# Horizontal Loading Performance of Offshore Wind Turbine Pile Foundation Based on DPP-BOTDA
## 3 Results and Discussion
### Distributed Displacement Data  
The steel pipe pile can be simplified as a cantilever beam structure. According to the flat section assumption, after the microsegment beam is deformed, the left (m-m') and right (n-n') cross-sections in Figure 8 remain flat, but are rotated by an angle. The longitudinal strain at each point is equal  
Figure 6: Strain values of the corresponding elevation from 0 kN to 900 kN. Figure 7: Load and maximum strain (\(F\)-\(\epsilon_{\rm max}\)) curve at \(-\)19.63 m.  
where the neutral layer O\({}_{1}\)O\({}_{2}\) is \(y\), and the calculation equation of the strain can be obtained as in the Equation (4),  
\[\varepsilon(y_{0},x_{0})=\frac{\widehat{ab}-\overline{ab}}{\overline{ab}}=\frac{ (\rho+y_{0})d\theta-\rho d\theta}{\rho d\theta}=\frac{y_{0}}{\rho}, \tag{4}\]  
where \(\varepsilon\) is the axial strain, \(\rho\) is the radius of curvature, \(d\theta\) is the central angle formed by the micro-section bending, \(x_{0}\) is the distance from the neutral axis to the starting point, and \(y_{0}\) is the distance from the neutral axis along the radius of curvature. The equation for calculating the curvature is:  
\[\kappa=\left|\frac{d\alpha}{dl}\right|=\left|\frac{d\alpha/dx}{dl/dx}\right|= \left|\frac{\frac{y^{\prime\prime}}{1+\left(y^{\prime}_{x}\right)^{2}}}{ \sqrt{1+\left(y^{\prime}_{x}\right)^{2}}}\right|=\frac{-y^{\prime\prime}{}_{x }}{\left(1+\left(y^{\prime}_{x}\right)^{2}\right)^{3/2}}, \tag{5}\]  
where \(K\) is the curvature; \(x\), \(y\) are the abscissa and ordinate of the point on the neutral axis in the Cartesian coordinate system, respectively; \(\alpha\) is the angle between the tangent of the \((x,y)\) point and the axis of abscissa; \(l\) is the arc length from the starting point to the \((x,y)\) point along the neutral axis. The relationship between curvature and radius is  
\[\kappa=\frac{1}{\rho}. \tag{6}\]  
From Equations (4)-(6),  
\[\frac{\varepsilon(y_{0},x_{0})}{y_{0}}=\frac{-y^{\prime\prime}{}_{x}}{\left(1+ \left(y^{\prime}_{x}\right)^{2}\right)^{3/2}}. \tag{7}\]  
In the actual project, \(\left(y^{\prime}_{x}\right)^{2}\) can be ignored, then the relationship between displacement and strain is  
\[y^{\prime\prime}{}_{x}=-\frac{\varepsilon(y_{0},x_{0})}{y_{0}}. \tag{8}\]  
As shown in Figure 9, the deflection of the pile is obtained by the second integral:  
\[w=\int\left[\int\frac{\varepsilon(y_{0},x_{0})}{d}dx\right]dx+C_{1}x+C_{2}, \tag{9}\]  
where \(w\) is the deflection and \(d\) is the radius of the steel pipe pile--here, \(d\) equals 1 m.  
Figure 8: Pure curved structure deformation. The boundary conditions of steel pipe piles are: \(x=0\), \(w=0\); \(x=0\), \(w^{\prime}=\theta=0\). Then, the calculated \(C_{1}=0\), \(C_{2}=0\). Finally, the deflection equation of the steel pipe pile is:  
\[w=\int\!\!\!\left[\int\frac{\varepsilon(y_{0},x_{0})}{d}dx\right]\!dx \tag{10}\]  
The strain data collected by the distributed optical fiber sensor are dense and continuous, and the distributed displacement curve of the pile was obtained by the second integral of the corresponding curve. The calculated displacement is shown in Figure 7:0 As the load increased, the absolute value of the displacement measured by the fiber increased. The force on the fiber at FUT1 was the tensile, and the displacement value was positive. The force on the fiber at FUT3 was the pressure, and the displacement value was negative. The maximum displacement occurred at the uppermost end of the fiber, the elevation was +10.0 m, and the maximum displacement generated under the 900 kN load was 0.79 m.  
During the test, the horizontal displacement of the pile was measured at the elevation of +10.5 m by the dial indicators, as shown in Figure 7:1 The dial indicators base was fixed to the reference beam, and its direction parallel to horizontal load direction.