# Error Analysis and Numeric Simulation Research of Parallel Seismic Method

Authors: Chao Lin; Cheng-Lin Zhang; Li Rao; Chuang Yang; Bin An; Bao–Feng Zhao; Tao Zhan; Xian-Jun Tan
Journal: Geofluids
 (2022)

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## Abstract

The parallel seismic method is an effective method to detect the length of pile. In the paper, based on the ray theory of Snell’s law, the two methods of determining the pile length by the first transmission P-wave in the side hole are discussed and modified. The error influencing factors of both two methods are then analysed. Finally, the finite element software ABAQUS was used to simulate the inspection of pile foundations by the parallel seismic method. The numerical simulation results were compared with the results of the error calculation theory to verify the applicability of the error calculation theory. The error calculation theory was then tested and applied in an engineering example. The results show that (1) in order to reduce errors, unreasonable data points need to be removed when fitting the inspection data points and (2) the larger the pile slenderness ratio, the smaller the distance between the side holes and the pile, and the more accurate the pile length results obtained.


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## Body

## 1. Introduction

For existing buildings with incomplete original design and construction information, it is necessary to carry out on-site tests on the foundation depth or pile length and the pile body quality when strengthening and renovating or assessing the remaining service life in order to extrapolate the existing load-bearing capacity of the foundation; for buildings that have been built and used for a short time, there will also be the issue of testing the length and quality of such foundation piles that have been connected to the superstructure when their foundation settlement appears abnormally large, or when the quality of their pile foundation is in doubt during the construction of the superstructure.For piles in newly built projects, as the tops of the piles are exposed, the more simple and convenient method of sonic transmission, core drilling method, and low strain reflected wave method can be used to test the integrity of the piles [1]. However, for the existing building foundation piles, where the top of the pile is connected to the superstructure, it is difficult to accurately detect the pile length and pile quality by conventional testing methods. Parallel seismic method is an effective means of detecting the length of existing building pile foundations, which has started to emerge in recent years [2–4], and its related theoretical research has also been steadily developing. Liao et al. [5] proposed an estimate method to derive the depth of pile foundations through the intersection of 2 fitted straight lines by the means of Snell’s law; Huang and Chen [6, 7] established a three-dimensional finite element model of the pile-soil system and analysed the application of this method in cement mixing piles; Chen and Zhao [8] established a simplified model of the first to P-wave transmission path under the homogeneous foundation and thus modified the evaluation method of the depth of existing pile foundations. Subsequently, Du et al. [9] further verified the reasonableness of the simplified model proposed by Chen and Zhao [8] through finite element simulation analysis. Based on the three-dimensional finite element model, Zhang et al. [10–12] investigated the effect of different excitation methods on the test results of parallel seismic method; Du et al. [13] proposed an improved parallel seismic test method to detect the unknown length and integrity of piles and derived an analytical solution of the t–d relationship for the initial arrival of PS waves under homogeneous and stratified soil conditions. Rashidyan et al. [14, 15] used the parallel seismic wave method to detect the depth of wooden bridge foundations and investigated the effect of striking method and foundation stiffness on the detection results. Huang and Ni [16] found that the pile length estimated by the PS method increased with the distance of the side hole to the pile, but this error could be reduced by applying a length correction factor.There are a variety of factors that can affect the results of pile testing. Yuan et al. [17] present a method of visualizing the displacement field of the soil around the laterally loaded pile by using transparent soil technology. Bai et al. [18] believe that the changes in the stress states are closely related to soil particle rearrangement and the transitions between different matter phases and derived a generalized effective stress principle. Yuan et al. [19] investigated the role of glass fibers and liquid-modified polyvinyl alcohol polymers in reinforcement of granite residual soils. Bai et al. [20] employed the smoothed particle hydrodynamics (SPH) method to simulate the heat transfer process in porous media at the pore scale and proposed a concise method to produce unsaturated media by simulating the wetting process in dry media. Yuan et al. [21] carried out analyzing the mechanical and deformational behavior of both the material of the laterally loaded piles and soil with groundwater level as a variable.How to analyse the pile length and pile body quality information from the measured depth-wave-time graph and how to improve the accuracy of the analytical solution require further in-depth study. This research team has carried out some preliminary research work related to concrete quality testing and loading damage in pile foundations [22]. Starting from the transfer law of stress waves in the pile body and soil, this article analyses the pile length information through the measured depth-wave-time graph to improve the accuracy of the analysis, focuses on the sources of error through the test judgment method to verify the validity, validates the theory through the finite element model, and verifies it in the practical engineering applications.


