: ∼ ∼∼ Q (s ) = GUz (s )

γw D DEv1 uz (0) , the V DE v1 uz ∗ w = V , and the

wh∗ =

dimensionless

vertical

displacement

σ∗

of

pile

2πDLσ (z ) . V

= dimensionless excess pore pressure Here Ev1 and k v1 are the vertical Young's modulus and the permeability coefficient of the shallow soil, respectively, γw is the unit weight of groundwater, uz (0) is the settlement of pile top and σ (z ) is the excess pore pressure at arbitrary depth;

(22)

∼ ∼−1 where G = TH .

3.1. Verification 2.3. The coupling equation of single pile and layered foundation 3.1.1. Comparison with existing solutions To the authors’ knowledge, no solutions have been proposed for the research of vertically loaded piles socketed into transversely isotropic saturated rock mass based on the continuum media model. In order to validate the feasibility of the theory in this study, the present regressive solutions are compared to the existing solutions [6] for the rocksocketed piles embedded in homogeneous rock mass. As depicted in the illustration of Fig. 4, Donald et al. [6] have investigated the load-deformation behavior of a pile socketed in the homogeneous rock mass by the FEM method. It can be observed that excellent agreements have been observed between the results in this study and those in Ref. [6]. Therefore, the reliability of the presented method is verified.

According to the force equilibrium and displacement continuity conditions of pile-soil-rock mass interface, we can obtain:

Up (t ) = Uz (t ) = U (t )

(23)

By applying Laplace transform to the Eqs. (23) and (5), and combining with the Eq. (22), the relationship between the vertical displacement vector and the external load vector can be expressed as

∼∼ ∼ (Kp + G) U (s ) = Fp (s )

(24)

For the assumption that the external load is constant, the above equation can be rewritten as

∼ Fp ∼ U (s ) = (Kp + G)−1 s

3.1.2. Comparison with the finite element method In order to further verify the accuracy of the proposed method, the solutions computed by the proposed method is compared with the solutions of the vertically loaded rock-socketed pile in layered transversely isotropic saturated foundation by ABAQUS. As shown in Fig. 5, in the ABAQUS simulations, the rock masses are modelled by linear brick elements with pore pressure C3D8P and the number of the elements is 16100. The type of pile is C3D8 and the number of the elements is 60. The normal behavior of the pile and the foundation is selected as 'hard contact'. The tangential behavior is set by the 'penalty function', and the coefficient of friction is selected as tan φ in the penalty function. Then we have

(25)

T where Fp = [pz , 0, ⋯, 0](2 m + 1) , in which pz is the vertical load acting on the pile top. Then, the actual vertical displacement can be calculated through the Laplace inverse transform using the Schapery method [42]. The axial force is further obtained through the FEM theory [26]. Finally, through the change of pile's internal force, the boundary force along the pile shaft can be calculated, and then the excess pore pressure can be further obtained through Eq. (19).

3. Numerical examples and analysis

f = σe·K 0·tan φ

According to the theory in the above section, the behavior of the vertically loaded rock-socketed pile in multilayered transversely isotropic porous media is computed by compiling a MATLAB program. The

(27)

where f is the side friction resistance of the pile, σe is the effective stress of foundation around pile, K 0 is the lateral coefficient of foundation and 5

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Z.Y. Ai and Y.F. Chen

w∗ 0

0.1

0.2

0.3

0.4

0

0.2

z L

0.4

0.6 ABAQUS (IJ=0.001) ABAQUS (IJ=1000)

0.8

The present solution (IJ=0.001) The present solution (IJ=1000)

1

Fig. 6. Comparison of the vertical displacement of the pile with ABAQUS.

Table 2 The parameters for the analysis of rock-socketed pile.

Fig. 5. The FEM model.

φ is the angle of internal friction of foundation. The material parameters of the rock masses are listed in Table 1. The other parameters are: Ep Ev1 = 25, L D = 10 , γw = 0.01MN m3 , and Ep is the elastic modulus of the pile. As shown in Fig. 6, the vertical displacements along depth at initial time τ = 0.001 and τ = 1000 are calculated by the two methods. It can be found that the solutions of the present method match well with the calculated results of ABAQUS, which proves that the proposed method is applicable in dealing with the interaction between saturated foundation and rock-socketed pile. Therefore, the accuracy and reliability of the present method is further verified. Additionally, the computation time taken by ABAQUS (16100 elements) are about 10 times longer than the present method, so the computational efficiency of the coupling approach is higher than the finite element method. On the other hand, since the solution by the proposed method is close to the finite element method, it may be taken as a benchmark to the fully discrete mesh-based method.

