Improved prediction of pile bending moment and deflection due to adjacent braced excavation

Abstract

Deep excavations in dense urban areas have caused damage to nearby existing structures in numerous past construction cases. Proper assessment is crucial in the initial design stages. This study develops equations to predict the existing pile bending moment and deflection produced by adjacent braced excavations. Influential parameters (i.e., the excavation geometry, diaphragm wall thickness, pile geometry, strength and small-strain stiffness of the soil, and soft clay thickness) were considered and employed in the developed equations. It is practically unfeasible to obtain measurement data; hence, artificial data for the bending moment and deflection of existing piles were produced from well-calibrated numerical analyses of hypothetical cases, using the three-dimensional finite element method. The developed equations were established through a multiple linear regression analysis of the artificial data, using the transformation technique. In addition, the three-dimensional nature of the excavation work was characterized by considering the excavation corner effect, using the plane strain ratio parameter. The estimation results of the developed equations can provide satisfactory pile bending moment and deflection data and are more accurate than those found in previous studies.

Graphical abstract

Keywords

pile responses / excavation / prediction / deflection / bending moments

1 Introduction

Urban development in densely packed environments often involves the construction of underground spaces (e.g., basements). Braced excavations are widely employed for the basement construction of buildings and the development of underground structures (e.g., metro stations and cut-and-cover tunnels) near to the existing foundations of adjacent structures (e.g., high-rise buildings, flyovers, and elevated train lines [1–6]). When existing structures are supported by pile foundations, it is imperative to suitably evaluate the responses produced in the existing piles of adjacent braced excavations, to ensure the stability and integrity of both the structures and piles. In addition, the pile foundations must be fully functional without changing the way they respond to loads. In current practice, impact assessments must include the stability and integrity of both the overall structure and individual piles. Although excavations can induce soil movements in both the vertical and horizontal directions, the latter component is characterized as being more critical, because the piles are commonly designed to bear vertical loads [7]. Several previous studies (e.g., [8–16]) have shown that basement excavations for new structures unavoidably produce excessive soil movements behind the retaining wall system; consequently, the soil movements can produce an additional lateral loading on the nearby existing pile foundation, resulting in an augmentative bending moment and deflection therein; this may result in structural distress or even failure. Hence, the risk to nearby pile structures is a primary design concern in such projects. To this end, reasonable evaluations are essential to preventing or minimizing damage to the adjacent piles.

Numerous studies have concentrated upon the deflection of the retaining structure and the prediction of the soil surface settlement behind the retaining wall. Soil movement should be smaller than the allowable limit, to avoid damage to adjacent structures. Additionally, numerous studies have sought to understand the mechanism of deep excavation work and the key parameters influencing soil movements. (i) The quantity of soil surface settlement behind the wall, (ii) horizontal wall movement, and (iii) respective distributions of both (i) and (ii) depend upon several factors, including the geological profile and soil properties, the stiffness of the retaining structure and support system, the excavation process and methods, the workmanship and duration of excavations, and the boundary conditions of the excavation area (e.g., [13,17–22]). Moreover, the excavation-induced soil movement behind the retaining structure produces three-dimensional deformations (the so-called “influence of excavation corner effect”) that depend on the limited excavation areas and the length–width ratios of the excavation (e.g., [23,24]). The corner effects lead to smaller ground surface settlements behind the retaining wall as well as horizontal wall movements at corner sections [1,2]. However, the mobilized stress and movements of the adjacent structure, and the surface settlement of nearby buildings and structures, remain significant [25–27]. Thus, the excavation corner effect complicates the characteristics of the deformation behind the retaining wall and should be comprehensively studied, to reduce the braced-excavation-induced risks to adjacent piles.

Many previous publications have shown that existing pile foundations can be affected by horizontal soil movements attributable to nearby excavation activities. Finno et al. [28] and Goh et al. [8] reported case studies of pile groups and single piles adjacent to deep excavations, and they illustrated that excavation-induced lateral soil movement can degrade the existing adjacent pile structures. In addition to actual case studies, centrifuge tests were also carried out to assess the responses of existing pile groups and single piles under the actions of adjacent excavations [9–11,14,29]. They concluded that the excavation-induced lateral deflection and bending moment of existing pile structures are significantly influenced by the distance from the pile to the retaining structure, as well as the pile-head condition. These two key influential factors regarding excavation-induced existing pile responses were confirmed by later studies using analytical and numerical methods [15,16,30–34]. Based on these past studies (e.g., [3,7–12,15,16,28,30–32,35–38]), it was found that an additional bending moment and deflection within the existing piles adjacent to excavations were significantly influenced by numerous factors (e.g., excavation depth and width, retaining wall stiffness, supporting system stiffness and spacing, diameter and length of pile, distance of pile from the excavation face, fixity conditions of pile head, and undrained shear strength of soil layer). Therefore, it is imperative to assess pile responses, to maintain the serviceability of existing structures. Several researchers have proposed and developed design charts (i.e., [7,31,38]) to evaluate pile responses (both pile bending moment and deflection) to adjacent excavations. Poulos and Chen [7] proposed simplified equations using design charts, to preliminarily estimate the bending moment and deflection of adjacent piles. These design charts were separately constructed via a two-stage analysis process using finite element (FE) and boundary element techniques. The free-field soil movements (neglecting the adjacent structural pile) were first obtained using the FE method (under a two-dimensional (2D) plain strain condition); then, a boundary element method (simplified as springs) was employed to estimate the pile responses using the computed free-field soil movements. A similar study was conducted by Liyanapathirana and Nishanthan [38], who developed a set of design charts using a three-dimensional (3D) FE analysis in which the excavation activities were simulated along with the adjacent pile. However, these methods for predicting the pile response (i.e., [7,38]) considered only the single soil layer condition and neglected the nonlinear and small-strain behaviors of the soil, which should be considered in excavation analysis involving rigid retaining walls (as documented in, e.g., [15,19,21,30,39]). Zhang et al. [31] conducted 2D FE simulations considering the nonlinear characteristics of soils at small strain levels using the hardening soil (HS) constitutive soil model, to investigate excavation-induced pile behavior; they proposed a set of modification factors for estimating pile responses in terms of both the maximum bending moment and deflection. In that approach, the pile responses were estimated approximately using the axial loadings borne by the pile, pile distance from the excavation, excavation depth and width, diameter and length of pile, and pile-head fixity as modification factors. However, the stiffness and strength of the soil, soft soil thickness, and system stiffness of the retaining wall were neglected. In previous studies [14,19,20], the results obtained from centrifugal and numerical modeling of braced excavations shows that the soft soil thickness and system stiffness of the retaining wall are crucial factors that reflect the magnitude and patterns of soil movements behind the wall. Moreover, it is commonly understood that the highly nonlinear stress–strain characteristics of soils strongly depend upon the stress path, stress level, strain level [40,41], and soil stiffness; these can change under strain variations (even at low strain levels). The displacement characteristics of braced excavations are determined by the stress path, which in turn depends on the stress history and stress state [42]. Hence, these previous studies indicate that the soil surface settlement behind the wall and the horizontal wall movement (which directly affect the nearby existing piles) are considerably dominated by the small-strain characteristics of the soils [20,39,43,44]. Therefore, an improved method that considers these shortcomings when predicting the pile responses attributable to nearby excavations is required.

In this study, as an assessment tool for individual piles, semi-empirical equations for predicting the maximum bending moment and deflection of existing piles, as produced by adjacent braced excavations in soft deposits, were developed (using a diaphragm wall as a supporting structure) from reliable artificial data. The artificial data were obtained via 3D FE analyses through suitable modeling of the nonlinear stress–strain characteristic of soils at small strain levels (verified by measurement data). The characteristics of the excavation, wall system, pile, and subsoil foundation varied across 632 hypothetical cases. Furthermore, the influence of the pile-head boundary conditions was examined and considered in the data generation. Then, semi-empirical equations were proposed using multiple-variable regression analysis combined with the transformation technique. In this study, all significant parameters (i.e., the excavation depth and width, system stiffness of the diaphragm wall, diameter and length of the pile, pile location with respect to the excavation, pile-head fixity conditions, shear strength and small-strain stiffness of both the soft and stiff clays, and soft clay thickness) were employed to construct semi-empirical equations for predicting the maximum bending moment and deflection of the existing pile. The proposed equations were also validated by comparing the predicted results against the values obtained from a previous study. Moreover, this study developed a simplified equation to estimate the plane strain ratio (PSR) values used to evaluate the maximum pile deflection, by considering the excavation corner effect. The proposed procedure can be considered a solution of interest for assessing pile deflections when the investigated piles are located near to the excavation corners.

2 Factors affecting pile bending moment and deflection

Fig.1 shows a deep excavation adjacent to an existing pile, as well as the relevant notations. The effects of these factors were extensively assessed using the FE method, via sensitivity analyses and a series of experimental model tests. The influencing factors included the excavation geometry, wall dimension and stiffness, supporting system (i.e., strut or floor slab), pile characteristics (i.e., the pile geometry, stiffness, and location and the pile-head fixity condition), and undrained soil strength; these have been frequently considered in the sensitivity analyses of previous studies and are also included in existing semi-empirical methods for predicting the maximum pile bending moment (M^{max}) and maximum pile deflection (d^{max}). The impact of each influencing factor (caused by nearby excavation) upon the behavior of a single pile in previous studies is derived under the assumption of a free-head pile condition.

