A simplified method for investigating the bending behavior of piles supporting embankments on soft ground

Abstract

In recent years, concrete and reinforced concrete piles have been widely used to stabilize soft ground under embankments. Previous research has shown that bending failure, particularly during rapid filling on soft ground, is the critical failure mode for pile-supported embankments. Here, we propose an efficient two-stage method that combines a test-verified soil deformation mechanism and Poulos’ solution for pile–soil interaction to investigate the bending behavior of piles supporting embankments on soft ground. The results reveal that there are three possible bending failure scenarios for such piles: at the interface between the soft and firm ground layers, at mid-depths of the fan zone, and at the boundary of the soil deformation mechanism. The location of the bending failure depends on the position and relative stiffness of the given pile. Furthermore, the effect of embedding a pile into a firm ground layer on the bending behavior was investigated. When the embedded length of a pile exceeded a critical value, the bending moment at the interface between the soft and firm ground layers reached a limiting value. In addition, floating piles that are not embedded exhibit an overturning pattern of movement in the soft ground layer, and a potential failure is located in the upper part of these piles.

Graphical abstract

Keywords

bending behavior / pile / embankment / soil−structure interaction / failure mode

1 Introduction

Stability is one of the most important factors affecting rapid embankment construction on soft ground. Granular or deep-mixed columns are commonly used to enhance stability. Recently, in China, there has been an increasing use of concrete or reinforced concrete piles to fulfil the strict requirements of post-construction settlement associated with the development of high-speed railways [1,2]. Although concrete or reinforced concrete piles have higher strength and stiffness than granular or deep-mixed columns, many case histories have shown that deep-seated failure of soft ground can still occur and lead to pile failure [1,3]. Concrete piles were observed with bending failure, which caused the failure of the embankment. The failure modes of the piles at different positions were different. Thus, it is necessary to predict the potential failure locations of piles in engineering.

In most cases where failure occurred, conventional methods had been adopted, which assumed that simultaneous shear failure represented the critical failure mode of the piles. This assumption is valid for granular columns but may not apply to concrete piles; bending failure appears to be the critical failure mode for these piles [3,4]. Even for deep-mixed columns that have lower strength than piles, Sagaseta [5], Han et al. [6], and Navin and Filz [7] concluded that the safety factor cannot be reliably determined by adopting a shear failure approach.

Although the bending behavior of piles subjected to soil movements has been widely investigated in the case of excavation [8–10] and slope [11], few simplified and efficient approaches exist to capture the bending behavior of piles supporting embankments on soft ground. The British Standard BS 8006 [12] considered the deep-seated failure of a piled soft ground under embankments. However, only vertical penetration was considered instead of bending failure. A shear strength reduction method combined with numerical methods, such as the finite element method, can be adopted [13–16]. The DEM is also an effective method [17] but is usually time-consuming.

As discussed above, the bending behavior of piles supporting embankments on soft ground requires further investigation. This study proposes a two-stage method based on a soft ground deformation mechanism that considers soil‒pile interactions. Single- and multiple-row of piles with various pile positions and relative stiffnesses were studied to determine their deformation and bending characteristics. Subsequently, the effects of the embankment maximum settlement, pile embedment length, and ratio of the moduli of the soft and firm ground layers were studied.

2 Materials and methods

A two-stage method is proposed in this study. The soil displacement during deep-seated failure of soft ground was calculated using a deformation mechanism validated by centrifuge tests [18]. Subsequently, the response of the piles under the calculated displacement field of the soil was obtained using Poulos’ solution of the pile–soil interaction.

2.1 Deformation field of deep-seated failure

As reported in many case histories [19,20], centrifuge tests [18,21], and numerical analyses [22,23], the displacement of soft ground associated with deep-seated failure mainly occurs in a curved zone in the soft layer. Based on these observations, a simplified deformation mechanism for deep-seated failure is proposed, as shown in Fig.1. The soil was assumed to deform compatibly and continuously in a kinematically admissible mechanism, with the displacement mainly occurring in a curved zone (ABCDE) in the soft layer. The ABCDE zone was divided into three parts: an active zone (AEG), fan zone (CDEGF), and passive zone (BCF). Outside these zones, the soil was assumed to be rigid owing to its small displacement. This assumption has been widely adopted in previous research [24–27]. Both the AEG and BCF zones comprise isosceles right triangles. The fan zone is bounded by an arc (CDE) with a center (O) immediately above the embankment toe at a height of h_{\text{o}} to the ground surface. Many test and numerical simulation results have shown that h_{\text{o}} can be taken as one-third of the soft layer thickness h_{\text{s}} [28]. The CDE arc is tangential to the interface between the soft and firm soil layers. The BC and AE were smoothly connected to the CDE arc.

