Finite element prediction on the response of non-uniformly arranged pile groups considering progressive failure of pile-soil system

Abstract

A uniform arrangement of individual piles is commonly adopted in the conventional pile group foundation, and basin-shaped settlement is often observed in practice. Large differential settlement of pile groups will decrease the use-safety requirements of building, even cause the whole-building tilt or collapse. To reduce differential settlement among individual piles, non-uniformly arranged pile groups can be adopted. This paper presents a finite element analysis on the response of pile groups with different layouts of individual piles in pile groups. Using the user-defined subroutine FRIC as the secondary development platform, a softening model of skin friction and a hyperbolic model of end resistance are introduced into the contact pair calculation of ABAQUS software. As to the response analysis of a single pile, the reliability of the proposed secondary development method of ABAQUS software is verified using an iterative computer program. The reinforcing effects of individual piles is then analyzed using the present finite element analysis. Furthermore, the response of non-uniformly arranged pile groups, e.g., individual piles with variable length and individual piles with variable diameter, is analyzed using the proposed numerical analysis method. Some suggestions on the layout of individual piles are proposed to reduce differential settlement and make full use of the bearing capacity of individual piles in pile groups for practical purposes.

Keywords

numerical simulation / non-uniformly arranged pile groups / differential settlement / pile-soil interaction

Introduction

In the conventional pile group foundation, a uniform arrangement of individual piles is commonly adopted according to the entire load of superstructure and the designed bearing capacity of each individual pile, and basin-shaped settlement is often observed in practice [¹–⁴]. Previous studies [⁵–⁷] showed that the settlement of individual piles installed at the central part of pile groups is larger than that installed at the peripheral part of pile groups due to the interactive effects among individual piles, leading to basin-shaped settlement of pile groups. Large differential settlement of pile groups will decrease the use-safety requirements of building, even cause the whole-building tilt or collapse. The researchers always focus on the optimum design of layout of pile groups to reduce differential settlement and make full use of the bearing capacity of each individual pile in pile groups. In general, non-uniformly arranged pile groups, e.g., individual piles with variable length, individual piles with variable diameter, or individual piles arranged with variable pile spacing, can be used to diminish differential settlement among individual piles. The analysis of the vertical bearing characteristics of non-uniformly arranged pile groups is a hot issue in geotechnical engineering [⁸–¹¹].

In situ field test, model test, theoretical approach and numerical analysis can be used to capture the response of non-uniformly arranged pile groups. However, the interactive effects among piles cannot be easily and accurately estimated because of the complicated interactions of piles, pile caps, and surrounding soils. Due to the development of finite element analysis software, numerical analysis method is playing a more and more important role in the response evaluation of axially loaded pile groups, and can be considered as a powerful approach for the analysis of pile behavior. Based on the finite element analysis method, an accurate estimation of pile response can be obtained using appropriate parameter value and reasonable load transfer model, and nonlinear soil behavior or complete history of the pile construction procedure can also be simulated. Field test results [²,³,¹²,¹³] showed that the failure of pile-soil system is affected by loading level and follows a progressive process from pile top to end. Therefore, rational load-transfer functions are required to describe the progressive failure behavior of pile-soil system, and there is a need to select appropriate load-transfer models for the accurate prediction on pile response using the numerical simulation method.

To consider nonlinear response developed along the pile-soil interface and at the pile end, softening model [¹⁴–¹⁶], exponential function model [¹³,¹⁷], and hyperbolic model [¹⁸–²⁰] can be used for practical purposes. A hyperbolic model is commonly used to simulate the relationship between mobilized skin friction and pile-soil relative displacement, and the stress-displacement relationship mobilized at the pile base. In practice, the degradation of skin friction is observed under large loading level [²,²¹–²⁵], and an exponential function model or a hyperbolic model is not suitable for the response analysis of pile-soil interface after the skin friction is fully mobilized. A softening model can be conveniently adopted to describe the nonlinear response developed along the pile-soil interface [¹⁴–¹⁶]. As to the relationship between toe stress and pile end displacement, a hyperbolic model can be used because a fully-developed end resistance is not commonly observed in practice [²⁰,²³,²⁶], and the corresponding reliability has been assessed using the instrumented pile results [²⁶].

As stated previously, numerical analysis, finite element methods, artificial neural networks, or material nonlinearity have been broadly adopted to solve practical engineering problems [²⁷–³⁰], the finite element analysis is an effective way to capture the response of pile groups non-uniformly arranged. ABAQUS software has been widely used to simulate pile response [³¹–³³]. Note that in ABAQUS software the contact problem is often simulated using the zero-thickness contact-element algorithm, and an ideal elastoplastic Coulomb Friction Model is commonly used to capture the pile-soil interface response. This is not in accordance with the actual shear behavior of pile-soil interface, leading to an unreasonable response evaluation on the non-uniformly arranged pile groups. As to the simulation of pile end response, a hard contact model is often used in ABAQUS software to capture the normal contact behavior in a compacted condition. However, using a hard contact model in ABAQUS software a sharp change in tip resistance of the pile is often observed in the initial mobilization of pile tip resistance, and the normal contact behavior of the interface between pile tip and soil beneath pile end cannot be well simulated, leading to an inappropriate prediction on the pile end response or non-convergence of the calculation. Therefore, there is a need to incorporate a softening model and a hyperbolic model into ABAQUS software to get an accurate estimation on pile response.

In this paper, the user-defined subroutine FRIC provided by ABAQUS software is used as the secondary development platform, and a softening model and a hyperbolic model are introduced into the calculation of contact pair in ABAQUS software to simulate the nonlinear response developed along the pile-soil interface and at the pile end, respectively. The reliability of the simulation result of a single pile response is verified using an iterative computer program, suggesting the rationality of the load transfer models imbedded in ABAQUS software. The reinforcing and sheltering effect, e.g., the loaded pile displacement will reduce due to the presence of the non-loaded pile, is then analyzed using the proposed secondary development method of ABAQUS software. Considering the complicated interactive effects among piles, the response of pile groups with different layouts of individual piles, e.g., pile groups composed of identical pile, piles with variable length or piles with variable diameter, is captured using the finite element analysis. Some suggestions on the layout of individual piles are proposed to reduce differential settlement and make full use of the bearing capacity of individual piles in pile groups for practical purposes.

Load transfer model of skin friction

As mentioned earlier, the hyperbolic model has some limitations in the analysis of pile response after the skin friction is fully mobilized, and the softening model can be conveniently adopted to describe the nonlinear response developed along the pile-soil interface, especially under large loading level. As suggested by Zhang and Zhang [¹⁵], the softening model can be expressed as Eq. (1), and the reliability of the softening model adopted has been verified.

(1){\tau }_{\mathrm{s}}\left(z\right)={\displaystyle \frac{S_{\mathrm{s}}(z)\left[a+c{S}_{\mathrm{s}}\left(z\right)\right]}{{\left[a+b{S}_{\mathrm{s}}\left(z\right)\right]}^{\mathrm{2}}}},

where τ_s(z) is the unit skin friction at a given depth z; S_s(z) is the pile-soil relative displacement at a given depth z; a, b, and c are the model parameters, and can be calculated as:

(2)a={\displaystyle \frac{{\beta }_{\mathrm{s}}-\mathrm{1}+\sqrt{\mathrm{1}-{\beta }_{\mathrm{s}}}}{\mathrm{2}{\beta }_{\mathrm{s}}}}\frac{S_{\mathrm{su}}}{{\tau }_{\mathrm{su}}},

(3)b={\displaystyle \frac{\mathrm{1}-\sqrt{\mathrm{1}-{\beta }_{\mathrm{s}}}}{\mathrm{2}{\beta }_{\mathrm{s}}}}\frac{\mathrm{1}}{{\tau }_{\mathrm{su}}},

(4)c={\displaystyle \frac{\mathrm{2}-{\beta }_{\mathrm{s}}-\mathrm{2}\sqrt{\mathrm{1}-{\beta }_{\mathrm{s}}}}{\mathrm{4}{\beta }_{\mathrm{s}}}}\frac{\mathrm{1}}{{\tau }_{\mathrm{su}}},

where β_s is defined as the ratio of the residual unit skin friction τ_s to the limiting unit shaft resistance τ_su; and S_su is the pile-soil relative displacement corresponding to the ultimate unit skin friction. The values of S_su and β_s can be determined following the suggestion of Zhang Q Q and Zhang Z M [¹⁵].

