A Semianalytical Solution for Passively Loaded Piles Adjacent to Surcharge Load

Piles adjacent to a surcharge load commonly support not only active loads from superstructures but also the passive loads caused by soil lateral movement. To investigate the influence of passive load and the response along pile shafts of existing actively loaded piles, a load transfer model for analyzing the soil-pile interaction was developed based on plastic deformation theory and the triparameter soil model. An analytical solution for the deformation and internal force of such piles was proposed using the transfer matrixmethod, in which the transfer matrix coefficients for piles in free, plastic, and elastic zones were analytically obtained by considering the second-order axial force effect caused by lateral loading and soil yielding based on the triparameter soil model. (e proposed methodology was validated by comparing its predictions with field measurements and previously published results. A good match between model predictions, field measurements, and previously published results implies that the proposed method can be used to evaluate the response of passive piles adjacent to a surcharge load. Parametric studies were also carried out to investigate the influence of surcharge pressure, soil resistance, and boundary conditions on the behavior of passively loaded piles adjacent to a surcharge load.


Introduction
Piles below existing buildings are commonly subjected to additional passive loading from moving soil caused by, for example, the construction of road embankments or fills adjacent to buildings. ese piles are commonly classified as passive piles [1,2]. Passive loading can be significant, particularly for soft soil deposits and large lateral soil displacement which can induce changes in pile bearing and deformation characteristics. Vertically loaded piles originally designed according to the pile bearing capacity and settlement control may become piles controlled by lateral passive loading, which is commonly neglected and often causes engineering accidents, such as the collapse of the Shanghai BaoGang Steel Warehouse caused by a steel slag surcharge in 1998 [3,4], and the toppling of a nearly finished 13-story residential building in Shanghai due to an excess surface load to the north and an excavation to the south in 2009 [5].
Proper consideration of the reaction of passive piles to surcharge load is the key to avoid such accidents. Numerous numerical and theoretical methods have been developed to analyze passive piles [6][7][8]. Numerical methods usually utilize the finite-element method [9,10] or the finite difference method [11], which can consider complex boundary conditions and geological profiles but requires significant computational capacity and longer time. eoretical methods can be categorized as displacement-based methods and pressure-based methods [12]. e displacement-based methods can better reflect the field behavior of piles due to soil movements by using free-field soil displacements to determine the soil-pile interaction forces [9,13,14]. However, a complex numerical simulation or a large number of field data is needed to determine the free-field soil displacements in advance, which increases the difficulty of applying the displacement-based method in engineering practice [1,15]. e pressure-based methods assume the distribution of passive loading induced by lateral soil displacements based on laboratory tests or theoretical analyses and calculate the deflection and bending moment of the laterally loaded pile [16][17][18][19]. e availability of analytical solutions for laterally loaded piles makes the pressure-based methods more practical. e capacity of the pressure-based methods to reflect the real passive pile behavior is closely related to the load transfer model employed for predicting the soil-pile interactions. Different soil-pile interaction models have been proposed to evaluate the responses of laterally loaded pile [20][21][22][23][24]. For passively loaded piles, Ito et al. proposed a theoretical method for analyzing the growth mechanism of lateral forces acting on stabilizing piles based on plastic deformation theory [25]. A special boundary condition (the lateral pressure between piles was assumed to be the active Earth pressure) was used to derive the passive load, which may result in the lateral soil pressure in the plastic zone different from the actual soil pressure [26]. Shen presented a limit analysis method in which the soil-pile interaction was analyzed using a plane-strain slip-line theory to consider the soil plastic flow around the piles [27]. Ashour et al. extended the strain wedge (SW) model and deduced the mobilized nonuniformly distributed passive load based on soil-pile interaction in an incremental fashion [16]. However, these studies are primarily about stabilizing piles which have no active loading and cannot be applied to the passively loaded piles adjacent to surcharge load. erefore, this paper aims to develop a semianalytical solution for passively loaded piles adjacent to surcharge load (i.e., the piles subjected to both active loading from superstructure and passive loading from adjacent surcharge load). Specifically, a load transfer model for soil-pile interactions is proposed using plastic deformation theory and the triparameter soil model. e passive load induced by a neighboring surcharge load was determined in an incremental fashion. en, the deformation and internal force solutions for piles under combined active and passive loads were derived using the transfer matrix method. e proposed methodology was validated by comparing its predictions with field measurements and previously published results. Finally, parametric studies were carried out to investigate the influence of surcharge pressure, soil resistance, and boundary conditions.