As illustrated in Fig.2, in the Cartesian coordinate system xOy, every point in the fan zone has a direction vector, {\overline{D}}_{\text{f}}, the length of which is the distance between O and the point (i.e., \sqrt{x^2+y^2}). At each point, the displacement of soil perpendicular to the vector {\overline{D}}_{\text{f}} follows a cosine function with a maximum magnitude, w_{\text{max}}, and a wavelength, \lambda. Similarly, the displacements of the active and passive zones are determined by the active zone vector, {\overline{D}}_{\text{a}}, and the passive zone vector, {\overline{D}}_{\text{p}}, respectively, together with w_{\text{max}} and \lambda. In the active zone, the length of {\overline{D}}_{\text{a}} is the distance between the point and the line y=x. In the passive zone, the length of {\overline{D}}_{\text{p}} is the distance between the point and the line y=-x. The horizontal (u) and vertical (v) components of the displacement field are shown in Tab.1. It means that with the equation in Tab.1, once the maximum soil displacement (w_{\text{max}}) is measured, the horizontal displacements and vertical displacements of the soil in the deformation zones can be obtained.

2.2 Responses of passive piles to soil movements

The displacement of the soil causes a horizontal deflection of the pile in the soft ground. The piles in this case were passive piles, the response of which has been successfully predicted using Poulos’ solution [29–31]. Therefore, the bending behavior of passive piles induced by the proposed soil mechanism was investigated using a solution for soil−pile interaction, as proposed by Poulos and Davis [29].

Fig.3 illustrates a slice of soft ground, with a thickness of one pile spacing along the longitudinal direction of the embankment, which was taken as an element of calculation. The displacement of the soil immediately adjacent to the pile was considered to be the same as the pile displacement in \left\{u_{\text{p}}\right\}. The displacement of the soil immediately adjacent to the pile can also be regarded as the difference between the ground displacement, \left\{u_{\text{s}}\right\}, and opposing displacement, \left\{u_{\text{sp}}\right\}, caused by the resistance of the pile, \left\{p\right\}. The relationship above is expressed as follows:

(1)\left\{u_{\text{p}}\right\}=\left\{u_{\text{s}}\right\}-\left\{u_{\text{sp}}\right\},

where \left\{u_{\text{sp}}\right\} can be solved by using Mindlin’s solution. Furthermore, Douglas and Davis [30] integrated Mindlin’s solution to facilitate the calculation and obtain a more accurate solution for the pile–soil interaction. The relationship between \left\{u_{\text{sp}}\right\} and \left\{p\right\} is given by

(2)\left\{u_{\text{sp}}\right\}=\frac{d_{\text{p}}}{E_{\text{soil}}}\left[I\right]\left\{p\right\},

where d_{\text{p}} is the diameter of the pile, E_{\text{soil}} is the equivalent Young’s modulus of the soil, and \left[I\right] is the interaction matrix that reflects the displacement response of the soil to the pressure exerted on the ground.

The pile displacement, \left\{u_{\text{p}}\right\}, and resistance of the pile, \left\{p\right\}, should satisfy the following bending equation in discrete form:

(3)\left[S\right]\left\{u_{\text{p}}\right\}+\frac{d_{\text{p}}{L}_{\text{p}}^4}{E_{\text{p}}{I}_{\text{p}}{n}^4}\left\{p\right\}=0,

where E_{\text{p}} is the Young’s modulus of the pile, I_{\text{p}} is the moment of inertia of the pile, L_{\text{p}} is the pile length, n is the number of pile elements, and [S] is the matrix of finite-difference coefficients for the horizontal pile‒soil interaction. Additional details can be found in Poulos and Davis [29].

Combining Eqs. (1)‒(3), the pile displacement, \left\{u_{\text{p}}\right\}, can be solved:

(4)\left\{u_{\text{p}}\right\}={\left[[S]+\frac{{\left[I\right]}^{-1}}{K_{\text{r}}{n}^4}\right]}^{-1}\frac{{\left[I\right]}^{-1}}{K_{\text{r}}{n}^4}\left\{u_{\text{s}}\right\},

where K_{\text{r}}=E_{\text{p}}{I}_{\text{p}}/E_{\text{soil}}{L}_{\text{p}}^4 denotes the relative stiffness of the pile. It can be observed that the displacement of a pile is closely related to its relative stiffness.