The limiting shear stress τ_su of the pile-soil interface can be expressed as [³⁴]:

(5){\tau }_{\mathrm{su}}=K{\sigma }_{\mathrm{v}z}^{\prime }\tan \delta ,

where {\sigma }_{\mathrm{v}z}^{\prime } is the vertical effective stress at a given depth z; δ is the friction angle of pile-soil interface, which is related to the effective angle of shearing resistance of surrounding soils ϕ′, and can be estimated by δ = arctan[sinϕ′ × cosϕ′/(1+ sin²ϕ′)] as suggested by Cho et al. [³⁵]; and K is the lateral earth pressure coefficient, for the normally consolidated soil, K=\mathrm{1}-\sin {\varphi }^{\prime } [³⁶], while for the overconsolidated soil, K=\left(\mathrm{1}-\sin {\varphi }^{\prime }\right){\mathit{OCR}}^{\sin {\varphi }^{\prime }} (OCR repre-sents overconsolidation ratio) [³⁷].

Load transfer model of end resistance

Previous studies [³⁸–⁴⁰] demonstrated that the relationship between toe stress and pile end displacement can be simulated using a hyperbolic model, because pile end resistance cannot often be fully mobilized under large loading level. The hyperbolic model can be expressed as Ref. [²⁶]:

(6){\tau }_{\mathrm{b}}={\displaystyle \frac{S_{\mathrm{b}}}{A+B{S}_{\mathrm{b}}}}={\displaystyle \frac{S_{\mathrm{b}}}{{\displaystyle \frac{\mathrm{\pi }{r}_{\mathrm{0}}\left(\mathrm{1}-{\upsilon }_{\mathrm{b}}\right)}{\mathrm{4}{G}_{\mathrm{b}}}}+{\displaystyle \frac{R_{\mathrm{bf}}}{{\tau }_{\mathrm{bu}}}}{S}_{\mathrm{b}}}},

where τ_b is the unit end resistance; S_b is the pile end displacement; r₀ is the pile radius; G_b is the shear modulus of soil below the pile base; υ_b is the Poisson’s ratio of the base soil; R_bf is an empirical coefficient ranged from 0.80 to 0.95 [⁴¹]. A series of field tests conducted by the authors [²,¹⁹,⁴²–⁴⁴] demonstrated that a single pile response could be evaluated with sufficient accuracy when the value of R_bf was adopted as 0.9. Therefore, a default value of 0.9 is chosen for R_bf in the present study; τ_bu is the maximum possible value of unit end resistance, and can be computed following the suggestion of Zhang et al. [⁴⁵]; and A and B are the model parameters, e.g., A={\displaystyle \frac{\mathrm{\pi }{r}_{\mathrm{0}}\left(\mathrm{1}-{\upsilon }_{\mathrm{b}}\right)}{\mathrm{4}{G}_{\mathrm{b}}}},and B={\displaystyle \frac{R_{\mathrm{bf}}}{{\tau }_{\mathrm{bu}}}}.

Secondary development of pile-soil interface model in ABAQUS software

In practical applications, load-transfer approach is an efficient method for the nonlinear response analysis of a single pile embedded into multilayered soils [³⁸–⁴⁰,⁴⁶–⁴⁸]. However, load-transfer method has some limitations in the response evaluation of pile groups because the pile-soil interactive effects cannot be considered in the conventional load-transfer curve. Using the finite element analysis, the pile-soil interaction and the effect of the variation of normal stress along pile-soil interface on shear behavior can be simulated. The response of pile groups can be captured using the secondary development of finite element analysis, especially for large pile groups.

Using the user-defined subroutine FRIC, often adopted to define the frictional behavior of contact surface, provided by ABAQUS software as the secondary development platform, the softening model of skin friction (see Eq. (1)) and the hyperbolic model of end resistance (see Eq. (6)) are introduced into the contact calculation of ABAQUS software to achieve the analysis on pile response (Note that the proposed models are limited in the conditions that the shaft resistance shows softening behavior and the base resistance shows hardening behavior). The process of the secondary development of pile-soil contact calculation in ABAQUS software can be summarized as follows (see Fig. 1).

1) Input the parameters of load transfer model of skin friction, such as a, b, and c computed using Eqs. (2), (3), and (4), respectively.

2) When the slip is observed in the pile-soil interface, the value of LM is assumed to be 0, where LM is a marking variable used for describing the state of relative displacement in the subroutine FRIC. Compute the relative displacement in two orthogonal directions parallel to the interface plane, e.g., s(1) and s(2), using the following equations: s(1) = s(1) + dgam(1), and s(2) = s(2) + dgam(2), respectively, where dgam(1) and dgam(2) are the increment of the relative displacement in two orthogonal directions, respectively.

3) When the value of LM is adopted as 2, the contact surface is assumed to be in a perfectly smooth state. The pile-soil contact points are separate, no extra variables are needed to be defined, and the subroutine FRIC is then exited.

4) Calculate the skin friction along two orthogonal directions of pile shaft using Eq. (1), i.e.,τ_s(1) and τ_s(2).

5) Assign the value of {\displaystyle \frac{\partial {\tau }_{\mathrm{s}}\left(\mathrm{1}\right)}{\partial s\left(\mathrm{1}\right)}} and {\displaystyle \frac{\partial {\tau }_{\mathrm{s}}\left(\mathrm{2}\right)}{\partial s\left(\mathrm{2}\right)}} to the stiffness coefficient matrix of pile shaft, i.e., \left[\begin{array}{cc}\hfill {\displaystyle \frac{\partial {\tau }_{\mathrm{s}}\left(\mathrm{1}\right)}{\partial s\left(\mathrm{1}\right)}}\hfill & \hfill \mathrm{0}\hfill \\ \hfill \mathrm{0}\hfill & \hfill {\displaystyle \frac{\partial {\tau }_{\mathrm{s}}\left(\mathrm{2}\right)}{\partial s\left(\mathrm{2}\right)}}\hfill \end{array}\right].

6) Repeat steps 1 to 5 to obtain the stiffness coefficient matrix of the load transfer model of pile end resistance.

7) Update the stiffness coefficient matrixes of pile shaft and base by exchanging data of state variables between ABAQUS and FRIC (Parametric variables related to the constitutive models are stored in the state variables and can be updated with the calculating process). The updated stiffness coefficient matrixes are then used in the ABAQUS main program to complete the following incremental step.

User subroutine FRIC can be used when the extended versions of the classical Coulomb friction model provided in ABAQUS are too restrictive, and a more complex definition of the shear transmission between two contacting surfaces is required. User must provide the entire definition of the shear interaction between two contacting surfaces, and assign values to variables associated with the friction definition. FRIC is called only if the contact pressure is positive or the contact point is open in the previous iteration.

