Nonlinear Analysis of Flexible Pile near Undrained Clay Slope under Lateral Loading

In this paper, a method is developed for nonlinear analysis of laterally loaded long-flexible pile near undrained clay slope.-e ideal elastic-plastic p-y curve model is used as the basic calculationmodel to study the pile-soil interaction system. To consider the slope effect on the soil resistance along with pile length, related reduction function expressions of soil resistance are selected by assessing the existing methods. A finite difference iteration scheme is proposed to solve the pile’s deflection curve equations to obtain nonlinear response of pile. -e developed method is validated by comparing its results with the existing method, which shows a good agreement. A number of parameters analyses are carried out, and effects of different influence factors on laterally loaded pile are further discussed.


Introduction
Pile foundations are widely used to support structures such as bridges, high-rise buildings, transmission line towers, and traffic signs that are frequently constructed near or in a natural or man-made slope [1].e pile foundations are often subjected to static lateral loading in many cases.For the design needs of pile foundations, many scholars have conducted a lot of research on laterally loaded piles in horizontal ground.e most commonly used theoretical method for laterally loaded pile in the current design approach is the well-known subgrade reaction method, which considers the pile as a beam supported by a series of nonlinear soil springs.e lateral load transfer curves (i.e., p-y curves) are often used to describe the loaddeformation characteristics of nonlinear soil springs.Different criteria for developing p-y curves under many cases, including soil types, soil strength, loading conditions, pore water pressure conditions in soil, etc., have been proposed by several investigators (Matlock [2], Reese et al. [3,4], Reese and Welch [5], Gabr et al. [6], Fan and Long [7], and some others) mainly through the back analyses of lateral load tests, and these typical p-y curves have been incorporated into the API specification as a reference for the design of pile foundation structure in offshore engineering [8].Subsequently, Wang et al. [9,10] proposed a new p-y curve method of laterally loaded pile in clay based on the analyses and calculation of the existing field test data through the stress-strain relationship.Wang and Yang [11] used the construction parameters to normalize the different forms of p-y curves in sand and then proposed a hyperbolic model.An ideal elastic-plastic p-y calculation model for clay was also proposed based on a comprehensive analysis of the measured p-y curves in clay [12].Zhou et al. [13] proposed a new empirical formula for the p-y curves of ultralong PHC pipe piles in soft clay based on the analyses of the relationship between ultimate soil resistance and some geotechnical parameters from lateral pile load test.
For laterally loaded pile near slops, the stress environment of the pile foundation becomes, due to the lack of soil volume in front of the pile, more complex, and the lateral load-bearing characteristics and failure model are different from that pile in horizontal ground [14][15][16].
us, the existing theoretical calculation methods for laterally loaded pile in horizontal ground are not suitable for pile near slopes.Available studies about the lateral loading capacity behavior of piles in sloped ground mainly focus on model tests and numerical simulation (Georgiadis [17,18]; Muthukkumaran et al. [19]; Zhang et al. [20]; Yin et al. [21]; Gao et al. [22]), but rare in the theoretical calculations.
is paper attempts to present a method for the nonlinear analysis of laterally loaded long-exible piles near sloped ground.For the developed method, the ideal elastic-plastic p-y curve model is used as the basic calculation model to study the pile-soil interaction system.To consider the slope e ect on the soil resistance along with pile length, related reduction function expressions of soil resistance are selected by assessing the existing methods.A nite di erence iteration scheme is then proposed to solve the pile's de ection curve equations to obtain nonlinear response of pile.To validate the developed method, it is rst compared to pile test and three-dimensional numerical nite element analysis results, and the results show that the developed method can provide satisfactory prediction of the response of laterally loaded pile near clay slopes.en, it is applied to analyze the parametric in uence law, and the nonlinear response results of pile near clay slope under di erent in uence factors are obtained.