Case

LR

Ep Ev2

L D

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

2.5 5 7.5 10 15 20 5 5 5 5 5 5 5 5 5

10 10 10 10 10 10 6 8 12 10 10 10 10 10 10

20 20 20 20 20 20 20 20 20 10 15 25 50 100 150

Table 3 The parameters of soil and rock mass. Material

Gv

m = Gv E v

n = Eh Ev

v vh = vh

kv = kh (m s)

Soil Rock mass

Gv1 500Gv1

0.3 0.3

1 1

0.35 0.3

1E−8 1E−9

to the literature [11]; Ep Ev1 = 5000 , in which Ev1 is the modulus of the shallow soil. In the analysis of vertically loaded piles socketed into rock, it is of great significance to reasonably design the socketed length, so that both the cost and the settlement of pile foundation could be well controlled. In the following, the effect of socketed length is studied by conducting a numerical example. As depicted in Fig. 7, a vertically loaded pile is

3.2. The influence of socketed length In the following, the problems of pile-soil-rock mass interaction are investigated by conducting some numerical examples. In order to facilitate the analysis, fifteen cases are set up to reveal the effect of socketed length, pile-rock mass modulus ratio and length-diameter ratio on the performance of pile socketed in rock, as shown in Table 2. The parameters of foundation are selected based on the field tests [43,44] and listed in Table 3. In these field tests, the socketed rock masses are different degrees of weathered rock and the elastic modulus usually locates at the range between 30 MPa and 5GPa. Here, LR = LR D is the dimensional socketed length, in which LR denotes the socketed length of the pile; Ep Ev2 is the pile-rock mass modulus ratio and is selected with reference

V

x

o soil

L B D

LR

g

rock mass

Table 1 The parameters of the foundation. Material

Gvi (MPa)

Evi (MPa)

Ehi (MPa)

v vhi = vhi

kvi = khi (m s)

hi D

Rock mass 1 Rock mass 2

60 120

200 400

200 400

0.35 0.3

1E−9 5E−9

10 10

z Fig. 7. A vertically loaded pile embedded in the half space. 6

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Z.Y. Ai and Y.F. Chen 0.006

N (z) V

0

IJ=0.001 0.005

0.2

0.4

0.6

0.8

1

1.2

1.4

0 Case 1

IJ=1000

Case 2

0.2

0.004

Case 3

wh∗ 0.003

z Lp

0.002

Case 4

0.4

Case 5 Case 6

0.6

0.001 0.8

0 2.5

5

7.5

10

12.5

15

17.5

20

1

LR

Fig. 10. The influence of socketed length on the pile’s axial force.

Fig. 8. The influence of socketed length on the vertical displacement of pile top.

w∗

installed in the half space, where the lower part of the pile is socketed into the rock mass. Fig. 8 demonstrates the influence of socketed length on the vertical displacement of pile top at initial time (τ = 0.001) and τ = 1000 time, respectively. It can be observed that with the socketed length increasing, the vertical displacement of pile top is linearly decreasing. For any socketed length, the settlement of pile top at time of τ = 1000 is relatively larger than that at initial time. The reason for this phenomenon is that with the increase of socketed length, more loads transferred from the pile top are mainly undertaken by the rock-socketed segment, and thus the settlement of pile top is reduced. Due to the dissipation of excess pore water pressure, the settlement of pile gradually raises over time. Therefore, the settlement of pile top at time τ = 1000 is relatively larger than that at initial time. Fig. 9 revealed the variation of the excess pore pressure at pile base over time. It can be concluded that with the consolidation of foundation completed, the pressure at pile base gradually approaches to zero. The initial pressure at pile base grows smaller with the increase of socketed length and dissipates at the same time. The reason is that with the increase of socketed length, the load transferred from the pile top at pile base is decreasing. Therefore, the initial pressure decreases. After the excess pore pressure is dissipated, the influences of socketed length on the axial force and settlement of pile shaft are also demonstrated, respectively. It can be observed from the Fig. 10 that the load transferred from the pile top is mainly undertaken by rocksocketed section and few side resistances are provided by the shallow soils. Then, the axial force at the interface of soils and rock mass produces a significant turning point. When the pile is entirely embedded in the rock mass, the side resistance is totally provided by the rock mass, the axial force curve is relatively smooth. With the increase of socketed length, more loads are undertaken by the socketed rock mass, and thus the axial load of pile shaft at the same depth is decreasing. As shown in Fig. 11, the pile’s vertical displacement is also decreasing with the

0 0

0.001

0.002

0.003

0.004

0.005

0.2 Case 1

z Lp

Case 2

0.4

Case 3 Case 4

0.6

Case 5 0.8

Case 6

1

Fig. 11. The influence of socketed length on the pile’s vertical displacement.

increase of socketed length. Therefore, in engineering practice, the socketed length of pile should be reasonably designed so that the settlement of upper structures could be better controlled.