The following case was designed to check the reliability of the previously user-defined subroutine FRIC. The pile diameter was adopted as 0.8 m, and the pile length was taken as 20 m. Elastic modulus of the single pile used was 30 GPa. Elastic modulus of the soil around the pile shaft and beneath the pile base were adopts as 40 MPa and 80 MPa, respectively. Uniform load was applied on the whole pile head, and increased step by step. To avoid boundary effects, the dimensions of the model boundaries were kept as 25 times the pile radius in the lateral extent and 1 times the pile length below the pile length in the vertical extent, respectively. Mesh discretization was done by using the C3D8R element. Considering the progressive failure process of pile-soil system (see Fig. 2), a highly effective computer program can be used to analyze the nonlinear load-settlement behavior of a single pile (see Fig. 3) [¹⁵] using the load transfer method.

To verify the validity of the implementation flow chart developed in ABAQUS software, the load-displacement relationship derived from the present finite element analysis is compared with the calculation result estimated from the iterative algorithm, as shown in Fig. 4.

The second case history of validation was reported by O’Neill et al. [⁴⁹] on a closed-ended steel pipe pile in stiff overconsolidated clays. The pile had an external radius of 137 mm with a wall thickness of 9.3 mm and was driven to a penetration of 13.1 m. The soil compression modulus back-calculated from the test results was taken as 195 MPa, and the elastic modulus for the steel pipe pile was adopted as 210 GPa (Castelli and Maugeri [⁵⁰]). The load-displacement relationship derived from the present finite element analysis is compared with measured single-pile load-settlement curve given by O’Neill et al. [⁴⁹], as shown in Fig. 5.

Figure 4 shows that using the softening model of skin friction and the hyperbolic model of end resistance mentioned above, the calculated load-displacement relationship at the pile head derived from the secondary developed finite element analysis is generally consistent with the calculation result of the iterative algorithm suggested by Zhang Q Q and Zhang Z M [¹⁵]. Figure 5 shows that at the loading levels, the load-displacement curve at the pile head plotted from the present finite element analysis is generally consistent with the measured results given by O’Neill et al. [⁴⁹]. This demonstrates that the softening model of skin friction and the hyperbolic model of end resistance have been successfully developed in ABAQUS software, which can be used for the subsequent analyses.

Finite element analysis on the response of a single pile and pile groups

A single pile response

As previously stated, the pile length adopted was 20 m and the pile diameter used was 0.8 m in the finite element analysis. The single pile was embedded into a homogeneous soil layer with an elastic modulus of 40 MPa, a Poisson’s ratio of 0.30, an internal friction angle of 35° and a dilation angle of 5°. Sand has stiffness that increases with depth and is certainly not homogeneous in terms of the stiffness, but for simplification, soil is treated as homogeneous medium in this analysis and the elastic modulus is considered as constant. The calculation model size was taken as 30 m (width) × 30 m (length) × 40 m (depth). The single pile response computed using the secondary developed finite element simulation is shown in Fig. 6(a). A series of numerical simulations has been done to study the influence of the discretization. The results of the load-settlement curves was shown in Fig. 6(b). Note that “EN” means the element numbers. Taking an applied load of 400 kN as an example, the calculated surface settlement of soil around the single pile is shown in Fig. 7.

Figure 6(a) shows a steep-descending load-displacement curve, and the distinct plunging point corresponding to the load-displacement curve of the axially loaded single pile can be taken as the ultimate bearing capacity of a single pile. The settlement of a single pile linearly increases with increasing applied load up to an applied load of 400 kN, and there is a significant increase in the pile head displacement when the applied pile head load is larger than 500 kN. The ultimate bearing capacity of the axially loaded single pile can be adopted as 500 kN.

As shown in Fig. 6(b), there is an optimal number of elements in the simulation analysis. When the number of elements is smaller than the optimal amounts, the results are comparable. When the number of elements is larger than the optimal amounts, great errors are existed in the calculation results.

It can be seen from Fig. 7 that the surface settlement of soil around the axially loaded single pile decreases with increasing distance from pile center, and the vertical settlement of soil around the axially loaded single pile caused by the skin friction can be assumed to be a logarithmic relationship of the radial distance away from pile shaft. A large settlement of soil around the axially loaded single pile, e.g., 8 mm, is observed within 3 m distance from the pile center, approximately 3.75 times pile diameter.

An axially loaded single pile response and a non-loaded single pile response caused by the loaded pile

To clarify the influence of the loaded single pile on the non-loaded single pile response, two individual piles with pile length of 20 m and pile diameter of 0.8 m were set in the finite element analysis. The distance from the loaded pile center to the non-loaded pile center was adopted as 3.2 m (4 times pile diameter). The calculation model size and the soil properties were considered to be identical to those adopted in the finite element analysis of an axially loaded single pile in Section 5.1. The axially loaded single pile response and the non-loaded single pile response caused by the loaded pile are shown in Fig. 8. Taking an applied load of 400 kN as an example, the computed surface settlement of soil around the single pile considering the influence of existence of the non-loaded single pile is shown in Fig. 9.

Figure 8 shows that during the whole loading process the settlement of a load-free pile increases linearly with increasing load applied on an adjacent pile, whereas the load-displacement relationship of the loaded single pile is highly nonlinear, especially at large loading level, i.e., the pile-pile interaction remains essentially elastic. This finding is consistent with a well-documented case history on the interaction between two identical piles reported by Caputo and Viggiani [⁵¹], and can be used in the response analysis of pile groups considering the elastic pile-pile interaction. It can also be seen from Fig. 9 that the loaded pile displacement affected by the presence of a non-loaded pile is smaller than that without the influence of a non-loaded pile under the same loading level, suggesting that the presence of a non-loaded pile has a reinforcing effect on the loaded pile bearing capacity. This finding may provide an inspiration for the analysis of the modified pile interaction factor used in the response calculation on pile groups.

Figure 9 shows that as previously stated the vertical settlement of soil around the axially loaded single pile can be considered as a logarithmic relationship of the radial distance away from pile shaft. However, due to the presence of a load-free pile, the soil settlement around the axially loaded pile is smaller than that without a non-loaded pile. Due to the reinforcing effect of non-loaded pile, a large soil settlement around the axially loaded single pile, e.g., 8 mm, is observed within 1.5 m distance from the pile center, approximately 2 times pile diameter. It can be concluded that the presence of a non-loaded pile hinders free soil displacement, and leads to a reduction of soil settlement around the axially loaded pile. This indicates that the presence of a non-loaded pile has a reinforcing effect on the bearing capacity of soil around the loaded pile. This finding can be used to modify the interaction effects of pile groups considering the reinforcing or sheltering effect of individual piles in pile groups, e.g., the reduced displacement caused by the reinforcing effect of adjacent piles should be considered in the displacement of the soil away from an individual pile in pile groups.

Response of pile groups

To eliminate the boundary effect presented in the finite element analysis, the calculation model size was taken as 30 m (width) × 30 m (length) × 50 m (depth). The pile groups composed of nine identical piles with pile length of 20 m and pile diameter of 0.8 m was designed in the present finite analysis simulation. Nine piles embedded in a homogeneous soil layer, e.g., medium dense sand, were placed at a spacing of 4 times pile diameter and connected to a pile cap in contact with the soil. The pile cap size was adopted as 8 m (width) × 8 m (length) × 0.5 m (thickness). To avoid boundary effects, the dimensions of the model boundaries were kept as 5 times the cap width in the lateral extent and 1 times the pile length below the pile length in the vertical extent, respectively. Mesh discretization was done by using the C3D8R element. The parameters of pile cap, pile and soil used in the present finite element simulation are listed in Table 1.