Method of Analysis
2.1.Mechanical Analysis Model.Currently, p-y curve methods were used by many researchers to study the nonlinear response characteristics of the laterally loaded pile in horizontal ground.Zhu [23] conducted extensive analysis of laterally loaded pile in horizontal ground by using the elastic-plastic p-y curve model and suggested that choosing reasonable model parameters can obtain realistic results.In succession, the general calculation form of the ideal elasticplastic p-y curve method in clay (Figure 1), based on this, was given by Wang and Yang [12] as follows: where K initial slope of the p-y curve, y lateral displacement of pile, p u ultimate soil resistance (per unit pile length), and y u allowable lateral displacement for soil in elastic state.
In order to consider the in uence of slope e ects, the analysis model for laterally loaded pile near sloped ground is set up as shown in Figure 2.
Where L buried pile length, H 0 lateral load acting on the pile top, B distance from the pile shaft to the slope crest, θ slope angle, D pile diameter, and e height of the load above the clay surface.

Assumptions.
To facilitate the establishment of method, assumptions and statements are made in advance.
(1) e cross-sectional shape of pile at all depths is equal; besides, both the pile and the soil are homogeneous and isotropic materials (2) e ultimate lateral soil resistance (p u ) varies nonlinearly with depth (3) Considering the slope e ect, the initial sti ness (K) and the ultimate lateral soil resistance (p u ) should be reduced within a certain depth (4) Assume that clay slope is stable, and the slope destruction and instability are not taken into consideration during the process of calculation

p-y Curve Model of Laterally Loaded Pile near Slope.
With the basic assumptions in Section 2.2, equation ( 1) is then modi ed to obtain the new ideal elastic-plastic p-y curves as follows: where μ(z) is the reduction function of the soil lateral initial sti ness (K); p su (θ, B, z) is the ultimate lateral soil resistance per unit pile length under the condition of pile near slope.Allowable lateral displacement of y u can be obtained according to equation (2):

Initial Stiffness K
e problem of choosing the initial sti ness of the soil in the p-y curve has always been focused on by investigators.Many scholars believed that the lateral initial sti ness K of clay remains constant in all depth.is paper adopts the initial  Advances in Civil Engineering stiffness recommended by Rajeshree and Sitharam [24] as follows: where v s , E p I p is Poisson's ratio of soil and pile bending rigidity, respectively.According to the Kondner [25], the initial elasticity modulus can be related to elasticity modulus E 50 under the 50% failure stress, and the elasticity modulus at deviatoric stress can be expressed by where σ is deviation stress, σ f is deviatoric failure stress, R f is the ratio of the deviatoric failure stress over the ultimate deviatoric stress, generally taken as 0.8, E i is the initial elastic modulus, and σ/σ f � 0.5.For undrained loading, we have v s � 0.5.e elastic modulus of the clay is represented by the deformation modulus E 50 under the 50% failure stress of the reaction soil, and E i � 1.67 E 50 is satisfied.e above expression can be used to derive the initial stiffness (K) of clay under undrained conditions:

Reduction Function of Initial Stiffness u(z)
For near-slope laterally loaded pile, the initial stiffness of the soil is also affected by the slope [17,18,22].Georgiadis and Georgiadis [18] gave the reduction function for the initial stiffness of clay within the shallow foundation under undrained loading condition:

Ultimate Soil Resistance of Clay p su
Several methods are available for determining the ultimate soil lateral resistance p u to piles in cohesive soil [2, 4-6, 10, 12].It is generally believed that the ultimate resistance (p u ) of clay depends mainly on the type of failure mechanism at a certain depth of the soil.e normative formula in API design code for horizontal ground is given in equation (8): where N p is ultimate capacity coefficient of clay, c u is undrained shear strength of clay, c ′ is effective bulk density of clay, and J is dimensionless experience coefficient, its value ranges from 0.25 to 0.5.
To consider slope effect on the ultimate soil resistance, the following expression proposed by [18] is used to calculate the ultimate soil resistance N sp : where z c is the certain critical depth, α θ is the parameter related to the slope, N pu is the ultimate lateral bearing capacity factor.Besides, Δ � sin −1 α and λ � 0.55 − 0.15α, both are dimensionless coefficients; N p0 is the lateral bearing capacity factor at ground surface, N p0 � 2 + 1.5 α. α is the adhesion coefficient, and its value ranges from 0∼1; N pc is the ultimate soil resistance at a certain critical depth.
According to equations ( 9)-( 13), the ultimate soil resistance of clay p su for undrained conditions can be expressed as