2.1 Effect of excavation geometry

By referring to previous studies [7–9,11,31,38], the excavation depth (H_{\text{e}}) was found to be an essential factor in the variations in the excavation-induced M^{max} and d^{max} of the existing pile. An increase in H_{\text{e}} had a considerable impact on the increase in both M^{max} and d^{max}, with an approximately linear trend. Although the effect of the excavation width (B) upon nearby responses has never been investigated in previous studies, Kung et al. [19] highlighted that B exerts a significant influence on the lateral wall deflection. The lateral wall deflection increases linearly with B [45]; hence, the additional M^{max} and d^{max} of existing piles would certainly be affected by this factor. Another important factor that influences M^{max} and d^{max} is the parameter B.

2.2 Effect of wall dimension and stiffness

Poulos and Chen [7] showed that the wall stiffness E_{\text{w}}{I}_{\text{w}} (where E_{\text{w}} = Young’s modulus of the retaining wall and I_{\text{w}} = the moment of inertia of the wall cross-sectional area) plays a crucial role in excavation-induced pile responses. Both the M^{max} and d^{max} of piles decrease under an increase in E_{\text{w}}{I}_{\text{w}}. In contrast, the wall length is found to be an insignificant parameter in the deformations of braced excavations in soft deposits [18], provided a sufficient length is applied (as here). Hence, this factor can be neglected when developing a semi-empirical equation, as described in previous studies [7,31,38].

2.3 Effect of supporting system

A bracing system is generally used to enhance the rigidity of retaining walls. In Ref. [38], both the induced M^{max} and d^{max} attributable to the adjacent excavation slightly decreased under a decrease and increase in the vertical spacing and stiffness of strut, respectively. A similar trend was also obtained from the results of Poulos and Chen [7]. However, it seems that both strut spacing and stiffness have a less significant impact upon the excavation-induced M^{max} and d^{max} [38].

2.4 Effect of pile characteristics

Poulos and Chen [7] and Zhang et al. [31] stated that the excavation-induced d^{max} of an existing circular pile tends to increase substantially as the pile diameter (D_{\text{p}}) decreases. Conversely, the induced M^{max} considerably decreases when D_{\text{p}} decreases. This is mainly attributable to the flexibility of the pile. A consistent result is also obtained from the study of Liyanapathirana and Nishanthan [38], who investigated the width effect of the square pile. The impact of pile length (L_{\text{p}}) upon pile response was also examined by Zhang et al. [31], who found that d^{max} is strongly affected by L_{\text{p}}. A greater L_{\text{p}} can effectively prevent the occurrence of large d^{max}. In addition, the pile bending moment profile is dominated by L_{\text{p}}. When a longer pile is considered, the pile bending moment profile is homologous to the cantilever beam; meanwhile, the bending moment distribution with respect to depth for a short pile is identical to that of the truss beam. Zhang et al. [31] also indicated that L_{\text{p}} has a significant influence upon M^{max}, especially for the short pile (L_{\text{p}} < wall length). In an attempt to ascertain the influence of pile distance from excavation face (X_{\text{p}}) upon pile behavior, Leung et al. [9] performed centrifuge model tests by altering the pile location behind the retaining wall from X_{\text{p}} = 1−9 m. They concluded that both M^{max} and d^{max} were significantly influenced by the X_{\text{p}}, and their results showed that these pile responses exponentially increase as X_{\text{p}} decreases. This experimental result is consistent with numerical results obtained by other researchers [7,31,38]. Furthermore, the effect of the axial load exerted upon the pile head has been extensively investigated in many studies [3,31,38], and the results show that the axial load does not significantly alter M^{max} or d^{max}.

The numerical analyses and centrifuge model tests conducted in previous studies [9,31,35,38] show that both M^{max} and d^{max} are strongly influenced by the pile-head fixity condition (especially M^{max}); thus, the existing pile should be carefully analyzed using a reasonable pile-head boundary condition. Four different pile-head fixity conditions have been frequently assessed in previous investigations: (i) free rotation and free deflection (free), (ii) free rotation and restrained deflection (pinned), (iii) restrained rotation and free deflection (translatable), and (iv) fixed translation and fixed rotation (fixed). In previous studies, a larger d^{max} was achieved when the pile head was free. However, a smaller M^{max} was identified for this free pile-head fixity condition when compared against the other fixity conditions. On the other hand, a fixed pile-head condition tends to require a larger M^{max} but leads to a small d^{max}. Zhang et al. [31] found that the fixed pile-head condition can greatly reduce (by ~23%) the excavation-induced d^{max} compared to the free pile-head condition, under which a larger M^{max} (~1.76 times) is obtained. Their results also showed that the excavation-induced M^{max} and d^{max} for the three cases of free, pinned, and translatable conditions do not significantly differ, although the pile bending moment and deflection patterns are dissimilar. Hence, both free and fixed pile-head conditions should be cautiously considered and included in the semi-empirical equations for predicting the M^{max} and d^{max} induced by adjacent excavation within the existing pile.

2.5 Effect of soil characteristics

In previous studies, the undrained soil shear strength (s_{\text{u}}) parameter was the only factor employed to evaluate the effects of soil characteristics upon excavation-induced pile responses [7,35,38]. As mentioned by Poulos and Chen [35] and Liyanapathirana and Nishanthan [38], both M^{max} and d^{max} significantly increase under a decrease in s_{\text{u}}, owing to the larger soil movements attributable to weaker clays.

Over the past three decades, several studies (e.g., [16,18,19,22,30,39,41,43,44]) have attempted to improve the estimations of wall deflection and soil movement, by using an advanced soil constitutive model. Kung et al. [43] employed a modified cam-clay model and small-strain constitutive model for soft and medium clays, to predict the deflection of the wall and soil surface settlement of braced excavations in Taipei, by calibrating against well-documented data. Their results showed that to precisely estimate the wall deflection and ground surface settlement, a small-strain constitutive model is required. In this situation, the HS with small-strain stiffness (HSS) model has been widely adopted for simulating excavations in soft-to-stiff clays [15,21,31,42]. The HSS model represents an improvement over the traditional HS model; it considers the stiffness increase of soils at small strain levels. The HSS model introduces two additional parameters to model this higher stiffness (where the soil stiffness is nonlinearly proportional to the strain): the reference small-strain stiffness (G_{\text{0}}^{\text{ref}}) and the reference shear strain ({\gamma }_{0.7}) at which the secant shear stiffness is reduced to 70% of G_0.

According to previous studies [14,20], the soft clay thickness (T_{\text{s}}) is a significant factor in excavation-induced wall deflection and soil surface settlement. Zhang et al. [20] considered T_{\text{s}} as an input parameter in a polynomial regression model, to estimate the maximum wall deflection. According to the centrifuge model tests of excavations adjacent to a single pile, performed by Leung et al. [14], a larger lateral wall movement and greater M^{max} arise in the case of a thicker soft clay layer.

To summarize, ten influential factors should be considered to obtain a comprehensive prediction of excavation-induced M^{max} and d^{max} for an existing single pile behind the retaining wall: excavation depth (H_{\text{e}}), excavation width (B), retaining wall thickness (t_{\text{w}}), pile diameter (D_{\text{p}}), pile length (L_{\text{p}}), pile distance from excavation (X_{\text{p}}), pile-head fixity, soil undrained shear strength (s_{\text{u}}), reference small strain stiffness (G_{\text{0}}^{\text{ref}}) of soil, and thickness of soft clay layer (T_{\text{s}}).

3 Generation of artificial data for hypothetical cases

To develop semi-empirical equations for predicting the pile responses, artificial data for the excavation-induced M^{max} and d^{max} (taking into account several key influencing factors) are required. In this study, 632 hypothetical analyses of excavations adjacent to an existing pile were employed to obtain data for various excavation scenarios, as presented in Fig.1.

The excavation length was fixed at 100 m for all hypothetic analyses with variations of H_{\text{e}} and B. The excavation is supported by the diaphragm walls with various t_{\text{w}} while a constant wall length of 20 m from the ground surface is considered throughout this study. The stiffness value of diaphragm wall (E_{\text{w}}) was assumed to be a constant at of 2\times {10}^7 kPa. The basement was finished by the top-down excavation method. The excavation depths with four staged constructions are considered to be 2, 5, 8, and 11 m. The bracing locations (0.3-m-thick floor slab) are assumed to be at 1.85, 4.85, and 7.85 m (constant vertical strut spacing of 3 m). An existing pile (circular concrete pile) with constant stiffness (E_{\text{p}}) of 2\times {10}^7 kPa and varying dimensions of D_{\text{p}} and L_{\text{p}} were horizontally located at various distances X_{\text{p}} and X_{\text{p}}^{\text{cn}} from the primary and complementary walls, respectively. The parameter X_{\text{p}}^{\text{cn}} was only employed to assess the impact of the excavation corner effect upon pile deflection.