To clarify the response of non-uniformly arranged pile groups, a series of pile groups composed of dissimilar piles with variable pile length or pile diameter were established in the present finite element simulation. Under the same conditions of pile, pile cap and soil profile, different layouts of individual piles were designed considering the same or approximate amount of reinforced concrete used, as summarized in Tables 2 and 3.

Response of pile groups composed of identical piles

In the response analysis on the pile groups composed of identical piles, uniform load was applied on the whole pile cap. The uniform load was increased step by step, and the magnitude of total load at each step was computed as 1600 kN according to the pile cap dimension. Taking a total load of 4800 kN applied on the pile cap as an example, the response of the pile groups composed of 9 identical piles is shown in Fig. 10. Under different loading levels, the response of individual piles arranged at different locations of pile groups is shown in Fig. 11. Note that Fig. 11 is used to assess the differential settlement of each individual pile in pile groups under the same applied load. According to the deformation of different positions of pile cap and surrounding soil, a three-dimensional graph of settlement can be depicted using SURFER software. Figure 12 shows the three-dimensional graph of settlement of pile cap and surrounding soil under a total load of 4800 kN.

It can be seen from Figs. 10, 11, and 12 that under the same loading level, the largest, the second largest and the smallest pile head settlement are observed at the center, edge, and corner pile, respectively, due to the interactive effects of individual piles in pile groups. This finding is consistent with the computed result of Zhang et al. [⁶]. The differential settlement showed as a basin shape is observed in the pile groups composed of identical piles (see Fig. 12), which is consistent with the field measurement result [¹–⁴]. This is due to the fact that the individual piles arranged at different locations are subjected to different cumulative interactions, and the center pile response is most significantly affected by other adjacent piles, followed by the edge pile and the corner pile. For practical purposes, to reduce differential settlement of pile groups, dissimilar piles can be installed in different locations of pile groups to eliminate the discrepancy in interactive effects.

Response of pile groups composed of individual piles with variable length

As previously stated, uniform load was applied on the whole pile cap, and increased step by step. The magnitude of total load at each step was adopted as 1600 kN computed according to the dimension of pile cap. Taking a total load of 6400 kN applied on the pile cap as an example, the response of the pile groups composed of individual piles with variable length is shown in Fig. 13. Under different loads applied on the pile cap, the response of each individual pile in the pile groups with variable length is shown in Fig. 14. Based on the calculated results of pile groups shown in Fig. 14, the differential settlement among individual piles with different layouts can be evidently shown in Fig. 15. Note that in Fig. 15 “center pile (L=24 m)-corner pile (L=19 m)” means the differential settlement between 24 m length center pile and 19 m length corner pile.

From Fig. 13 it can be seen that basin-shaped settlement is also observed in the pile groups composed of individual piles with variable length, however, the differential settlement of individual piles is smaller than that of the pile groups with variable length (see Fig. 14), suggesting that the settlement of each individual pile with variable length arranged at different locations of pile groups tends to be uniform.

Figure 15 shows that the differential settlement of individual piles can be reduced by appropriately increasing the center pile length and properly decreasing the length of edge pile and corner pile, e.g., when the length of center pile, edge pile, and corner pile used are adopted as 28, 22, and 16 m, respectively, the largest differential settlement of the pile groups with variable length is 1.84 mm (see Fig. 15(c)) under a total load of 9600 kN, which is smaller than that of the pile groups composed of identical piles at the same loading level (e.g., 2.84 mm). It can be concluded from the numerical simulation result that using the same reinforced concrete amount the differential settlement of pile groups can be diminished by varying pile length of individual piles at different positions. For practical purposes, the economic benefits will be brought by using the pile groups with variable length. Due to the decrease of edge pile length or corner pile length, the construction difficulties of piles will be weakened and the corresponding cost will be reduced. In practice, pile groups with variable length, e.g., properly increasing the center pile length, and appropriately decreasing the edge pile length or the corner pile length, can be used to reduce differential settlement and make full use of the bearing capacity of each individual pile in pile groups.

Response of pile groups composed of individual piles with variable diameter

As above-mentioned, in the response calculation on the pile groups with variable diameter, uniform load was applied on the whole pile cap, and increased step by step. The magnitude of total load at each step was adopted as 1600 kN computed according to the dimension of pile cap. Taking a total load of 4800 kN applied on the pile cap as an example, the response of the pile groups composed of individual piles with variable diameter is shown in Fig. 16. Under different loads applied on the pile cap, the response of each individual pile in the pile groups with variable diameter is shown in Fig. 17. Based on the calculated results of the pile groups shown in Fig. 17, the differential settlement among individual piles with different layouts can be evidently shown in Fig. 18. Note that in Fig. 18 “center pile (d=0.9 m)-corner pile (d=0.7 m)” means the differential settlement between 0.9 m diameter center pile and 0.7 m diameter corner pile.

Figure 17 shows that the differential settlement of the pile groups with variable diameter decreases comparing with that of the pile groups composed of identical piles, and the settlement of each individual pile of pile groups with variable diameter tends to be uniform, e.g., when the diameter of center pile, edge pile, and corner pile are adopted as 1.2, 0.8, and 0.6 m, respectively, the largest differential settlement of the pile groups with variable diameter is 1.73 mm (see Fig. 18(d)) under a total load of 9600 kN, which is smaller than that of the pile groups composed of individual piles at the same loading level (e.g., 2.84 mm). The numerical simulation result shows that using proper reduction of reinforced concrete amount the differential settlement of pile groups can be diminished by varying pile diameter of individual piles at different positions, e.g., properly increasing the center pile diameter, and appropriately decreasing the edge or the corner pile diameter in practice, and the bearing capacity of the soil beneath pile cap can be mobilized in a large degree. For practical purposes, the construction difficulties and the cost of pile groups can be decreased by using the non-uniformly arranged pile groups with variable diameter.

Load-sharing ratio of different individual piles

Under different loading levels, the load acted on each individual pile head in pile groups can be obtained using the present analysis results. The load-sharing behavior of different individual piles can be described using a load-sharing ratio [⁵²], which is defined as the ratio of the load shared by each individual pile to the total load applied on the whole pile cap, as follows:

(7){\alpha }_{\mathrm{p}}={\displaystyle \frac{Q_{\mathrm{ip}}}{Q_{\mathrm{pr}}}},

where α_p is the load-sharing ratio, Q_pr is the load applied on the whole pile cap, and Q_ip is the loads shared by each individual pile.

Figure 19 shows the load shared by piles (0.8 m pile diameter and 20 m pile length) arranged at different locations of pile groups composed of identical piles. The load-sharing ratio of different individual piles in the pile groups composed of identical piles is shown in Fig. 20.

Figure 19 shows that in the pile groups composed of identical piles, the center and corner pile share the maximum and minimum load under the same loading level, respectively. The discrepancy of the load shared by different individual piles decreases with increasing total load applied on the pile cap, suggesting a full mobilization of axial bearing capacity of each individual pile at large loading level. The ultimate bearing capacity of each individual pile can be adopted as about 500 kN.

Figure 20 shows that the load-sharing ratio of different individual piles increases with increasing total load applied on the pile cap up to a certain applied load, e.g., 4000 kN, and then afterward decreases with an increase in total load. It can also be observed from Fig. 20 that the load-sharing ratio of the center pile attains the maximum value earliest, suggesting the center pile bearing capacity is fully mobilized at the earliest possible time. This is due to the fact that the center pile response is most significantly affect by other adjacent piles due to the pile-pile interactions.

Figure 21 shows the load shared by identical piles (0.8 m diameter and 20 m length) and individual piles with variable length (0.8 m diameter, 28 length center pile, 22 m length edge pile, and 16 m length corner pile) arranged at different locations of pile groups. Figure 22 presents the load-sharing ratio of identical piles and individual piles with variable length arranged at different locations of pile groups.