Establishment of Calculation Model.
A simplified calculation model can be established as shown in Figure 3. Select a microunit of the pile for force balance analysis.It is stated here that the bending moment is positive in the counterclockwise rotation direction, and the normal of the shear force is positive in the positive direction of the z axis.
e shear force is positive in the negative direction of the y axis.en the equation for the deflection of the pile can be derived as follows: where E p I p is the pile bending stiffness, p(y, z) is the soil resistance distributed along pile length, which can be obtained from the previous p-y curve, and the positive direction is positive direction of the y axis.Boundary restrictions must be determined to solve the above-mentioned deflection differential equation.In this paper, the boundary conditions of the pile are determined: the pile top is free and the pile bottom is fixed, and the corresponding deflection differential equations are as follows: Free pile head: Fixed pile bottom: Advances in Civil Engineering

Di erential Format Derivation and Solution.
e nite di erence method principle is used to discrete the buried section L into n segments, each of which is s L/n. e top node is denoted as 0, and the bottom of the pile is marked as n.In order to improve the simplicity of the nite di erence operation, four virtual nodes, −2, −1, n + 1, n + 2 are added at the top of pile and the bottom of the pile, respectively.e displacement of the ith node is y i is shown in Figure 4.
For a pile, there exists (n + 5) di erential equations after the discrete division, and four extra known di erential equations can be derived in combination with the boundary conditions, thereby establishing the pile displacement node vector y i and further solving the pile displacement.
Substituting the fourth derivative of pile microelement in equation ( 15), the di erential equation for each element of the pile can be derived: In combination with equation (2), equation ( 18) can be transformed into the following two forms according to the allowable displacement y u : (1) y ≤ y u (2) y > y u Similarly, in terms of equations ( 16) and ( 17), the differential form of the boundary conditions can be derived: According to the di erential equations from equation ( 21) of the boundary conditions, (n + 1) unknown di erential equations are combined to obtain the displacement of the pile matrix, the corresponding pile displacement solution equations can be expressed as where [K n ] is the sti ness matrix of pile, which is divided into two type sti ness matrix of [K 1 ] and [K 2 ] according to the elastic state and plastic state of the soil along the pile length, respectively; y is the pile displacement node vector; F { } is external load vector, which is controlled by boundary conditions and is also divided into two type vector of F 1 and F 2 correspond to [K 1 ] and [K 2 ], respectively.
In most practical projects, the maximum bending moment section is regarded as the most dangerous cross section  4 Advances in Civil Engineering of the pile design.Here, the di erential form of the solution of the bending moment is listed as According to the basic solution ideas of equation ( 22), a MATLAB iterative code is developed to solve the response behavior of pile, and the ow chart of iterative calculation is shown in Figure 5.

Validation to Previous Studies
e feasibility of the proposed method is validated in this section to some previous studies, which are well documented in the literature. is method is introduced as input into MATLAB to calculate the response of each pile.e geometrical characteristics (pile length L, pile diameter D, distance from the pile shaft to the slope crest B, bending sti ness of the pile E p I p , and slope angleθ) and the soil properties (undrained shear strength c u , secant modulus E 50 , e ective bulk density c or c ′ , and cohesion coe cient α.) are summarized in Table 1 for each previous study considered.
Case 1. e computed pile responses are compared to the pile test results and to the pile responses computed with the same computer code using as input the Matlock [2], Reese and Welch [5], Bhushan et al. [26], Stevens and Audibert [27].e Shanghai pile tests, as shown in Figure 6, reported by the literature [28], involved lateral loading of piles in level ground.It can be seen that despite the small discrepancy measured and predicted between the proposed p-y curve load-displacement relationships, the general shape of the curves is very similar, indicating that the small discrepancy may be due to overestimation of secant modulus E 50 .
Case 2. e results calculated by the proposed method and the results calculated by Georgiadis and Georgiadis [17], which presented 3D nite-element analyses of pile in sloping ground, are presented in Figure 7 for comparison purposes.
is paper selects two di erent slope angles (θ 20 °, 40 °, respectively) for veri cation analysis.As the slope angle increases, the results of this method are getting closer to the results of the literature [17].
Case 3. e calculating results given by Georgiadis and Georgiadis [18] are used as examples for comparison veri cation.Pile foundation is completely buried in the undrained clay slope.us, the section of the pile head at ground surface only has the initial horizontal force H 0 .is paper selects four di erent pile's distance from slope crest (B 0.3 m, 0.6 m, 1.2 m, 2.4 m, respectively) for veri cation analysis.
e H 0 -y 0 curves and H 0 -maxM curves of pile head obtained from the proposed method and literature [18] are plotted in Figure 8.A good agreement has been observed between the calculating results from the present study and literature results.