New Method for Analysis of Piles under Both
Active and Passive Loadings 2.1. Problem Definition. e problem addressed in this study is schematically shown in Figure 1(a) where the pile is subjected to both the active loading from the superstructure and the passive loading from the adjacent surcharge load, such as an embankment. e pile is embedded in a multilayered soil deposit with n sublayers and has length L, diameter d, and bending stiffness EI. e pile has length L a and L b above and below the ground surface, respectively. e embedded part is divided into n segments according to the n soil layers. e active loads from the structure include the horizontal force V 0 , moment M 0 , and axial load N 0 acting on the pile head. A distributed load q(z) acting on the pile above the ground surface (called the free zone) is considered to simulate the extra external loading from, for example, wind or wave. e pile is also subjected to a passive load P s (z) below the ground surface induced by nearby surcharge loading, which will be determined in Section 2.2.
Under the combined active and passive loads, the pile deforms continuously and may push the surrounding soil to its limit state. e soil begins to yield at the ground surface and extends downward, although the rest of the soil along the pile is still in an elastic state, and thus the soil state along the pile shaft can be divided into plastic and elastic zones (Figure 1(b)) [22,28]. e reaction of the soil is denoted by a series of spring-slider elements (which are deactivated in the elastic state) along the shaft and each element is described by an elastic-perfectly plastic curve. e influence of axial loads on the response of this passively loaded pile is also considered. An empirical axial force distribution on the pile shaft [29][30][31] is used to simulate the distribution of axial forces along the pile shaft in a multilayered soil deposit, For the above ground portion of the pile, the axial force is expressed by where η (� A p c G ) is a coefficient describing the increase in axial force along the pile in the free zone due to the weight of the pile and A p and c p represent the cross-sectional area and the unit weight of the pile, respectively. For the pile below the ground, the axial force is assumed to vary with depth as follows: where N i− 1 is the axial force at the bottom of the pile section in the (i − 1)th soil layer and ψ i is a coefficient dependent on skin friction mobilization in the soil.

Load Transfer Model of Soil-Pile Interaction.
e model for analyzing soil-pile interactions of existing piles adjacent to a surcharge load is developed following the model of Ito et al. [25]. e soil-pile interaction growth mechanism is shown in Figure 2. With increasing adjacent surcharge pressure, the lateral stress P BB′ acting on surface BB′ in front of pile increases and causes plastic deformation of the soil. For the load transfer between moving soil and piles, the plastic soil is assumed to start from surface BB′ and extends to surface OO′ where the lateral stress may equal the static Earth pressure P 0 in considering long-term soil consolidation [26], as shown in Figure 2. e plastic deformation zone in the soil continues to develop with increasing lateral stress P BB′ acting on surface BB′ (Figures 2(a) and 2(c)). When the lateral stress acting on surface BB′ increases sufficiently, soil is plastically deformed at the front and back of the piles (Figure 2(d)).
In order to analyze the load transfer between the moving soil and pile, the following assumptions are made (Figure 2 e lateral stress at any point in BB′E′E, EE′A′A, AA′D′D, and DD′G′G can be obtained from "Appendix A" and expressed as follows: In the BB′E′E zone, In the EE′A′A zone, In the AA′D′D zone, In the DD′G′G zone,