Figure 21 shows that in the pile group composed of individual piles with variable length, the center pile with a largest length of 28 m shares the maximum load, and the load shared by the corner pile with a minimum length of 16 m is the smallest. The increase rate of the load shared by individual piles decreases with increasing total load, and the axial bearing capacity of each individual pile at large loading level is fully mobilized. Based on the calculation result of Fig. 21, the ultimate bearing capacity of the center pile, the edge pile and the corner pile can be adopted as about 800, 600, and 400 kN, respectively.

Figure 22 shows that under the same loading level, the load shared by the center pile in the pile groups with variable length is larger than that in the pile groups composed of identical piles, while the load shared by the corner pile in the pile groups with variable length is smaller than that in the pile groups with identical piles. The load-sharing ratio of individual piles with variable length is greatly influenced by loading level. The load-sharing ratio attains a maximum value at a certain applied load, and then afterward decreases with increasing total load applied on the pile cap. It can be observed that the center pile in the pile groups with identical piles is fully mobilized earliest, and will be destroyed with further increased load, because the center pile is most significantly affect by other adjacent piles. However, the bearing capacity of the edge pile or the corner pile has not been fully mobilized, and cannot be completely utilized in the conventional pile groups with identical piles. For practical purposes, considering the same or approximate amount of reinforced concrete used, the ultimate bearing capacity of each individual pile can be achieved simultaneously by adjusting individual pile length, e.g., properly increasing the center pile length, and appropriately decreasing the edge or corner pile length. In the pile groups with variable length, more load will be shared by the center pile, causing a large compression of the center pile. In practice, the center pile compression should be evaluated to achieve the safety and the serviceability requirements.

Figure 23 shows the load shared by identical piles (0.8 m diameter and 20 m length) and individual piles with variable diameter (20 m length, 1.2 m diameter center pile, 0.8 m diameter edge pile, and 0.6 m diameter corner pile) arranged at different locations of pile groups. Figure 24 presents the load-sharing ratio of identical piles and individual piles with variable diameter arranged at different locations of pile groups.

Figure 23 shows that in the pile group composed of individual piles with variable diameter, the center pile with a largest diameter of 1.2 m shares the maximum load, and the load shared by the corner pile with a minimum diameter of 0.6 m is the smallest. The increase rate of the load shared by individual piles decreases with increasing total load, and the axial bearing capacity of each individual pile at large loading level is fully mobilized. Based on the calculation result of Fig. 23, the ultimate bearing capacity of the center pile, the edge pile and the corner pile can be adopted as about 1200, 500, and 300 kN, respectively.

Figure 24 shows that the load-sharing ratio attains a maximum value at a certain applied load, and then afterward decreases with increasing total load applied on the pile cap. This is consistent with the findings of the pile groups composed of identical piles and individual piles with variable length. Compared with the pile groups with variable length, there is an advantage of the pile groups with variable diameter that the center pile compressive stiffness can be improved by increasing pile diameter, leading to a decrease of pile compression. It can also be concluded from Fig. 24 that in the pile groups with variable diameter the load-sharing ratio of the center pile reaches the maximum value latest, suggesting a full mobilization of the bearing capacity of the center pile is later than that of the corner pile and the edge pile.

The largest, the second largest, and the smallest pile settlements are supposed to be presented in the corner, edge, and center piles, respectively. This is due to the fact that the individual piles arranged at different locations are subjected to different cumulative interactions, the most serious interactive effect is supposed to occur in the center pile, and the smallest interactive effect is presented in the corner pile. In the pile groups with variable length or diameter, the ultimate bearing capacity of center pile increases, therefore, the settlement of each individual pile tends to be uniform.

Discussion

Reinforcing effect of an adjacent pile on the axially loaded single pile response

The previous analysis shows that the pile-pile interaction remains essentially elastic, and the presence of a non-loaded pile has a reinforcing effect on the loaded pile capacity, leading to a reduction of the loaded pile settlement. This finding can be used to modify the traditional calculation method for the response analysis on pile groups considering the reinforcing effect of piles.

The total shaft displacement can be assumed to be consisted of nonlinear displacement of a narrow disturbed zone of soil around the pile shaft and purely elastic displacements of soil away from the pile shaft (see Fig. 25(a)). Following the suggestion of Lee and Xiao [⁵³], the nonlinear displacement can be simplified as a displacement discontinuity with no physical size around the pile shaft to provide a mathematical treatment (see Fig. 25(b)), and the elastic vertical soil displacement is assumed to develop outside the displacement discontinuity. Considering the reinforcing effect caused by the presence of adjacent pile, the elastic vertical soil displacement will be reduced. The total shaft displacement of the loaded pile A, W_sA (see the solid line in Fig. 25), is composed of three parts: 1) the nonlinear displacement around the pile shaft, W_snA; 2) the elastic soil displacement outside the displacement discontinuity, W_seA; 3) the elastic soil settlement reduced by the adjacent load-free pile due to the reinforcing effects, W_sAB, and can be written in the following form:

(8)W_{\mathrm{sA}}=W_{\mathrm{snA}}+W_{\mathrm{seA}}-W_{\mathrm{sAB}}.

Based on Eq. (1), the nonlinear displacement W_snA can be computed as:

(9)W_{\mathrm{snA}}={\displaystyle \frac{\left(a-\mathrm{2}ab{\tau }_{\mathrm{s0}}\right)-a\sqrt{\mathrm{1}-\mathrm{4}b{\tau }_{\mathrm{s0}}+\mathrm{4}c{\tau }_{\mathrm{s0}}}}{2\left(b^{\mathrm{2}}{\tau }_{\mathrm{s0}}-c\right)}},

where τ_s0 is the shear stress along the loaded pile shaft.

Following the elastic solution proposed by Randolph and Wroth [⁵⁴], the elastic soil displacement W_seA induced by the shaft shear stress can be expressed by:

(10)W_{\mathrm{seA}}={\displaystyle \frac{{\tau }_{\mathrm{s}\mathrm{0}}{r}_{\mathrm{0}}}{G_{\mathrm{s}}}}\ln \left({\displaystyle \frac{r_{\mathrm{m}}}{r_{\mathrm{0}}}}\right),

where G_s is the shear modulus of soil around the pile shaft, r₀ is the pile radius, r_m is the radial distance from the pile center to a point where the shaft shear stress induced by the pile is negligible, the value of which can be adopted following the suggestion of Zhang et al. [⁴⁰].

The elastic soil settlement reduced by the adjacent load-free pile W_sAB due to the reinforcing effects can be computed in the following form:

(11)W_{\mathrm{sAB}}={\displaystyle \frac{{\tau }_{\mathrm{s}\mathrm{0}}{r}_{\mathrm{0}}^{\mathrm{2}}}{S_{\mathrm{a}}{G}_{\mathrm{s}}}}\left[\ln \left({\displaystyle \frac{r_{\mathrm{m}}}{r_{\mathrm{0}}}}\right)-\ln \left({\displaystyle \frac{S}{r_{\mathrm{0}}}}\right)\right],

where S_a is the distance from the loaded pile center to the non-loaded pile center.