For a given soil displacement field [22], pile displacement is solved using Eq. (4), and the bending moment of a pile can easily be obtained based on its displacement. Notably, when the limiting lateral soil pressure is reached, the equivalent Young’s modulus of the soil is zero, and Eq. (4) becomes invalid. However, in this case, the limiting lateral soil pressure {p} = c_u and the pile response can be directly solved by Eq. (3).

2.3 Material parameters for the soil and pile

Owing to the rapid construction of embankments, the soil of the soft ground can be considered to be in an undrained condition [22,23]. Extensive research has been conducted on the strength properties of undrained soils [32–35]. Based on data from typical soft soils presented in these previous studies, we proposed a trilinear model to describe the shear strain‒stress curve of undrained soft soil. As the curve in Fig.4 shows, this model has three stages of shearing, each of which has a constant undrained shearing modulus, G_{\text{u}}, as follows:

(5)

where {\gamma }_{\text{mob}} is the mobilized shear strain of the soil corresponding to the mobilized shear stress, c_{\text{mob}}. One advantage of the trilinear model is that the shear modulus of each stage can be easily transferred to an equivalent Young’s modulus of the soil; hence, the pile displacement can be calculated using Eq. (4).

The space-averaged mobilized shear strain [26] and {\gamma }_{\text{mob}} of soft soil are calculated as follows:

(6){\gamma }_{\text{mob}}=\frac{\int \sqrt{{\left({\displaystyle \frac{\text{d}u}{\text{d}y}}+{\displaystyle \frac{\text{d}v}{\text{d}x}}\right)}^2-4{\displaystyle \frac{\text{d}u\text{d}v}{\text{d}x\text{d}y}}}\text{d}V}{\int \text{d}V}.

In this study, the maximum soil displacement, w_{\text{max}}, was set as 0.05 m. The {\gamma }_{\text{mob}} was 0.5%, which was calculated using Eq. (6). As the soil strain was mobilized from 0 to 0.5%, the undrained shear modulus G_{\text{u}} = 100c_{\text{u}}, which is the same as the value used in Ladd and Lambe [36]. A typical undrained shear strength, c_{\text{u}} = 10 kPa, for the soft soil was taken, and thus, the undrained shear modulus, G_{\text{u}} = 100c_{\text{u}} = 1 MPa. Owing to the undrained condition, Poisson’s ratio was 0.5; hence, the equivalent Young’s modulus of the soft soil, E_{\text{s}}, was 3 MPa. These parameters of the undrained shear strength and modulus are very close to those of typical soft soils [1,31].

For the pile, we focused on a prestressed high-strength concrete (PHC) pipe-pile because this type of pile has been widely applied in practice. The outer diameter of the PHC pile was 0.8 m, and the wall thickness was 0.11 m [18]. The length of pile buried in the soft layer was 15 m, while the length of pile embedded in the firm layer was 5 m. The Young’s modulus of the PHC pile, E_{\text{p}}, was 3.8 × 10⁴ MPa. The ultimate bending moment, M_{\text{u}}, was 589 kN·m. The PHC pipe-pile had a pile relative stiffness, K_{\text{r}}, of 3.64 × 10⁻³.

2.4 Calculation procedure

As shown in Fig.5, the calculation procedure for the proposed method involves the following four steps.

(1) Input the maximum soil displacement w_{\text{max}} and calculate the displacement field.

(2) Obtain the horizontal displacement of the soil and determine its modulus.

(3) Calculate the displacement of the piles using the equations of displacement compatibility, bending equilibrium, and soil−pile interaction.

(4) Calculate the distribution of the pile bending moment at the base of the pile displacement.

2.5 Validation of the proposed method

To verify the proposed method, the displacement field in the method was compared with that of centrifuge tests by Zheng et al. [18], as shown in Fig.6. Compared to the observed results, the proposed displacement field can successfully capture the characteristics of ground movements under a given embankment load. Furthermore, Fig.7 shows the bending moments and horizontal displacements of the pile obtained by the centrifuge test and proposed method on a prototype scale. The measured and calculated pile responses are consistent.

3 Parameter study and discussion

To investigate the influence of K_{\text{r}}, a series of K_{\text{r}} values were used in parametric studies, considering the common type (pipe and solid), diameter (0.3‒0.8 m), material (concrete, steel, and reinforced concrete), and length (10‒20 m) of the piles in practice. The K_{\text{r}} values of 1 × 10⁻¹, 1 × 10⁻², 1 × 10⁻³, and 1 × 10⁻⁴ were selected, as suggested by Poulos and Davis [29].