## 2. Theory of Parallel Seismic Method

Parallel seismic method utilizes the refraction waves carrying information to determine the pile length. A compression wave (P-wave) or shear wave (S-wave) is generated by striking the pile vertically or horizontally with a hand hammer on the top surface (or superstructure such as a bearing or pile cap attached to the top of the pile). The stress wave propagates along the pile body downwards, as shown in Figure1, and encounters the surrounding soil layer for transmission. A sensor is placed in a hole predrilled next to the pile to receive the transmitted wave signal, from which wave times at different depths are read and from which the first arrival time-depth relationship graph is plotted. When the sensor is below the bottom of the pile, the wave velocity will change and an inflection point will be shown on the time-depth diagram. Therefore, the length of the pile can be deduced from the position in the graph where the slope of the line changes. Additionally, the wave velocities obtained from the slope of the straight line above and below the time-depth diagram can be used to analyse the properties of the pile body material and the pile bottom soil, respectively.Figure 1
Schematic diagram of the principle of parallel seismic method.Currently, there are two main methods for determining the depth of the pile in the relevant scientific and technical literature at home and abroad, as shown in Figure2.Figure 2
Schematic diagram of the pile bottom depth determination method.Method 1 is to first fit the upper and lower regions of the time-depth relationship diagram into a straight line (L1 and L2) and then treat the depth of their intersection point as the depth of the pile bottom (Z in the figure).Method 2 is to make a parallel line to the lineL1 through the coordinate origin (0,0) to obtain the line L1′ and then treat the depth of the intersection of the line L1′ and the line L2 as the depth of the pile bottom (Z′ in the figure).It has been suggested that the slopes of the upper and lower lines about the time axis are the P-wave velocities of the pile body and the soil below the pile bottom, respectively. For the same test data, it is clear that the depth to the bottom of the pile determined by method 1 is generally greater than that of method 2. Whether the depth to the bottom of the pile can be accurately determined by the straight line intersection method requires more research.


## 3. Analysis of Stress Wave Transmission Law

### 3.1. Theoretical Formula for Determining Pile Length

Shu-Tao Liao et al. carried out the theoretical analysis of the transfer of stress waves, on the basis of which this paper further discusses the theoretical error in the method of pile length determination and considers the error caused by the tilt factor of the side hole. Suppose there is a pile partially or completely buried in the soil, as shown in Figure3, with the pile diameter r, the total pile length L, the pile length above ground level LA, the distance D from the side hole to the pile body, and the inclination angle α (assuming the side hole is only inclined to one direction). When the external excitation force acts right in the centre of the pile top, the excitation force is assumed to be a point source as the contact area is small relative to the pile body interface. The signals received in the side hole can be separated into two groups, signal group 1 above the bottom of the pile and signal group 2 below the bottom of the pile. The distance from signal B1 (signal group 1) to the top of the pile is denoted by H1, and the distance from signal Bn (signal group 2) to the top of the pile is denoted by Hn.Figure 3
Schematic diagram of the stress wave transmission law.According to the wave transmission law, the stress wave starts from the top of the pileO, passes through the lateral point A1 of the pile and finally reaches the point B1. The propagation distance of stress wave O⟶A1⟶B1 is denoted by S1. The propagation velocities of the stress wave in the pile and the soil are V1 and V2, respectively. The propagation of the stress wave at the intersection of the pile body and the soil should be in accordance with Snell’s law, which means
(1)sinθ1V1=sinθ2V2.In the above formula,θ1 and θ2 are the refraction angles of the stress wave in the pile medium and soil medium, respectively.The shortest distanceDa from inspection point B1 of the side hole to the side of the pile is given by
(2)Dα=D+H1−LAtanα.Then, there is(3)S1=H1−X2+r2+Dαsecθ2=H1−X1+rH1−X2+Dαsecθ2.In formula (3),
(4)X=Dα⋅tanθ2,(5)θ2=sin−1sinθ1⋅V2V1≈sin−1V2V1.Then, the total time for the P-wave to travel the distanceS1 to bottom point B1 is given by
(6)T1=H1−X1+r/H1−X2V1+Dαsecθ2V2.The related relationship between P-wave arrival time and depth is shown in Figure2. L1 is the signal reception point above the bottom of the pile, and the time formula of L1 is
(7)L1:T=H−X1+r/H−X2V1+Dαsecθ2V2.Similarly, assuming thatHn is the distance from the signal reception point Bn (signal group 2) to the top of the pile, the stress wave transmission distance Sn is O⟶An⟶Bn, which is given by
(8)Sn=L2+r2+Dα2+Hn−L2=L1+rL2+Hn−L1+DαHn−L2.Therefore, the propagation time of the distanceSn is
(9)Tn=L1+r/L2V1+Hn−L1+Dα/Hn−L2V2.Time formula ofL2 in Figure 2 is given by
(10)L2:T=L1+r/L2V1+H−L1+Dα/H−L2V2.The depth of the signal reception point at the intersection ofL1 and L2 can be obtained by combining equations (7) and (10) as follows:
(11)L1:T=H−X1+r/H−X2V1+Dαsecθ2V2,L2:T=L1+r/L2V1+H−L1+Dα/H−L2V2.