Substitute Eqs. (9), (10), and (11) into Eq. (8), the total shaft displacement W_sA can be written as:

(12)W_{\mathrm{sA}}={\displaystyle \frac{\left(a-\mathrm{2}ab{\tau }_{\mathrm{s0}}\right)-a\sqrt{\mathrm{1}-\mathrm{4}b{\tau }_{\mathrm{s0}}+\mathrm{4}c{\tau }_{\mathrm{s0}}}}{2\left(b^{\mathrm{2}}{\tau }_{\mathrm{s0}}-c\right)}}+\left(1-{\lambda }_{\mathrm{s}}\right){\displaystyle \frac{{\tau }_{s\mathrm{0}}{r}_{\mathrm{0}}}{G_{\mathrm{s}}}}\ln \left({\displaystyle \frac{r_{\mathrm{m}}}{r_{\mathrm{0}}}}\right),

where l_s is a non-dimensional parameter named a reduction coefficient due to the reinforcing effect of the adjacent load-free pile, and can be expressed as:

(13){\lambda }_{\mathrm{s}}={\displaystyle \frac{r_0}{S_{\mathrm{a}}}}\left[1-{\displaystyle \frac{\ln \left({\displaystyle \frac{S_{\mathrm{a}}}{r_{\mathrm{0}}}}\right)}{\ln \left({\displaystyle \frac{r_{\mathrm{m}}}{r_{\mathrm{0}}}}\right)}}\right].

Equation (12) is the load-transfer function for a single pile, and can be used to describe the reinforcing effect between the loaded pile and the adjacent load-free pile (see Fig. 26). The computed reduction coefficient l_s decreases with increasing center to center distance between two piles S_a (see Fig. 27). This indicates that the reinforcing effect of piles will decrease by increasing center to center distance between two piles. In practice, the center to center distance between two piles is commonly adopted as 6 to 12 times pile radius, and the reinforcing effect of piles is obvious. For large pile groups, the reinforcing effect of piles cannot be neglected.

Reinforcing effect of adjacent piles on the response of axially loaded individual pile in pile groups

Note that the relationship between total shaft displacement W_sA and shear stress along pile shaft τ_s0 has been established with Eq. (12), and the shear stress τ_s0 can be expressed using the total shaft displacement W_sA at a given depth below the ground surface. Equation (12) used to modify the traditional calculation method for the response analysis of a single pile can be extended to cater for the response of pile groups. Considering pile-pile interaction and reinforcing effect of piles, the total shaft displacement W_si of an arbitrarily individual pile i in pile groups can be written as:

W_{\mathrm{s}i}={\displaystyle \frac{\left(a-\mathrm{2}ab{\tau }_{\mathrm{s0}i}\right)-a\sqrt{\mathrm{1}-\mathrm{4}b{\tau }_{\mathrm{s0}i}+\mathrm{4}c{\tau }_{\mathrm{s0}i}}}{2\left(b^{\mathrm{2}}{\tau }_{\mathrm{s0}i}-c\right)}}+{\displaystyle \frac{{\tau }_{s\mathrm{0}i}{r}_{\mathrm{0}}}{G_{\mathrm{s}}}}\ln \left({\displaystyle \frac{r_{\mathrm{m}}}{r_{\mathrm{0}}}}\right)

(14)+{\displaystyle \sum _{j=\mathrm{1},j\ne i}^{n_{\mathrm{p}}}{\displaystyle \frac{{\tau }_{\mathrm{s0}j}{r}_{\mathrm{0}}}{G_{\mathrm{s}}}}\ln \left({\displaystyle \frac{r_{\mathrm{m}}}{r_{\mathrm{0}}}}\right)}-{\displaystyle \sum _{j=\mathrm{1},j\ne i}^{n_{\mathrm{p}}}{\displaystyle \frac{{\lambda }_{\mathrm{s}ij}{\tau }_{\mathrm{s0}j}{r}_{\mathrm{0}}}{G_{\mathrm{s}}}}\ln \left({\displaystyle \frac{r_{\mathrm{m}}}{r_{\mathrm{0}}}}\right)},

where τ_s0i and τ_s0j are the shaft shear stress for pile i and pile j, respectively; n_p is the quantity of individual piles; S_aij is the center to center distance between pile i and pile j, and S_aij = r₀, when i = j; and l_sij is a reduction coefficient between pile i and pile j due to the reinforcing effect.

l_sij can be computed by:

(15){\lambda }_{\mathrm{s}ij}={\displaystyle \frac{r_0}{S_{\mathrm{a}ij}}}\left[1-{\displaystyle \frac{\ln \left({\displaystyle \frac{S_{\mathrm{a}ij}}{r_{\mathrm{0}}}}\right)}{\ln \left({\displaystyle \frac{r_{\mathrm{m}}}{r_{\mathrm{0}}}}\right)}}\right].

To simplify the calculation procedure, τ_s0i = τ_s0j (for i = 1 to n_p) is assumed for all individual piles in the group, and the reliability of this simplification has been verified by Lee and Xiao [⁵³]. Equation (14) can be simplified as:

W_{\mathrm{s}i}={\displaystyle \frac{\left(a-\mathrm{2}ab{\tau }_{\mathrm{s0}i}\right)-a\sqrt{\mathrm{1}-\mathrm{4}b{\tau }_{\mathrm{s0}i}+\mathrm{4}c{\tau }_{\mathrm{s0}i}}}{2\left(b^{\mathrm{2}}{\tau }_{\mathrm{s0}i}-c\right)}}

(16)+{\displaystyle \sum _{j=\mathrm{1}}^{n_{\mathrm{p}}}{\displaystyle \frac{{\tau }_{\mathrm{s0}j}{r}_{\mathrm{0}}}{G_{\mathrm{s}}}}\ln \left({\displaystyle \frac{r_{\mathrm{m}}}{r_{\mathrm{0}}}}\right)}-{\displaystyle \sum _{j=\mathrm{1},j\ne i}^{n_{\mathrm{p}}}{\displaystyle \frac{{\lambda }_{\mathrm{s}ij}{\tau }_{\mathrm{s0}j}{r}_{\mathrm{0}}}{G_{\mathrm{s}}}}\ln \left({\displaystyle \frac{r_{\mathrm{m}}}{r_{\mathrm{0}}}}\right)}.

Equation (16) is the load-transfer function for an arbitrarily individual pile in a group of n_p piles, and can be considered to be a more realistic model in simulating the pile-soil-pile interaction.

Pile-soil-pile interaction of pile groups composed of individual piles with variable diameter or length

In the response analysis of the pile groups composed of identical piles, the interaction between two piles can be assumed to be consisted of pile shaft and base interactions, and the pile shaft and base interactions are independent of each other [³⁸]. The pile head displacement of an individual pile in pile groups can be described as the sum of the settlement due to its own loading plus that due to the neighboring piles considering the pile-soil-pile interaction. In the prediction of an arbitrary pile response, the shaft displacement can be assumed to be composed of three parts: 1) the pile shaft displacement due to its own loading; 2) the shaft displacement induced by shear stress of other adjacent loaded piles; 3) the reduced shaft displacement due to the reinforcing effects of other adjacent piles. The pile end displacement can be assumed to be composed of two parts: the pile end displacement due to its own loading and the pile end displacement induced by end resistance of the neighboring loaded piles.

As to the response analysis of the pile groups composed of individual piles with variable diameter, the pile shaft and base interactions can also be assumed to be independent of each other. Although the determination method of the pile-soil-pile interaction of the individual piles with variable diameter arranged at different locations of pile groups is identical to that of the pile groups composed of identical piles, the computed pile shaft and base interactions are different due to variable pile diameter.