3.1 Single-row piles

A slice of soft ground with a thickness of one pile spacing in the longitudinal direction can be regarded as the basic element of the pile‒soil interaction. Equation (4) shows that the horizontal deformation of a pile is not only affected by the displacement of soil but is also related to the relative stiffness of the pile (K_{\text{r}}). As discussed above, different values of K_{\text{r}} were taken (1 × 10⁻¹, 1 × 10⁻², 1 × 10⁻³, and 1 × 10⁻⁴) to represent typical values of the pile relative stiffness in soft ground.

Piles that improve soft ground are usually embedded in firm layers. Thus, in this section, piles were assumed to be ideally fixed. In other words, the node of the pile at the interface between soft and firm soil was fixed for all degrees of freedom. This means that the part of the pile embedded in the firm layer had no effect on the results (the effect of embedment is discussed in Subsection 3.3). In this case, the length of the pile, L_{\text{p}}, was equal to the thickness of the soft layer, h_{\text{s}} = 15 m. The modulus of the soil, E_{\text{soil}}, was a constant and equal to the modulus of the soft layer, E_{\text{s}}. In addition, the height of the fan zone center was h_{\text{o}} = h_{\text{s}}/3 = 5 m, and the wavelength of the soil mechanism was \lambda = h_{\text{o}} + h_{\text{s}} = 20 m.

Fig.8 shows the final displacement of the pile and soil beneath the embankment toe (i.e., x/x_{\text{max}} = 0) when using different K_{\text{r}} values. Notably, when the pile stiffness is relatively high (K_{\text{r}} > 1 × 10⁻²), the pile can substantially prevent the movement of soft ground. There is an apparent difference between the soil displacement and pile displacement. In addition, the distribution of the pile displacement approximates an inclined line, indicating that the maximum displacement is at the pile top. This indicates that a pile with a high stiffness behaves like a cantilever beam.

When the pile stiffness is relatively low (K_{\text{r}} ≤ 1 × 10⁻⁴), the resistance of the pile to the horizontal movement of the soil can be neglected, and the displacement of the pile is very close to that of the soil. Moreover, the shape of the deformed pile is similar to a cosine curve, which has a peak toward its center. This implies that a pile with low stiffness behaves as a simply supported beam under the soil deformation mechanism.

Fig.9 presents the variation in the M_{\text{max}} of piles with different positions (x/x_{\text{max}}) and relative stiffnesses (K_{\text{r}}). The maximum bending moment of the pile was normalized using E_{\text{s}}{L}_{\text{p}}^3. With a larger K_{\text{r}} value, the normalized maximum bending moment of the pile increased. The normalized maximum bending moment was equal to 0.014 for a pile with high stiffness (K_{\text{r}} = 1 × 10⁻¹), which was approximately 16 times that of a pile with low stiffness (K_{\text{r}} = 1 × 10⁻³). For a pile with a very low stiffness (K_{\text{r}} = 1 × 10⁻⁴), the normalized maximum bending moment was nearly zero.

Although the maximum bending moment may occur at different locations for different piles, the M_{\text{max}} decreases as the x/x_{\text{max}} increases. As shown in Fig.9, three regions within the curves relate to the interaction between the pile and different zones of the soil mechanism: (1) when the pile is near the centerline of the fan zone (x/x_{\text{max}} < 0.2), M_{\text{max}} decreases slowly because the soil in this range has a very large horizontal displacement, and hence, has a marked bending action on the pile; (2) when the pile is between the center range and the boundary of the fan zone (0.2 < x/x_{\text{max}} < 0.6), M_{\text{max}} sharply decreases because the horizontal component of the soil displacement decreases; and (3) when the pile is outside the fan zone and mainly in the active zone (x/x_{\text{max}} > 0.6), M_{\text{max}} approaches zero at a low rate because the direction of the soil displacement does not change, and the length of the pile in the active zone decreases gradually.

Fig.10 shows the locations of the maximum bending moment, M_{\text{max}}, along a single-row pile at different positions (x/x_{\text{max}}). The maximum bending moment indicates the potential failure locations within the piles. There are three types of failure locations that vary with the stiffness and position of the pile.

Type I: M_{\text{max}} occurs at the interface between soft and firm layers. This type of failure is a bending failure of the pile owing to the strong embedment of the pile toe in the firm layer. When the pile stiffness is high (K_{\text{r}} = 1 × 10⁻¹), this type of failure is significant.

Type II: M_{\text{max}} occurs in the fan zone of the soil-deformation mechanism. When the relative stiffness of the pile is low (K_{\text{r}} = 1 × 10⁻⁴), the pile may suffer from this type of failure near the center of the fan zone, where the largest horizontal displacement of the soil exists.