### 3.2. Error Analysis of Pile Lengths Determined by Method 1

#### 3.2.1. Discussion ofL1

WhenH−X>>r and 1+r/H−X2≈1, the equation for L1 can be simplified to a linear equation as follows:
(12)L1:T=H−XV1+Dαsecθ2V2.For the side hole test point near the ground,L1 is the curve equation at this point as the condition H−X>>r does not hold. Assuming r=1, LA=0.5, α=0°, D=1, V1=4000, and V2=1000, the error curve is obtained as shown in Figure 4.Figure 4
Simplify item error curve.From Figure4, it can be seen that the error is already less than 2% when H≥5r. Also the fitting of a straight line L1 from the test data points in Figure 2 does not cause too much error.


#### 3.2.2. Discussion ofL2

WhenL>>r, H−L>>Dα, and 1+Dα/H−L2≈1, the equation for L2 can be simplified to a linear equation as follows:
(13)L2:T=LV1+H−LV2.For the side hole test point near the pile bottom, the equation simplification condition does not hold. The error is calculated to be less than 2% whenH−L≥5Da. Therefore, the fitting of L2 straight line should be made with the test data points far away from the pile bottom.


#### 3.2.3. Error of Method 1

After simplification, bothL1 and L2 are straight lines. The intersection of these two straight lines, Z1, is the depth of the pile bottom, Hp, as determined by method 1. So the formula of Hp is obtained as follows:
(14)Hp=L+V1V1−V2Dαsecθ2−V2V1−V2X.The actual pile lengthL can be derived from the above equation:
(15)L=Hp−V1V1−V2Dαsecθ2−V2V1−V2X.In the equation above,V1/V1−V2Dαsecθ2−V2/V1−V2X is defined as the correction factor for the depth of the pile bottom determined by method 1. The more slender the pile to be tested and the smaller the side hole distance, the more accurate the test result will be.


#### 3.2.4. Field Operation Errors due to the Tilted Side Hole

In the field inspection, a time-depth diagram can be plotted through method 1 based on the inspection data. When the actual side hole is inclined at an angleα and the plotted inspection depth is determined as if the side hole were not inclined, the intersection depth value in the plotted time-depth diagram is larger than the depth value calculated by equation (13), as shown in Figure 5. Also in the field site inspection, the depth of the test point is calculated from the ground level, ignoring the height of the pile foundations above the ground level LA. Combining the above factors, the intersection depth obtained by method 1 plotting has the following conversion relationship with the theoretically calculated intersection depth:
(16)Hp=Hp,inclined×cosα+LA,where Hp is the theoretical intersection depth and Hp,inclined is the actual intersection depth obtained by plotting through method 1.Figure 5
The schematic diagram of field operation errors due to tilted side holes.


### 3.3. Error Analysis of Pile Lengths Determined by Method 2

In method 2, the lineL1′ is obtained by translating the straight line L1 along the time axis to the origin of the coordinates in the time-depth diagram. And formula (11) can be transformed into
(17)L1′:T=H1−tanaV1+Htanαsecθ2V2.When equations (13) and (17) are combined, the depth of point Z1′, the intersection of line L1′ with line L2, can be obtained. The depth of point Z1′ determined through method 2 is the depth of the pile bottom Hp′.
(18)Hp′=LV2−V1V21−tanα+V1tanαsecθ2−1,(19)L=Hp′V21−tanα+V1tanαsecθ2−1V2−V1.L is defined as the correction factor for the depth of the pile bottom determined through method 2. When α=0, L=Hp′. This shows that the pile bottom depth determined through method 2 is correct.