For the pile groups composed of individual piles with variable length, the pile shaft and base interactions will not be independent of each other. The short pile end displacement has influence on the shaft displacement of long pile, while the shaft displacement of long pile will affect the short pile end displacement. In the prediction of an arbitrarily long pile response, the shaft displacement of long pile can be assumed to be composed of six parts: 1) the long pile shaft displacement due to its own loading; 2) the long pile shaft displacement induced by shear stress of adjacent loaded long piles; 3) the reduced shaft displacement of the long pile due to the reinforcing effects of adjacent long piles; 4) the long pile shaft displacement induced by shear stress of the neighboring loaded short piles; 5) the reduced shaft displacement of the long pile due to the reinforcing effects of adjacent short piles; 6) the long pile shaft displacement induced by end resistance of the neighboring loaded short piles. The long pile end displacement can be assumed to be composed of two parts: the long pile end displacement due to its own loading the long pile end displacement induced by end resistance of the neighboring loaded long piles. As to the analysis of an arbitrarily short pile response, the shaft displacement of short pile can be assumed to be composed of five parts: 1) the short pile shaft displacement due to its own loading; 2) the short pile shaft displacement induced by shear stress of the neighboring loaded short piles; 3) the reduced shaft displacement of the short pile due to the reinforcing effects of adjacent short piles; 4) the short pile shaft displacement induced by shear stress of the neighboring loaded long piles; 5) the reduced shaft displacement of the short pile due to the reinforcing effects of adjacent long piles. The short pile end displacement can be assumed to be composed of three parts: 1) the short pile end displacement due to its own loading; 2) the short pile end displacement induced by end resistance of the neighboring loaded short piles; 3) the short pile end displacement induced by end resistance of the neighboring loaded long piles.

Note that in practice the pile groups are commonly connected with a cap embedded beneath the ground surface, the relative stiffness of piles, cap and soil will affect the distribution of load on piles, and the pile-pile, pile-cap, and cap-soil interactions should be taken into account in the response prediction of pile groups. Due to the complicated interactions among pile, pile cap, and soil, reasonable methods should be established under reasonable and justified assumptions to determine the pile-pile interaction, cap-pile interaction, and pile-soil interaction.

Conclusions

This paper presents a finite element analysis on the response of pile groups with different layouts of individual piles in pile groups. Using the user-defined subroutine FRIC as the secondary development platform, a softening model of skin friction and a hyperbolic model of end resistance are introduced into the contact pair calculation of ABAQUS software. The response analyses of the uniformly and non-uniformly arranged pile groups, e.g., individual piles with variable length and variable diameter, are conducted using the proposed numerical analysis method. Some suggestions on the layout of individual piles are proposed to reduce differential settlement of pile groups for practical purposes. Based on the results of the present study, the following conclusions are drawn.

1) During the whole loading process, the pile-pile interaction remains essentially elastic, which can be used in the response analysis of pile groups. The presence of a non-loaded pile hinders free soil displacement, and leads to a reduction of soil settlement around the axially loaded pile. The presence of a non-loaded pile has a reinforcing effect on the loaded pile bearing capacity. This finding can be used to modify the interaction effects of pile groups considering the reinforcing or sheltering effect of piles.

2) Due to the interactive effects of piles, basin-shaped settlement is observed in pile groups. For practical purposes, to reduce differential settlement of pile groups, dissimilar piles can be installed at different locations of pile groups to eliminate the discrepancy in interactive effects. Pile groups with variable length, e.g., properly increasing the center pile length, and appropriately decreasing the edge pile length or the corner pile length, can be used to reduce differential settlement and make full use of the bearing capacity of each individual pile in pile groups. Furthermore, differential settlement of pile groups can also be diminished by varying pile diameter of individual piles at different positions, e.g., properly increasing the center pile diameter, and appropriately decreasing the edge or the corner pile diameter in practice.

3) The load-sharing ratio attains a maximum value at a certain applied load, and then afterward decreases with increasing total load applied on the pile cap. In the conventional pile groups with identical piles, the center pile is fully mobilized at the earliest possible time, and the bearing capacity of the edge pile or the corner pile cannot be completely utilized.

4) For practical purposes, the ultimate bearing capacity of each individual pile can be achieved simultaneously by adjusting length or diameter of each individual pile. In the pile groups with variable length, more load will be shared by the center pile, causing a large compression of the center pile. In practice, the compression of the center pile in pile groups with variable length should be evaluated to achieve the safety and the serviceability requirements. As to the pile groups with variable diameter, the load-sharing ratio of the center pile reaches the maximum value latest, and a full mobilization of the bearing capacity of the center pile is later than that of the corner pile and the edge pile. Compared with the pile groups with variable length, there is an advantage of the pile groups with variable diameter that the center pile compressive stiffness can be improved by increasing pile diameter, leading to a decrease of pile compression.

References

[1] Dai B B, Ai Z Y, Zhao X H, Fan Q G, Deng W L. Field experimental studies on super-tall buildings, super-long piles & super-thick raft in Shanghai. Chinese Journal of Geotechnical Engineering, 2008, 30(3): 406–413 (in Chinese)

[2] Zhang Q Q, Zhang Z M, Yu F, Liu J W. Field performance of long bored piles within piled rafts. Proceedings of the Institution of Civil Engineers: Geotechnical Engineering, 2010, 163(6): 293–305

[3] Chao S, Zhao X H, Zhang B L, Jiang W H, Kong J, Xiao J H, Yuan J Y. Study of computation of load on pile top of piled raft foundation for superhigh buildings. Rock and Soil Mechanics, 2011, 32(4): 1138–1142, 1148 (in Chinese)

[4] Nguyen D D C, Kim D S, Jo S B. Parametric study for optimal design of large piled raft foundations on sand. Computers and Geotechnics, 2014, 55: 14–26

[5] Taghavi Ghalesari A, Janalizadeh Choobbasti A. Numerical analysis of settlement and bearing behaviour of piled raft in Babol clay. European Journal of Environmental and Civil Engineering, 2018, 22(8): 978–1003

[6] Zhang Q Q, Zhang S M, Liang F Y, Zhang Q, Xu F. Some observations of the influence factors on the response of pile groups. KSCE Journal of Civil Engineering, 2015, 19(6): 1667–1674

[7] Park D, Lee J. Comparative analysis of various interaction effects for piled rafts in sands using centrifuge tests. Journal of Geotechnical and Geoenvironmental Engineering, 2015, 141(1): 04014082

[8] Wong S C, Poulos H G. Approximate pile-to-pile interaction factors between two dissimilar piles. Computers and Geotechnics, 2005, 32(8): 613–618

[9] Liang F Y, Chen L Z, Han J. Integral equation method for analysis of piled rafts with dissimilar piles under vertical loading. Computers and Geotechnics, 2009, 36(3): 419–426

[10] Zhang Q Q, Zhang Z M. Study on interaction between dissimilar piles in layered soils. International Journal for Numerical and Analytical Methods in Geomechanics, 2011, 35(1): 67–81

[11] Liang F Y, Song Z. BEM analysis of the interaction factor for vertically loaded dissimilar piles in saturated poroelastic soil. Computers and Geotechnics, 2014, 62: 223–231

[12] Kou H L, Guo W, Zhang M Y, Xu Y Q. Axial resistance of long rock-socketed bored piles in stratified soils. Ocean Engineering, 2016, 114: 58–65

[13] Li S C, Zhang Q, Zhang Q Q, Li L P. Field and theoretical study of the response of super-long bored pile subjected to compressive load. Marine Georesources and Geotechnology, 2016, 34(1): 71–78

[14] Xu X, Wang X M, Cai C Y, Yao W J. Improved calculation method of super-long pile in deep soft soil area. International Journal of Geomechanics, 2018, 18(10): 04018117

[15] Zhang Q Q, Zhang Z M. A simplified nonlinear approach for single pile settlement analysis. Canadian Geotechnical Journal, 2012, 49(11): 1256–1266

