Probabilistic analysis of secant piles with random geometric imperfections

Abstract

The failure to achieve minimum design overlap between secant piles compromises the ability of a structure to perform as designed, resulting in water leakage or even ground collapse. To establish a more realistic simulation and provide guidelines for designing a safe and cost-effective secant-pile wall, a three-dimensional model of a secant pile, considering the geometric imperfections of the diameter and direction of the borehole, is introduced. An ultrasonic cross-hole test was performed during the construction of secant piles in a launching shaft in Beijing, China. Based on the test results, the statistical characteristics of the pile diameters and orientation parameters were obtained. By taking the pile diameter D, inclination angle β, and azimuth angle α as random variables, Monte Carlo simulations were performed to discuss the influence of different design parameters on the probability density functions of the overlap of secant piles. The obtained results show that the randomness of the inclination angle and pile diameter can be well described by a normal distribution, whereas the azimuth angle is more consistent with a uniform distribution. The integrity of the secant-pile wall can be overestimated without considering the uncertainty of geometric imperfections. The failure of the secant-pile wall increases substantially with increasing spatial variability in drilling inclination and diameter. A design flowchart for pile spacing under the target safety level is proposed to help engineers design a safe and economical pile wall.

Graphical abstract

Keywords

secant piles / ultrasonic cross-hole testing / probabilistic analysis / reliability-based design / random imperfections

1 Introduction

Secant-pile walls are formed by the construction of alternating primary unreinforced piles and secondary reinforced piles that interlock. Typically, primary (female) piles are cast in situ, leaving space between them. This is followed by the cutting of secondary (male) piles into the primary piles to form a continuous wall. Primary piles are usually made of plastic concrete, which is usually weaker than ordinary piles, to facilitate drilling. There are two typical types of cast-in-place secant-pile walls: (a) hard-firm type, which is usually formed by reinforced concrete piles (RCPs) and plastic concrete piles (PCPs) arranged alternatively; (b) hard-hard type, which is formed by RCPs arranged sequentially [1]. In the past 20 years, secant piles have been used as retaining walls in a wide variety of civil engineering applications including embankment stabilization [2] and the construction of basement walls [3–6], dams [7–10], and other buried structures.

Because secant-pile construction is a relatively quiet and vibration-free method that provides continuous support to the ground during excavation and reduces the inflow of groundwater, it is being used more frequently in deep excavations in urban areas. Based on engineering practices, some studies have investigated the mechanical behavior of secant piles using the finite-element method and obtained the deformation and stress distribution of the pile wall, which were considered to be of interest to designers and construction managers [11,12]. It is worth noting that the piles were usually considered to be ideal cylinders in these studies. However, even if a high construction standard is adopted, secant piles sometimes fail in deep foundation pit excavation owing to geological and mechanical uncertainties. Poulos [13] classified the imperfections of piles into two categories according to their causes: (a) natural (geological) imperfections and (b) imperfections related to construction techniques.

The verticality, continuity, and closure of some secant piles with a depth of 90 m were tested using a biaxial inclinometer in Arapuni Dam in New Zealand [14]. The results showed that some of the overlaps between piles no longer existed with increasing depth, and there were some gaps between piles owing to the inclination of the pile holes. Continuous flight auger (CFA) bored piles were used in the shafts of Abbotts Way and Nikau Street in Auckland, and some piles were found to be misaligned and defective [15]. Generally, the geometric faultiness of secant piles can be divided into (a) randomness of pile diameter and (b) inclination error of the pile. Several recent studies suggested that geometric imperfections considerably affect the quality of the pile wall [16–18]. The existence of geometric imperfections significantly reduces the stiffness and capacity of piles when the piles are used as the foundation [19,20]. The overlap failure in the secant piles may result in severe seepage and ground collapse.

However, most previous studies considered the pile geometry as a complete vertical cylinder without any imperfections, which may significantly overestimate the ability of the structure. To provide a more effective method of evaluating the integrity of secant piles, a numerical model considering geometric faultiness is established based on random field theory. Based on an engineering case in Beijing, China, the spatial randomness of both pile diameter and inclination are illustrated. The influences of the statistical parameters of geometric defects and design parameters on the probability density functions (PDFs) of the overlap between adjacent piles are explored through Monte Carlo simulations. At the end of this paper, a reliability-based pile-spacing calculation process is described.