The geological profile used for the hypothetical analysis cases was a typical Bangkok subsoil foundation, which includes four soil layers with a total depth of up to 36 m. The uppermost soil layer was a 2-m-thick sedimentary crust. The next soil layer was soft clay, with a thickness of 9–14.5 m; this was placed on the first stiff clay layer (thickness: 7.5–13 m). The deepest soil layer was the first sand layer, with a constant thickness of 12 m. To analyze the effects of soil characteristics upon excavation-induced pile responses, hypothetical excavations were performed by considering the influences of s_{\text{u}} and G_{\text{0}}^{\text{ref}} for both soft and stiff clays. s_{\text{u}} and G_{\text{0}}^{\text{ref}} were divided by the vertical effective stress ({\sigma }_{\text{v}}^{\prime }) to give the s_{\text{u}}/\phantom{s_{\text{u}}{\sigma }_{\text{v}}^{\prime }}{\sigma }_{\text{v}}^{\prime } and G_{\text{0}}^{\text{ref}}/\phantom{G_{\text{0}}^{\text{ref}}{\sigma }_{\text{v}}^{\prime }}{\sigma }_{\text{v}}^{\prime } ratios [20]. In the numerical analyses, both soft and stiff clay layers were divided into four layers, to assess the effect of both s_{\text{u}} and G_{\text{0}}^{\text{ref}} with respect to the variation in {\sigma }_{\text{v}}^{\prime }, as well as to investigate the influence of soft clay thickness. The soft clay layers were (1) 2–5 m, (2) 5–8 m, (3) 8–11 m, and (4) 11–14 m from the ground surface; meanwhile, for the stiff clay layer, the layers were (1) 14–16.5 m, (2) 16.5–19 m, (3) 19–21.5 m, and (4) 21.5–24 m from the ground surface. To assess the effect of soft clay thickness, soft clay thicknesses of 9, 12, and 14.5 m were considered. To analyze this effect, the soft clay thickness was increased concurrently with the decrease in the stiff clay layer thickness, to maintain a total clay thickness of 22 m. The properties of the subsoil layers employed in this study (as listed in Tab.1) were similar to those used in several previous studies [46–51]. The behavior of the sedimentary crust and lowermost sand layers was simulated using the Mohr–Coulomb (MC) model. This model has been widely used to model the behaviors of sandy soils under braced excavation work in several previous studies (e.g., [52–56]). According to the literature, the MC model is suitable and sufficient for capturing the behavior of sandy soils and obtaining satisfactory characteristics for the soil–structure interactions of the excavation problem. In addition, because the considered excavation depth and length of the adjacent existing pile were not extended into the lowermost sand layer, it can be postulated that the excavation-induced responses of the adjacent existing pile were marginally influenced by the underlying sand layer. The soft and stiff clays were simulated using the HSS model, in which the small-strain behaviors of the soil were considered. The properties of the adopted subsoils have been successfully applied in Refs. [46–49], and the input soil parameters were well calibrated with triaxial testing results for the soil samples, as reported by Rukdeechuai et al. [47]. Therefore, the HSS and MC models adopted in the present study can be considered suitable soil constitutive models for the following numerical analyses. It should be noted that all stiffnesses of soft and stiff clays (i.e., E_{\text{50}}^{\text{ref}}, E_{\text{oed}}^{\text{ref}}=E_{\text{50}}^{\text{ref}}, and E_{\text{ur}}^{\text{ref}}=3E_{\text{50}}^{\text{ref}} [21,24,48,57–59]) vary with s_{\text{u}}, where E_{\text{50}}^{\text{ref}}/s_{\text{u}} = 265 and 400 are considered for soft and stiff clays, respectively [20].

Using the commercial FE software PLAXIS 3D, numerical analyses were performed to assess the complicated behaviors of excavation–pile interactions. 3D FE modeling was performed for one-quarter of the excavation problem, owing to the symmetry in both the longitudinal and transverse directions, as shown in Fig.2 (example case). The adopted FE modeling dimensions were 36 m (depth along the Z-axis), B/2 + 50 m in the X-direction, and L/2+50 m in the Y-direction, where L is the primary wall length, which is sufficiently large to minimize the effect of boundary restraints. For the displacement boundary conditions, a pin support was applied at the bottom of the FE model to restrict the displacement in all directions, and roller supports were applied for all vertical sides to restrain only the horizontal displacement. The top of the FE model was allowed to move freely in all directions. The pore water pressure profile was considered hydrostatic, in which the groundwater level was situated 2 m below the soil surface. The drainage boundary was allowed to flow at the top of the FE model, whereas four vertical sides and the base of the FE model, were set as fully impermeable. Note that because of the large distance between the excavation faces and two vertical boundaries at X = B/2+50 m and Y = L/2+50 m (more than three times the largest excavation depth (i.e., 11 m) in this study), no excess pore water pressure arose in the vicinity of the vertical boundaries. In all analyses, the initial soil stress was simulated using the coefficient of lateral earth pressure at rest and the soil unit weight. Each analysis was conducted in three phases. A wished-in-place concrete pile and diaphragm wall were simulated in the first phase. The pile was modeled using an embedded beam. For the fixed-head pile condition, a plate element with sufficiently high stiffness was used to restrain the rotation of the pile head. The second phase regards the application of pile axial loading; here, the working load of the pile is defined as the pile ultimate capacity divided by a safety factor of 2.5 (for a concrete pile). The excess pore pressure mobilized in the soil is allowed to dissipate perfectly before excavation in the next phase. The third phase considers excavation activity, which is simulated by consecutively deactivating the soil elements located in the excavation zone and activating the support elements as an actual excavation sequence.

It is practically unfeasible to collect historical pile responses induced by nearby excavations; hence, the FE model used to establish the artificial data could only be verified using green field data for soil movements caused by excavations. The FE procedures using the adopted soil constitutive models (both HSS and MC models, as listed in Tab.1) were validated with lateral wall movements attributable to braced excavation in the Bangkok subsoil. A case history of basement excavation (in which the soil testing data and well-documented field monitoring data were recorded perfectly [46,47]) was employed to calibrate the FE procedures implemented in this study, as shown in Fig.3. This figure shows that the employed FE procedures with calibrated soil parameters well captured the distributions of lateral wall movement with respect to depth for all staged excavations. Consequently, the verified FE modeling procedures using the calibrated soil parameters were considered acceptable for analyzing hypothetical cases.

When obtaining artificial data, the major factors influencing the responses of existing piles adjacent to excavations, including H_{\text{e}}, B, t_{\text{w}}, D_{\text{p}}, L_{\text{p}}, X_{\text{p}}, X_{\text{p}}^{\text{cn}}, s_{\text{u}}/\phantom{s_{\text{u}}{\sigma }_{\text{v}}^{\prime }}{\sigma }_{\text{v}}^{\prime }, G_{\text{0}}^{\text{ref}}/\phantom{G_{\text{0}}^{\text{ref}}{\sigma }_{\text{v}}^{\prime }}{\sigma }_{\text{v}}^{\prime }, and T_{\text{s}}, were altered in the hypothetical excavation analyses. The s_{\text{u}}/\phantom{s_{\text{u}}{\sigma }_{\text{v}}^{\prime }}{\sigma }_{\text{v}}^{\prime } and G_{\text{0}}^{\text{ref}}/\phantom{G_{\text{0}}^{\text{ref}}{\sigma }_{\text{v}}^{\prime }}{\sigma }_{\text{v}}^{\prime } parameters for soft clay (s_{\text{u-soft}}/\phantom{s_{\text{u-soft}}{\sigma }_{\text{v}}^{\prime }}{\sigma }_{\text{v}}^{\prime } = 0.27−0.40 and G_{\text{0-soft}}^{\text{ref}}/\phantom{G_{\text{0-soft}}^{\text{ref}}{\sigma }_{\text{v}}^{\prime }}{\sigma }_{\text{v}}^{\prime } = 123−315) and stiff clay (s_{\text{u-stiff}}/\phantom{s_{\text{u-stiff}}{\sigma }_{\text{v}}^{\prime }}{\sigma }_{\text{v}}^{\prime } = 0.45−1.05 and G_{\text{0-stiff}}^{\text{ref}}/\phantom{G_{\text{0-stiff}}^{\text{ref}}{\sigma }_{\text{v}}^{\prime }}{\sigma }_{\text{v}}^{\prime } = 375−975) were considered. For simplicity, the T_{\text{s}} parameter considered in this study is defined as the distance from the soil surface to the base level of the soft clay layer (including the sedimentary crust layer). Tab.2 summarizes these input factors and the ranges involved in the FE analyses of hypothetical excavation cases, used to generate artificial data for M^{max} and d^{max}. Referring to the variations in input factors (as described in Tab.2), a total of 632 hypothetical excavations were conducted (i.e., 136 cases for the free-head pile condition to obtain d^{max}, 136 cases for the fixed-head pile condition to obtain M^{max}, and 360 cases for the corner effect evaluation to obtain the PSR value).

4 Sensitivity analyses

4.1 Influence of pile head condition

In this section, the impact of the pile-head fixity conditions is examined, to assess the pile responses attributable to adjacent excavations. A hypothetic excavation case with H_{\text{e}} = 8 m, B = 60 m, t_{\text{w}} = 1 m, D_{\text{p}} = 1 m, L_{\text{p}} = 18 m, X_{\text{p}} = 3 m, s_{\text{u-soft}}/\phantom{s_{\text{u-soft}}{\sigma }_{\text{v}}^{\prime }}{\sigma }_{\text{v}}^{\prime } = 0.35, s_{\text{u-stiff}}/\phantom{s_{\text{u-stiff}}{\sigma }_{\text{v}}^{\prime }}{\sigma }_{\text{v}}^{\prime } = 0.75, G_{\text{0-soft}}^{\text{ref}}/\phantom{G_{\text{0-soft}}^{\text{ref}}{\sigma }_{\text{v}}^{\prime }}{\sigma }_{\text{v}}^{\prime } = 210, G_{\text{0-stiff}}^{\text{ref}}/\phantom{G_{\text{0-stiff}}^{\text{ref}}{\sigma }_{\text{v}}^{\prime }}{\sigma }_{\text{v}}^{\prime } = 675, and T_{\text{s}} = 14 m (including a 2-m-thick sedimentary crust and 12-m-thick soft clay) was used for this investigation. As mentioned in Subsection 2.4, two critical types of pile-head conditions were considered: (1) free-head and (2) fixed-head.