## 3.1. Theoretical Formula for Determining Pile Length

Shu-Tao Liao et al. carried out the theoretical analysis of the transfer of stress waves, on the basis of which this paper further discusses the theoretical error in the method of pile length determination and considers the error caused by the tilt factor of the side hole. Suppose there is a pile partially or completely buried in the soil, as shown in Figure3, with the pile diameter r, the total pile length L, the pile length above ground level LA, the distance D from the side hole to the pile body, and the inclination angle α (assuming the side hole is only inclined to one direction). When the external excitation force acts right in the centre of the pile top, the excitation force is assumed to be a point source as the contact area is small relative to the pile body interface. The signals received in the side hole can be separated into two groups, signal group 1 above the bottom of the pile and signal group 2 below the bottom of the pile. The distance from signal B1 (signal group 1) to the top of the pile is denoted by H1, and the distance from signal Bn (signal group 2) to the top of the pile is denoted by Hn.Figure 3
Schematic diagram of the stress wave transmission law.According to the wave transmission law, the stress wave starts from the top of the pileO, passes through the lateral point A1 of the pile and finally reaches the point B1. The propagation distance of stress wave O⟶A1⟶B1 is denoted by S1. The propagation velocities of the stress wave in the pile and the soil are V1 and V2, respectively. The propagation of the stress wave at the intersection of the pile body and the soil should be in accordance with Snell’s law, which means
(1)sinθ1V1=sinθ2V2.In the above formula,θ1 and θ2 are the refraction angles of the stress wave in the pile medium and soil medium, respectively.The shortest distanceDa from inspection point B1 of the side hole to the side of the pile is given by
(2)Dα=D+H1−LAtanα.Then, there is(3)S1=H1−X2+r2+Dαsecθ2=H1−X1+rH1−X2+Dαsecθ2.In formula (3),
(4)X=Dα⋅tanθ2,(5)θ2=sin−1sinθ1⋅V2V1≈sin−1V2V1.Then, the total time for the P-wave to travel the distanceS1 to bottom point B1 is given by
(6)T1=H1−X1+r/H1−X2V1+Dαsecθ2V2.The related relationship between P-wave arrival time and depth is shown in Figure2. L1 is the signal reception point above the bottom of the pile, and the time formula of L1 is
(7)L1:T=H−X1+r/H−X2V1+Dαsecθ2V2.Similarly, assuming thatHn is the distance from the signal reception point Bn (signal group 2) to the top of the pile, the stress wave transmission distance Sn is O⟶An⟶Bn, which is given by
(8)Sn=L2+r2+Dα2+Hn−L2=L1+rL2+Hn−L1+DαHn−L2.Therefore, the propagation time of the distanceSn is
(9)Tn=L1+r/L2V1+Hn−L1+Dα/Hn−L2V2.Time formula ofL2 in Figure 2 is given by
(10)L2:T=L1+r/L2V1+H−L1+Dα/H−L2V2.The depth of the signal reception point at the intersection ofL1 and L2 can be obtained by combining equations (7) and (10) as follows:
(11)L1:T=H−X1+r/H−X2V1+Dαsecθ2V2,L2:T=L1+r/L2V1+H−L1+Dα/H−L2V2.


## 3.2. Error Analysis of Pile Lengths Determined by Method 1

### 3.2.1. Discussion ofL1

WhenH−X>>r and 1+r/H−X2≈1, the equation for L1 can be simplified to a linear equation as follows:
(12)L1:T=H−XV1+Dαsecθ2V2.For the side hole test point near the ground,L1 is the curve equation at this point as the condition H−X>>r does not hold. Assuming r=1, LA=0.5, α=0°, D=1, V1=4000, and V2=1000, the error curve is obtained as shown in Figure 4.Figure 4
Simplify item error curve.From Figure4, it can be seen that the error is already less than 2% when H≥5r. Also the fitting of a straight line L1 from the test data points in Figure 2 does not cause too much error.


### 3.2.2. Discussion ofL2

WhenL>>r, H−L>>Dα, and 1+Dα/H−L2≈1, the equation for L2 can be simplified to a linear equation as follows:
(13)L2:T=LV1+H−LV2.For the side hole test point near the pile bottom, the equation simplification condition does not hold. The error is calculated to be less than 2% whenH−L≥5Da. Therefore, the fitting of L2 straight line should be made with the test data points far away from the pile bottom.