3.3. The influence of pile-rock mass modulus ratio The effect of pile-rock mass modulus ratio on the performance of vertically loaded rock-socketed pile is investigated in this section. The model of pile-soil-rock mass interaction is shown in Fig. 7. Here, the modulus ratio increases with the decrease of rock mass modulus Ev2 , while the pile’s modulus Ep is kept constant. Fig. 12 demonstrates the effect of pile-rock mass modulus ratio on the settlement of pile top. It can be concluded that with pile-rock mass modulus ratio growing, the vertical displacement of pile top grows larger. The excess pore pressure at pile base is also growing, as shown in Fig. 13. The reason is that with pile-rock mass modulus ratio increasing, 0.0052

Case 2

1.5 Case 1

Case 7

0.005

Case 2

1.2

Case 8

Case 3

0.0048

Case 9

Case 4

0.9

∗ h

w

Case 5

σ∗

0.006

0.0046

Case 6 0.6

0.0044 0.0042

0.3

0 0.001

0.004 0.001 0.01

0.1

1

10

100

1000

τ

0.01

0.1

1

τ

10

100

1000

Fig. 12. The influence of pile-rock mass modulus ratio on the vertical displacement of pile top.

Fig. 9. The influence of socketed length on the excess pore pressure at pile base. 7

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Z.Y. Ai and Y.F. Chen 1

w∗ Case 2

0

0.002

0.003

0.004

0.005

0

Case 7

0.8

0.001

Case 8

σ

0.2

Case 9

0.6 ∗

z Lp

0.4

Case 2

0.6

0.2

0 0.001

0.4

Case 7 Case 8

0.8 0.01

0.1

1

10

100

Case 9

1000

τ

1

Fig. 13. The influence of pile-rock mass modulus ratio on the excess pore pressure at pile base.

Fig. 15. The influence of pile-rock mass modulus ratio on the pile’s vertical displacement.

the modulus of rock mass is decreasing. As we know, the socketed rock mass undertakes most of the load transferred from the pile top. When the modulus of rock mass is decreasing, the overlying soils undertake more loads, which leads to the increase of settlement of pile top. Similarly, the load of pile base transferred from the pile top grows larger with pile-rock mass modulus ratio increasing, which leads to the increase of excess pore pressure at pile base. After the excess pore pressure is dissipated, Figs. 14 and 15 demonstrate the change law of the axial force and displacement of pile shaft under the impact of modulus ratio, respectively. As shown in Figs. 14 and 15, with pile-rock mass modulus ratio increasing, the axial force and displacement at any depth are increasing. The reason is that with pile-rock mass modulus ratio growing, the modulus of rock mass is decreasing, and thus the shear resistances provided by the rock mass is decreasing. Therefore, the axial force and vertical displacement increase. It can be concluded from this study that the vertical displacement of pile top is minimum when the pile-rock mass modulus ratio Ep Ev2 = 6, and the elastic modulus of the socketed rock mass in this working condition is the largest among the four cases. However, it should be pointed out that, in the process of design and construction of rock-socketed piles, with the increase of the elastic modulus of the socketed rock mass, the settlement of pile top will decrease but the construction difficulty and economic investment are also increasing. Therefore, the pile-rock mass modulus ratio should be reasonably selected by combining the building settlement needs with the economic requirements, so that the security of the upper structure supported by the pile foundation can be better improved, while the engineering investment can be reasonably optimized.

