# Application Study of the High-Strain Direct Dynamic Testing Method
## 2 Theory of the High-Strain Direct Dynamic Testing Method
### Computation Theory
For any pile segment between two adjacent nodes \(x_{i}\) and \(x_{i+1}\), a force analysis can be conducted, as shown in \[u_{0,j+1} =u_{0,j}+\frac{\left(v_{0,j}+v_{0,j+1}\right)}{2}dt \tag{2}\]1 Based on the basic assumptions of this method, the dynamic resistance \(R_{dt,j}\) experienced by this pile segment at any given time can be determined as follows:  
\[u_{0,j+1} =u_{0,j}+\frac{\left(v_{0,j}+v_{0,j+1}\right)}{2}dt \tag{2}\]2  
where \(i\in[0,N-1];j\in[0,M-1]\). According to the basic assumptions of this method, the total soil resistance \(R\) during the vibration of any pile segment unit is composed of static soil resistance \(R_{S}\) and dynamic soil resistance \(R_{d}\). Affected by the soil resistance, the vertical vibration of the pile segment will decrease over time to a relatively static state. Therefore, the displacement of the pile segment unit reaches its peak value at the moment when the initial velocity drops to zero. In addition, the dynamic resistance of the pile segment also drops to zero and the static soil resistance at this moment will also reach its maximum. For a certain pile section with node depths from \(x_{i}\) to \(x_{i+1}\) in the pile, the average velocity of the pile section at any time can be calculated from the velocities of the upper and lower nodes:  
\[u_{0,j+1} =u_{0,j}+\frac{\left(v_{0,j}+v_{0,j+1}\right)}{2}dt \tag{2}\]3  
where \(i\in[0,N-1];j\in[1,M]\). At the moment of the first time \(\overline{v}_{i,j}=0\), the dynamic resistance \(R_{dt,j}\) at the moment obtained by Equation (10) is the pile side ultimate static resistance of this pile segment. By  
\[u_{0,j+1} =u_{0,j}+\frac{\left(v_{0,j}+v_{0,j+1}\right)}{2}dt \tag{2}\]4 Pile segment force diagram. adding the lateral ultimate static resistance of each pile segment, the lateral ultimate static resistance of the entire pile body \(R_{S}\) can be obtained as:  
\[u_{0,j+1} =u_{0,j}+\frac{\left(v_{0,j}+v_{0,j+1}\right)}{2}dt \tag{2}\]5  
For the pile segment unit located at the bottom of the pile, after obtaining the moment \(t=j\) when \(v_{N,i}=0\), the axial force \(F_{N,i}\) calculated at the moment is regarded as the ultimate static resistance \(R_{b}\) of this pile segment unit. In summary, the ultimate bearing capacity \(R\) of a single pile is the sum of the ultimate static resistance of the pile side \(R_{s}\) and the ultimate static resistance of the pile end \(R_{b}\):  
\[u_{0,j+1} =u_{0,j}+\frac{\left(v_{0,j}+v_{0,j+1}\right)}{2}dt \tag{2}\]6  
The high-strain direct dynamic testing method utilizes strain sensors distributed at various depths along the pile shaft and an accelerometer positioned at the pile head to directly determine the ultimate bearing capacity of a single pile based on the results of field hammer impact tests. The calculated bearing capacity is influenced solely by the material properties and dimensions of the pile itself. This method avoids the impact on the objectivity and reliability of results caused by relying on the experience of testers to select pile-soil interaction parameters, as seen in traditional techniques. Based on solving straightforward dynamic equilibrium equations, this method eliminates the need for the cumbersome steps of repeatedly solving and fitting the upward reflective wave from the pile base, thereby addressing the issue of multiple solutions present in traditional techniques.