Energy‐based analysis of seismic damage mechanism of multi‐anchor piles in tunnel crossing landslide area

Abstract

To study the damage mechanism of multi‐anchor piles in tunnel crossing landslide area under earthquake, the damping performance of multi‐anchor piles was discussed. The energy dissipation springs were used as the optimization device of the anchor head to carry out the shaking table comparison test on the reinforced slope. The Hilbert spectrum and Hilbert marginal spectrum were proposed to analyze the seismic damage mechanism of the multi‐anchor piles, and the peak Fourier spectrum amplitude (PFSA) was used to verify the effectiveness of the method. The results show that the seismic energy is concentrated in the high‐frequency component (30–40 Hz) of the Hilbert spectrum and the low‐frequency component (12–30 Hz) of the marginal spectrum. This indicates that they can be combined with the distribution law of the PFSA to identify the overall and local dynamic responses of the multi‐anchored piles, respectively. The stretchable deformation of the energy‐dissipation springs improves the coordination of the multi‐anchor piles, resulting in better pile integrity. The damage mechanism of the multi‐anchor piles is elucidated based on the energy method: local damage at the top and middle areas of the multi‐anchor piles is mainly caused by the low‐frequency component (12–30 Hz) of the marginal spectrum under the action of 0.15g and 0.20g seismic intensities. As the seismic intensity increases to 0.30g, the dynamic response of the slope is further amplified by the high‐frequency component (30–40 Hz) of the Hilbert energy spectrum, which leads to the overall damage of the multi‐anchor piles.

INTRODUCTION

With China's “the Belt and Road Initiative” policy, more and more railroads, highways and national defense tunnels are to be constructed through the western high-intensity mountainous areas. The high-intensity mountainous areas are prone to natural disasters, which poses a great threat to the smooth construction and safe operation of tunnel projects (Yuan, 2015). The design level of the pile anchor retaining structure directly affects the safety of slope treatment. Once the pile anchor retaining structure fails, it will bring serious economic losses (Chen & Wang, 2012). Therefore, it is necessary to research the damage mechanism and seismic optimization design of pile-anchor retaining structures in the tunnel landslide area.

Antislide piles with a single anchor point are flexible retaining structure, which is widely used in geotechnical engineering because of its advantages of safety, reliability, fast construction, and good social benefits (Hu et al., 2021). However, most of the research on single anchor antislide piles is mainly limited to numerical research and optimization design (Poulos, 1973; Won et al., 2005). Won et al. (2005) and Martin and Chen (2005) used the finite difference software FLAC3D to numerically simulate the reinforcing effect of prestressed anchor cable antislide piles. Some scholars have done related research on single anchor antislide piles based on numerical simulation and static conditions. For example, Degrande et al. (2002) proposed a linear elastic seismic analysis method for anchored sheet piles. Bilgin (2010) studied the influence of different soil conditions and wall heights on the type of wall construction by numerical calculations.

The shaking table test is one of the most reliable measures to study the dynamic response of the soil-prevention structures. At present, scholars have improved the seismic design theories and methods of the various retaining structures by this method (Hong et al., 2005; Lin et al., 2015; Matsuo et al., 1998). Multi-anchor cable antislide piles (hereinafter referred to as multi-anchor piles) are one of the effective measures applied to geotechnical engineering reinforcement. According to a survey of the Wenchuan earthquake in 2008, this retaining structure showed good seismic performance in the earthquake (Zhang et al., 2012). However, there is little research reference on multi-anchor piles at home and abroad. The earliest use of the multi-anchor piles was in the landslide control project of the Jietai Temple in Beijing, China (Wang et al., 2007). By comparing the multi-anchor piles with the single-anchor piles, Deng (2007) concluded that the multi-anchor piles have great advantages in force, deformation control and economy. Wu et al. (2021) and Pai and Wu (2021) did basic research on the optimal seismic performance of the multi-anchor piles by shaking table tests, but their research objects mainly focused on the acceleration and dynamic earth pressure response of the multi-anchor piles. No research on the damage mechanism of the multi-anchor piles from the perspective of energy has ever been reported in the previous literatures.