Fig.4(a) presents the profiles of the bending moment (M_{\text{p}}) with respect to the pile depth. It indicates that the M_{\text{p}} profiles provided by the two-different pile-head conditions are distinct. For the fixed-head pile condition, M_{\text{p}} developed at the head pile, which showed a large negative value because the pile head was prevented from rotating. A positive M^{max} [M^{\text{max}(+)}] of 1770 kN·m was obtained at the pile head and then rapidly reduced with increasing depth. Thus, the moment became negative. At the level of excavation (−8 m), a negative M^{max} [M^{\text{max}(-)}] was observed (−570 kN·m). Subsequently, M_{\text{p}} gradually decreased in magnitude and became positive at a pile depth of approximately −13 m. A second turning point in the M_{\text{p}} curve was observed at the interface between the soft and stiff clay layers, attributable to different soil stiffnesses. Within the stiff clay layer, M_{\text{p}} has a positive value and gradually decays with respect to depth until it reaches the pile tip (where M_{\text{p}} = 0). By inspecting the M_{\text{p}} distribution of the pile under the free-head condition, we see that no M_{\text{p}} developed at the pile head, and a negative moment arose for most of the entire length (from 0 to −13 m). Consequently, an M^{\text{max}(-)} of −484 kN·m was obtained at a depth of approximately −10.4 m. At the bottom level (below −13 m), the M_{\text{p}} distribution for the free-head pile condition was comparable to that of the fixed-head pile condition, in which the M^{\text{max}(+)} (250 kN·m) occurred at the interfacial soil layers. By comparing the magnitude of the maximum moment (both positive and negative values) between these two pile-head conditions, it was found that the excavation-induced M^{\text{max}(+)} and M^{\text{max}(-)} for the existing pile under the fixed-head pile condition exceeded those under the free-head pile one. The considered fixed-head pile condition achieves M^{\text{max}(+)} and M^{\text{max}(-)} values approximately 7 and 1.2 times larger, respectively, than the considered free-head pile condition.

The pile deflections (d_{\text{p}}) and the depths corresponding to different pile-head fixity conditions are presented in Fig.4(b). The distributions of d_{\text{p}} for both the fixed- and free-head pile conditions show deep inward movements; however, a large difference in the d_{\text{p}} value is obtained, particularly in the upper region. Because the pile head can move freely under the free-head pile condition, a larger d_{\text{p}} is obtained at the pile head compared with the fixed-head pile condition. In this case, the d_{\text{p}} increases with respect to depth until it approaches the maximum value (d^{max} = 14.7 mm) at a depth of approximately −8.8 m, close to the final excavation level (−8 m). The d_{\text{p}} then gently decreases with depth until it reaches the pile tip. However, the d_{\text{p}} at the pile head under the fixed-head pile condition is apparently zero; in contrast, the d_{\text{p}} curve resembles that of the free-head pile but with smaller magnitudes. Note that although the pile head can allow movement with the adopted modeling of the fixed-head pile condition in this study, the pile deflection at the upper part is very small owing to the strong stiffness provided by the uppermost crust layer and restraint from the upper bracing system. Consequently, a reduction of ~10% (13.3 mm) is obtained. The locations of d^{max} between these two pile-head conditions differ slightly. Therefore, the free-head pile condition can induce a larger pile deflection, which is a concern in real cases.

According to the previously presented results, the rotation constraint of the pile head can produce significant variations in excavation-induced M^{max} and d^{max} within the existing pile behind the wall. The critical M^{max} arises when the fixed-head pile condition is considered, whereas the free-head pile condition provides a greater d^{max} owing to an unrestrained translation at the pile head. Thus, to obtain a conservative estimate of the pile responses produced by braced excavation, M^{\text{max}(+)} and M^{\text{max}(-)} were used to develop semi-empirical equations by considering hypothetical excavations in the presence of existing piles under fixed-head pile conditions. To generate artificial data that account for conservative d^{max}, an existing pile with a free-head pile condition was considered. These assumptions were adopted for further investigation, in which 136 hypothetical excavation cases were considered to obtain each response of M^{max} (both M^{\text{max}(+)} and M^{\text{max}(-)}) and d^{max}. In these cases, the X_{\text{p}}^{\text{cn}} was sufficiently large to ensure the plane strain condition.

4.2 Influence of input factors

Computational results obtained from FE analyses for 272 artificial excavation cases were collected and used to evaluate the influence of all input factors upon M^{\text{max}(+)}, M^{\text{max}(-)}, and d^{max} (output variables). By applying the Statistical Package for Social Sciences (SPSS) software to the response variables and input factors, the results of the influence contributions were obtained, as presented in Fig.5. To determine the contribution of an individual input factor upon each response variable, the percentage contribution (P_{\text{C}}) was used. The summation of P_{\text{C}} values of all input factors was equal to 100%. Greater P_{\text{C}} values indicate a higher contribution of the input factor to the considered response. The results in Fig.5 show that the input factor D_{\text{p}} has the highest contribution (P_{\text{C}} = 52.4%) to both M^{\text{max}(+)} and M^{\text{max}(-)} but a low one to d^{max} (P_{\text{C}} = 0.5%). The input factor H_{\text{e}} also strongly influences M^{\text{max}(+)} and M^{\text{max}(-)} (P_{\text{C}} = ~18%). The contributions of other input factors (i.e., B, t_{\text{w}}, L_{\text{p}}, X_{\text{p}}, s_{\text{u-soft}}/\phantom{s_{\text{u-soft}}{\sigma }_{\text{v}}^{\prime }}{\sigma }_{\text{v}}^{\prime }, s_{\text{u-stiff}}/\phantom{s_{\text{u-stiff}}{\sigma }_{\text{v}}^{\prime }}{\sigma }_{\text{v}}^{\prime }, G_{\text{0-soft}}^{\text{ref}}/\phantom{G_{\text{0-soft}}^{\text{ref}}{\sigma }_{\text{v}}^{\prime }}{\sigma }_{\text{v}}^{\prime }, G_{\text{0-stiff}}^{\text{ref}}/\phantom{G_{\text{0-stiff}}^{\text{ref}}{\sigma }_{\text{v}}^{\prime }}{\sigma }_{\text{v}}^{\prime }, and T_{\text{s}}) to M^{\text{max}(+)} and M^{\text{max}(-)} responses are characterized as less significant (P_{\text{C}} < 8%).

Several input factors (i.e., B, L_{\text{p}}, s_{\text{u-soft}}/\phantom{s_{\text{u-soft}}{\sigma }_{\text{v}}^{\prime }}{\sigma }_{\text{v}}^{\prime }, s_{\text{u-stiff}}/\phantom{s_{\text{u-stiff}}{\sigma }_{\text{v}}^{\prime }}{\sigma }_{\text{v}}^{\prime }, and G_{\text{0-stiff}}^{\text{ref}}/\phantom{G_{\text{0-stiff}}^{\text{ref}}{\sigma }_{\text{v}}^{\prime }}{\sigma }_{\text{v}}^{\prime }) are also found to contribute less to d^{max}; for these, P_{\text{C}} = 1%–5% (see Fig.5). For d^{max}, a strongly contributing parameter is H_{\text{e}}, for which a P_{\text{C}} of 33.5% is obtained. The input factors t_{\text{w}}, X_{\text{p}}, G_{\text{0-soft}}^{\text{ref}}/\phantom{G_{\text{0-soft}}^{\text{ref}}{\sigma }_{\text{v}}^{\prime }}{\sigma }_{\text{v}}^{\prime }, and T_{\text{s}} also have a strong influence on d^{max}, yielding P_{\text{C}} values of 10%–17%. Note that this investigation aims to examine the influence of each input factor upon the excavation-induced pile responses.

All input factors considered in this study influence the M^{\text{max}(+)}, M^{\text{max}(-)}, and d^{max} variations; thus, all these factors are required in the semi-empirical equations used to precisely predict the pile responses for braced excavations.

5 Improved equations for predicting pile responses

5.1 Overview and evaluation of existing methods

As mentioned above, three previous studies [7,38,31] developed simplified methods to predict the responses of an existing pile to adjacent braced excavations. These studies proposed semi-empirical equations alongside design charts for assessing the excavation-induced M^{max} and d^{max}, which are typically based on the pile responses of the reference case (M^{\text{ref}} and d^{\text{ref}}) and a set of modified factors obtained from the design charts, as follows:

\begin{array}{}\text{(1a)}& M^{max}=M^{\text{ref}}\times k_1\times k_2\times k_3\times \cdots \times k_n,\end{array}

\begin{array}{}\text{(1b)}& d^{max}=d^{\text{ref}}\times k_1^{\prime }\times k_2^{\prime }\times k_3^{\prime }\times \cdots \times k_{\text{n}}^{\prime }.\end{array}

In these equations, M^{\text{ref}} and d^{\text{ref}} refer respectively to M^{max} and d^{max} of the reference excavation case. The k_1,k_2,k_3,...,k_n and k_1^{\prime },k_2^{\prime },k_3^{\prime },...,k_n^{\prime } are modified factors that take into account the significant influencers of M^{max} and d^{max}, respectively. According to the previous study by Poulos and Chen [7], both M^{\text{ref}} and d^{\text{ref}} are proportional to the distance of the existing pile from the retaining wall and can be obtained from design charts. Referring to the work of Liyanapathirana and Nishanthan [38], they proposed the approximate equations M^{\text{ref}}=140H_{\mathrm{e}}{\mathrm{e}}^{-0.05X_{\text{p}}} (kN·m) and d^{\text{ref}}=0.01H_{\mathrm{e}}{\mathrm{e}}^{-0.015X_{\text{p}}} (m) to estimate the pile responses of the reference excavation case. Following Zhang et al. [31], M^{max} and d^{max} were obtained from numerical analysis of historical cases for braced excavations in Singapore and were chosen as M^{\text{ref}} and d^{\text{ref}}.