Type III: M_{\text{max}} occurs near the boundary of the soil-deformation mechanism. When K_{\text{r}} is lower than 1 × 10⁻¹, the soil outside the soil deformation mechanism has sufficient stiffness to restrain the pile. Therefore, a pile outside the soil deformation mechanism can be regarded as being embedded in the soil. This failure type is similar to a type I failure, except that the location of the failure is near the boundary of the mechanism.

It should be noted that the bending moment of the type II failure, M_{\text{max}}, is in the opposite direction to those of types I and III. As mentioned above, in the central position of the fan zone, the pile bent as a simply supported beam, which had a bending moment opposite to that of a cantilever beam.

All three types of failures were found in the centrifuge tests [37]. The failure mode is a combination of types I and II. This means that these three types of failures are not independent, usually occurring one after the other. The combination was usually type I–II or type II–III. This method can effectively predict the failure modes of the piles.

3.2 Multiple-row piles

The parameters of multiple-row piles were the same as those of a single-row piles. The distribution of the bending moment and the location of the maximum bending moment for 6-row and 10-row of piles were studied.

For multiple-row piles, the method was essentially the same as that for single-row piles. However, some of the elements in \left[I\right] can be simplified using Mindlin’s solution rather than an integrated solution [29,30]. For example, considering the element I_{ij}, where i is an element of Pile 1 and j is another element of Pile 2, I_{ij} represents the displacement of soil near i caused by the force provided by element j. Because the elements i and j were from different piles, Mindlin’s solution was used to obtain I_{ij}.

Fig.11 shows the normalized bending moment distribution of piles in a single-row, 6-row, and 10-row when w_{\text{max}} = 0.05 m and x/x_{\text{max}} = 0. The bending moment was normalized by the respective M_{\text{max}} values. It can be seen that the bending moment distributions of piles at a comparable location (x/x_{\text{max}}) are nearly independent of both the number of rows and the relative stiffness of the pile. For the piles in 6-row, the maximum difference is only 5.1% of the bending moment of the single-row piles; for the piles in 10-row, the difference is 10.4%. This implies that any interaction between the piles in different rows is minor. This finding is further supported by the interaction matrix \left[I\right]. The interaction coefficient, I_{ij}, between pile elements i and j in the same pile is approximately ten times larger than that between elements in different piles.

Furthermore, the bending moment of the multiple-row piles can be calculated under the condition of single-row piles, neglecting the interaction of the piles. This simplified method reduces the sizes of \left[S\right] and \left[I\right] in Eq. (4). Hence, the calculation can be accelerated substantially while retaining a maximum error of approximately 10%.

Fig.12 shows the locations of M_{\text{max}} in piles arranged in a single-row, 6-row, and 10-row. As shown in Fig.11, changing K_{\text{r}} does not result in differences in the pile bending moment between single- and multiple-row; hence, Fig.12 only presents the case of K_{\text{r}} = 1 × 10⁻² to illustrate the effect of the number of rows on the M_{\text{max}} location. It can be observed that the maximum bending moment locations of the multiple-row piles closely correspond to those of the single-row piles. This indicates that the influence of different pile rows on each other is minor. For other values of pile relative stiffness (K_{\text{r}}), the M_{\text{max}} locations of the multiple-row piles are distributed in nearly the same manner as those of the single-row piles, as shown in Fig.10. Three types of failure modes can also be obtained in both the centrifuge tests and numerical results [37,38], and the failure modes are almost the same as those of the single-row pile.

3.3 Effect of embedment

Notably, in the previous section, the piles were simplified to be fixed at the interface between the soft and firm layers. However, in practice, even though the pile toe is embedded in the firm soil layer, there is still translation and rotation at the interface between the soft and firm layers; thus, it is difficult to realize the ideal fixed condition. Therefore, the bending moment at the interface was lower than that under ideal fixed conditions.

To consider the effect of embedment, we made three modifications to Eq. (4). First, L_{\text{p}} was considered as the total length of the pile rather than the length of the soft layer. Second, the modulus of the soil, E_{\text{soil}}, was spatially varied; E_{\text{soil}} was set as the modulus of the soft layer, E_{\text{s}}, for pile elements in the soft layer, while it was set as the modulus of the firm layer, E_{\text{f}}, for pile elements in the firm layer. Third, the boundary of the pile toe was assumed to be a hinge that cannot translate but can rotate freely.

The bending moment ratio, R_{\text{m}}, was introduced and is defined as the ratio of the pile bending moment at the interface to that at the ideally fixed pile node. Fig.13 shows the effect of the embedded ratio, R_{\text{e}}, and modulus ratio, E_{\text{f}}/E_{\text{s}}, on R_{\text{m}} for various values of K_{\text{r}}; R_{\text{e}} is the ratio of the pile length in the firm layer to the total pile length, and E_{\text{f}}/E_{\text{s}} is the ratio of the moduli of the firm and soft layers. The pile position was taken as the centerline of the fan zone (x/x_{\text{max}} = 0), where the maximum bending moment occurred at the interface of the soft and firm layers (Fig.10) for all values of K_{\text{r}}.