At present, there are several methods to identify structural damage, such as wavelet analysis and intrinsic mode analysis, but each of these methods has different drawbacks when dealing with nonstationary signals (Fan et al., 2017). The Hilbert–Huang transform (HHT) has excellent time-frequency resolution, which can make up the defects of the traditional signal processing methods (Fan et al., 2017; Jia, 2008). In addition, HHT has been widely used in structural damage detection because of its excellent time-frequency analysis ability (Frank Pai and Palazotto, 2008; Song et al., 2020; Xin-Quan et al., 2017), ocean engineering (Veltcheva & Guedes Soares, 2016), and damage identification of earthquake landslide (Fan et al., 2016; Song et al., 2021). Jia (2008) used the HHT method to propose a general calculation program for structural damage caused by sea waves. WenQin et al. (2016) studied the damage modes of carbon fiber-reinforced plastics (CFRP) using HHT. In general, the HHT method has good applicability and efficiency in processing seismic signals. However, the analysis of the seismic damage mechanism of multi-anchor piles based on an energy perspective needs to be further discussed.

Therefore, in this study, the shaking table test was conducted with energy-dissipation springs as the optimal design scheme of anchor heads, and HHT and marginal spectrum energy were proposed to analyze the seismic damage mechanism of the multi-anchor piles. Based on the seismic energy response of the multi-anchor piles at different elevations, the energy spectrum difference of seismic damage of multi-anchor piles was analyzed, and the damping performance of multi-anchor piles was clarified. Combined with the distribution law of PFSA, the applicability of the Hilbert spectrum and Hilbert marginal spectrum in analyzing the seismic damage of multi-anchor piles was discussed in depth.

ENERGY-BASED METHOD OF SEISMIC MULTI-ANCHOR ANTISLIP PILES DAMAGE

HHT and the theory of marginal spectrum

The HHT is a stochastic signal processing method proposed by Huang in 1998, which mainly consists of two parts: the empirical modal decomposition (EMD) method and Hilbert spectrum analysis (HSA) (Huang et al., 1998), the whole process is as follows Figure 1. EMD can decompose the signal into a finite number of intrinsic mode functions and residual terms r (t), whose mathematical expression is as follows: 1$x(t)=\sum _{i=1}^n{f}_{\mathrm{im}i}\left(t\right)+r(t).$

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After EMD decomposition, the instantaneous frequency $w_i\left(t\right)$ corresponding to each order $f_{\mathrm{im}i}$ (intrinsic mode function) is calculated by the Hilbert transform. Then $a_i(t,w_i)$ represents the amplitude of the i-th order $f_{\mathrm{im}}$ at t time corresponding to the frequency of $w_i\left(t\right)$. All $a_i(t,w_i)$ are combined to realize the time-frequency distribution $\mathit{H}(t,\mathrm{w})$ of the whole signal amplitude (energy), which is called the HHT spectrum. Its mathematical expression is: 2$H(t,w)=\sum _{i=1}^n{a}_i(t,w_i).$

The HHT marginal spectrum $h\left(w\right)$ can be obtained by integrating H (w, t) in the time domain. The expression of $h\left(w\right)$ is: 3$h(w)={\int }_0^{T}H(w,t)\mathrm{d}t.$

The marginal spectrum quantitatively characterizes the distribution of the signal energy on the frequency axis. The amplitude corresponding to a certain frequency of the Hilbert marginal spectrum represents the vibration at that frequency throughout the signal, and the specific moment when the vibration at that frequency appears is determined by the Hilbert spectrum. The damage characteristics of the engineering structures can be elucidated by the Hilbert marginal spectrum from the perspective of energy transfer characteristics.

Identification method for the seismic failure of multi-anchor antislip piles

Referring to the energy-based method of slope failure identification, when under a certain level of seismic load, the seismic damage at any position of the slope will lead to the abnormal transfer of energy. In other words, the peak marginal spectrum amplitude (PMSA) will decrease rapidly (Li et al., 2012). Therefore, for multi-anchor piles, if the PMSA increases with the rise of pile height, it indicates that there is no dynamic damage to the multi-anchor piles. On the contrary, if the PMSA at a certain position of the pile decreases significantly, it indicates that seismic damage occurs at that position of the pile. Combined with the dynamic failure phenomenon of the slope, the seismic damage evolution process and mechanism of the multi-anchor piles can be elucidated.