Parametric studies adopting either a combination of the FE and boundary element methods (i.e., [7]) or the FE method alone (i.e., [31,38]) have been performed to establish design charts for determining modified factors. The existing methods developed in these previous studies consider different sets of influential factors, as presented in Tab.3. Moreover, the determination of modified factors using design charts may result in errors attributable to human customization, which directly affect the estimated results for M^{max} and d^{max}. Furthermore, there is no practical method for estimating d^{max} considering the excavation corner effect.

The proposed methods of Poulos and Chen [7] and Liyanapathirana and Nishanthan [38] (hereafter “PC” and “LN” methods, respectively) do not account for the effects of the fixed-head pile condition; thus, only the simplified equation proposed by Zhang et al. [31] (hereafter the “ZZG” method) can be used to compare the excavation-induced M^{max} obtained in this study. To summarize, from a total of 272 hypothetical cases, 136 cases with fixed-head pile conditions were used to assess the M^{max} prediction performance of the ZZG method, while the remaining 136 data (under free-head pile conditions) were used to assess the d^{max} prediction performance of the PC, LN, and ZZG methods. The predicted pile responses obtained from the existing methods were evaluated via statistical analyses using the coefficient of determination (R²), the root mean squared error (RMSE), mean absolute error (MAE) and bias factors (BFs) (mean value ({\mu }_{\text{BF}}) and standard deviation ({\sigma }_{\text{BF}}) values). The BF can be calculated as the ratio of the observed value (i.e., numerical result) to the estimated value (i.e., that obtained using the equation). The closer the R² value is to 1 and lower the values of both RMSE and MAE (i.e., the closer to 0), the more accurate the prediction. The closer {\mu }_{\text{BF}} is to 1 and the lower the {\sigma }_{\text{BF}} value, the more precise the model.

Fig.6 compares the values of M^{max} and d^{max} obtained from FE analyses, as well as the values predicted using existing methods. Note that both the M^{\text{max}(+)} and M^{\text{max}(-)} values observed from the hypothetical cases considered in this study were adopted separately for comparison. From Fig.6(a), we see that the overall predictions using the ZZG method tended to provide lower values than the FE analysis results, particularly for M^{\text{max}(+)}. Moreover, most of the estimated M^{max} data provided by the ZZG method did not agree with the target values, particularly for M^{\text{max}(+)}. The optimal prediction performance of the ZZG method was obtained from the estimated results of M^{\text{max}(-)}, where RMSE = 292 kN·m, MAE = 257 kN·m, {\mu }_{\text{BF}} = 3.17, and {\sigma }_{\text{BF}} = 2.25. The large error may arise from the adoption of the bending moment for a specific excavation case as the reference value, the M^{\text{ref}} used for predicting M^{max}, and the neglection of the soil parameters. As shown in in Fig.6(b), for the predicted d^{max}, most of the values computed by the ZZG and LN methods were overpredicted (particularly the LN method) and fell more than 30% outside of the target values. An unsatisfactory prediction performance was obtained for these existing methods (RMSE = 12 and 63 mm, MAE = 9.3 and 54 mm, and {\mu }_{\text{BF}} = 0.84 and 0.23 for the ZZG and LN methods, respectively). Again, this error was most likely caused by the neglect of the soil characteristics and the use of a d^{\text{ref}} value obtained from a specific excavation case, as described above. Note that for the LN method, the influences of the small-strain behavior of the soil and the excavation width were neglected. The predicted results of the PC method indicate that this method can estimate the excavation-induced d^{max} of the existing pile better than other methods. Most of the estimated results fell within ±30% of the target values (Fig.6(b)). However, in the statistical results, the predictions of the PC method still show a relatively large RMSE (3.5 mm), MAE (2.9 mm), {\mu }_{\text{BF}} (1.37), and {\sigma }_{\text{BF}} (0.36). These errors may arise from the interpolation of the curves in the design charts used to specify the modified factors. Therefore, improved predictions of excavation-induced pile response are required. The prediction performance of the semi-empirical equations proposed in this study is presented and discussed in the following section.

5.2 Semi-empirical equations for predicting pile responses

The main objective of the current study is to develop semi-empirical equations for estimating pile responses (comprising M^{\text{max}(+)}, M^{\text{max}(-)}, and d^{max}) induced by adjacent braced excavation in clayey ground. There are two motivations for developing the method: (1) the prediction performance must be improved and (2) the errors of human customization when obtaining the modified factors from graphical charts should be reduced. Based on these two requirements, the multiple-variable regression analysis previously used by Kung et al. [19] to predict wall movement and ground surface settlement was adopted in this study. The detailed development of the semi-empirical equations is described below.

Before performing the multiple-variable regression analysis, several considered input variables were transformed to ensure a linear correlation with the pile responses where necessary. The pattern of the transformation function was achieved via numerous trial-and-error analyses to maintain a uniform and simple functional form across all input variables. This suggests that some relationships between input variables and outputs (i.e., pile responses) have a parabolic relationship. Therefore, the following parabolic function is appropriate for transforming the variables [19]:

(2)X=t\left[X\right]=a{x}^2+bx+c,

where x refers to each of the considered input variables and t\left[X\right] is a transformed variable. The three coefficients (a, b, and c) for each transformed variable were obtained through error minimization considering the artificial data obtained via FE analyses. Using the transformed variables, semi-empirical equations for predicting the pile responses were obtained via multiple-variable regression analysis.

To summarize, the semi-empirical equations used for predicting pile responses were obtained from the input variables (i.e., H_{\text{e}}, B, t_{\text{w}}, D_{\text{p}}, L_{\text{p}}, X_{\text{p}}, X_{\text{p}}^{\text{cn}}, s_{\text{u-soft}}/\phantom{s_{\text{u-soft}}{\sigma }_{\text{v}}^{\prime }}{\sigma }_{\text{v}}^{\prime }, s_{\text{u-stiff}}/\phantom{s_{\text{u-stiff}}{\sigma }_{\text{v}}^{\prime }}{\sigma }_{\text{v}}^{\prime }, G_{\text{0-soft}}^{\text{ref}}/\phantom{G_{\text{0-soft}}^{\text{ref}}{\sigma }_{\text{v}}^{\prime }}{\sigma }_{\text{v}}^{\prime }, G_{\text{0-stiff}}^{\text{ref}}/\phantom{G_{\text{0-stiff}}^{\text{ref}}{\sigma }_{\text{v}}^{\prime }}{\sigma }_{\text{v}}^{\prime }, and T_{\text{s}}), as detailed in the previous section. In this study, both the pile responses and input variables were considered in terms of dimensionless values. Accordingly, Models A1, A2, and B, respectively, were established for estimating the normalized values of M^{\text{max}(+)}, M^{\text{max}(-)} and d^{max} (referred to as M_{\text{n}}^{\text{max}(+)}, M_{\text{n}}^{\text{max}(-)}, and d_{\text{n}}^{\text{max}}, respectively). These normalized values are defined as

(3)M_{\text{n}}^{\text{max (}+,-)}=\frac{M^{\text{max (}+,-)}}{p_{\mathrm{a}}{D}_{\text{p}}^{\text{3}}},

(4)d_{\text{n}}^{\text{max}}=\frac{d^{max}}{D_{\text{p}}},

where p_{\mathrm{a}} is atmospheric pressure (101.325 kPa), D_{\text{p}} and p_{\mathrm{a}} were included to provide a non-dimensional value of M^{max} [60].

Most of the input variables considered here are simply normalized by the referenced excavation width (B^{\text{ref}}, which is 20 m in this study) [31], as illustrated in Tab.2; the only exceptions were t_{\text{w}} and D_{\text{p}}. In the regression analysis, t_{\text{w}} is represented in the context of wall system stiffness (following previous studies [17,19,20]) in conjunction with the B^{\text{ref}} parameter, to normalize this input variable in the form of E_{\text{w}}{I}_{\text{w}}/(B^{\text{ref}}{\gamma }_{\text{w}}{h}_{\text{avg}}^4). This dimensionless parameter E_{\text{w}}{I}_{\text{w}} refers to wall stiffness, {\gamma }_{\text{w}} is the unit weight of water (9.81 kN/m³), and h_{\text{avg}} is the average vertical strut spacing. To express the pile diameter input variable in terms of a dimensionless parameter, the normalized pile stiffness (E_{\text{p}}{I}_{\text{p}}) concerning D_{\text{p}} and {\gamma }_{\text{w}}, as represented in the form E_{\text{p}}{I}_{\text{p}}/{\gamma }_{\text{w}}{D}_{\text{p}}^{\text{5}}, was adopted in the regression analysis. Both E_{\text{w}}{I}_{\text{w}}/B^{\text{ref}}{\gamma }_{\text{w}}{h}_{\text{avg}}^4 and E_{\text{p}}{I}_{\text{p}}/{\gamma }_{\text{w}}{D}_{\text{p}}^{\text{5}} were considered using their natural logarithmic values. Soil characteristics input variables (i.e., s_{\text{u-soft}}/{\sigma }_{\text{v}}^{\prime }, G_{\text{0-soft}}^{\text{ref}}/{\sigma }_{\text{v}}^{\prime }, s_{\text{u-stiff}}/{\sigma }_{\text{v}}^{\prime }, and G_{\text{0-stiff}}^{\text{ref}}/{\sigma }_{\text{v}}^{\prime }) were directly utilized in the regression model [20].