2 Statistical characteristics

A bored cast-in-situ pile is constructed by digging a hole in the ground by means of a percussive or rotary method with the use of temporary or permanent casing or drilling mud. In the actual installation process, the pile body may have a certain inclination owing to the construction accuracy, which has a significant relationship with the soil conditions and the operation deviation of the drilling machine. Because shrinkage and even collapse of the bored holes often occur during the construction in soft soil, the pile diameter also has strong variability in its longitudinal direction. Therefore, the geometric defects of bored piles usually involve two aspects: (a) diameter and (b) random direction of the column axis [21]. Herein, the geometric imperfection distribution of secant piles is illustrated based on an ultrasonic cross-hole test in an engineering case in Beijing. In this case, the secant piles are used as water stop walls for the launching shaft of Subway Line 27. The retaining walls consisted of 48 primary piles with a length of 35.7 m and 48 secondary piles with a length of 38.7 m, which were constructed by a ZR200A rotary drilling rig, as shown in Fig. 1. The pile-hole profile was obtained using a Japanese KODEN DM-604 ultrasonic detector, as shown in Fig. 2.

As shown in Fig. 3, the orientation defect of the bored piles is described by two independent parameters: inclination angle β and azimuth angle α. It is clear that every time a pile hole is drilled, the drill pipe is repositioned during the construction of the bored pile. Consequently, it is reasonable to assume that the inclination angle b and azimuth angle α of each pile are independent identically distributed variables and independent of each other. By transmitting ultrasonic and receiving reflection information at different depths, borehole parameters such as the diameter, inclination, and azimuth are obtained. As shown in Fig. 4, O is the center of the hole, O′ is the center of the ultrasonic probe, and D is the diameter of the hole. By using an ultrasonic detector, the distance from the ultrasonic probe to the hole wall at any depth can be obtained from four directions. L_W, L_E, L_N, and L_S are the monitored distances from O′ to W, E, N, and S, respectively. The diameter of the pile was measured every 1.0 m along the depth. When the hole at a certain depth is assumed to be a perfect circle, the diameter D of the pile can be written as

(1)D=[\sqrt{{(L_{\mathrm{W}}+L_{\mathrm{E}})}^{\mathrm{2}}+{(L_{\mathrm{N}}-L_{\mathrm{S}})}^{\mathrm{2}}}+\sqrt{{(L_{\mathrm{W}}-L_{\mathrm{E}})}^{\mathrm{2}}+{(L_{\mathrm{N}}+L_{\mathrm{S}})}^{\mathrm{2}}}]/\mathrm{2.}

The inclination β of the pile can be obtained by calculating the offset of the center, which can be represented as

(2)\beta =\arctan \hspace{0.17em}(\mathrm{\Delta }L/H)=\arctan \hspace{0.17em}{\displaystyle \frac{\sqrt{{(L_{\mathrm{N}}-L_{\mathrm{S}})}^{\mathrm{2}}+{(L_{\mathrm{W}}-L_{\mathrm{E}})}^{\mathrm{2}}}}{\mathrm{2}}}.

Figure 5(a) shows a comparison of different distribution types including half-normal, lognormal, and Weibull distributions. All of them can well describe the variability of pile inclination, and their coefficients of determination (R²) are nearly the same. The lognormal distribution can best fit the test results. Because the pile inclination is theoretically symmetrical about the arrangement direction of the pile wall, a half-normal distribution is selected herein to describe its variability. As shown in Fig. 5(a), the inclination β follows a half-normal distribution, which is a fold at the mean of an ordinary normal distribution with mean zero. Figure 5(b) shows the measured distribution of azimuth α. The north direction is defined as 0° in this paper. It is acceptable to assume that the distribution of azimuth α is uniform within [–180°,180°]. Since the azimuth α is uniformly distributed and the distribution of inclination β is folded at the mean zero, the inclination β is also regarded as normally distributed with the same standard deviation S.D.(β) as shown in Fig. 5(a). A negative inclination angle indicates the opposite direction.

Figure 6 presents the statistical results of pile diameter D collected from the engineering case in Beijing. The diameter D can be described by a normal distribution with a mean value of 0.983 m, which is close to the design value of 1 m. The coefficient of variation (COV) of diameter D is approximately 0.09. Additionally, the distributions of diameter D at depths H of 0, 10, 20, and 30 m are further calculated, and the results are presented in Fig. 7, wherein four groups of data have similar COVs. The COV at H = 30 m is slightly larger than the other three, which is mainly owing to soil collapsing near the bottom of the borehole. By comparing the distributions of diameter D at different depths, it is found that the mean value of the diameter decreases with an increase in depth H, which is mainly induced by the shrinkage of the borehole. The support pressure is usually lower than the lateral soil pressure, and the amount of borehole shrinkage is proportional to depth H and diameter D [22]. The variation in diameter D at different depths H is given in Fig. 8. The borehole shrinkage presents an approximately linear relationship with the depth, and the shrinkage factor is 2.39‰ in this case.