E ect of Slope Angle θ.
e size of the slope angle directly determines the soil volume in front of the pile to participate in the pile-soil interaction.In order to better show the e ect of slope angles, the pile head de ection of slope was normalized by dividing the pile head de ection of slope (y 0,s ) by that of horizontal (y 0,H ).Figures 9-12, respectively, show the lateral load-de ection curves at pile head, the normalized de ection (y 0,s /y 0,H )-load (H 0 ) curves at pile head y 0 , the maximum bending moment (M max )load (H 0 ) curves, and the bending moment(M)-depth(z) distribution curves with di erent slope angles of 0 °, 10 °, 20 As shown in Figure 9, as the slope angle increases, the lateral de ection of the pile head increases signi cantly, and Advances in Civil Engineering this trend becomes much more obvious with the increase of the lateral load.It is also observed that when ground surface changes from horizontal ground to 10 °slope, the behavior of pile is almost like horizontal ground surface, indicating the e ect of slope is almost negligible when the slope angle is <10 °.
Figure 10 further shows the distribution curves of normalized de ection (y 0,s /y 0,H ) with the lateral load.It can be seen that di erent xed ratios of y 0,s /y 0,H are almost maintained at low load levels, but as the lateral load continue to increase, the values of y 0,s /y 0,H start to increase unequally for di erent slope angle with the increase of load.When the lateral load is at 750 kN, the values of y 0,s /y 0,H for 40 °slope and 50 °slope are 1.065, 2.157, respectively, indicating that the steeper the slope angle is, the more de ection of pile head grows.It can be seen from Figure 11 that the maximum bending moment values of the pile also increase as the slope angle increases, but the rate of increase of maximum bending moment with lateral load is more moderate.Figures 12 and 13 show the bending moment distribution with depth and the depth of xity (where the maximum bending moment occurs) of the pile for all slope angles with a lateral load at 300 kN and 600 kN, respectively.It is clear that the maximum bending moment of the pile increases with the increase of slope angle at the same load, but the rst zero location of bending moment curve and depth of xity increase correspondingly.With the increase of the load, the di erence in the maximum bending moment growth at each slope angle becomes obvious, and the rst zero location of bending moment curve and depth of xity of the pile also increase more quickly.For example, as is shown in Figure 13, the depth of xity for 30 °and 50 °slope angle increased by 4.46%, 17.34% compared with that in horizontal ground at the load of 300 kN, respectively, but the ratio further increases to 11.32%, 32.03%, respectively, when the load increases to 600 kN.It can also be seen from Figure 13 that the depth of pile xity almost occurs at a depth of 4∼8D (D diameter of pile).