[16] Ni P P, Song L H, Mei G X, Zhao Y L. Generalized nonlinear softening load-transfer model for axially loaded piles. International Journal of Geomechanics, 2017, 17(8): 04017019

[17] Wang Z J, Xie X Y, Wang J C. A new nonlinear method for vertical settlement prediction of a single pile and pile groups in layered soils. Computers and Geotechnics, 2012, 45(9): 118–126

[18] Fleming W G K. A new method for single pile settlement prediction and analysis. Geotechnique, 1992, 42(3): 411–425

[19] Zhang Q Q, Li L P, Chen Y J. Analysis of compression pile response using a softening model, a hyperbolic model of skin friction, and a bilinear model of end resistance. Journal of Engineering Mechanics, 2014, 140(1): 102–111

[20] Bohn C, Lopes dos Santos A, Frank R. Development of axial pile load transfer curves based on instrumented load tests. Journal of Geotechnical and Geoenvironmental Engineering, 2017, 143(1): 04016081

[21] Knodel P C, Airey D W, Al-Douri R H, Poulos H G. Estimation of pile friction degradation from shearbox tests. Geotechnical Testing Journal, 1992, 15(4): 388–392

[22] Abbasa J M, Chik Z, Taha M R. Influence of axial load on the lateral pile groups response in cohesionless and cohesive soil. Frontiers of Structural and Civil Engineering, 2015, 9(2): 176–193

[23] Tamura S, Higuchi Y, Hayashi Y, Yamzasaki M. Centrifuge studies on the effects of existing piles on the end resistance and shaft friction of a new pile. Soil and Foundation, 2012, 52(6): 1062–1072

[24] Zhang Q Q, Li S C, Li L P. Field study on the behavior of destructive and non-destructive piles under compression. Marine Georesources and Geotechnology, 2014, 32(1): 18–37

[25] Pelecanos L, Soga K, Elshafie M Z E B, de Battista N, Kechavarzi C, Gue C Y, Ouyang Y, Seo H J. Distributed fiber optic sensing of axially loaded bored piles. Journal of Geotechnical and Geoenvironmental Engineering, 2018, 144(3): 04017122

[26] Zhang Q Q, Li S C, Liang F Y, Yang M, Zhang Q. Simplified method for settlement prediction of single pile and pile group using a hyperbolic model. International Journal of Civil Engineering, 2014, 12(2B): 179–192

[27] Zhou S W, Zhuang X Y, Rabczuk T. Phase-field modeling of fluid-driven dynamic cracking in porous media. Computer Methods in Applied Mechanics and Engineering, 2019, 350: 169–198

[28] Zhou S W, Zhuang X Y, Rabczuk T. A phase-field modeling approach of fracture propagation in poroelastic media. Engineering Geology, 2018, 240: 189–203

[29] Anitescu C, Atroshchenko E, Alajlan N, Rabczuk T. Artificial neural network methods for the solution of second order boundary value problems. Computers Materials & Continua, 2019, 59(1): 345–359

[30] Rabczuk T, Ren H L, Zhuang X Y. A nonlocal operator method for partial differential equations with application to electromagnetic waveguide problem. Computers Materials & Continua, 2019, 59(1): 31–55

[31] Rose A V, Taylor R N, El Naggar M H. Numerical modelling of perimeter pile groups in clay. Canadian Geotechnical Journal, 2013, 50(3): 250–258

[32] Mendoza C C, Cunha R, Lizcano A. Mechanical and numerical behavior of groups of screw (type) piles founded in a tropical soil of the Midwestern Brazil. Computers and Geotechnics, 2015, 67: 187–203

[33] Sinha A, Hanna A M. 3D numerical model for piled raft foundation. International Journal of Geomechanics, 2017, 17(2): 04016055

[34] Fleming W G K, Weltman A J, Randolph M F, Elson W K. Piling Engineering. New York: Surrey University Press, Halsted Press, 1985

[35] Cho J, Lee J H, Jeong S, Lee J. The settlement behavior of piled raft in clay soils. Ocean Engineering, 2012, 53: 153–163

[36] Jaky J. Pressure in soils. In: Proceedings of the 2nd International Conference on Soil Mechanics and Foundation Engineering. Rotterdam, The Netherlands, 1948, 103–107

[37] Mayne P W, Kulhawy F H. K0-OCR relationships in soil. Journal of the Geotechnical Engineering Division, 1982, 108(GT6): 851–872

[38] Zhang Q Q, Zhang Z M, He J Y. A simplified approach for settlement analysis of single pile and pile groups considering interaction between identical piles in multilayered soils. Computers and Geotechnics, 2010, 37(7–8): 969–976

[39] Zhang Q Q, Zhang Z M. Simplified calculation approach for settlement of single pile and pile groups. Journal of Computing in Civil Engineering, 2012, 26(6): 750–758

[40] Zhang Q Q, Liu S W, Zhang S M, Zhang J, Wang K. Simplified non-linear approaches for response of a single pile and pile groups considering progressive deformation of pile-soil system. Soil and Foundations, 2016, 56(3): 473–484

[41] Clough G W, Duncan J M. Finite element analysis of retaining wall behavior. Journal of the Soil Mechanics and Foundations Division, 1971, 97(12): 1657–1673

[42] Zhang Z M, Zhang Q Q. Field tests on behavior of destructive and non-destructive base-grouting piles under compression. Chinese Journal of Geotechnical Engineering, 2011, 33(10): 1601–1608 (in Chinese)

[43] Zhang Z M, Zhang Q Q, Yu F. A destructive field study on the behavior of piles under tension and compression. Journal of Zhejiang University. Science A, 2011, 12(4): 291–300

[44] Zhang Z M, Zhang Q Q, Zhang G X, Shi M F. Large tonnage tests on super-long piles in soft soil area. Chinese Journal of Geotechnical Engineering, 2011, 33(4): 535–543 (in Chinese)

[45] Zhang Q Q, Liu S W, Feng R F, Li X M. Analytical method for prediction of progressive deformation mechanism of existing piles due to excavation beneath a pile-supported building. International Journal of Civil Engineering, 2019, 17(6): 751–763

[46] Coyle H M, Reese L C. Load transfer for axially loaded pile in clay. Journal of the Soil Mechanics and Foundations Division, 1966, 92(SM2): 1–26

[47] Kraft L M, Ray R P, Kagawa T. Theoretical T-Z curves. Journal of the Geotechnical Engineering Division, 1981, 107(11): 1543–1561

[48] Sheil B B, McCabe B A. An analytical approach for the prediction of single pile and pile group behavior in clay. Computers and Geotechnics, 2016, 75: 145–158

[49] O’Neill M W, Hawkins R A, Mahar L J. Load transfer mechanisms in piles and pile groups. Journal of the Geotechnical Engineering Division, 1982, 108(12): 1605–1623

[50] Castelli F, Maugeri M. Simplified nonlinear analysis for settlement prediction of pile groups. Journal of Geotechnical and Geoenvironmental Engineering, 2002, 128(1): 76–84

[51] Caputo V, Viggiani C. Pile foundation analysis: A simple approach to nonlinearity effects. Rivista Italiana di Geotecnica, 1984, 18(1): 32–51

[52] Lee J, Park D, Park D, Park K. Estimation of load-sharing ratios for piled rafts in sands that includes interaction effects. Computers and Geotechnics, 2015, 63(2): 306–314

[53] Lee K M, Xiao Z R. A simplified nonlinear approach for pile group settlement analysis in multilayered soils. Canadian Geotechnical Journal, 2001, 38(5): 1063–1080

[54] Randolph M F, Wroth C P. Analysis of deformation of vertically loaded piles. Journal of the Geotechnical Engineering Division, 1978, 104(GT12): 1465–1488

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