## 2 Methods  
### Low-Strain Pile Integrity Test  
LSPIT, a nondestructive testing technique with the advantages of simple operation, lightweight equipment, and low cost, has rapidly developed since the 1980s [25]. Sonic Echo (SE) and Impulse Response (IR) testing are two of LSPIT's testing methodologies, and one-dimensional wave theory is the theoretical basis for their damage detection [26].  
This study used the impulse response method. The hammer hits the pile head, causing the stress wave to propagate along the pile. When some defects in the pile body change the pile cross-sectional impedance, the stress wave reflects. Figure 1 shows the principle of LSPIT. A sensor placed at the top of the pile receives and collects the stress wave signal inside the pile, and analyzing this signal identifies pile defects (where r0: radius of the model pile, r1: radius of the first defect, r2: radius of the second defect, h1: distance of the first defect location from the top of the pile, h2: distance of the second defect location from the top of the pile, l1: length of the first defect, l2: length of the second defect, and L: pile length).  
Figure 1: Low-strain pile integrity test principle.  
The pile can be reduced to a one-dimensional elastomer, presuming it is formed of a homogeneous material and has a length-to-slenderness ratio larger than or equal to 5 [27]. Consequently, the one-dimensional wave propagation equation can represent the stress wave propagation process (1).  
\[\frac{\partial^{2}u(x,t)}{\partial^{2}t}=c^{2}\frac{\partial^{2}u(x,t)}{ \partial^{2}x} \tag{1}\]  
where \(u(x,t)\) is the axial displacement of the mass point at depth \(x\) and time \(t\). \(c\) signifies the wave's propagation velocity, obtained from Equation (2), with \(E\) and \(\rho\) standing for the pile material's Young's modulus and mass density, respectively.  
\[c=\sqrt{E/\rho} \tag{2}\]  
If the pile met assumptions, the specified beginning circumstances [2] are introduced into Equation (1) to apply the one-dimensional wave propagation equation to examine the dynamic properties of the foundation pile. Following is the result of using the one-dimensional elastic rod equation of motion to derive the time domain curve of the base pile vibration velocity.  
\[V(x,t)=\frac{1}{2}\bigg{[}\frac{\partial\varphi}{\partial t}(x-ct)-c\varphi_{1 }(x-ct)\bigg{]}+\frac{1}{2}\bigg{[}\frac{\partial\varphi}{\partial t}(x+ct)+ c\varphi_{1}(x+ct)\bigg{]} \tag{3}\]  
The pile defect is idealized as a break in the wave impedance, and the wave impedance \(Z\) is calculated as \(Z=\) EA/C, where A is the pile's cross-sectional area. Impedance is a measure of how pile resistance changes with velocity; changes in both A and E cause impedance changes or discontinuities. When a stress wave encounters such a discontinuity, some of the stress waves are reflected, and some are transmitted. \(Z_{1}\) denotes the wave impedance of the entire section of the pile, and \(Z_{2}\) denotes the wave impedance of the defective section. Equation (4) is the relationship between the mass velocity of the reflected wave \(v_{r}\) and the mass velocity of the incident wave \(v_{i}\)[28].  
\[v_{r}=\frac{Z_{1}-Z_{2}}{Z_{1}+Z_{2}}v_{i} \tag{4}\]