E ect of Normalized Distance of Pile B/D.
Selecting the normalized distance of pile B/D as 0.5, 2, 4, 6, and consider the slope angle of 10 °, 30 °, 50 °, respectively, as the existing slope conditions.e other calculation parameters are the same as in Section 8.1.
Figures 14-16 show the load-de ection curve for different normalized distance B/D of pile head at above three slope conditions.From these gures, it is observed that the increase in normalized distance B/D ratio decreases the de ection of pile head.When normalized distance B/D ratio exceeds 6 in all slope angle conditions, the de ection at pile head is basically the same as that in horizontal ground, which means the in uence of the slope e ect is almost negligible in this case.It is also observed that the load-de ection curves in di erent slope angle conditions have di erent response results, and the increase in slope angle can increase the discreteness of the curves.Figure 17 further shows the e ect of normalized distance of pile on Present study LPILE (Georgiadis and Georgiadis [18]) Present study LPILE (Georgiadis and Georgiadis [18]) Advances in Civil Engineering lateral-load capacity on horizontal ground and sloping ground (θ 10 °, 30 °, 50 °).It can be seen that when the normalized distance B/D of pile changes from 0.5 to 6, the lateral-load capacity is increased by 2.27%, 11.1%, 29.87% on 10 °, 30 °, 50 °sloping ground, respectively, indicating that the increase in slope angle increases the in uence of B/D on the lateral-load capacity of the pile.show the bending moment variation along pile length of the pile for di erent normalized distance B/D on horizontal ground and sloping ground (θ 10 °, 30 °, 50 °).From these gures, it is observed that the maximum bending moment increases with the increase in the normalized distance B/D of the pile and the increase in the applied lateral load.Also, for the same normalized distance B/D of pile and applied lateral load, the increase in slope angle of θ increases the maximum bending moment of pile.When ground surface changes from horizontal to 10 °, 30 °, and 50 °sloping ground, the maximum bending moment of pile in sloping ground is increased by 2.8%, 12.6%, and 31.9%,respectively, under lateral load of 600 kN and normalized distance B/D of 0.5.It is also noted that when the pile is nearer to slope crest, the slope angle becomes a more dominant factor to a ect the lateral capacity of the pile, especially at a high-load level.Advances in Civil Engineering

E ect of Adhesion Coe cient α.
In order to study the e ect of adhesion coe cient α on the lateral behavior of pile in sloped ground, this paper analyzes the parameter α from 0 to 1, with increment of 0.1, and chooses lateral load H 0 500 kN and the bending moment M 0 0 (kN•m) as the xed loading condition to calculate and analyze the de ection of pile head.Consider the slope angle θ as the same in Section 8.1, and other basic calculation parameters are the same as Section 8.1.
Figures 21 and 22 show pile head de ection (y 0 ) and the maximum bending moment (M max ) of the pile variation with the adhesion coe cient α for di erent slope angle, respectively.From these gures, it can be seen that increasing the adhesion coe cient α at the pile-soil interface can effectively reduce the pile head de ection and the maximum bending moment of the pile at the same time.Also, the increase in slope angle almost does not change the corresponding curve growth model but will signi cantly increase   Advances in Civil Engineering the pile head de ection and maximum bending moment of the pile especially when the slope angle exceeds 40 °.An example is used for pile head de ection, when the value of α is equal to 0, the pile head de ection values of pile in sloping ground with θ 10 °, 30 °, 50 °is increased by 7.4%, 39.8%, 137%, respectively, compared to that in horizontal ground.

E ect of Undrained Shear Strength c u .
Here, lateral load H 0 500 kN and the initial bending moment M 0 0 (kN•m) were taken as the initial loading conditions, choosing undrained shear strengths c u of 20 kPa, 40 kPa, 60 , 80 kPa, 100 kPa, respectively, to calculate and analyze.Other basic calculation parameters are the same as in Section 8.1.
Figures 23 and 24 show pile head de ection (y 0 ) and the maximum bending moment (M max ) of the pile variation with the undrained shear strength c u for di erent slope angles, respectively.From the curves given in Figures 23 and  24, it is observed that the increase in undrained shear strength c u decreases the pile head de ection and the maximum bending moment.is is because of the increase in the strength of soil and the relative sti ness of the pile-soil system.Also, it is obvious that the presence of slopes and horizontal ground surfaces, from which can conclude that the weaker soil parameters are, the more signi cant slope e ect becomes.Besides, the curve of the maximum bending moment variation with undrained shear strength shown in Figure 24 is smoother, and the range of the maximum bending moment of the pile with the increase of the slope is relatively stable.