loaded pile embedded in soil. However, due to the lack of theoretical research and field test, the influence of the factor for the research of rock-socketed pile is seldom studied. Therefore, the effect of this factor is studied by conducting some numerical examples. The model is shown in Fig. 7 as before. Figs. 16 and 17 reveal the variation of the settlement of pile top and the excess pore pressure at pile base, respectively. It can be observed that the settlement of pile top is proportional to the ratio, and the rate of displacement growth is gradually decreasing. The settlement of pile top is relatively larger at τ = 1000 than that at initial time (τ = 0.001) due to the dissipation of excess pore pressure. The pressure at pile base gradually dissipates over time. For the situation that the length-diameter ratio is relatively small (L D ⩽ 25), the initial pressure at pile base is basically the same with length-diameter ratio growing. When the ratio is relatively large (L D ⩾ 50 ), the initial pressure at pile base is bigger than that of relatively small length-diameter ratio. With the ratio growing, the initial pressure at pile base is decreasing. The reason is that when the length-diameter ratio is relatively large (L D ⩾ 50 ), the pile base undertakes more loads than that of relatively small ratio (L D ⩽ 25), so the initial pore pressure is relatively larger. On the other hand, the increase of length-diameter ratio means that more side resistance has been provided by the shallow soils and less force is undertaken by the pile base, so the initial pressure at pile base is decreasing with the growth of length-diameter ratio. After the excess pore pressure is dissipated, the variation of the axial force and vertical displacement of pile under the impact of lengthdiameter ratio is also studied by Figs. 18 and 19, respectively. It can be observed that with the increase of length-diameter ratio, the shallow soils provide more side friction resistances, and the axial force of the pile shaft is inversely proportional to the length-diameter ratio within any depth of the soil layer. While the vertical displacement is also

3.4. The influence of length-diameter ratio In engineering practice, the influence of length-diameter ratio has been widely investigated by many scholars for the research of vertically 0 0

0.2

0.4

N (z) V

0.6

0.8

0.03 0.025

1

0.02 Case 2

0.2

z Lp

0.4

wh∗ 0.015

Case 7 Case 8

τ =0.001 τ =1000

0.01

Case 9

0.005

0.6

0 10

0.8

30

50

70

90

110

130

150

L D

1

Fig. 16. The influence of pile length-diameter ratio on the vertical displacement of pile top.

Fig. 14. The influence of pile-rock mass modulus ratio on the pile’s axial force. 8

Computers and Geotechnics 120 (2020) 103437

Z.Y. Ai and Y.F. Chen

V

6 5

Case 2

σ∗

o

Case 10

4

x

Case 11 Case 12

3

L

Case 13

soil

Case 14

2

Case 15 1

rock mass 1

LR

0 0.001

0.01

0.1

1

10

100

1000

τ

D

Fig. 17. The influence of pile length-diameter ratio on the excess pore pressure at pile base.

0.2

0.4

rock mass 2

z

N (z) V 0

g B

Fig. 20. A vertically loaded pile socketed in multilayered rock mass. 0.6

0.8

1

0

be designed to a relatively small value when the foundation settlement and bearing capacity requirements are satisfied. In view of such engineering geological conditions, the pile bottom can be slightly socketed or not socketed into the bedrock.

Case 2 0.2

Case 10 Case 11

z Lp

0.4

Case 12 Case 13

0.6

3.5. The influence of stratification of the socketed rock mass

Case 14 Case 15

In the natural environment, due to the difference in sedimentary history, the rock mass always has the property of stratification. Meanwhile, the socketed rock mass plays a major role on the performance of rock-socketed pile, as revealed in the above analysis. Therefore, the study for the factor of stratification of rock mass is necessary. As shown in Fig. 20, a vertically loaded pile is socketed in the multilayered rock mass. In the following, three cases are conducted to study the influence of rock mass stratification, as shown in Table 4. And thickness of the first, the second and the third layer is 15D, 2.5D, 50D, respectively. Table 5 gives the parameters of foundation. The other parameters are: Ep Ev1 = 5000 , L D = 20 . Figs. 21 and 22 reveal the influence of stratification of rock mass on the settlement of pile top and excess pore pressure at pile base, respectively. From the Fig. 21, we can observe that the displacement of pile top gradually increases over time. Meanwhile, the bottom rock mass exerts a more important influence on the pile top's displacement rather than that of the upper rock mass. The displacement of pile top is decreasing with the increase of modulus of bottom rock mass. The displacement of pile top is the smallest when the rock-socketed section is embedded in the relatively harder rock strata. The reason is that the harder rock mass can provide more side resistances than those of the softer rock mass, and less force is undertaken by the shallow soil. Then the displacement of pile top decreases. As shown in Fig. 22, since more loads are undertaken by the relatively harder rock mass, the initial pressure decreases with the modulus of bottom rock mass decreasing. With the dissipation of pore water pressure gradually completed, the variation of the axial force and vertical displacement of pile shaft under the impact of stratification of rock mass are also described, as shown in Figs. 23 and 24. It can be observed that the side resistance provided by the shallow soil is also limited. The axial force is decreasing