In addition, this study proposes a semi-empirical equation to account for the excavation corner effect upon the deflection of the existing pile situated near the corner of the excavation (Model C). The PSR concept developed by Ou et al. [25] was adapted to this situation; in it, the PSR value can be calculated as d_{\text{3D}}^{\text{max}}/d_{\text{2D}}^{\text{max}}, where d_{\text{3D}}^{\text{max}} is the existing pile maximum deflection at a certain distance (X_{\text{p}}^{\text{cn}}) along the longitudinal wall (primary wall) away from the corner, and d_{\text{2D}}^{\text{max}} is the normalized value of the maximum deflection of the existing pile, as measured under plane strain conditions (equivalent to the d^{max}=d_{\text{n}}^{\text{max}}\times D_{\text{p}} obtained by 3D FE analysis, in which the existing pile is located sufficiently far from the corner (Model B)). Thus, to obtain artificial PSR value data for establishing the multiple-variable regression analysis (which will provide the assessment of d_{\text{3D}}^{\text{max}}), an additional 360 FE analyses involving various X_{\text{p}}^{\text{cn}} (Tab.2) were performed.

5.3 Models A1 and A2 for predicting the pile bending moment

In this subsection, a method for establishing semi-empirical equations to predict M_{\text{n}}^{\text{max}(+)} and M_{\text{n}}^{\text{max}(-)} is presented, along with a discussion on the model prediction performances. Note that the collected M_{\text{n}}^{\text{max}(+)} and M_{\text{n}}^{\text{max}(-)} data for the existing pile, as obtained from 136 FE analysis cases, are based on the assumption of fixed-head pile conditions. A total of 11 input variables (i.e., H_{\text{e}}/B^{\text{ref}}, B/B^{\text{ref}}, \ln \left(E_{\text{w}}{I}_{\text{w}}/B^{\text{ref}}{\gamma }_{\text{w}}{h}_{\text{avg}}^{\text{4}}\right), \ln \left(E_{\text{p}}{I}_{\text{p}}/{\gamma }_{\text{w}}{D}_{\text{p}}^{\text{5}}\right), L_{\text{p}}/B^{\text{ref}}, X_{\text{p}}/B^{\text{ref}}, s_{\text{u-soft}}/{\sigma }_{\text{v}}^{\prime }, s_{\text{u-stiff}}/{\sigma }_{\text{v}}^{\prime }, G_{\text{0-soft}}^{\text{ref}}/{\sigma }_{\text{v}}^{\prime }, G_{\text{0-stiff}}^{\text{ref}}/{\sigma }_{\text{v}}^{\prime }, and T_{\text{s}}/B^{\text{ref}}) are employed for the multiple-variable regression analysis, to establish the prediction equations of M_{\text{n}}^{\text{max}(+)} and M_{\text{n}}^{\text{max}(-)}. Using these parameters as the input variables, and both M_{\text{n}}^{\text{max}(+)} and M_{\text{n}}^{\text{max}(-)} (Eq. (3)) as output variables, the a, b, and c coefficients for each transformed variable t\left[X\right] were obtained as listed in Tab.4. It should be noted that, for the output M_{\text{n}}^{\text{max}(+)}, parabolic relationships between the input and output variables were found for H_{\text{e}}/B^{\text{ref}}, \ln \left(E_{\text{w}}{I}_{\text{w}}/B^{\text{ref}}{\gamma }_{\text{w}}{h}_{\text{avg}}^{\text{4}}\right), \ln \left(E_{\text{p}}{I}_{\text{p}}/{\gamma }_{\text{w}}{D}_{\text{p}}^{\text{5}}\right), and X_{\text{p}}/B^{\text{ref}}.

Meanwhile, five input variables, H_{\text{e}}/B^{\text{ref}}, B/B^{\text{ref}}, \ln \left(E_{\text{w}}{I}_{\text{w}}/B^{\text{ref}}{\gamma }_{\text{w}}{h}_{\text{avg}}^{\text{4}}\right), \ln \left(E_{\text{p}}{I}_{\text{p}}/{\gamma }_{\text{w}}{D}_{\text{p}}^{\text{5}}\right), and X_{\text{p}}/B^{\text{ref}} have a parabolic correlation with the output M_{\text{n}}^{\text{max}(-)}. Using least-squares regression analysis, the semi-empirical equations for predicting M_{\text{n}}^{\text{max}(+)} and M_{\text{n}}^{\text{max}(-)} were expressed as

(5)\begin{array}{rl}{M}_{\text{n}}^{\text{max}(+)}=& {\beta }_0+{\beta }_{1}t\left(H_{\text{e}}/B^{\text{ref}}\right)+{\beta }_2\left(B/B^{\text{ref}}\right)\\ & +{\beta }_{3}t\ln \left(E_{\text{w}}{I}_{\text{w}}/B^{\text{ref}}{\gamma }_{\text{w}}{h}_{\text{avg}}^{\text{4}}\right)+{\beta }_{4}t\ln \left(E_{\text{p}}{I}_{\text{p}}/{\gamma }_{\text{w}}{D}_{\text{p}}^{\text{5}}\right)\\ & +{\beta }_5\left(L_{\text{p}}/B^{\text{ref}}\right)+{\beta }_{6}t\left[X_{\text{p}}/B^{\text{ref}}\right]+{\beta }_7\left(s_{\text{u-soft}}/{\sigma }_{\text{v}}^{\prime }\right)\\ & +{\beta }_8\left(s_{\text{u-stiff}}/{\sigma }_{\text{v}}^{\prime }\right)+{\beta }_9\left(G_{\text{0-soft}}^{\text{ref}}/{\sigma }_{\text{v}}^{\prime }\right)\\ & +{\beta }_{10}\left(G_{\text{0-stiff}}^{\text{ref}}/{\sigma }_{\text{v}}^{\prime }\right)+{\beta }_{11}\left(T_{\text{s}}/B^{\text{ref}}\right),\end{array}

(6)\begin{array}{rl}{M}_{\text{n}}^{\text{max}(-)}=& {\alpha }_0+{\alpha }_{1}t\left(H_{\text{e}}/B^{\text{ref}}\right)+{\alpha }_{2}t\left(B/B^{\text{ref}}\right)\\ & +{\alpha }_{3}t\ln \left(E_{\text{w}}{I}_{\text{w}}/B^{\text{ref}}{\gamma }_{\text{w}}{h}_{\text{avg}}^{\text{4}}\right)+{\alpha }_{4}t\ln \left(E_{\text{p}}{I}_{\text{p}}/{\gamma }_{\text{w}}{D}_{\text{p}}^{\text{5}}\right)\\ & +{\alpha }_5\left(L_{\text{p}}/B^{\text{ref}}\right)+{\alpha }_{6}t\left(X_{\text{p}}/B^{\text{ref}}\right)+{\alpha }_7\left(s_{\text{u-soft}}/{\sigma }_{\text{v}}^{\prime }\right)\\ & +{\alpha }_8\left(s_{\text{u-stiff}}/{\sigma }_{\text{v}}^{\prime }\right)+{\alpha }_9\left(G_{\text{0-soft}}^{\text{ref}}/{\sigma }_{\text{v}}^{\prime }\right)\\ & +{\alpha }_{10}\left(G_{\text{0-stiff}}^{\text{ref}}/{\sigma }_{\text{v}}^{\prime }\right)+{\alpha }_{11}\left(T_{\text{s}}/B^{\text{ref}}\right).\end{array}

The \beta and \alpha coefficients for Eqs. (5) and (6), as specified in the least-squares regression analysis, are listed in Tab.4.

Fig.7 presents the predicted values of M_{\text{n}}^{\text{max}(+)} (obtained using Eq. (5)) and M_{\text{n}}^{\text{max}(-)} (obtained using Eq. (6)) with respect to the FE analysis results for 136 hypothetical cases. As shown in the figure, the errors between the predicted moment (both M_{\text{n}}^{\text{max}(+)} and M_{\text{n}}^{\text{max}(-)}) and the analyzed moment primarily fell within the range of ± 10%. The R² values for the predicted M_{\text{n}}^{\text{max}(+)} and M_{\text{n}}^{\text{max}(-)} data were 0.96 and 0.90, respectively. Low values of RMSE = 0.85 and 0.50 and MAE = 0.74 and 0.42 were obtained for M_{\text{n}}^{\text{max}(+)} and M_{\text{n}}^{\text{max}(-)}, respectively. In addition, {\mu }_{\text{BF}} values were very close to 1 (1.013 and 1.054 for the predicted M_{\text{n}}^{\text{max}(+)} and M_{\text{n}}^{\text{max}(-)}, respectively), and {\sigma }_{\text{BF}} values for both M_{\text{n}}^{\text{max}(+)} and M_{\text{n}}^{\text{max}(-)} estimations were very small. Accordingly, consistency was observed between the predicted and numerical results, demonstrating that the semi-empirical equations (Eqs. (5) and (6)) proposed in this study yield a high precision (high R² value of at least 0.9 and satisfactory values of RMSE, MAE, {\mu }_{\text{BF}}, and {\sigma }_{\text{BF}} as compared to the existing method; see Fig.6(a)) for the prediction of the excavation-induced bending moments of the existing pile.