Conclusions
e purpose of this paper is to study the behavior of laterally loaded long-exible pile in undrained clay sloping ground.Based on the basic model of ideal elastic-plastic p-y curve method, this paper derives the di erential equations of the laterally loaded single pile in undrained clay sloped ground by introducing the existing related reduction variable function, and the corresponding nite di erence method for the pile de ection and internal force is obtained.A series of parameters analyses are further performed for slope angle, normalized distance of pile, adhesion coe cient, and undrained shear strength.e conclusions obtained from this study are summarized as follows: (1) A nonlinear analysis method for laterally loaded longexible pile in undrained clay sloped ground is established.rough the veri cation of examples, the correctness and feasibility of the method is proved.
(2) Slope angle is the most important factor that a ecting the lateral bearing capacity of the near-slope pile.e increase in slope angle increases the de ection and the rate of increase of de ection, and this trend becomes more obvious as the lateral load increased.When the slope angle is small enough (θ < 10 o ), the e ect of slope is almost negligible.Besides, the depth of xity of the pile is about 4∼8 times the size of the pile diameter.
(3) e distance away from pile to slope crest is also an important factor that a ects the lateral-load capacity of the pile.e increase in normalized distance B/D of the pile decreases the de ection of pile head but increases the lateral-load capacity, and when normalized distance B/D ratio exceeds 6, the slope e ect can be avoided.When laterally loaded pile is closer to the slope crest, the in uence of the slope angle seems more dominant.

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(4) e adhesion coefficient α at the pile-soil interface can effectively reduce the pile head deflection and the maximum bending moment of the pile at the same time.e increase in slope angle will significantly increase the pile head deflection and maximum bending moment of the pile especially when the slope angle exceeds 40 °.
(5) e increase in undrained shear strength c u decreases the pile head deflection and the maximum bending moment of the pile. is is because of the increase in the strength of soil and the relative stiffness of the pile-soil system.Also, the presence of slopes aggravates the rate of change of pile head deflection, especially when the c u is in a mall value (ranges from 20 kPa to 40 kPa).In contrast, the maximum bending moment variation with undrained shear strength affected by the slope effect appears more moderate.( 6) is study researches the lateral bearing behavior of single pile in undrained clay sloping ground.In practical projects, the situation of pile foundations on slope ground surface is not uncommon, and the lateral bearing characteristics of pile foundations are more complicated.erefore, the influence of slope effect on the lateral bearing capacity of pile which is on slope ground surface is worth further study.

Figure 6 :
Figure 6: Veri cation analysis of H 0 -y 0 curves relationship for Shanghai pile tests.

Figure 12 :
Figure 12: Bending moment variation along the depth of pile for di erent slope angle.

Figure 9 :
Figure 9: Lateral load-de ection curves for di erent slope angle at pile head.

Figure 11 :
Figure 11: Maximum bending moment-load curves for di erent slope angle.

Figure 14 :igure 17 :
Figure 14: Lateral load-de ection curves for θ 10 °for di erent normalized distance of pile at pile head.
, 40 °, 50 °.e relevant calculation parameters here are pile length L 14 m, pile diameter D 0.6 m, bending sti ness of pile E p I p 184.49 (MN•m 2 ), normalized distance of pile B/D 0.5, adhesion coe cient α 1; piles are completely buried in undrained clay, and Poisson's ratio of clay ] s 0.49, undrained shear strength c u 40 kPa, and secant modulus E 50 14 MPa. ° Calculate the values of K, p sui , and μ(z i ) for each node of the pile Calculate the allowable displacement value of y u for each node of the pile Calculate pile lateral displacement y i for each node by using the elastic matrix of K i and the external load vector of F i Judge y i > y u Resolve by replacing K 2 and F 2 with K i and F i correspondingly where needed Output lateral displacement value of y i for each node and calculate the corresponding bending moment value of M i End Yes No Figure 5: Flowchart of iterative calculation.

Table 1 :
Summary of pile load test.