SHAKING TABLE TEST DESIGN

Purpose of the test

A seismic damage identification method of the multi-anchor piles based on HHT and marginal spectrum was proposed to discuss the damping performance of the multi-anchor piles. The damage mechanism of the multi-anchor piles was elucidated by using the energy transfer characteristics of the Hilbert marginal spectrum.

Overview of shaking table

The shaking table test was carried out at China Railway Science and Technology Research and Development Center in Gansu Province. The table size is 2 m × 1 m, and the model box size is 150 cm × 96 cm × 120 cm (length × width × height). The maximum load capacity of the shaking table is 3000 kg, with the maximum speed being 0.7 m/s, the frequency 0.5–50 Hz, the displacement range ±75 mm, and the maximum acceleration 1.0 m/s². A 64-channel data acquisition system was used in this test, with a maximum error of ≤0.5%, and data acquisition and signal monitoring are performed simultaneously.

In this study, the following measures were taken to reduce the effect of the model box boundary effect on the test results (Dou & Byrne, 1997): A polyethylene foam with a thickness of 5 cm was placed between the boundary of the model box and the soil as a flexible boundary to reduce the influence of seismic wave reflection on the test results (Kostyukov, 1977). A layer of vaseline was applied to the surface of the box on both sides parallel to the excitation direction, which could reduce the contact friction between the model and the surface of the box on both sides (Shaw, 2013). A polystyrene foam board was placed between the tunnel and the two side walls to reduce the effect of boundary seismic reflection on the tunnel structure. A layer of gravel soil with a particle size of 2 cm and a thickness of 5 cm was laid at the bottom of the model box to reduce the relative sliding between the soil and the bottom plate of the box (Xu, 2009).

Design and manufacture of slope model

Due to the fact that there are few cases of tunnels orthogonally crossing the sliding surface under the action of the earthquake, the test was not designed according to the actual engineering prototype. However, considering the geometric similarity relationship and the possible application of this model in some practical projects, this study also referred to the previous scholars' research on the stress mode of the orthogonal system of tunnel landslide (Wu et al., 2016), and the test model was designed according to the geometric similarity ratio C_L = 100. Figure 2 shows the complete test model before the test.

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In the study of the dynamic response law of the soil-pile system, its additional counterweight is difficult to determine, so the gravity distortion model is adopted (Ramu et al., 2013). On the basis of comprehensive consideration of various factors, the similarity of gravity acceleration is neglected. With physical dimension, density, and acceleration as the basic physical quantities, the similarity ratio of other physical quantities is derived according to $\text{Buckingham}\mathrm{\pi }$ theorem. Some basic similar parameters are listed in Table 1.

Table 1 Similarity ratio of shaking table test

Attribute types	Physical quantity	Similarity ration	Dimension (MLT)	Ratio of similitude
Material properties	Elastic modulus (E)	$C_E=C_{\rho }{C}_L$	[M][L]−1[T]−2	1/100
Density ($\rho$)	$C_{\rho }=1$	[M][L]−3	1/1
Stress ($\sigma$)	$C_{\mathrm{\sigma }}=C_L$	[M][L]−1[T]−2	1/100
Strain ($\epsilon$)	$C_{\epsilon }=1$	-	1
Force of cohesion (c)	$C_c=C_L$	[M][L]−1[T]−2	1/100
Angle of internal friction (φ)	$C_{\phi }=1$	-	1
Poisson ratio ($\mu$)	$C_{\mu }=1$	-	1
Geometric characteristic	physical dimension (L)	$C_L=100$	[L]	1/100
Dynamic characteristic	Acceleration (a)	$C_a=1$	-	1
Time (t)	$C_t=1$	[T]	1
Velocity (v)	$C_v=C_{\rho }{C}_L{C}_a$	[L][T]−1	1/100
Frequency ($f$)	$C_f={C_t}^{-1}$	[T]−1	1