### 3.2.3. Error of Method 1

After simplification, bothL1 and L2 are straight lines. The intersection of these two straight lines, Z1, is the depth of the pile bottom, Hp, as determined by method 1. So the formula of Hp is obtained as follows:
(14)Hp=L+V1V1−V2Dαsecθ2−V2V1−V2X.The actual pile lengthL can be derived from the above equation:
(15)L=Hp−V1V1−V2Dαsecθ2−V2V1−V2X.In the equation above,V1/V1−V2Dαsecθ2−V2/V1−V2X is defined as the correction factor for the depth of the pile bottom determined by method 1. The more slender the pile to be tested and the smaller the side hole distance, the more accurate the test result will be.


### 3.2.4. Field Operation Errors due to the Tilted Side Hole

In the field inspection, a time-depth diagram can be plotted through method 1 based on the inspection data. When the actual side hole is inclined at an angleα and the plotted inspection depth is determined as if the side hole were not inclined, the intersection depth value in the plotted time-depth diagram is larger than the depth value calculated by equation (13), as shown in Figure 5. Also in the field site inspection, the depth of the test point is calculated from the ground level, ignoring the height of the pile foundations above the ground level LA. Combining the above factors, the intersection depth obtained by method 1 plotting has the following conversion relationship with the theoretically calculated intersection depth:
(16)Hp=Hp,inclined×cosα+LA,where Hp is the theoretical intersection depth and Hp,inclined is the actual intersection depth obtained by plotting through method 1.Figure 5
The schematic diagram of field operation errors due to tilted side holes.


## 3.2.1. Discussion ofL1

WhenH−X>>r and 1+r/H−X2≈1, the equation for L1 can be simplified to a linear equation as follows:
(12)L1:T=H−XV1+Dαsecθ2V2.For the side hole test point near the ground,L1 is the curve equation at this point as the condition H−X>>r does not hold. Assuming r=1, LA=0.5, α=0°, D=1, V1=4000, and V2=1000, the error curve is obtained as shown in Figure 4.Figure 4
Simplify item error curve.From Figure4, it can be seen that the error is already less than 2% when H≥5r. Also the fitting of a straight line L1 from the test data points in Figure 2 does not cause too much error.


## 3.2.2. Discussion ofL2

WhenL>>r, H−L>>Dα, and 1+Dα/H−L2≈1, the equation for L2 can be simplified to a linear equation as follows:
(13)L2:T=LV1+H−LV2.For the side hole test point near the pile bottom, the equation simplification condition does not hold. The error is calculated to be less than 2% whenH−L≥5Da. Therefore, the fitting of L2 straight line should be made with the test data points far away from the pile bottom.


## 3.2.3. Error of Method 1

After simplification, bothL1 and L2 are straight lines. The intersection of these two straight lines, Z1, is the depth of the pile bottom, Hp, as determined by method 1. So the formula of Hp is obtained as follows:
(14)Hp=L+V1V1−V2Dαsecθ2−V2V1−V2X.The actual pile lengthL can be derived from the above equation:
(15)L=Hp−V1V1−V2Dαsecθ2−V2V1−V2X.In the equation above,V1/V1−V2Dαsecθ2−V2/V1−V2X is defined as the correction factor for the depth of the pile bottom determined by method 1. The more slender the pile to be tested and the smaller the side hole distance, the more accurate the test result will be.


## 3.2.4. Field Operation Errors due to the Tilted Side Hole

In the field inspection, a time-depth diagram can be plotted through method 1 based on the inspection data. When the actual side hole is inclined at an angleα and the plotted inspection depth is determined as if the side hole were not inclined, the intersection depth value in the plotted time-depth diagram is larger than the depth value calculated by equation (13), as shown in Figure 5. Also in the field site inspection, the depth of the test point is calculated from the ground level, ignoring the height of the pile foundations above the ground level LA. Combining the above factors, the intersection depth obtained by method 1 plotting has the following conversion relationship with the theoretically calculated intersection depth:
(16)Hp=Hp,inclined×cosα+LA,where Hp is the theoretical intersection depth and Hp,inclined is the actual intersection depth obtained by plotting through method 1.Figure 5
The schematic diagram of field operation errors due to tilted side holes.