3 Methodology

3.1 Random model of imperfections

In this study, the relevant parameters of the pile diameter and verticality were taken as random variables. The boreholes usually shrink or even collapse during the drilling process, depending on the soil properties. Owing to the natural spatial variability of soil, the cross section along a pile varies with the encountered soil property, and the pile diameter is spatially dependent rather than dependent on uncorrelated variables. Therefore, the variation in diameter along a pile is represented using a random field. The pile is divided into small segments, and their lengths are shorter than the SOF. Based on the statistical characteristics, a normally distributed random field is employed to describe the diameter variability in a pile. It is assumed that the diameter of a pile has similar scales of fluctuation as the soil. This is reasonable because the cross sections of the pile vary according to the encountered soil conditions. The drilling of each borehole is an independent process. Therefore, the verticality parameters and pile diameter are independent random variables. Based on the statistical characteristics, the azimuth α is considered to have a uniform distribution ranging from −180° to 180°. The inclination β is considered a half-normal distribution.

Herein, the correlation between different spatial variables is expressed by an autocorrelation function. Among the correlation functions, the squared exponential function is the most widely used and expressed as

(3){\rho }_{\mathrm{c}}=\exp [-{\left({\displaystyle \frac{x_i-x_j}{{\theta }_{\mathrm{h}}}}\right)}^{\mathrm{2}}],

where x_i and x_j define the space center coordinates of two segments, and {\theta }_{\mathrm{h}} denotes the scale of fluctuation.

As the generation of a pile diameter only needs to consider the longitudinal direction, a one-dimensional random field was employed to represent the variability of the pile diameter. The K–L expansion, consisting of a linear combination of orthogonal functions, is adopted here to obtain a random field owing to its high computing efficiency. More details about this method can be found in Ref. [23]. The random field of standard normal distribution can be simplified to

(4)G\left(x\right)\text{'}\approx {\displaystyle \sum _{i=\mathrm{1}}^N\sqrt{{\mathit{\kappa }}_i}}{\zeta }_i{\chi }_i\left(x\right),

where {\zeta }_i is the independent random variable that is standard normally distributed, N is the expansion term, {\mathit{\kappa }}_i is the eigenvalue of the exponential autocorrelation function, and {\chi }_i\left(x\right) is the eigenvector of the autocorrelation function.

The normally distributed random field of the pile diameter is written as

(5)G\left(x\right)=u_{\mathrm{d}}+{\sigma }_{\mathrm{d}}G\left(x\right)\text{'},

where u_{\mathrm{d}} is the mean of the pile diameter, and {\sigma }_{\mathrm{d}} is the standard deviation of the pile diameter.

In this study, it was assumed that the pile diameter consists of a nondeterministic term and a deterministic term. The nondeterministic term corresponds to the spatial randomness, whereas the deterministic term corresponds to the borehole shrinkage, as described in Fig. 9. Based on reports by Wang et al. [22], the amount of shrinkage is assumed to be directly proportional to the depth of the pile cross sections, and the shrinkage factor k can be introduced to characterize this determination term. Therefore, the diameter of the pile can be written as a superposition of a random term and a deterministic term, as shown in Eq. (6):

(6)D^{\mathrm{H}}=G\left(x\right)-k{u}_{\mathrm{d}}H,

where k is the borehole shrinkage factor, which represents the shrinkage with the unit increase in depth. Thus, the wall defects can be determined based on the random geometry of the secant piles.

3.2 Secant-pile overlap

Once the uncertainties of geometric imperfections, including the pile diameter and random orientations of the column axis, are considered, a three-dimensional model can be established based on random field theory. The full modeling procedure is as follows. (a) The adjacent primary and secondary piles are divided into small segments in the vertical direction. The geometric parameters, including the pile orientations, inclinations, and cross sections for each segment, can be generated using random field theory. (b) A 3D geometric model for secant piles can be established by connecting the segment boundaries. (c) The secant-pile overlap at any depth can be determined based on the spatial geometric relationship. (d) Failures of the secant piles will be identified if the overlap at a certain depth is less than the minimum design overlap.

Figure 10(a) shows the typical realization of secant piles. The slices (A and B) at the top and any depth of the pile are shown in Figs. 10(b) and 10(c). As shown in Fig. 10(b), if the center coordinate of the left circle was assumed to be (x, y) = (0, 0), then the center coordinate of the right circle is (x, y) = (0, L). By using the inclination angle and azimuth angle models shown in Fig. 3, the center coordinates of the two circles in Fig. 10(c) can be expressed as (H\tan {\beta }_i\sin {\alpha }_i, H\tan {\beta }_i\cos {\alpha }_i) and (H\tan {\beta }_j\sin {\alpha }_j, L+H\tan {\beta }_j\cos {\alpha }_j). The distance between the two circle centers can be easily obtained through their coordinates. After considering the effect of variation in pile diameter, the overlap at a certain depth of secant piles can be expressed as