Similar materials in this test were selected mainly with reference to previous studies (Lai et al., 2014). Based on the similar relationship between the test model and the prototype materials, the ratio of bedrock materials was determined as m_quartz : m_clay : m_barite : m_gypsum : m_water = 80 : 30 : 120 : 5 : 10; the ratio of sliding surface material was m_quartz : m_clay : m_talcum : m_water = 15 : 40 : 30 : 10; the ratio of slip material was m_clay : m_water = 45 : 6; the ratio of multi-anchor piles material was m_gypsum : m_water = 2 : 8. Based on the similar material properties, gypsum was used as a simulation material for C30 tunnel lining with a ratio of m_gypsum : m_water = 1.1 : 1.0 (Li et al., 2021). The anchor cable was simulated by a screw rod with a diameter of 0.6 cm and a length of 70 cm (30 cm in the anchorage section and 40 cm in the free section). The specific locations of the multi-anchor piles, anchor cables, and tunnel structure are shown in Figure 3a–c, and the complete test model is shown in Figure 4. The appropriate ratio of similar materials in the model is mainly determined by indoor direct shear tests, which mainly satisfies c and φ similarity (Liu, 2012). The specific parameters of the test materials are shown in Table 2.

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Table 2 Physical and mechanical parameters of simulated materials

Material category	Weight capacity (g/cm3)	Modulus of elasticity E (GPa)	Compressive strength ${\sigma }_{\mathrm{c}}$ (MPa)	Tensile strength ${\sigma }_{\mathrm{t}}$ (MPa)	Force of cohesion c (kPa)	Angle of internal friction φ (°)
Bedrock	15.65	0.03	0.12		25.07	39.32
Sliding belt	12.51				5.68	39.81
Sliding mass	13.77				17.94	38.10
Antislide pile	20.40		4.50
Lining		0.77	0.58	0.05

The optimized multi-anchor piles are to set the energy-dissipation springs device at the anchor head, and the unoptimized multi-anchor piles are not optimized by the energy-dissipation springs. In this test, Alloy Steel Die Spring 65 Mn springs were used as the optimized structure of the anchor head. The outer diameter and inner diameter of the springs are 16 and 8 mm respectively, with the length being 25 mm, the ultimate compression ratio being 50%, and the ultimate pressure being 206 N. Due to the characteristics of fatigue resistance and good toughness, it is a commonly used optimization mold for antislide pile anchor heads in shaking table tests, as shown in Figure 3a.

Measurements

To reduce the side boundary effect, acceleration sensors were arranged in the middle antislide piles on both sides of the model box. The acceleration sensors are manufactured by Donghua Testing Co., Ltd. The frequency range and sensitivity in the horizontal direction are 0–900 Hz and 193 mV/g, respectively, and the measurement range (peak) is ±5 g. The layout of the acceleration sensor measuring points is shown in Figure 5.

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Input motions

EL Centro wave was selected as the loading type of seismic wave in this test (Towhata, 1996). The seismic wave acceleration was set to 0.10g, 0.15g, 0.20g, and 0.30g, and the loading direction was set to the horizontal direction (x-direction).

Before loading, a 0.05g white noise was input to test the damage to the system, that is, sine sweep (Chang et al., 2016). According to the previous research, the influence of the original waveform on the test results can be neglected when studying the dynamic response between soil-structure. Therefore, the input seismic wave is not compressed in this test (Cesca & Grigoli, 2015). The acceleration amplitude-time curve and Fourier spectrum are shown in Figure 6.