## 3.3. Error Analysis of Pile Lengths Determined by Method 2

In method 2, the lineL1′ is obtained by translating the straight line L1 along the time axis to the origin of the coordinates in the time-depth diagram. And formula (11) can be transformed into
(17)L1′:T=H1−tanaV1+Htanαsecθ2V2.When equations (13) and (17) are combined, the depth of point Z1′, the intersection of line L1′ with line L2, can be obtained. The depth of point Z1′ determined through method 2 is the depth of the pile bottom Hp′.
(18)Hp′=LV2−V1V21−tanα+V1tanαsecθ2−1,(19)L=Hp′V21−tanα+V1tanαsecθ2−1V2−V1.L is defined as the correction factor for the depth of the pile bottom determined through method 2. When α=0, L=Hp′. This shows that the pile bottom depth determined through method 2 is correct.


## 4. Finite Element Simulation Analysis and Verification

### 4.1. Model and Parameters

In recent years, the commercial software ABAQUS has been well used in the numerical simulation of underground engineering [23]. Therefore, we used ABAQUS to build a finite element model to numerically simulate the pile foundation inspection process. A reasonable artificial boundary was set to simulate and analyse the time-depth relationship between the first and P-waves in the parallel seismic wave signal. In the finite element model, the pile diameter D is taken as 1 m and the pile length L is taken as 20 m. After trial calculations, the following grid areas were selected: 7.5 times the pile diameter in the horizontal radial direction, 2 times the pile length in the vertical direction, and the side holes were arranged in the range of 0-5 m on the pile side. CAX4R cells were used for the finite elements and CINX4 cells were used for the boundary infinite elements [5, 6]. Based on symmetry, half of the model was taken along the centreline of the hammer load range for analysis, as shown in Figure 6.Figure 6
Schematic diagram of the finite element mesh for pile foundation inspection by parallel seismic method.In order to obtain a clear transmission waveform, both the pile and the soil are simplified to a homogeneous linear elastic material in the analysis. For a normal section of the pile, the density is taken as 2300 kg/m3 with the elasticity modulus of 33.1 GPa and the Poisson’s ratio of 0.2. For homogeneous soils, the density is taken as 1900 kg/m3 with the elasticity modulus of 0.213 GPa and the Poisson’s ratio of 0.4. The corresponding wave velocities calculated from the above material properties are shown in Table 1.Table 1
Pile-soil parameters.
MaterialDensityρ (kg/m3)Elasticity modulus (GPa)Poisson ratioP-wave speed (m/s)S-wave speed (m/s)Pile shaft230033.10.23998.792448.75Homogeneous soils19000.2130.4490.13200.09The P-wave is the fastest, the S-wave is slightly slower, and the R-wave is the slowest. The monitoring point of the side hole is first stimulated by the P-wave, followed by the S-wave and R-wave. P-wave masses move vertically at a small speed, decay quickly, and take up a small amount of excitation energy. For the plasmas at a distance from the excitation point, the excitation velocity is mainly influenced by the slower R- and S-waves. Theoretically, the large amplitude of the S-wave plasmas makes the S-wave easier to be identified in the waveform diagram, but it is still difficult to judge the waveform arrival time due to the influence of waveform superposition. A line can be formed by connecting the time points at which the P-waveform starts. The depth corresponding to the turning point of the line is the depth of the pile bottom, as shown by the arrow at the fold in Figure7.Figure 7
Waveform combination time-depth diagram.


### 4.2. Analysis of the Effect of Distance from the Side Hole

The effect of the side hole distance on the pile length test results was assessed through simulation analysis of a finite element model. The parameters of the finite element model were taken as follows: the distance between the side hole and the edge of the pileD=0.1 m, 0.2 m, 0.5 m, 1 m, 1.5 m, 2.0 m, and 3.0 m in that order; the inclination angle of the side hole a=0; the distance of the vertical excitation point from the ground is 0.5 m; no defects in the pile body; homogeneous soil quality; and the rest of the parameters are the same as before.As shown in Figure8, the depth of the curve turning point increases as the distance between the side borehole and the pile side increases. Near depth H=0, the data points are distributed in a curved line, which does not conform to the assumption of a straight line L1. Near the depth H=20 m, the data points are curved and the curve becomes full as the distance D increases, which does not conform to the assumption of straight line L2. The results of the numerical simulations are in accordance with the aforementioned theoretical analysis.Figure 8
Time-depth comparison for different hole distance.By comparing method 1 and method 2 with the theoretical correction values, as shown in Figure9, it can be found that the error of method 1 increases with the increase of the side hole distance, while method 2 can better eliminate the influence of the error caused by the side hole distance through the use of translating straight lines, and the correction effect of the correction method is close to that of method 2.Figure 9
Comparison of the error-side hole distance for different methods of predicting pile bottom depth.