(7){\delta }_{\mathrm{c}}=(D_i^{\mathrm{H}}+D_j^{\mathrm{H}})/\mathrm{2}-\sqrt{{(L+H(\tan {\beta }_i\cos {\alpha }_i-\tan {\beta }_j\cos {\alpha }_j\left)\right)}^{\mathrm{2}}+{\left(H\right(\tan {\beta }_i\sin {\alpha }_i-\tan {\beta }_j\sin {\alpha }_j\left)\right)}^{\mathrm{2}}},

where D_i^{\mathrm{H}} and D_j^{\mathrm{H}} are the diameters of the primary and secondary piles, respectively; L is the distance from the primary pile to the secondary pile; H is the depth of the specific slice; {\alpha }_i and {\alpha }_j are the azimuths of the primary and secondary piles, respectively; and {\beta }_i and {\beta }_j are the inclinations of the primary and secondary piles, respectively. δ is the initial overlap at the top, L is the pile spacing, and δ_c is the overlap value at the calculated depth.

4 Results and discussion

4.1 Input parameters

In this study, the input parameters should be selected as accurately as possible. The codes for the vertical control of bored piles in various countries are listed in Table 1. Among them, the China code is formulated for secant piles of boreholes supported with a casing, whereas the American and Singaporean codes are for the bored pile. The allowable range of verticality tolerance ranges from 1/75 to 1/300. Table 1 shows the S.D.(β) corresponding to the allowable vertical tolerance, ranging from 0.1 to 0.4, where the allowable vertical tolerance corresponds to a 5% probability of exceedance.

According to the classification of the construction machinery of a secant-pile wall, the hole-forming methods can be divided into standard hole-forming methods (including boreholes supported with casing and boreholes supported with slurry) and CFA piles [27]. Different construction machinery will lead to differences in statistical parameters. More statistical parameters of the geometric imperfections of the piles in the reference cases are listed in Table 2. Amos et al. [14] reported on the application of a 400-mm-diameter, 90-m-deep secant-pile wall in Arapuni Dam, and a continuity test result of secant-pile wall was obtained. In this project, computer-controlled directional drilling was adopted, and a hole with a diameter of 150 mm was drilled first. Then, the hole was expanded to 400 mm to ensure that the S.D.(β) was only 0.15. For a borehole supported with slurry, the S.D.(b) of the results in this study (Fig. 5(a)) and Fetzer [28] both exceeded 0.3. Wang et al. [29] tested the variability of the pile diameter and found that the COV of pile diameter COV(D) ranged from 0.012 to 0.029. The results obtained in this study (Fig. 5(b)) show that the COV(D) of a borehole supported with slurry is only 0.091. It is easy to believe that the COV(D) of the borehole supported with a casing may be smaller owing to the protection of the casing. As reported by Wang et al. [22], there is a significant difference in the hole diameter shrinkage factor k in sand and clay, and k is relatively smaller in the sand layer. ^aS.D.(β) is calculated by assuming that 5% of pile inclination exceeds vertical limit.

4.2 Parametric study

To examine the effects of various factors on the PDF of the overlap value at a certain excavation depth, a detailed parametric study is conducted. The influencing factors are summarized in Table 3 and divided into three groups. The first group, including the COV of diameter COV(D), standard deviation of inclination S.D.(β), hole diameter shrinkage factor k, and diameter SOF, represents the statistical parameters. The second group represents the functional design parameters including the excavation depth H. The third group consists of the design parameters of secant-pile walls such as the diameter of piles D and initial overlap δ between two adjacent secant piles.

Table 3 lists the model calculation parameters according to the reference cases in Tables 1 and 2. The azimuth α is assumed to be uniformly distributed at [−180°, 180°], and it is an independent random variable. The inclination β is taken as semi-normally distributed. The standard deviation is 0.1°–0.4°, whereas the mean is 0. Compared to the cases in Tables 1 and 2, the value of 0.4° is conservative and ensures that the range can cover most of the cases. In the simulation process, the diameter of the secant piles is assumed to be a vertical random term. The design diameter value can be used as the mean value, and COV(D) can be taken as 0.05–0.15. These input parameters are the same for the primary and secondary piles. The hole diameter shrinkage factor k is assumed to be a constant parameter along the vertical direction with a value of 0–2‰, which depends on the soil conditions and hole wall-protection technology. The excavation depth H represents the excavation dimensions, which are common within 40 m in foundation pit engineering. There are many different combinations of design diameter D and initial overlap δ. The ultimate purpose of the design is to ensure that the overlap at a certain depth δ_c is as concentrated as possible and close to the design value so that the column element can be connected to a seamless wall.