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ENERGY IDENTIFICATION OF MULTI-ANCHOR PILE DAMAGE

The PSHEA analysis of multi-anchor piles

To study the relationship between the frequency components of seismic waves and the dynamic response of the multi-anchor piles, the EL Centro wave action of 0.10g was taken as an example to obtain the acceleration amplitude-time curves and the corresponding Fourier spectrums of multi-anchor piles, as shown in Figure 7. Three main frequencies can be identified from Figure 7, which are f₁(8–10 Hz), f₂(12–16 Hz), and f₃(41–44 Hz). However, f₁ is similar to the input EL Centro wave (Figure 6b). In other words, f₁ is mainly caused by the input waveform. Accordingly, it can be inferred that f₂ and f₃ are the natural frequencies of the multi-anchor piles. The PFSA of all measuring points of multi-anchor piles at different natural frequencies were plotted using Surfer10 software, as shown in Figure 8. It can be seen from Figure 8 that the low-frequency component f₂(12–16 Hz) mainly induces the local response of the multi-anchor piles. The dynamic response of the multi-anchor piles on the optimized side is mainly concentrated in the top area of the pile, while the dynamic response on the unoptimized side is mainly concentrated in the top and middle areas of the pile. The high-frequency component f₃(41–44 Hz) mainly induces the overall response of the multi-anchor piles.

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The acceleration data of the optimized side S01 measuring point under the action of 0.10g EL Centro wave was selected for EMD decomposition, and the corresponding instantaneous spectrum was obtained and shown in Figure 9. The Hilbert energy of the input waveform can be obtained by combining the instantaneous spectrum of all IMF components. By analyzing the Hilbert energy spectrum of seismic wave, the dynamic response characteristics of the multi-anchor piles in the time-frequency domain can be elucidated. The Hilbert energy spectrum of typical monitoring points (S01/S01', S03/S03', and S05/S05' within the bedrock, sliding surface, and sliding mass) of the pile when a 0.10g seismic wave is input are shown in Figure 10. As can be seen, the shape and scale of the peak seismic Hilbert energy amplitude (PSHEA) change significantly as the elevation of the multi-anchor pile measurement point on the optimized side increases (S01 → S05), specifically in the number and size of the peak at measurement point S05, showing a significant elevation amplification effect. With the increase of the elevation of the multi-anchor pile measuring points on the unoptimized side (S01' → S05'), the peak seismic Hilbert energy amplitudes gradually transform from a single peak to multi-peaks, and the peak value increases first and then decreases. A certain energy loss occurs in the top area of the pile, which is caused by the damage of the pile top area of the multi-anchor piles on the unoptimized side. The PSHEA of both piles are mainly concentrated in the time domain at 20–40 s and in the frequency domain at 30–50 Hz, which indicates that the energy-dissipation springs only have an effect on the distribution model of PSHEA along the pile, but less on the time and frequency.

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According to Figure 11, the PSHEA generally increases with the rise of pile elevation for all working conditions except for the 0.10g seismic intensity. It can also be seen that PSHEA displays a slow growth rate in the area below the sliding surface and a rapid growth in the area above the sliding surface. This indicates that the difference in physical properties between the sliding surface and bedrock leads to a redistribution of energy that amplifies the dynamic characteristics of the slope (Kumar & Kaur, 2014), resulting in a significant difference in the energy transfer between the multi-anchor pile within the sliding body and bedrock. In addition, as can be seen from Figure 11, the PSHEA of the multi-anchor pile on the unoptimized side is larger than that on the optimized side, indicating that the scalable deformation of the energy-dissipation springs changes the PSHEA distribution of multi-anchor pile. Accordingly, the multi-anchor pile presents a better damping performance.

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Since the damage of multi-anchor piles is triggered by slope sliding, the analysis of damage characteristics of multi-anchor piles can clearly identify the disaster evolution process of the slope. According to the frequency distribution range (30–50 Hz) of the seismic Hilbert energy spectrum and the analysis results of Figure 8, it can be obtained that the seismic Hilbert energy spectrum of the high-frequency component mainly reflects the overall damage characteristics of the pile. The variation of PSHEA with seismic excitation for multi-anchor piles was plotted as shown in Figure 12. As can be seen, under the excitation of small amplitude seismic wave (0.10g–0.15g), the PSHEA of each measuring point of multi-anchor piles increases linearly and gently with the seismic excitation, that is, the slope is in the stage of cumulative deformation. When the seismic excitation is 0.15g–0.20g, the PSHEA of each measuring point of the multi-anchor piles increases rapidly, and the cumulative damage of the pile continues to increase. At this time, the existing tensile cracks and seismic damage cracks at the top of the slope gradually deepen, expand and penetrate, and the whole slope is in the accelerated deformation stage. When the seismic excitation is 0.20g–0.30g, the PSHEA of each measuring point increases rapidly and reaches the peak value. The growth rate is obviously larger than that of the previous two stages, and the damage rate of the pile body increases significantly. The top of the slope collapses and dislocations, while the local collapse occurs on the surface of the slope, and the whole slope is in the stage of sliding failure. In summary, under the reciprocal action of different seismic intensities, the dynamic cumulative damage of the slope gradually increases, and the slope model shows the continuous effect of regional failure.