0.8

1

Fig. 18. The influence of pile length-diameter ratio on the pile’s axial force.

w∗ 0

0.005

0.01

0.015

0.02

0.025

0.03

0

0.2 Case 2 0.4 z Lp

Case 10 Case 11

0.6

Case 12 Case 13

0.8

Case 14 Case 15

1

Fig. 19. The influence of pile length-diameter ratio on the pile’s vertical displacement.

increasing, the rate of displacement growth is gradually decreasing. This observed phenomenon is also consistent with that in Figs. 16 and 17. It can be observed from this study that for the condition that the socketed length and pile-rock mass modulus ratio remain the same, the settlement of pile top is the smallest when the pile length-diameter ratio L D = 10 . With the increase of length-diameter ratio, the settlement of pile top is increasing and the rate of displacement growth is gradually decreasing. Therefore, in engineering practice, when the socketed rock layer is shallow, the pile length-diameter ratio can be selected as a small value so that the settlement of pile top and engineering investment can be well controlled. For the case that the rock mass is deep in the stratum, the side friction is mainly provided by the upper soil, and at this moment the lower bedrock has little effect on the load-displacement behavior of a single pile. Therefore, the length-diameter ratio can

Table 4 Stratification design for the soil and rock mass.

9

Cases

Layer 1

Layer 2

Layer 3

1 2 3

soil Soil Soil

rock mass 1 rock mass 2 rock mass 2

rock mass 2 rock mass 1 rock mass 2

Computers and Geotechnics 120 (2020) 103437

Z.Y. Ai and Y.F. Chen

Table 5 The parameters of soil and rock mass. Material

Gv

m = Gv E v

n = Eh Ev

v vh (vh)

kv = kh (m s)

soil rock mass 1 rock mass 2

Gv1 300Gv1 600Gv1

0.3 0.3 0.3

1 1 1

0.35 0.3 0.25

1E−8 5E−9 1E−9

Fig. 24. The influence of stratification of rock mass on the pile’s vertical displacement.

relatively harder rock strata so that the settlement of pile foundation could be well controlled. 4. Conclusion A coupled FEM-BEM approach is proposed for the research of vertically loaded rock-socketed pile embedded in multilayered transversely isotropic porous media. The results calculated by the presented theory agree well with those from the existing solutions and ABAQUS. The effects of the socketed length, pile-rock mass modulus ratio, lengthdiameter ratio and the stratification of the socketed rock mass on the rock-socketed pile are studied by conducting some numerical examples. Numerical results show that the settlement of pile top is inversely proportional to the rock-socketed length ratio, and is proportional to the pile-rock mass modulus ratio and pile length-diameter ratio. For the situation of stratification of the socketed rock mass, the settlement of pile top is significantly affected by the bottom rock mass. In addition, the displacement of pile top is almost unaffected by the excess pore pressure. Therefore, in the engineering application of rock-socketed pile, the socketed length should be relatively longer and the lengthdiameter ratio should be relatively smaller according to engineering requirement, while the socketed rock mass should be relatively complete and rigid, so that the settlement of upper structures could be well controlled.

Fig. 21. The influence of stratification of rock mass on the vertical displacement of pile top.

Fig. 22. The influence of stratification of rock mass on the excess pore pressure at pile base.

CRediT authorship contribution statement Zhi Yong Ai: Conceptualization, Methodology, Writing - review & editing, Supervision. Yuan Feng Chen: Data curation, Validation, Investigation, Writing - original draft. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgements This research is supported by the National Natural Science Foundation of China (Grant No. 41672275). The authors would like to express their gratitude to the respected editor, Prof. J.S. McCartney, and the anonymous reviewers for their insightful comments on this article.

Fig. 23. The influence of stratification of rock mass on the pile’s axial force.

with the increase of upper rock mass modulus. When the upper rock mass modulus remains the same, the distribution of pile's axial force is similar, as revealed by case 2 and case 3 in Fig. 23. The influence of stratification of rock mass on the vertical displacement of pile in Fig. 24 is the same as the law revealed from Fig. 21. Therefore, in engineering practice, the stratification of socketed section of rock mass should be taken into consideration and the pile base should be socketed into

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