Fig.13 shows that the R_{\text{e}}‒R_{\text{m}} curves can be characterized by two parameters, a critical R_{\text{m}} and a limiting R_{\text{e}}. When R_{\text{e}} is smaller than the critical value, R_{\text{m}} increases with increasing R_{\text{e}}. Provided that R_{\text{e}} exceeds the critical value, R_{\text{m}} reaches a limiting value. Both the critical R_{\text{e}} and the limiting R_{\text{m}} are affected by E_{\text{f}}/E_{\text{s}} and K_{\text{r}}. It should be noted that K_{\text{r}} here still represents the relative pile stiffness compared with the soft layer; this is because, although the firm layer and embedment were considered, the overall bending behavior of the pile was still dominated by the soft layer and the part of the pile in this layer. The critical R_{\text{e}} becomes smaller in cases with a high E_{\text{f}}/E_{\text{s}} and low K_{\text{r}}. A high E_{\text{f}}/E_{\text{s}} equates to a firmer embedment layer, which helps the pile resist higher bending moments. In this situation, only a short length of embedded pile is required for the pile to behave like in the ideally fixed condition. This result implies that, when the modulus of the embedded layer is sufficiently high, the embedded length of a pile with a relatively low stiffness does not need to be very long to sustain the bending moment at the soft‒firm layer interface. In Fig.13(d), for a pile with K_{\text{r}} = 1 × 10⁻⁴, an embedded length accounting for only 10% of the total pile length allows > 70% of the bending moment to be mobilized in the case of the ideally fixed condition. The limiting R_{\text{m}} is more sensitive to E_{\text{f}}/E_{\text{s}} when K_r is high. For instance, for the case of K_{\text{r}} = 1 × 10⁻¹, the limiting R_{\text{m}} ranges from 0.2 to 0.7, while for the case of K_{\text{r}} = 1 × 10⁻⁴, the limiting R_{\text{m}} ranges from 0.4 to 0.5. This indicates that a pile with a high stiffness will lose much of its resistance to a bending moment if the embedment layer becomes soft.

Because we considered a differential embedment, the bending moment at the interface between the soft and firm layers was lower than that under the ideally fixed boundary condition. Furthermore, as shown in Fig.14, the position of the M_{\text{max}} also changed. Regardless of K_{\text{r}}, when x/x_{\text{max}} > 0.4, a type III failure may occur. In the fan zone, type II failure may occur on piles with K_{\text{r}} = 1 × 10⁻³ when R_{\text{m}} = 0.33 and R_{\text{m}} = 0.58. However, for piles with K_{\text{r}} > 1 × 10⁻³, the tendency for type I failure remains.

3.4 Effect of a floating pile

A pile located entirely in the soft layer is typically referred to as a floating pile. A floating pile can be considered to have an embedded ratio, R_{\text{e}} = 0, and thus, the pile toe does not restrain the bending moment (R_{\text{m}} = 0). This situation corresponds to the origin of each case, as shown in Fig.15. A typical case of x/x_{\text{max}} = 0 and K_{\text{r}} = 1 × 10⁻² was taken as an example to compare the behaviors of the floating and embedded piles.

Fig.15 shows the normalized bending moments of the piles under various embedment conditions. The bending moment at the soft‒firm layer interface decreased to zero as the embedment condition was relaxed from fully fixed to no rotation restraint. However, it is worth noting that a change in the embedment condition had a small influence on the negative bending moment of the upper part of the pile. The maximum negative bending moment retained the same value and location under each embedment condition. For the floating pile, the negative bending moment corresponded to the maximum bending moment along the pile, whereas for the embedded piles, the positive bending moment at the soft‒firm layer interface corresponded to the maximum bending value. This indicates that a floating pile has a different potential failure location than an embedded pile.

Fig.16 shows the displacement distributions of the piles under various embedment conditions, together with the soil movements at the corresponding position (x/x_{\text{max}} = 0). It can be observed that the movement pattern of the floating pile is overturning, similar to that of the embedded piles, although the toe of the floating pile has a much weaker restraint than the embedded piles. This can be explained by the zero displacement of the soil at the interface between the soft and firm layers, which induces a very small displacement for the lower parts of both the floating and embedded piles (Fig.15). At the same time, the soil has a very large movement along the upper part of the piles leading to an overturn of the piles. The overturning of the floating pile is consistent with observations from the centrifuge tests conducted by Kitazume et al. [39].