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The PMSA analysis of multi-anchor piles

To further study the variation characteristics of seismic wave energy of multi-anchor piles, the PMSA of multi-anchor piles was analyzed and shown in Figure 13. According to Figure 13, the marginal spectral amplitude is much larger in the range of 12–30 Hz, which indicates that the energy is concentrated on the low-frequency component. According to the analysis results in Figure 8, the local damage of the multi-anchor pile is mainly caused by the low-frequency component of the seismic wave. Therefore, the damage evolution process of multi-anchor piles can be clearly identified by using low-order natural frequency from the perspective of local damage. The PMSA with elevation is shown in Figure 14. As can be seen from Figure 14a, under the action of 0.10g ground motion intensity, the PMSA of the optimized side multi-anchor pile maintains a nearly linear growth trend with the increase of elevation, indicating that the pile body remains intact and does not show damage under this ground vibration intensity. Subsequently, the PMSA shows a decreasing trend at the measurement points with relative heights of 0.65 → 0.94 (S04 → S05) under the action of 0.15g and 0.20g ground motion intensity, which indicates that the optimized multi-anchor piles are damaged at the relative height of 0.65 → 0.94 (S04 → S05). From Figure 14b, it can be seen that the unoptimized multi-anchor pile's PMSA maintains a nearly linear growth trend below a relative height of 0.42 (S03') when the seismic excitation is 0.10g, 0.15g, and 0.20g. When the relative height exceeds 0.42 (S03'), the PMSA shows a nearly linear decreasing trend, which indicates that the pile is damaged at a relative height of 0.42 (S03'). In addition, when the input seismic intensity is 0.30g, the PMSA of the measuring point S02' has a cliff-like decline. Therefore, the damage of the unoptimized multi-anchor piles under the ground motion intensity first appears at a relative height of 0.42 (S03'), and the pile damage area further develops to the relative height of 0.22 (S02') when the ground motion intensity increases to 0.30g.

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In addition, the distribution of the multi-anchor pile's PMSA when inputting EL Centro waves of different intensities is shown in Figure 15. According to Figure 15a, when the seismic wave of 0.15g is input, the PMSA of multi-anchor pile maintains an upward trend with the increase of pile height, which indicates that there is no damage of the multi-anchor pile on the optimized side. However, when 0.20g and 0.30g seismic waves are input, the PMSA at the top of the pile is smaller than that at the anchor head of the pile, indicating that the damage has occurred at the top of the pile and thus resulted in abnormal energy transfer near the top of the pile. As shown in Figure 15b, the PMSA of the multi-anchor pile on the unoptimized side under 0.15g seismic intensity exhibits a local amplification effect in the middle of the pile. In other words, the PMSA in the top area of the pile is smaller than that in the middle of the pile, which indicates that the multi-anchor pile on the unoptimized side has produced the damage.

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As is revealed by the comparison of Figure 15a,b, the optimized side multi-anchor pile remains largely intact when the 0.15g seismic wave is input. Subsequently, damage occurs in the top area of the pile under 0.20g and 0.30g seismic waves. Accordingly, the optimized side multi-anchor pile shows the progressive cumulative damage in the top area of the pile. The multi-anchor pile on the unoptimized side is first damaged near the sliding zone under the action of a 0.15g seismic wave. However, due to the small ground motion intensity, the pile body is not damaged and still maintains certain pile integrity. However, when a 0.20g seismic wave is input, the damage area of the multi-anchor pile is shifting from the middle of the pile to the top direction of the pile. Then under the action of a 0.30g seismic wave, the damage area of the pile body extends downward to the bottom of the pile. In other words, the multi-anchor pile on the unoptimized side shows the progressive cumulative damage in the direction of the middle of the pile → top of the pile → bottom of the pile.