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where σ I x , σ II x , σ III x , and σ IV x are the lateral stress at any point x in the BB′E′E zone, EE′A′A zone, AA′D′D zone, and DD′G′G zone, respectively; D 1 is pile spacing; D 2 is net spacing between piles; N φ � tan 2 ((π/4) + (φ/2)); N φ ′ � tan 2 ((π/4) − (φ/2)); P BB′ (z) is the total soil pressure acting on surface BB′ which is given by where σ x (z) is the additional lateral stress acting on surface BB′ induced by the surcharge load and can be estimated using Boussinesq's solution and the Flamant integral solution according to the distribution characteristics of    Advances in Civil Engineering surcharge load [26,32]; K 0 is the static Earth pressure coefficient.
is the lateral stress acting on the surface AA′ which is given by (Appendix A) e passive load P s (z) acting on piles induced by moving soil can be obtained by the following steps: (a) Using (7), the total soil pressure P BB′ (z) acting on surface BB′ can be obtained. en, the lateral stress P AA′ (z) acting on surface AA′ can be obtained using (8).
, as given by the condition in (3), the passive load acting on piles can be calculated by Otherwise, x � x o can be obtained by substituting σ II x � K 0 cz into (4). e passive load acting on piles can be calculated by , as given by the condition in (5), the passive load acting on piles can be calculated by Otherwise, x � x o can be obtained by substituting σ IV cos((π/4) + (φ/2)), as given by the condition in (6), the passive load acting on piles can be calculated by If x o > (d/2)/cos((π/4) + (φ/2)), the passive load acting on piles can be calculated by e soil resistance behind the piles can be represented by springs for evaluating pile response, which is simulated using a perfectly elastic-plastic model. e reaction of the soil behind piles can be described as where k is the soil reaction modulus. k can be described using a triparameter soil model k(z) � m(z 0 + z) n , in which m and n are soil reaction modulus coefficients; z 0 is the initial height of the ground surface; b 1 is the calculated pile width; p u is ultimate soil resistance; y is pile deflection; y * is ultimate soil displacement. According to Matlock, y * � 20ε c d for clay and y * � 3d/80 for sand. ε c is the soil strain, and ε c � 0.005-0.02 [20]. δ is a correction coefficient for considering the influence of plastic soil deformation on the reaction calculated width of the soil behind the piles. When plastic soil deformation enters the DD′G′G zone, the expansion of plastic deformation may decrease the reaction calculated width of the soil behind the piles (Figures 2(c) and 2(d)):

Governing Differential Equations.
Assuming the pile behaves as a Euler-Bernoulli beam, the differential equations governing pile deflection in the free, plastic, and elastic zones ( Figure 1(b)) can be obtained by discretizing each pile segment into subslices. e accuracy of the solution is improved with finer pile slices.
For the pile in free zone above the ground surface, each pile slice has an individual coordinate system marked with the subscript a (Figure 3(a)). e pile length L a is spilt into n a subslices with length l a � L a /n a . e distributed lateral load acting on the κth pile slice can be calculated as the average of the load acting on the slice: e average axial force of the κth slice for the pile in the free zone can be calculated using equation (1): For the pile beneath the ground surface, the pile length L b is divided into n segments by the soil layers. Each pile segment has its individual coordinate system marked with the subscript i (Figures 3(b) and 3(c)) and is split into n i subslices. e average axial force on the pile slice can be calculated using equation (2): where the subscripts i and w refer to the wth pile slice of the ith pile segment in the ith soil layer. Based on equations (9)∼(13), the average passive load (P s(i, w) ) acting on the wth pile slice of the ith pile segment in the ith soil layer can be calculated as the average load acting on the corresponding slice as follows: In the plastic zone, the average ultimate soil resistance behind the wth pile slice of the ith pile segment, p u(i, w) , can be regarded as the average ultimate soil resistance of the top and bottom of the corresponding slice: Similarly in the elastic zone, linear elastoplastic spring models are used, and the average soil reaction modulus (k (i, w) ) behind the wth pile slice of the ith pile segment can be calculated as the average soil reaction modulus of the top and bottom of the corresponding slice as For the κth pile slice in the free zone (Figure 3(a)), the governing differential equation is For the wth pile slice of the ith pile segment in the plastic zone (Figure 3(b)), the differential equation is where the subscript p indicates the plastic state.
When the soil resistance is in the elastic stage (Figure 3(c)), the differential equation can be expressed as where the subscript e indicates the plastic state.

Analysis for the Free Zone.
Solving (22), the deflection of the ith pile slice is given by where y a(κ) is the pile deflection on the κth pile slice at depth z a(κ) ; λ a(κ) 2 � N a(κ) /EI a ; a � q a(κ) /EI a ; C o1 , C o2 , C o3 , and C o4 are integration constants.
Using equation (25), pile slope φ a(κ) , bending moment M a(κ) , and shear force V a(κ) at depth z a(κ) are given by e transfer matrix function of the ith pile slice in the free zone can be expressed as , M a(κ) , and V a(κ) are the pile deflection, slope, bending moment, and shear force at the bottom of the κth pile slice in the free zone, respectively; are the pile deflection, slope, bending moment, and shear force at the top of the κth pile slice, respectively. S a(κ) is the transfer matrix coefficient of the κth pile slice: 6 Advances in Civil Engineering

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where A aj , B aj , C aj , D aj , and E aj (j � 1-4) are 20 dimensionless coefficients, which are a function of λ a(κ) l a . As a result, based on the deflection and stress continuity of the pile, the transfer matrix function for the pile segment in the free zone can be expressed as U a n a ( ) � S a n a ( ) S a n a − 1 , M a(n a ) , V a(n a ) , 1] T are 5-by-1 transposed matrices and represent pile deflection, slope, bending moment, and shear force for the head and bottom of the pile segment in the free zone, respectively; S a is the overall transfer matrix coefficient of the pile in the free zone.