To discuss the effects of relevant factors, three or four types of probability scenarios are considered for each pair of variables. Latin hypercube sampling (LHS) is adopted to obtain the input parameters. The sample size was determined by repeating Monte Carlo simulations until the mean and standard deviation of the obtained results converged. Based on Ref. [31], approximately 10^k+2 samples are needed to achieve a satisfactory accuracy for a failure probability of 10^−k (for a COV of a failure probability close to 0.1). Thus, each simulation was calculated 20000 times in this study, which is large enough to obtain statistically representative results. Figures 11, 12, and 13 show the corresponding results of PDFs plotted using the kernel smoothing technique. Figure 11(a) shows that a decrease in S.D.(β) produces a narrower and taller PDF curve, increasing the peak of the curve.

The variation in β is a key factor to be considered in the design. This is easy to understand because a smaller β makes the actual overlap more likely to meet the design overlap. The PDF curve also shows a narrower and higher trend when COV(D) decreases (Fig. 11(b)). When the variation in diameter is greater, a gap between the primary pile and secondary pile is more likely to occur. Fortunately, the variation in diameter will not be too large for the mechanical hole-forming method, and a COV(D) of 0.15 is a conservative assumption. The shapes of the PDF curves are basically similar under different k values. The larger the value of k, the smaller the overlap δ_c corresponding to the peak of the PDF curves (Fig. 11(c)). This is mainly because the existence of k causes a shorter pile diameter at the excavation depth, resulting in a higher possibility of discontinuity between adjacent piles. The PDF curves almost overlap when the SOF of the diameter varies within 0.5–2.0 m (Fig. 11(d)). Therefore, the SOF of the diameter can be assumed to be a fixed value in the design that does not require more attention.

Figure 12 presents PDFs of overlap δ_c at different excavation depths. The PDF curves exhibit a wider and shorter trend with an increase in excavation depth H. As the excavation depth H increases, the average value of the overlap decreases gradually. This is more likely to generate a larger vertical deviation of the piles, as shown in Fig. 10(c), thus making the overlap δ_c more unreliable. This reveals to the engineers that the failure risk may increase when conducting a deep excavation owing to the randomness of geometric imperfections. The more the excavation depth, the more difficult it is to ensure the continuity of the pile wall. When the geometric imperfections cannot be reduced, the probability of gaps can only be reduced by designing a reasonable diameter D and initial overlap value δ.

Figure 13(a) shows that an increase in diameter produces a narrower and taller PDF curve, increasing the peak of the curve. The mean value of the overlapping δ_c increases with a decrease in pile diameter D. Interestingly, with the same initial overlap δ, the longer the diameter D, the higher the probability of gaps. In other words, a small pile diameter is more beneficial to improving the impermeability of the secant-pile wall. Figure 13(b) shows the influence of initial overlap δ on PDFs of overlap at a certain depth δ_c. It can be seen that the mean of overlap δ_c increases with initial overlap δ, whereas the standard deviation stays unchanged. This can be attributed to the fact that a larger initial overlap δ implies a smaller pile spacing L, and the overlap δ_c decreases linearly with an increase in pile spacing L. This makes the design simpler; i.e., the probability of gaps can be considerably reduced by increasing initial overlap δ, which in turn leads to a higher cost with increasing cement consumption.

In engineering practice, the initial overlap δ cannot be increased by more than D/2. Therefore, the initial overlap range of large-diameter piles is larger than that of small-diameter piles. To consider both cost and engineering safety, it is necessary to select a suitable combination of diameter D and initial overlap δ. A basic principle is as follows: engineers should first determine a reasonable initial overlap δ, and then, a relatively small diameter of piles can be selected based on the overlap requirement. Owing to the page limit of this paper, the effects of input parameters on the secant-pile overlap are discussed based on a simple parametric study. The global sensitivity analysis method may help engineers better understand the parameter effects in engineering problems [32]. More global sensitivity analyses need to be performed in future work.

4.3 Failure analysis

When considering the occlusal effect of the secant piles, the allowable minimum overlap between adjacent piles can be regarded as the ultimate limit state. The performance function used to evaluate the failure probability at a given limit overlap δ_lim can be defined as

(8)G={\delta }_{\mathrm{c}}-{\delta }_{\lim },

where δ_lim represents the limit overlap of secant piles, δ_c represents the overlap at a certain depth, and G<0 represents the failure domain.

Using the Monte Carlo simulation method, the failure probability can be evaluated as

(9)P_{\mathrm{f}}={\displaystyle \frac{\mathrm{1}}{n_{\mathrm{MC}}}}{\displaystyle \sum _{i=\mathrm{1}}^{n_{\mathrm{MC}}}I\left(G\right)},

where n_{\mathrm{MC}} refers to the sample size of the Monte Carlo simulation. When G<0, I\left(G\right) = 1; otherwise, I\left(G\right) = 0.