5.4 Model B for predicting pile deflection

The d_{\text{n}}^{\text{max}} obtained from the FE analyses for 136 hypothetical excavation cases under free-head pile conditions was used to derive the semi-empirical equation. The d_{\text{n}}^{\text{max}} values used for multiple-variable regression analysis were obtained at various depths along the pile length (primarily depending on H_{\text{e}}). In the case of small H_{\text{e}} (i.e., 2 and 5 m in this study), d_{\text{n}}^{\text{max}} was located at the pile head, whereas for large H_{\text{e}} (i.e., 8 and 11 m in this study), the location of d_{\text{n}}^{\text{max}} was deeper. The 11 input variables used for the predicted equations of the pile bending moment were also adopted to construct the predicted model of d_{\text{n}}^{\text{max}}. Based on the obtained results, output d_{\text{n}}^{\text{max}} shows a nonlinear relationship with most of the input variables, except for H_{\text{e}}/B^{\text{ref}}, L_{\text{p}}/B^{\text{ref}} and T_{\text{s}}/B^{\text{ref}}. Therefore, the transformation of the input variables (obtained using Eq. (2)) is required to maintain a linear relationship between the input and output variables of the remaining input variables. Tab.4 shows the coefficients a, b, and c, which are used to calculate the transformed variables t\left[X\right] required for establishing the multiple-variable regression analysis. Consequently, the d_{\text{n}}^{\text{max}} of the existing pile, as induced by the adjacent excavation, can be calculated using the \eta coefficients (listed in Tab.4), as follows:

(7)\begin{array}{rl}{d}_{\text{n}}^{\text{max}}=& {\eta }_0+{\eta }_1\left(H_{\text{e}}/B^{\text{ref}}\right)+{\eta }_{2}t\left(B/B^{\text{ref}}\right)\\ & +{\eta }_{3}t\ln \left(E_{\text{w}}{I}_{\text{w}}/B^{\text{ref}}{\gamma }_{\text{w}}{h}_{\text{avg}}^{\text{4}}\right)+{\eta }_{4}t\ln \left(E_{\text{p}}{I}_{\text{p}}/{\gamma }_{\text{w}}{D}_{\text{p}}^{\text{5}}\right)\\ & +{\eta }_5\left(L_{\text{p}}/B^{\text{ref}}\right)+{\eta }_{6}t\left(X_{\text{p}}/B^{\text{ref}}\right)+{\eta }_{7}t\left(s_{\text{u-soft}}/{\sigma }_{\text{v}}^{\prime }\right)\\ & +{\eta }_{8}t\left(s_{\text{u-stiff}}/{\sigma }_{\text{v}}^{\prime }\right)+{\eta }_{9}t\left(G_{\text{0-soft}}^{\text{ref}}/{\sigma }_{\text{v}}^{\prime }\right)\\ & +{\eta }_{10}t\left(G_{\text{0-stiff}}^{\text{ref}}/{\sigma }_{\text{v}}^{\prime }\right)+{\eta }_{11}\left(T_{\text{s}}/B^{\text{ref}}\right).\end{array}

Fig.8 compares the d_{\text{n}}^{\text{max}} computed using Eq. (7) against those obtained from the FE analyses. From this figure, the proposed Eq. (7) is seen to be efficient for predicting d_{\text{n}}^{\text{max}}, with a high R² (0.93) and very low values of both RMSE (0.0011) and MAE (0.0008). Acceptable values of {\mu }_{\text{BF}} (0.989) and {\sigma }_{\text{BF}} (0.088) were also obtained. The errors between the predicted and numerical values were mostly within the range of ±10%, which indicates that the proposed semi-empirical equation for predicting d_{\text{n}}^{\text{max}} has a high accuracy. However, some points fell outside the agreement lines. Further investigation revealed that these points occurred where H_{\text{e}} = 2 m.

5.5 Model C for predicting pile deflection considering corner effect

The wall deflections at locations near the corner and center of the excavation (both primary and complementary walls) can differ significantly owing to the 3D corner effect. By accounting for this effect, the current work was extended to investigate the ratio between the maximum pile deflections in 3D (d_{\text{3D}}^{\text{max}}) and 2D (d_{\text{2D}}^{\text{max}}), the so-called PSR (= d_{\text{3D}}^{\text{max}}/d_{\text{2D}}^{\text{max}}). Previous studies have shown that the main factors in the PSR are the geometry of the excavation [24,52]. Considering the existing pile in this study, the L_{\text{p}}, X_{\text{p}}, and X_{\text{p}}^{\text{cn}} parameters were also included in the PSR evaluation. To generate artificial data for PSR values, 360 additional hypothetical excavation cases were performed using 3D FE analysis. In these cases, X_{\text{p}}^{\text{cn}} was sufficiently small to ensure the corner effect. The influential factors or input variables (including H_{\text{e}} (2–11 m), B/L (0.4–0.8), L_{\text{p}} (15–23 m), X_{\text{p}} (0.6–10 m), and distance of the existing pile along the primary wall away from the excavation corner (X_{\text{p}}^{\text{cn}} = 10–30 m)) were employed in the analyses. To this end, five input variables expressed in terms of dimensionless parameters (i.e., H_{\text{e}}/\phantom{H_{\text{e}}{B}^{\text{ref}}}{B}^{\text{ref}}, B/L, L_{\text{p}}/B^{\text{ref}}, X_{\text{p}}/B^{\text{ref}}, and X_{\text{p}}^{\text{cn}}/B^{\text{ref}}) were included in the PSR prediction equations. In multiple-variable regression analysis, the functional form shown in Eq. (2) was applied to transform the input variables. The a, b, and c coefficients for each input variable were obtained through error minimization. The PSR prediction equation is given in Eq. (8). Consequently, the d_{\text{3D}}^{\text{max}}, which considers the 3D corner effect of excavation, can be calculated using

(8)\begin{array}{rl}PSR=& {\omega }_0+{\omega }_{1}t\left(H_{\text{e}}/B^{\text{ref}}\right)+{\omega }_{2}t\left(B/L\right)+{\omega }_{3}t\left(L_{\text{p}}/B^{\text{ref}}\right)\\ & +{\omega }_{4}t\left(X_{\text{p}}/B^{\text{ref}}\right)+{\omega }_{5}t\left(X_{\text{p}}^{\text{cn}}/B^{\text{ref}}\right),\end{array}

(9)d_{\text{3D}}^{\text{max}}=PSR\times d_{\text{2D}}^{\text{max}},

where d_{\text{2D}}^{\text{max}} is equal to d^{max}=d_{\text{n}}^{\text{max}}\times D_{\text{p}} and d_{\text{n}}^{\text{max}} can be obtained using Eq. (7). The a, b, and c coefficients used for the transformed variables t\left[X\right] and the \omega coefficients used in Eq. (8) are determined from the least-squares regression analysis and are listed in Tab.5.

Fig.9 compares the PSR results computed using Eq. (8) against the numerical results obtained for the 360 hypothetical cases. From the results in this figure, Eqs. (2) and (8) were shown to yield a practical model for forecasting PSR values, with an acceptable R² value (0.97), low values of RMSE (0.04), MAE (0.05), and {\sigma }_{\text{BF}} (0.035). Moreover, the {\mu }_{\text{BF}} was very close to 1 (1.001). Most data points were within ±10% of 100% agreement. In addition, the prediction performance when using Eq. (8) to predict the PSR can be ensured by estimating d_{\text{3D}}^{\text{max}} from d_{\text{2D}}^{\text{max}}, as shown in Fig.10. For the 3D condition, we used Model B for evaluating the d_{\text{2D}}^{\text{max}} (via Eqs. (2) and (7)) and Model C for estimating the PSR (via Eqs. (2) and (6)); these were combined to predict d_{\text{3D}}^{\text{max}} (refer to Eq. (9)). Fig.10 compares the d_{\text{3D}}^{\text{max}} calculated using Eq. (9) against the numerical results obtained from the 360 hypothetical excavation cases. This figure indicates that the proposed equations can provide a reasonable agreement when predicting d_{\text{3D}}^{\text{max}}, by considering the 3D corner effect. Most of the predicted results fell within a ± 15% error, with a high R² value (0.92) and very low values in both RMSE (1.3 mm) and MAE (1.56 mm). The satisfaction of Eq. (9) can be ensured for {\mu }_{\mathrm{B}\mathrm{F}} = 0.992 and {\sigma }_{\mathrm{B}\mathrm{F}} = 0.106. Thus, by substituting Eqs. (7) and (8) into Eq. (9), d_{\text{3D}}^{\text{max}} can be estimated; this represents a convenient solution for engineers wishing to evaluate the maximum deflection of existing piles embedded near the corner of an excavation, provided the soil characteristics and the stiffness and geometry of both the pile and excavation are known. Note that, similar to Fig.8, certain predicted points fell outside the agreement lines. These points occurred for the same cases as those in Fig.8, indicating that the discrepancies arose from the d_{\text{n}}^{\text{max}} prediction, not the PSR.