### 4.3. Analysis of the Influence of the Side Hole Inclination

The influence of the side hole inclination on test results was assessed through simulation analysis of a finite element model. The parameters of the finite element model were taken as follows: inclination angles of the side hole taken as 0, ±0.50, ±10, and ±20 in that order; the distance of the side hole is 1 m; no defects in the pile body; and homogeneous soil quality.Figure10 shows the final simulation results. It can be seen from the figure that the direction of the side hole inclination is the same as the direction of the data point offset and that the side hole inclination has a greater effect on the data point distribution of straight line L1 and almost no effect on straight line L2, which is consistent with the results of the previous discussion on straight line L1 and straight line L2.Figure 10
Time-depth diagram under the inclination of the side hole.As shown in Figure11, the error caused by the angle of inclination of the side hole cannot be eliminated. The error of the inclination correction value decreases as the inclination angle increases. The reduction in the error of the inclination correction value better reflects the actual pile bottom depth.Figure 11
Comparison of the error-side hole inclination for different methods of predicting pile bottom depth.


## 4.1. Model and Parameters

In recent years, the commercial software ABAQUS has been well used in the numerical simulation of underground engineering [23]. Therefore, we used ABAQUS to build a finite element model to numerically simulate the pile foundation inspection process. A reasonable artificial boundary was set to simulate and analyse the time-depth relationship between the first and P-waves in the parallel seismic wave signal. In the finite element model, the pile diameter D is taken as 1 m and the pile length L is taken as 20 m. After trial calculations, the following grid areas were selected: 7.5 times the pile diameter in the horizontal radial direction, 2 times the pile length in the vertical direction, and the side holes were arranged in the range of 0-5 m on the pile side. CAX4R cells were used for the finite elements and CINX4 cells were used for the boundary infinite elements [5, 6]. Based on symmetry, half of the model was taken along the centreline of the hammer load range for analysis, as shown in Figure 6.Figure 6
Schematic diagram of the finite element mesh for pile foundation inspection by parallel seismic method.In order to obtain a clear transmission waveform, both the pile and the soil are simplified to a homogeneous linear elastic material in the analysis. For a normal section of the pile, the density is taken as 2300 kg/m3 with the elasticity modulus of 33.1 GPa and the Poisson’s ratio of 0.2. For homogeneous soils, the density is taken as 1900 kg/m3 with the elasticity modulus of 0.213 GPa and the Poisson’s ratio of 0.4. The corresponding wave velocities calculated from the above material properties are shown in Table 1.Table 1
Pile-soil parameters.
MaterialDensityρ (kg/m3)Elasticity modulus (GPa)Poisson ratioP-wave speed (m/s)S-wave speed (m/s)Pile shaft230033.10.23998.792448.75Homogeneous soils19000.2130.4490.13200.09The P-wave is the fastest, the S-wave is slightly slower, and the R-wave is the slowest. The monitoring point of the side hole is first stimulated by the P-wave, followed by the S-wave and R-wave. P-wave masses move vertically at a small speed, decay quickly, and take up a small amount of excitation energy. For the plasmas at a distance from the excitation point, the excitation velocity is mainly influenced by the slower R- and S-waves. Theoretically, the large amplitude of the S-wave plasmas makes the S-wave easier to be identified in the waveform diagram, but it is still difficult to judge the waveform arrival time due to the influence of waveform superposition. A line can be formed by connecting the time points at which the P-waveform starts. The depth corresponding to the turning point of the line is the depth of the pile bottom, as shown by the arrow at the fold in Figure7.Figure 7
Waveform combination time-depth diagram.