Figure 14 shows the relationship between the excavation depth H and failure probability P_f. The initial pile diameter D is taken as 1.00 m, and the pile spacing L is taken as 0.60 m, corresponding to an initial overlap δ of 0.40 m (i.e., D = L + δ). In this study, the minimum overlap value δ_lim, corresponding to the ultimate limit state, is 0.1 m, which is a conservative value for forming a continuous wall. As shown in Fig. 14, when the depth H exceeds 20 m, then the failure probability P_f significantly increases with depth H. The reason is that as depth H increases, the horizontal deviation increases, and simultaneously, the pile diameter decreases. The decrease in pile spacing L can reduce the failure probability P_f, which in turn increases the number of piles at the same length of the pile wall.

According to the recommendation of Eurocode 7 (BS EN 1997-2) [33], if it is defined as a limit state that the overlap value is lower than a certain value, then a prudent estimation can be taken as the amount of overlap of no more than 95% in all cases (i.e., a 95% fractile). If the excavation depth corresponding to a 5% failure probability is defined as the limit excavation depth H_max, then the changes in H_max with the variability of geometric imperfections [COV(D) and S.D. (β)] are shown in Fig. 15. The limit excavation depth H_max decreases with an increase in COV(D) and S.D. (β), which is a reasonable trend. However, the difference between them is that the decrease in H_max becomes more evident with an increase in COV(D), whereas the H_max decreases more slowly with an increase in S.D. (β). Therefore, ensuring a smaller variation of β is the key to obtain a deeper H_max, which deserves careful attention in the construction process.

According to the recommendation of Eurocode 7 (BS EN 1997-2) [33], if it is defined as a limit state that the overlap value is lower than a certain value, then a prudent estimation can be taken as the amount of overlap of no more than 95% in all cases (i.e., a 95% fractile). If the excavation depth corresponding to a 5% failure probability is defined as the limit excavation depth H_max, then the changes in H_max with the variability of geometric imperfections [COV(D) and S.D.(β)] are shown in Fig. 15. The limit excavation depth H_max decreases with an increase in COV(D) and S.D.(β), which is a reasonable trend. However, the difference between them is that the decrease in H_max becomes more evident with an increase in COV(D), whereas the H_max decreases more slowly with an increase in S.D.(β). Therefore, ensuring a smaller variation of β is the key to obtain a deeper H_max, which deserves careful attention in the construction process.

4.4 Design procedure

The purpose of this section is to provide guidelines for designing a safe and cost-effective secant-pile wall. The design procedure considers the random imperfections of piles induced by soil variability and mechanical drilling deviations. In general, the excavation depth H is usually determined by the functional requirements without much choice. According to the definition of H_max, the failure probability of secant piles will not exceed 5% if H is less than H_max. This makes the design relatively simple. Designers need to determine the optimal spacing and diameter of piles meeting H≤H_max. Figure 16 provides a detailed flowchart for the secant piles. The design procedure is as follows.

a) First, designers should choose the pile diameter D according to the excavation depth H and loading conditions. The size of the pile must be sufficient to maintain the stability of the surrounding soil.

b) The variability of the pile diameter and drilling orientation, including S.D.(β), COV(D), and shrinkage factor k, must be determined. The mechanical factors, the S.D.(β) in particular, can be conservatively estimated according to the specifications in Table 1. COV(D) can be estimated by referring to the experience of other local projects. The hole diameter shrinkage factor k can be estimated with the reference in Table 2. More detailed statistical parameters can be obtained through ultrasonic testing during the construction of secant piles.

c) For convenience, a maximum pile spacing L = D is selected as an initial input. The limit envelope surface can be calculated according to the soil and drilling conditions. Typical charts of the envelope surface are presented in Fig. 17.

d) The excavation depth H and limit depth H_max are compared. If H>H_max, then the pile spacing L decreases until H is almost equal to H_max. If the excavation depth H is still larger than H_max when the pile spacing L decreases to D/2, then the pile diameter should be increased. For example, considering a secant-pile wall with an excavation depth H of 20 m, H_max is 19.10 m at L = 0.65 m (Fig. 17(a)), and H_max = 24.75 m at L = 0.60 m (Fig. 17(b)).

e) Finally, a reasonable combination of pile diameter D and pile spacing L can be determined.