4 Conclusions

The bending behavior of piles supporting embankments on soft ground was investigated using a newly proposed two-stage design method. The bending moment distributions and displacement modes of single-row and multiple-row piles were obtained and analyzed. A parametric study was conducted to reveal the effects of the pile relative stiffness, pile position, and embedment conditions. The following conclusions were drawn.

(1) A simple mechanism for soft soil movement under embankments is proposed and verified using centrifuge tests. This mechanism, combined with Poulos’ solution to consider the pile−soil interaction, constitutes an efficient two-stage design method that can successfully predict the bending behavior of piles in soft ground during the rapid construction of embankments.

(2) The bending failure of piles mainly occurs at three potential locations: the interface between the soft and firm layers, the boundary of the soil deformation mechanism, and the mid-depth of the fan zone. The location of the bending failure depended on the position and relative stiffness of the pile.

(3) The bending moment distributions of the multiple-row piles are similar to those of a single-row pile. The maximum bending moment of the multiple-row pile is only a little lower than that of the single-row piles.

(4) The embedment condition of a pile significantly affects the bending moment value and distribution. When the embedded length of a pile exceeded a critical value, the bending moment at the interface between the soft and firm layers reached a limiting value.

(5) Floating piles in soft ground under embankment load exhibit an overturning movement pattern. A potential failure is located in the upper part of a floating pile owing to the lack of restraint of rotation at the pile toe.

References

[1] Zheng G, Yang X Y, Zhou H Z, Chai J C. Numerical modeling of progressive failure of rigid piles under embankment load. Canadian Geotechnical Journal, 2019, 56(1): 23–34

[2] Wang L, Wang P, Wei K, Dollevoet R, Li Z. Ground vibration induced by high speed trains on an embankment with pile-board foundation: Modelling and validation with in situ tests. Transportation Geotechnics, 2022, 34: 100734

[3] ZhengGLiuL. Numerical analysis of the pile lateral behavior and anti-slip mechanism of rigid pile supported embankments. In: Proceedings of U.S.−China Workshop on Ground Improvement Technologies 2009. Reston: American Society of Civil Engineers, 2009, 63–72

[4] ZhengGLiuLHanJ. Stability of embankment on soft subgrade reinforced by rigid inclusions (II)-group piles analysis. Chinese Journal of Geotechnical Engineering, 2010, 32: 1811−1820 (in Chinese)

[5] Sagaseta C. Analysis of undrained soil deformation due to ground loss. Géotechnique, 1987, 37(3): 301–320

[6] HanJChaiJ CLeshchinskyDShenS L. Evaluation of deep-seated slope stability of embankments over deep mixed foundations. In: Proceedings of GeoSupport Conference 2004. Reston: American Society of Civil Engineers, 2004

[7] NavinMFilzG. Numerical stability analyses of embankments supported on deep mixed columns. In: Proceedings of GeoShanghai International Conference 2006. Reston: American Society of Civil Engineers, 2006, 1–8

[8] Ong D E, Leung C E, Chow Y K. Pile behavior due to excavation-induced soil movement in clay. I: Stable wall. Journal of Geotechnical and Geoenvironmental Engineering, 2006, 132(1): 36–44

[9] Leung C F, Ong D E, Chow Y K. Pile behavior due to excavation-induced soil movement in clay. II: Collapsed wall. Journal of Geotechnical and Geoenvironmental Engineering, 2006, 132(1): 45–53

[10] Guo W D. Response and stability of piles subjected to excavation loading. International Journal of Geomechanics, 2021, 21(1): 04020238

[11] Ong D E L, Leung C F, Chow Y K, Ng T G. Severe damage of a pile group due to slope failure. Journal of Geotechnical and Geoenvironmental Engineering, 2015, 141(5): 04015014

[12] BS8006:1995. Code of Practice for Strengthened/Reinforced Soils and Other Fills. London: British Standards Institution, 1995

[13] Zhuang Y, Wang K. Finite element analysis on the dynamic behavior of soil arching effect in piled embankment. Transportation Geotechnics, 2018, 14: 8–21

[14] ZhouH ZXuH JYuX XGuoZ YZhengGYangX YTianY. Evaluation of the bending failure of columns under an embankment loading. International Journal of Geomechanics, 2021, 21(7): 4021112

[15] Indraratna B, Aljorany A, Rujikiatkamjorn C. Analytical and numerical modeling of consolidation by vertical drain beneath a circular embankment. International Journal of Geomechanics, 2008, 8(3): 199–206