Analysis for the Plastic Zone.
In the plastic soil zone, as the passive load and ultimate soil resistance can be determined using equations (19) and (20), respectively; the form of equation (23) is similar to equation (22), and the solution methods for the two differential equations are similar. e transfer matrix function of equation (23) of the wth pile slice of the ith pile segment can be expressed as where , and V p(i,w) are the pile deflection, slope, bending moment, and shear force at the bottom of the wth pile slice of the ith pile segment; , M p(i,w− 1) , and V p(i,w− 1) are the pile deflection, slope, bending moment, and shear force at the head of wth pile slice.
S p(i,w) is the transfer matrix coefficient of the wth pile slice of the ith pile segment when the pile section is in the plastic zone. When λ p(i,w) where A pj , B pj , C pj , D pj , and E pj (j � 1-4) are 20 dimensionless coefficients, which are function of λ p(i,w) l i . Assuming the pile in the plastic zone extends from the sth slice of the tth pile segment to the uth pile slice of the vth pile segment, the matrix transfer function for the pile in the plastic zone is given by

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Analysis of the Elastic Zone.
Similarly for piles in the elastic soil zone, the deflection of the wth pile slice of the ith pile segment in the elastic zone can be obtained by solving equation (24).
By substituting equation (33) or (34) into (26), the transfer matrix function of equation (24) for the wth pile slice of the ith pile segment can be expressed as where w) is the transfer matrix coefficient of the wth pile slice of the ith pile segment when the pile section is in the elastic zone: where A ej , B ej , C ej , D ej , and E ej (j � 1-4) are 20 dimensionless coefficients, which are a function of λ e(i,w) l i . Assuming that the pile range in the elastic zone is from the sth slice of the tth pile segment to the uth pile slice of the vth pile segment, the matrix transfer function for the pile in the elastic zone can be expressed as U e(v,u) � S e(v,u) S e(v,u− 1) · · · S e(t,s+1) S e(t,s) U e(t,s) � S e U e(t,s) , (37) where S e is the overall transfer matrix coefficient of the pile in the elastic zone. (27), (30), and (35) with the transfer matrix approach [33][34][35], the transfer matrix function for passively loaded piles adjacent to an embankment can be expressed as U n,n n ( ) � S n,n n ( ) S n,n n − 1 (

General Solutions. By combing equations
where U (n,n n ) � [y (n,n n ) , φ (n,n n ) , M (n,n n ) , V (n,n n ) , 1] T and U 0 � [y 0 , φ 0 , M 0 , V 0 , 1] T are pile deflection vectors, slope, bending moment, and shear force for the pile tip and pile head, respectively. S (i,w) is the transfer matrix coefficient, which is calculated by equations (31) and (36) for piles in the Advances in Civil Engineering plastic zone and elastic zone, respectively. S a is the overall transfer matrix coefficient of the pile in the free zone as deduced in equation (29). S is defined as the overall pile transfer matrix coefficient, which is the product of the transfer matrix coefficient of each pile slice in the sequence from pile head to pile toe. Equation (38) is composed of four equations with eight variables including the pile deflection, slope, bending moment, and shear force at the pile head and pile toe. e boundary conditions at the pile head and pile toe must be specified to solve this equation. Five types of boundary conditions are considered in this study: free-head, slip-head, fixed-head, free-base, and fixed-base.
For the free-head case, the boundary conditions at the pile head are Input applied load, soil resistance modulus, and basic parameters Utilize (38), (44), and (45) to solve pile response Yes Assume soil resistance is in elastic stage and calculate S a(κ) ; S e(i,w)

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For the slip-head case, the boundary conditions at the pile head are For the fixed-head case, the boundary conditions at the pile head are For the free-base case, the boundary conditions at the pile head are M n,n n ( ) � 0; V n,n n ( ) � 0.