5 Conclusions

In this study, a three-dimensional model of a secant pile, considering the geometric imperfections of the diameter and drilling orientation, was established. An ultrasonic cross-hole test was performed to obtain the statistical characteristics of the secant piles. The overlap failure of secant piles was analyzed using Monte Carlo simulations. The main conclusions are as follows.

a) An ultrasonic cross-hole test showed that the variation of the inclination angle and pile diameter can be well described by a Gaussian distribution, whereas that of the azimuth angle is more consistent with a uniform distribution.

b) A three-dimensional model, considering the geometric imperfections of the diameter and direction of the drilling hole, was introduced to estimate the overlap of secant piles. All the statistical parameters of pile geometric imperfections, including S.D.(β), COV(D), and k, affect the distribution of overlap δ_c, among which the S.D.(β) is the most significant. During the process of pile-hole drilling, more attention should be paid to the verticality control to reduce the influence of geometric imperfections.

c) The failure probability of secant piles increases with an increase in excavation depth H as well as the variability of the inclination angle and pile diameter. In particular, when the excavation depth H exceeds 20 m, the failure probability will increase significantly.

d) A design process within the probabilistic framework was given for engineers to select an economical pile spacing, which can also ensure the continuity of the pile wall. An economic combination of pile diameter D and pile spacing L can be obtained according to the design process provided herein. It is very important to ensure the accuracy of statistical parameters, which requires more statistical work on pile geometric imperfections in different strata.

Furthermore, the random field theory cannot completely reproduce the geometric imperfections of secant piles. Phase-field models (PFMs), which are rapidly developed in engineering problems, may be a useful tool for assessing secant piles [34–39]. The failure of secant piles using PFMs should be further explored in the future.

References

[1] Liao S M, Li W L, Fan Y Y, Sun X, Shi Z H. Model test on lateral loading performance of secant pile walls. Journal of Performance of Constructed Facilities, 2014, 28(2): 391–401

[2] Anderson S A, Williams J L. Road stabilization, reconstruction, and maintenance with a combined mechanically stabilized earth and secant pile wall: Zion National Park, Utah. Transportation Research Record: Journal of the Transportation Research Board, 2002, 1808(1): 76–83

[3] Finno R J, Bryson S, Calvello M. Performance of a stiff support system in soft clay. Journal of Geotechnical and Geoenvironmental Engineering, 2002, 128(8): 660–671

[4] Wang A H. Drilling occlusive piles—New style of metro protecting structure. Journal of Railway Engineering Society, 2003, 77(1): 53–59 (in Chinese)

[5] Chen B, Shi B, Lin M. A study on the scents pile in soft soil for Nanjing metro project. Chinese Journal of Geotechnical Engineering, 2005, 27(3): 354–357 (in Chinese)

[6] Zhou X L, Liao S M, Song B. Settlement analysis of adjacent buildings induced by secant pile construction in soft ground. Chinese Journal of Geotechnical Engineering, 2006, 28(S0): 1806–1810 (in Chinese)

[7] Anderson T C. Secant piles support access shafts for tunnel crossing in difficult geologic conditions. In: GeoSupport 2004: Drilled Shafts, Micropiling, Deep Mixing, Remedial Methods, and Specialty Foundation Systems. Orlando, FL: American Society of Civil Engineers (ASCE), 2004, 299–308

[8] Bruce D A, di Cervia A R, Amos-Venti J. Seepage remediation by positive cut-off walls: A compendium and analysis of North American case histories. Boston, MA: ASDSO Dam Safety, 2004, 10–14

[9] Simpson D E, Phipps M, di Cervia A L R. Constructing a cutoff wall in front of Walter F. George dam in 100 feet of water. Hydro Review, 2006, 25(1): 42, 44–46, 50–51

[10] Ketchem A J, Wright G W. The Use of Secant Pile Walls in the Rehabilitation of the Pohick Creek Dam No. 2 (Lake Barton) Auxiliary Spillway. Kansas City, Missouri: American Society of Agricultural and Biological Engineers, 2013

[11] Bryson L S, Zapata-Medina D G. Finite-element analysis of secant pile wall installation. Proceedings of the Institution of Civil Engineers-Geotechnical Engineering, 2010, 163(4): 209–219

[12] Chehadeh A, Turan A, Abed F. Numerical investigation of spatial aspects of soil structure interaction for secant pile wall circular shafts. Computers and Geotechnics, 2015, 69: 452–461

[13] Poulos H G. Pile behavior—Consequences of geological and construction imperfections. Journal of Geotechnical and Geoenvironmental Engineering, 2005, 131(5): 538–563

[14] Amos P D, Bruce D A, Lucchi M, Watkins N, Wharmby N. Design and construction of deep secant pile seepage cut-off walls under the Arapuni dam in New Zealand. In: The 28th USSD Conference: The Sustainability of Experience—Investing in the Human Factor. Portland, OR: United States Society on Dams (USSD), 2008, 5–6