[16] Zhou D Q, Feng C X, Li L X, Zhou Y, Zhu Q. Reinforcement effect of inclined prestressed concrete pipe piles on an inclined soft foundation. Advances in Civil Engineering, 2020, 2020: 5275903

[17] Han J, Bhandari A, Wang F. DEM analysis of stresses and deformations of geogrid-reinforced embankments over piles. International Journal of Geomechanics, 2012, 12(4): 340–350

[18] ZhengGLiSDiaoY. Centrifugal model tests on failure mechanisms of embankments on soft ground reinforced by rigid piles. Chinese Journal of Geotechnical Engineering, 2012, 34: 1977−1989 (in Chinese)

[19] Chai J C, Sakajo S, Miura N. Stability analysis of embankment on soft ground (A case study). Soil and Foundation, 1994, 34(2): 107–114

[20] Stark T D, Ricciardi P J, Sisk R D. Case study: Vertical drain and stability analyses for a compacted embankment on soft soils. Journal of Geotechnical and Geoenvironmental Engineering, 2018, 144(2): 05017007

[21] Guo W D. Elastic models for nonlinear response of rigid passive piles. International Journal for Numerical and Analytical Methods in Geomechanics, 2014, 38(18): 1969–1989

[22] AbushararS WHanJ. Two-dimensional deep-seated slope stability analysis of embankments over stone column-improved soft clay. Engineering Geology, 2011, 120(1−4): 103−110

[23] Zhang Z, Han J, Ye G. Numerical investigation on factors for deep-seated slope stability of stone column-supported embankments over soft clay. Engineering Geology, 2014, 168: 104–113

[24] Osman A, Bolton M. A new design method for retaining walls in clay. Canadian Geotechnical Journal, 2004, 41: 451–466

[25] Osman A, Bolton M. Simple plasticity-based prediction of the undrained settlement of shallow circular foundations on clay. Géotechnique, 2005, 55(6): 435–447

[26] Lam S Y, Bolton M D. Energy conservation as a principle underlying mobilizable strength design for deep excavations. Journal of Geotechnical and Geoenvironmental Engineering, 2011, 137(11): 1062–1074

[27] Zhang Z, Huang M, Xu C, Jiang Y, Wang W. Simplified solution for tunnel−soil−pile interaction in Pasternak’s foundation model. Tunnelling and Underground Space Technology, 2018, 78: 146–158

[28] JiaZ. A simplified method of soft grounds improved by rigid piles under construction of embankment. Thesis for the Master’s Degree. Tianjin: Tianjin University, 2019 (in Chinese)

[29] PoulosH GDavisE H. Pile Foundation Analysis and Design. New York: Wiley, 1980

[30] Douglas D J, Davis E H. The movement of buried footings due to moment and horizontal load and the movement of anchor plates. Géotechnique, 1964, 14(2): 115–132

[31] ZhangZHanJYeG. Numerical analysis of failure modes of deep mixed column-supported embankments on soft soils. In: Proceedings of Geo-Shanghai 2014. Reston: American Society of Civil Engineers, 2014: 78−87

[32] Dìaz-Rodrìguez J A, Leroueil S, Alemàn J D. Yielding of Mexico city clay and other natural clays. Journal of Geotechnical Engineering, 1992, 118(7): 981–995

[33] Vardanega P J, Bolton M D. Strength mobilization in clays and silts. Canadian Geotechnical Journal, 2011, 48(10): 1485–1503

[34] Futai M M, Almeida M S S, Lacerda W A. Yield, strength, and critical state behavior of a tropical saturated Soil. Journal of Geotechnical and Geoenvironmental Engineering, 2004, 130(11): 1169–1179

[35] Burland J, Rampello S, Georgiannou V, Calabresi G. A laboratory study of the strength of four stiff clays. Géotechnique, 1996, 46(3): 491–514

[36] Ladd C C, Lambe T W. The strength of “undisturbed” clay determined from undrained tests. American Society for Testing and Materials special technical publications, 1963, 361: 342–371

[37] ZhengGDiaoYLiSHanJ. Stability failure modes of rigid column-supported embankments. In: Proceedings of Geo-Congress 2013. Reston: American Society of Civil Engineers, 2013, 1814–1817

[38] Zheng G, Yang X, Zhou H, Chai J. Numerical modeling of progressive failure of rigid piles under an embankment load. Canadian Geotechnical Journal, 2018, 56(1): 23–34

[39] Kitazume M, Orano K, Miyajima S. Centrifuge model tests on failure envelope of column type deep mixing method improved ground. Soil and Foundation, 2000, 40(4): 43–55

RIGHTS & PERMISSIONS

Higher Education Press