6 Assessment of the proposed equations using case history

To assess the effectiveness of the proposed semi-empirical equations for predicting pile responses, the excavation-induced pile deflection estimations obtained in this study were compared with a case history documented by Goh et al. [8]. This case history provided well-monitored data for an actual full-scale field test that was conducted to assess the behavior of an existing pile subject to adjacent excavation activities. Braced excavation with L = 127.5 m, B = 20 m, and H_{\text{e}} = 16 m was undertaken for the basement of a cut-and-cover tunnel, as part of a mass rapid transit line in Singapore. This excavation was supported by an 0.8-m-thick and 31-m-deep reinforced diaphragm wall with six strut levels (h_{\text{avg}} = 2.5 m). The elastic modulus (E_{\text{w}}) of the diaphragm wall was 2.1\times {10}^7 kPa [31]. The existing pile (D_{\text{p}} of 1 m and L_{\text{p}} of 46 m) under the free-pile head condition was located ~3 m behind the wall. The subsoil profile at the test site consisted of 8 m of sand fill (average standard penetration test (SPT-N) value = 5.5), 10 m of soft marine clay (s_{\text{u}} = 10 kPa), 12 m of Old Alluvium I (average SPT-N value = 20), and 30 m of Old Alluvium II (SPT-N value > 100). The water table was ~4 m beneath the soil surface. More detailed information regarding this site can be found in the literature [8,31]. The pile deflection profile was observed using an inclinometer. The maximum pile deflection, d^{max} observed from this test site was employed to evaluate the accuracy of the proposed prediction equation (Eq. (7)). In this past study, the d^{max} at a H_{\text{e}} of 6.5 m (within the considered H_{\text{e}} range of this study) was 15 mm.

To estimate d^{max}, the 11 parameters included in Eq. (7) must first be determined; these parameters are listed in Tab.6. s_{\text{u-soft}}/{\sigma }_{\text{v}}^{\prime } and s_{\text{u-stiff}}/{\sigma }_{\text{v}}^{\prime } in the case history were determined from the empirical equation proposed by Wroth and Houlsby [61], using the effective friction angle ({\varphi }^{\prime }): s_{\text{u}}/{\sigma }_{\text{v}}^{\prime }=0.5743\left(3\sin {\varphi }^{\prime }/(3-\sin {\varphi }^{\prime })\right). According to Zhang et al. [31], {\varphi }^{\prime } values of 25° and 40° were considered for soft and stiff clays, respectively; consequently, s_{\text{u-soft}}/{\sigma }_{\text{v}}^{\prime } = 0.2825 and s_{\text{u-stiff}}/{\sigma }_{\text{v}}^{\prime } = 0.4698 were obtained for the case history. The relative shear stiffness ratio (G_{\text{0}}^{\text{ref}}/{\sigma }_{\text{v}}^{\prime }) was determined by multiplying the G_{\text{0}}^{\text{ref}}/s_{\text{u}} ratio by s_{\text{u}}/{\sigma }_{\text{v}}^{\prime }, as described above. G_{\text{0}}^{\text{ref}}/s_{\text{u}} was formulated from the general equation for the stress dependency of the shear modulus G_0 [62], together with the G_0/s_{\text{u}} ratio (i.e., the I_{\text{R}} parameter proposed by Anderson and Woods [63]), in accordance with the assumptions of m=1, c^{\prime }=0 (for clays) and {\sigma }_{\text{3}}^{\prime }=K_0{\sigma }_{\text{1}}^{\prime } [20]. Thus, G_{\text{0}}^{\text{ref}}/{\sigma }_{\text{v}}^{\prime } can be defined as G_{\text{0}}^{\text{ref}}/{\sigma }_{\text{v}}^{\prime } = \left(I_{\text{R}}{p}^{\text{ref}}/K_0{\sigma }_{\text{1}}^{\prime }\right)\times \left(s_{\text{u}}/{\sigma }_{\text{v}}^{\prime }\right). To this end, G_{\text{0-soft}}^{\text{ref}}/{\sigma }_{\text{v}}^{\prime } = 268 and G_{\text{0-stiff}}^{\text{ref}}/{\sigma }_{\text{v}}^{\prime } = 431 were obtained for the case history, by assuming p^{\text{ref}} = 100 kPa, K_0=1-\sin {\varphi }^{\prime }, and {\sigma }_{\text{1}}^{\prime }={\sigma }_{\text{v}}^{\prime } [20]. In this study, I_{\text{R}} = 700 was considered according to the braced excavation analysis in the Singapore subsoil foundation, as performed by Xuan [64].

A summary of the estimations for d^{max} at H_{\text{e}} = 6.5 m, as obtained from the proposed semi-empirical equation and existing methods (PC, LN, and ZZG methods), is presented in Tab.7. The d^{max} estimated according to the equations proposed in this study was 14.84 mm, in which a percentage error of 1.05% in the estimations was obtained. The estimated results obtained using the PC method were lower than the observed results and provided a relatively large percentage error (36.7%). The results show that the ZZG method can be used for evaluating the d^{max} attributable to nearby excavation, though it is found to underestimate d^{max} compared with the observed results (percentage error = 15.3%). For the LN method, an overestimation with a percentage error exceeding 330% was achieved. This overestimation was attributable to the fact that the parameter d^{\text{ref}} was substantially overestimated. By comparing these predicted results with the observed ones, we find that the predicted equations proposed in this study and developed by Zhang et al. [31] are capable of assessing excavation-induced pile deflection. These two methods can provide a relatively small percentage error in the predicted values as compared against the other methods. Interestingly, our proposed approach can be considered as an alternative method that minimizes the human customization errors when determining the modified factors from graphical charts (as often required for predicting pile responses). Therefore, we recommend that the proposed method and equations developed by this study be considered an efficient solution allowing engineers to capture the behavior of existing adjacent piles during excavation activities.

7 Conclusions

In this study, a variety of 3D FE analyses considering the small-strain behaviors of soil were conducted to obtain artificial data for the induced maximum pile bending moments and deflections attributable to adjacent braced excavations in soft deposits. Numerous factors affecting the maximum pile bending moments and deflections of the existing piles were investigated. Statistical analyses were performed to assess the influence of each factor. A set of simplified semi-empirical equations for predicting the maximum pile bending moments (both positive and negative moments) and pile deflections were derived via multiple-linear regression analysis. The required parameters in the proposed equations were the excavation characteristics (i.e., depth and width of excavation), system stiffness of the retaining wall, pile characteristics (i.e., pile stiffness, pile length, pile distance from the excavation face), and soil characteristics (i.e., undrained soil shear strength, soil stiffness at small strain levels, and soft clay thickness). In addition, we developed a simplified semi-empirical equation to predict the PSR value, to account for the 3D corner effect upon the deflection of existing piles near the braced excavation. The accuracy in the prediction results of the developed model was assessed according to different statistical results, including R², RMSE, MAE, and BF. According to the results, the following conclusions can be drawn.

1) In the conservative assessment of pile responses attributable to adjacent braced excavations, the conditions of fixed-head and free-head piles should be considered for estimating the pile bending moment and pile deflection, respectively. This result should help design engineers to evaluate the risks caused in existing piles by adjacent excavations.

2) Numerical and statistical findings show that D_{\text{p}} is the most influential parameter on M^{\text{max}(+)} and M^{\text{max}(-)} for existing piles under adjacent excavation. The H_{\text{e}} also significantly affects M^{\text{max}(+)} and M^{\text{max}(-)}. For the excavation-induced pile deflection, the effect of five factors from all 11 considered factors (i.e., H_{\text{e}}, X_{\text{p}}, G_{\text{0-soft}}^{\text{ref}}/{\sigma }_{\text{v}}^{\prime }, t_{\text{w}}, and T_{\text{s}}) significantly influence d^{max}.

3) The established semi-empirical equation can robustly capture complicated nonlinear relationships between the input factors and output responses. The prediction performance shows that the established semi-empirical equations are sufficiently reliable for predicting the maximum bending moment and deflection of existing piles. The influence of the excavation corner effect was also reflected in the prediction. These proposed equations are useful and convenient for engineers seeking to preliminarily estimate the existing pile responses to nearby excavations.

It should be mentioned that the developed equations are based on hypothetical cases of excavation-induced responses in a single passive pile. The ability of the developed equations is demonstrated through numerical analyses but is valid only for the parameter ranges given, as presented in Tab.2, particularly for the parameters that strongly contribute to pile responses, according to Fig.5. Therefore, several excavation-pile interaction schemes (e.g., for the case of an existing pile group near excavations) and a broader range of parameters to establish prediction equations are required to confirm and enhance the reliability of the findings presented in this study. However, the developed equations can readily be regarded as an “approximate method” for predicting existing single pile responses produced by adjacent excavations in the preliminary design stages of a project, and they can be systematically extended to cover other analysis problems. Moreover, although a specific soil profile was used in this study, by varying the soil parameters for every layer, we postulate that the developed equations are applicable to other similar soil profiles, as demonstrated in Section 6. It should be noted that by using the proposed equations for future studies under a small excavation depth (i.e., 2 m for this study), the relatively large error (exceeding ±20%) should be carefully considered.

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