[15] Wharmby N, Perry B, Waikato H. Development of Secant Pile Retaining Wall Construction in Urban New Zealand. Hamilton, Waikato: Brian Perry Civil, 2010

[16] Liu Y, Lee F H, Quek S T, Chen E J, Yi J T. Effect of spatial variation of strength and modulus on the lateral compression response of cement-admixed clay slab. Geotechnique, 2015, 65(10): 851–865

[17] Pan Y, Liu Y, Hu J, Sun M, Wang W. Probabilistic investigations on the watertightness of jet-grouted ground considering geometric imperfections in diameter and position. Canadian Geotechnical Journal, 2017, 54(10): 1447–1459

[18] Pan Y, Liu Y, Chen E J. Probabilistic investigation on defective jet-grouted cut-off wall with random geometric imperfections. Geotechnique, 2019, 69(5): 420–433

[19] Zhang L M, Wong E Y. Centrifuge modeling of large-diameter bored pile groups with defects. Journal of Geotechnical and Geoenvironmental Engineering, 2007, 133(9): 1091–1101

[20] Nadeem M, Chakraborty T, Matsagar V. Nonlinear buckling analysis of slender piles with geometric imperfections. Journal of Geotechnical and Geoenvironmental Engineering, 2015, 141(1): 06014014

[21] Croce P, Modoni G. Design of jet-grouting cut-offs. Proceedings of the Institution of Civil Engineers-Ground Improvement, 2007, 11(1): 11–19

[22] Wang Y, Zhang G, Hu Q. Analysis of stability of bored pile hole-wall. Chinese Journal of Rock Mechanics and Engineering, 2011, 30(S1): 3281–3287 (in Chinese)

[23] Huang S, Quek S T, Phoon K K. Convergence study of the truncated Karhunen–Loeve expansion for simulation of stochastic processes. International Journal for Numerical Methods in Engineering, 2001, 52(9): 1029–1043

[24] British Standard. ICE Specification for Piling and Embedded Retaining Walls. London: Institution of Civil Engineers (Great Britain), ICE Publishing, 1996

[25] Singapore Standard. Foundation Supervision Guide. Singapore: Association of Consulting Engineers & Institution of Engineers, 2012

[26] China Standard. Technical Standard for Secant Piles in Row. Beijing: China Architecture & Building Press, 2018 (in Chinese)

[27] Kempfert H G, Gebreselassie B. Excavations and Foundations in Soft Soils. Munich: Springer, 2006, 349–460

[28] Fetzer C A. Performance of the concrete wall at wolf creek dam. In: Transactions of 16th International Commission on Large Dams. San Francisco, CA: International Commission on Large Dams (ICOLD), 1988, 277–282

[29] Wang Q, Zhao C, Zhao C, Xue J X, Lou Y. Influence of normal and reverse mud circulations on hole diameter of cast-in-situ bored piles. Chinese Journal of Geotechnical Engineering, 2011, 33(S2): 205–208 (in Chinese)

[30] Zhao C, Chen H, Zhao C, Xue J. Research on change law of hole diameter over time and space in drilling cast-in-place piles considering unloading effect. Rock and Soil Mechanics, 2015, 36(S1): 573–576 (in Chinese)

[31] Pan Q, Dias D. Probabilistic analysis of a rock tunnel face using polynomial chaos expansion method. International Journal of Geomechanics, 2018, 18(4): 04018013

[32] Hamdia K M, Rabczuk T. Key parameters for fracture toughness of particle/polymer nanocomposites: Sensitivity analysis via XFEM modeling approach. In: Fracture, Fatigue and Wear. Singapore: Springer, 2018, 41–51

[33] BSI. BS EN 1997-2: 2007. Eurocode 7–Geotechnical Design, Part 2: Ground Investigation and Testing. London: BSI, 2007

[34] Zhuang X, Zhou S, Sheng M, Li G. On the hydraulic fracturing in naturally-layered porous media using the phase field method. Engineering Geology, 2020, 266: 105306

[35] Zhou S, Rabczuk T, Zhuang X. Phase field modeling of quasi-static and dynamic crack propagation: COMSOL implementation and case studies. Advances in Engineering Software, 2018, 122: 31–49

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

[37] Zhou S, Zhuang X, Rabczuk T. Phase field modeling of brittle compressive-shear fractures in rock-like materials: A new driving force and a hybrid formulation. Computer Methods in Applied Mechanics and Engineering, 2019, 355: 729–752

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

[39] Zhou S, Zhuang X, Zhu H, Rabczuk T. Phase field modelling of crack propagation, branching and coalescence in rocks. Theoretical and Applied Fracture Mechanics, 2018, 96: 174–192

RIGHTS & PERMISSIONS

Higher Education Press