In summary, it can be seen that the multi-anchor pile exhibits a continuum effect of regional damage under different levels of earthquake intensity. The optimized side multi-anchor pile experiences damage in the top area of the pile, while the middle and the bottom area of the pile remains intact. The damage area of the multi-anchor pile on the unoptimized side gradually extends from the middle of the pile to the anchor head and bottom area under the reciprocal action of different seismic intensities. It indicates that the retractable deformation of the energy-dissipation springs improves the coordination of the multi-anchor pile's PMSA along the elevation, leading to its better pile integrity. In addition, it can be seen from Figure 15 that the response of PMSA of the multi-anchor pile is more prominent in the sliding surface area; in other words, the sliding surface area is sensitive to earthquakes and easy to become a weak link in the seismic design of the multi-anchor pile.

DISCUSSION

It is well known that the damage of the multi-anchor piles is related to slope failure. Therefore, the damage characteristics of the multi-anchor piles were analyzed by the macroscopic deformation of the slope. Under the action of a 0.10g seismic wave, the surface of the slope has small feather tension (traction) cracks development, and there is no obvious failure phenomenon of the whole slope. Accordingly, the 0.10g seismic action was not analyzed. According to Figure 16a, the damage of the multi-anchor pile under the action of 0.15g seismic wave is mainly concentrated in the pile top area. Thus combined with the analysis results of Figure 8b, it can be seen that the local damage in the pile top area is mainly triggered by f₂. Then under the action of a 0.20g seismic wave, the collapse area further extends to the top of the slope, and the collapse area continues to expand. According to Figure 16b, it can be seen that the soil on the slope surface has a large number of collapses along the depth direction of the multi-anchor piles, which leads to the damage area of the multi-anchor piles gradually developing to the deep direction of the slope. Combined with Figure 8b, it can be seen that this local damage phenomenon of multi-anchor piles is also triggered by f₂. Under the action of a 0.30g seismic wave, the whole slope collapses on a large scale, and the landslide boundary extends to the platform area at the top of the slope. The landslide thrust is transmitted downward along the sliding surface, resulting in the overall failure of multi-anchor piles. According to Figure 8b, the damage of the pile in this phase is mainly triggered by f₃. Therefore, due to the influence of spatial position relationship and the seismic effect of multi-anchor piles, there are regional differences in the parts of pile failure.

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The damage mechanism of the multi-anchor piles is elucidated according to energy transfer characteristics under different seismic intensities. Under continuous seismic action, reflection occurs when seismic waves reach the slope surface, resulting in much greater Hilbert energy in the sliding mass than in the bedrock (Figure 11). This energy difference leads to a different degree of dynamic response between the sliding mass and bedrock. That is to say, the elevation amplification effect and trending surface amplification effect of Hilbert energy of seismic wave cause the damage of multi-anchor piles in the pile top area first. With the increase of seismic wave intensity, the damage of multi-anchor piles gradually develops to the deep part of the slope, which eventually leads to the damage of the whole pile. This reveals that the seismic damage of the multi-anchor piles is a gradual accumulation process. According to the marginal spectrum energy distribution of Figure 13, the marginal spectrum energy in the low-frequency component (12–30 Hz) mainly causes local damage of multi-anchor piles. As the local damage area continues to expand, the seismic Hilbert energy of the high-frequency component (30–40 Hz) further amplifies the seismic response of the slope surface, which in turn leads to overall damage of the multi-anchored piles.