(42)
For the fixed-base case, the boundary conditions at the pile head are      Advances in Civil Engineering 13 y n,n n ( ) � 0; φ n,n n ( ) � 0.

(43)
Substituting one of the pile head boundary conditions and one of the pile base boundary conditions into equation (38), the remaining pile head variables can be calculated.
us, based on the response of the pile head U 0 , the solution for the pile in the free zone at any depth can be calculated by Similarly, the pile response at any depth beneath the ground surface can be calculated by e algorithm for calculating U a(κ) and U (i,w) is summarized in Figure 4, in which the pile deflection convergence criteria ε � 0.003 of the (j − 1)th and jth iteration guarantee iteration accuracy.

Validation
In this section, the proposed analytical method is validated by analyzing two cases, one from existing literature and a new case. [9]. Based on the displacement-based method, a simplified numerical procedure for analyzing the pile response to lateral soil displacement was described in [9], in which free-field displacements were directly used to solve for the soil-pile load transfer mechanism.

Case from
is method was applied and verified using several full-scale and centrifuge model tests. Figure 5 shows the calculation model [9], in which the pile embedded in layered clay-sand was analyzed. Clay and sand physical properties were summarized in [9]. e pile was assumed to have been installed prior to construction of the embankment and was connected to a rigid cap such that the pile head could be deflected without rotating (φ c0 � 0°; V c0 � 0 kN). e pile has a length of L � 24 m, diameter d � 1.0 m, spacing s � 2.0 m, and modulus E � 28 GPa. e soil resistance was assumed to increase linearly with depth (n � 1; z 0 � 4.5 m), and the coefficients of the soil reaction modulus (m � 2 MN/m 4 for clay and 30 MN/m 4 for the sand) were evaluated using Skempton's empirical method [36]. Figure 6 shows the bending moment and pile deflection predicted by the numerical results [9] and the proposed method.
e variation of bending moment and pile deflection with pile depth by the two methods agreed well.
ere is an underprediction of the pile deflection of the proposed method in comparison to the numerical results [9]. e discrepancy may be because of the different soil models used by the two methods. e pile-soil interaction was modeled by hyperbolic soil springs and the soil resistance was assumed constant with depth in [9], while elasticperfectly plastic soil model was used and the soil resistance was assumed to increase linearly with depth in this paper. 14 Advances in Civil Engineering e soil resistance of the proposed method is larger than [9] under 9 m, which result in smaller pile deflection of the proposed method.

Bridge Pile Foundation near an Embankment.
e method described above is also employed to analyze the response of offset piles columns in the SX Bridge at Lianyungang, China. SX Bridge is one of the three bridges of the XinG Road interchange, a semicloverleaf semidirectional compounded interchange (Figure 7(a)). XinG Road passes beneath the SX Bridge between the #26 and #28 piers (Figure 7(b)). XinG Road has a width of 42 m and a slope of 3%. e road shoulder height adjacent to the #26 pier is 1.1 m, and the road shoulder height adjacent to the #28 pier is 2.3 m. e #26 and #28 piers are all composed of 8 sets of piles columns. e length and diameter of piles are 62 m and 1.6 m, respectively. e length and diameter of columns are 8 m and 1.2 m, respectively. e space between the #28-1, #28-2, #28-3, and #28-4 piles and the space between the #26-1, #26-2, #26-3, and #26-4 piles are all 6.06 m. e space between the #28-5, #28-6, #28-7, and #28-8 piles and the space between the #26-5, #26-6, #26-7, and #26-8 are all 4.85 m. e distance from the #26 pier (#26-1∼#26-8) to the XinG Road slope is 3.5 m. e distance from the #28 piers to the XinG Road slope increased from 2.37 m at #28-1 to 4.20 m at #28-8 ( Figure 8). e 16 columns of #26 and #28 piers were offset away from the embankment after the filling of the XinG was completed. e lateral displacement of column heads in the #26 pier range from 3.2 to 6.6 cm and the lateral displacement of column heads in the #28 piers are all greater than 10.0 cm. e maximum lateral displacement is approximately 13.3 cm located at the head of the 2th column of the #28 pier ( Figure 8). e average XinG Road embankment height is 1.7 m; however, due to the presence of a 1.0 m-1.5 m side trench outside the #28 pier, the effective embankment height in the calculation is 3.0 m. e embankment fill unit weight is 17 kN/m 3 . Soil mechanical indicators are based on the data from the investigation site and associated triaxial tests (Table 1). e soft soil resistance coefficient is recommended as 0.55 MN/m 4 based on the resistance coefficient reference value of the rock and soil provided by CRSSDIG [37]. In addition, sliding rubber bearings are used on the top of the columns, and the forces on the column heads are N 0 � 5379 kN, M 0 � 0 kN·m, and V 0 � 120 kN. e piles and columns are all composed of reinforce concrete, with a unit weight and modulus of 25 kN/m 3 and 28.5 GPa, respectively. Figure 9 shows the calculated column and pile lateral displacement compared to the measured data. It is noted that the calculated results of #28 piers are mainly smaller than the measured results, while the calculated results of #26 piers are mainly larger than the measured results. e reason is that the average embankment height is used in the proposed method, which is lower than the actual height at #28 piers and higher than the actual heights at #26 piers Advances in Civil Engineering ( Figure 7). Figure 10 shows the calculated bending moment along the column and pile for #28-2 and #28-6 from the proposed method and the improved strain wedge (SW) method [19]. e predictions of the improved SW method [19] are obtained from back analysis of measured pile head displacements which reflected real field behavior; however, it does not consider the effect of soil arching between piles and the rotation of pile head. is is the reason for the underprediction of pile bending moment of the improved SW method [19] in comparison to the proposed method.

