# Application Study of the High-Strain Direct Dynamic Testing Method
## 2 Theory of the High-Strain Direct Dynamic Testing Method
### Computation Theory  
For sensors \(x_{n}(n\in[0,N])\) at any given depth, the average strain \(\epsilon_{i,j}\) and average acceleration \(a_{i,j}\) of the pile cross-section at that depth are obtained by averaging the measured data from each row of the measurement lines (the subscript \(i\) represents position and the subscript \(j\) represents time). When \(j=0\), \(t=dt\cdot\ j=0\), it indicates that pilling has not begun, and at this moment, all the variables are zero. When \(j>0\), it indicates that the hammer impacted the pile top, and at this moment, by performing a single integration of the pile top acceleration \(\alpha_{0,j}\) over, the velocity \(dt\) of the pile top can be obtained. A second integration yields the displacement \(u_{0,j}\). \[v_{0,j+1} =v_{0,j}+\frac{\left(a_{0,j}+a_{0,j+1}\right)}{2}dt \tag{1}\] \[u_{0,j+1} =u_{0,j}+\frac{\left(v_{0,j}+v_{0,j+1}\right)}{2}dt \tag{2}\]  
where \(j\in[0,M-1]\). \(v_{0,j+1},v_{0,j}=\) the velocity of sensor \(x_{0}\) at the moment time \(=j+1\) and time \(=j\). \(a_{0,j+1},a_{0,j}=\) the acceleration of sensor \(x_{0}\) at the moment time \(=j\) and time \(=j\). \(u_{0,j}=\) the displacement of sensor \(x_{0}\) at the moment time \(=j\). However, measurement devices typically have zero-offset errors, leading to significant cumulative errors in time integration of acceleration. This error can be corrected using the actual settlement value of the pile top \(L_{0}\) measured at the end of the test. Where the measured settlement value at the end of sampling is \(P_{d}\) and the difference between the calculated displacement \(u_{0,j}\) by twice integrating the acceleration and \(P_{d}\) is \(s_{d}\). Then, the displacement can be corrected using the following equations:  
\[U_{0,j=u_{0,j}-u_{0,j}-s_{d}}.\frac{j}{M}=u_{0,j}-\left(u_{0,j}P_{d}\right).\frac{j}{M} \tag{3}\]  
where \(j\in[0,M-1]\). The corrected velocity \(V_{0,j}\) can be obtained by differentiating the corrected displacement \(U_{0,j}\) once with respect to the sampling time interval. Similarly, the corrected acceleration \(A_{0,j}\):  
\[V_{0,0}=0 \tag{4}\] \[V_{0,j}=\frac{U_{0,j}-U_{0,j-1}}{dt}\] (5) \[A_{0,0}=0\] \[A_{0,j}=\frac{V_{0,j}-V_{0,j-1}}{dt}\]  
where \(j\in[0,M]\). Then, the corrected displacement of the pile top \(U_{0,j}\), combined with the measured average strain \(\epsilon_{i,j}\) along the pile body, can be used to derive the displacement \(u_{i,j}\) at any depth nodes along the pile:  
\[u_{i,j}=u_{i-1,j}+\frac{\left(\epsilon_{i,j}+\epsilon_{i-1,j}\right)}{2}(x_{i }-x_{i-1}) \tag{6}\]  
where \(i\in[1,N];j\in[1,M-1]\). Based on the displacement \(u_{i,j}\) obtained for each depth node along the pile body, the velocity \(v_{i,j}\) at each node can be calculated using difference formulas. Subsequently, the acceleration \(a_{i,j}\) at each node can be determined from these velocity values:  
\[v_{i,j}=\frac{u_{i,j}-u_{i,j-1}}{dt} \tag{7}\]  
\[a_{i,j}=\frac{v_{i,j}-v_{i,j-1}}{dt} \tag{8}\]  
where \(i\in[1,N];j\in[1,M-1]\). For a chosen sampling time interval \(dt\), the axial force in the pile body at each depth node can be calculated using the measured average strain \(\epsilon_{ij}\) at that node, as the following equation:  
\[u_{0,j+1} =u_{0,j}+\frac{\left(v_{0,j}+v_{0,j+1}\right)}{2}dt \tag{2}\]0  
where \(E\) is the elastic modulus of the pile, \(E=\rho c^{2}\), \(\rho\) is the density of the pile, and \(c\) is the wave velocity of the stress wave within the pile. \(A\) is the cross-sectional area of the pile. In this paper, compressive stress is considered positive, while tensile strain is considered negative. Using the above equations, the axial force \(F_{i,j}\) and acceleration \(a_{i,j}\) at each depth node of the pile can be calculated.