The correctness of the energy method for analyzing the damage of the multi-anchor piles is verified by the peak Fourier spectrum. Based on the fast Fourier transform (FFT) and Savitzky-Golay filter fitting methods, the Fourier spectrum of each measuring point in the multi-anchor piles under different seismic intensities is shown in Figure 17. The first natural frequency (f₂) is selected for analysis, and the variation law of PFSA of f₂ with seismic wave loading conditions is shown in Figure 18. PFSA increases gradually from 0.10g to 0.15g, and the slope is in the cumulative deformation stage. Subsequently, it also maintains a nearly linear growth trend under 0.15g–0.20g seismic intensity, but the growth rate is slow, and the slope deformation gradually extends to the deep direction of the pile. Under 0.20g–0.30g seismic intensity, the PFSA grows rapidly and massive collapse of the slope surface occurs, resulting in overall damage of multi-anchor piles. Combined with the analysis results of Figure 12, it can be seen that PFSA and PSHEA have a similar trend, which indicates that PFSA and PSHEA are more beneficial to identify the overall damage of the multi-anchor piles. However, all IMF components are included in the Hilbert energy spectrum, thus increasing the difficulty in accurately identifying the local damage area of multi-anchor piles. On the contrary, the marginal spectrum has abundant frequency components and high discrimination, which can clearly reflect the energy transfer and local seismic damage characteristics of the multi-anchor piles. Therefore, the Hilbert and marginal spectrum curves of seismic waves can be used to reveal the instability mode of multi-anchor piles and the regional response characteristics of particle spatial position.

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CONCLUSIONS

To study the damage mechanism of multi-anchor piles in tunnel landslide area under earthquake, the damping performance of multi-anchor piles was discussed. A seismic damage identification method of multi-anchor piles based on HHT and marginal spectrum was proposed by shaking table test. On this basis, this study came to the following conclusions.

• 1.

  The marginal spectrum of the low-frequency component (12–30 Hz) and the Hilbert energy spectrum of the high-frequency component (30–40 Hz) have the largest amplitudes and thus can be used to identify the local and overall damage of multi-anchor piles, respectively. The energy-dissipation springs have a greater effect on the distribution pattern of PSHEA along the pile than on time and frequency.

• 2.

  Under the action of repeated seismic waves, the multi-anchor piles on the optimized side exhibit the progressive cumulative failure of the pile top area, and the multi-anchor piles on the unoptimized side exhibit the continuous effect of spatial coupling deformation in the middle of the pile → pile top → pile bottom.

• 3.

  The damage mechanism of the multi-anchor piles can be elucidated by using seismic Hilbert spectrum and marginal spectrum energy. Under the 0.15g earthquake, the marginal spectrum energy of the low-frequency component first triggers the local damage of the pile top area. As the seismic intensity increases, the multi-anchor piles' damage area gradually develops to the deep part of the slope, and finally, the overall damage is triggered by the Hilbert spectrum energy of the high-frequency component under the action of a 0.30g earthquake. That is, the local and overall damage of the multi-anchor piles are controlled by the marginal spectrum and Hilbert energy spectrum, respectively.

• 4.

  The overall damage of the multi-anchor pile can be identified by the seismic Hilbert spectrum. Compared with the Hilbert energy spectrum, the local damage characteristics of the multi-anchor piles can be clearly identified by the marginal spectrum energy. The combination of the two can provide a reliable basis for the seismic damage identification of the multi-anchor piles in the tunnel across the landslide area.

AUTHOR CONTRIBUTIONS

Hong Wei: Conceptualization; methodology; data curation; writing – original draft preparation. Honggang Wu: Visualization; investigation; resources; project administration; supervision; funding acquisition. Guojun Ren: Conceptualization; methodology; validation; formal analysis. Lin Tang: Formal analysis; validation; conceptualization. Kang Feng: Visualization; investigation.

ACKNOWLEDGMENTS

The authors gratefully acknowledge the financial support by the National Key R&D Program of China (No. 2018YFC1504901); Gansu Province Youth Science and Technology Fund program, China (Grant No. 22JR5RA777); Natural Science Foundation of Gansu Province, China (Grant No. 21JR7RA738); Science and Technology Development Project of China Railway Research Institute Co. Ltd (2017-KJ008-Z008-XB); Science and technology development project of China Railway Ninth Bureau Group Co. Ltd (DLF-ML-JSFW-2021-09).

CONFLICT OF INTEREST

The authors declare no conflict of interest.

DATA AVAILABILITY STATEMENT

The data used to support the findings of this research are available from the corresponding authors upon request.

References

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Zhang YP. HHT Analysis of Blasting Vibration and Its Application. Doctor Thesis. Central South University; 2006.