Factors Influencing Pile
Response. Taking #28-2 as an example, the effect of embankment pressure, soil resistance, and column boundary conditions on pile-column response is calculated. e influence of embankment pressure on the performance of neighboring piles columns is investigated by varying the embankment height between 1.0 m, 2.0 m, and 3.0 m. e other parameters are held constant as in previous sections. Increasing embankment height (or embankment pressure) causes an increase in bending moment and deflection of neighboring piles columns ( Figure 11). e maximum pile bending moment increased from 3333.7 kN·m for H � 1.0 m to 5254.8 kN·m for H � 3.0 m, a 57.6% increase. e column head displacement varied from 5.92 cm for H � 1.0 m to 13.85 cm for H � 3.0 m, an increase of 1.34 times. With increasing embankment height (or embankment pressure), more passive loads are transferred from moving soil to the piles, and the surcharge loading of the soft soil has a significant impact on nearby piles. e influence of soil resistance on pile-column performance is investigated by varying the mud clay reaction modulus coefficient between 2 MN/m 4 , 3 MN/m 4 , and 4 MN/m 4 . Figure 12 shows the effect of the clay reaction modulus coefficient on pile-column response. With increasing clay reaction modulus coefficient, the pile-column bending moment and deflection decrease significantly ( Figure 12). e maximum pile bending moment and column head displacement are reduced by 30 e influence of boundary conditions is investigated by comparing the free-head case (M c0 � 0, V c0 � 0), slip-head case (φ c0 � 0, V c0 � 0), and fixed-head case (y c0 � 0, φ c0 � 0). Figure 13 shows the effect of boundary conditions on the piles-columns responses. Bending moment and deflection along the pile and column for the fixed-head case are significantly different from the other two cases. By comparing this case to the free-head and slip-head cases, there is a negative moment in the upper part of the pile, and the maximum pile displacement occurs at 1/2 of the soft clay thickness. Comparing the free-head and slip-head cases, the difference in bending moment primarily occurs in the upper 2/3 s of the soft clay, and the difference in pile deflection primarily occurs within 7d depth beneath the pile head.

Conclusions
Based on plastic deformation theory, a load transfer model was presented to analyze the soil-pile interaction of passive piles adjacent to a surcharge load. A semianalytical solution was developed to assess the response of piles under active and passive loads using the transfer matrix approach. e proposed method was validated against the displacement-based method proposed by Goh et al. [9] and field measurements at SX Bridge, Lianyungang, China. Results showed that the proposed method can provide good response predictions (i.e., deflection, bending moment, and shear forces) for piles adjacent to a surcharge load.

⎧ ⎨ ⎩
(A.10) Using (A.6) and (A.10), the lateral stress σ II x at any point x in EE′A′A can be expressed using (4).
In regard to the AA′D′D zone, considering P AA′ (z) as the boundary condition of the surface AA′, the constant C III in (A.7) can be obtained: Using (A.7) and (A.11), the lateral stress σ III x at any point x in the AA′D′D can be expressed using (5).

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this article.