## 4.2. Analysis of the Effect of Distance from the Side Hole

The effect of the side hole distance on the pile length test results was assessed through simulation analysis of a finite element model. The parameters of the finite element model were taken as follows: the distance between the side hole and the edge of the pileD=0.1 m, 0.2 m, 0.5 m, 1 m, 1.5 m, 2.0 m, and 3.0 m in that order; the inclination angle of the side hole a=0; the distance of the vertical excitation point from the ground is 0.5 m; no defects in the pile body; homogeneous soil quality; and the rest of the parameters are the same as before.As shown in Figure8, the depth of the curve turning point increases as the distance between the side borehole and the pile side increases. Near depth H=0, the data points are distributed in a curved line, which does not conform to the assumption of a straight line L1. Near the depth H=20 m, the data points are curved and the curve becomes full as the distance D increases, which does not conform to the assumption of straight line L2. The results of the numerical simulations are in accordance with the aforementioned theoretical analysis.Figure 8
Time-depth comparison for different hole distance.By comparing method 1 and method 2 with the theoretical correction values, as shown in Figure9, it can be found that the error of method 1 increases with the increase of the side hole distance, while method 2 can better eliminate the influence of the error caused by the side hole distance through the use of translating straight lines, and the correction effect of the correction method is close to that of method 2.Figure 9
Comparison of the error-side hole distance for different methods of predicting pile bottom depth.


## 4.3. Analysis of the Influence of the Side Hole Inclination

The influence of the side hole inclination on test results was assessed through simulation analysis of a finite element model. The parameters of the finite element model were taken as follows: inclination angles of the side hole taken as 0, ±0.50, ±10, and ±20 in that order; the distance of the side hole is 1 m; no defects in the pile body; and homogeneous soil quality.Figure10 shows the final simulation results. It can be seen from the figure that the direction of the side hole inclination is the same as the direction of the data point offset and that the side hole inclination has a greater effect on the data point distribution of straight line L1 and almost no effect on straight line L2, which is consistent with the results of the previous discussion on straight line L1 and straight line L2.Figure 10
Time-depth diagram under the inclination of the side hole.As shown in Figure11, the error caused by the angle of inclination of the side hole cannot be eliminated. The error of the inclination correction value decreases as the inclination angle increases. The reduction in the error of the inclination correction value better reflects the actual pile bottom depth.Figure 11
Comparison of the error-side hole inclination for different methods of predicting pile bottom depth.


## 5. Engineering Measurements

There is a filling pile with a diameter of 800 mm and a pile length of 27 m located in a saturated clay layer at a site in Huadu District, Guangzhou. The field pile length inspection was carried out with a DAQlink III distributed seismometer manufactured by Seismic Source, USA. During the inspection process, the pile inclination angleα=0 and the distance between the side hole and the inspection pile D=1.0 m. Figure 12 shows the inspection process in the field.Figure 12
Parallel seismic wave method field testing process.
(a)
Commissioning the apparatus(b)
Pressurising the sensor(c)
Placement in the side hole(d)
Pile side excitationThe waveform diagram was plotted for the signals received at each depth point, with the vertical coordinate being the sampling time and the horizontal coordinate being the depth of each checkpoint, as shown in Figure13. Based on the aforementioned method 1, the pile length Hp was estimated to be 28.0 m.Figure 13
Waveform diagram.The correction volume was used to correct the pile length test results. In China, the one-dimensional P-wave velocity of common concrete piles is 3800~4200 m/s, and the common saturated foundation soil wave velocity is 1300~1700 m/s. Taking the pile wave velocityV1=4000 m/s and the soil wave velocity V2=1500 m/s, then the following is derived:
(20)X=Dtanθ2=Dtansin−1V2V1=1.0×tansin−115004000=0.3844,(21)L=Hp−V1V1−V2Dsecθ2−V2V1−V2X=27.56m.The error in pile length without correction is 3.7%. The error in the corrected pile length is 2.07%. It can be seen that the corrected test results could reflect the true pile length more accurately.


## 6. Conclusions

Based on the above theoretical analysis and the results of the finite element model validation, the following useful conclusions can be obtained:In the process of fitting a straight line from the measured point data, the fitted data points for straight lineL1 should be selected close to the bottom of the pile, and the fitted data points for straight line L2 should be selected at the bottom of the side hole. Such a selection method will maximise the theoretical assumptions of the straight line intersection method for determining the depth of the pile bottom in order to improve the accuracy of the judgement.The larger the length-diameter ratio of the pile and the smaller the distance between the side hole and the pile, the more accurate the test results. The inclination angle of the side hole has a greater effect on the distribution of data points in straight lineL1 and has almost no effect on the distribution of data points in straight line L2.Method 1, the straight line intersection method, yields the pile bottom depth value that is greater than the actual pile bottom depth value, so the correction factor needs to be introduced to correct the result. Method 2, the translational straight line method, produces a pile bottom depth value that is theoretically consistent with the actual pile bottom depth value. When the correction factor about the inclination of the side hole is introduced according to actual needs, the corrected test results can be better improve the accuracy of the test.



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*Source: 9396616-2022-08-12.xml*