## 6 Experimental and Numerical Results  
_Comparison between Predicted and Measured Displacements on the Prototype Test Bench_  
For the purpose of model validation, various experimental tests are performed. During each test, a landslide is induced artificially by means of actuators using different setup parameters of displacement. Table 1 shows strain measurements of the most significant tests, named test 1 and test 2, for each cross-section. Specifically, the strain value in microstrains (me) is reported for each of the three control points on the monitored sections along the pipeline. For example, \(B_{3}\) refers to sensor number 3 according to Figure 9 of the section labeled as \(B\) in Figure 8.  
A finite element model is constructed for the pipe sector delimited by C1 and C5 of length equal to 18 m and discretized with 27 beam finite elements. At each control section, the mean axial strain and the two bending curvatures are obtained from the values in Table 1 by solving the linear system in Equation (18). The corresponding values at the Gauss integration points of each finite element are obtained by interpolation using the pchip (piecewise cubic Hermite interpolating polynomial) function of the software Matlab. The predictive model needs six equations as boundary conditions since the strain field can define the kinematics except for a rigid motion of the body. These are obtained using the  
\begin{table}
\begin{tabular}{c c c} \hline \hline  & **Test 1 (\(\mu\epsilon\))** & **Test 2 (\(\mu\epsilon\))** \\ \hline \(A_{1}\) & 225 & 150 \\ \(A  
assumption that the displacement is restrained at the endpoints C1 and C5 external to the area directly affected by the landslide. Actually, these points have a non-null displacement as proved by the real measurements but with a small value compared to those of the inner part.  
The comparison between the measured and predicted values are reported in Tables 2 and 3 for the two tests. In particular, for each marker rod, the tables report:  
* label of the markers used to read the actual displacements: C1, C2, C3, C4, C5;
* the position X (mm) of each marker rod along the pipeline axis;
* the displacement Y (mm) measured by each marker rod in the thrust direction;
* the displacement Y (mm) at the marker rod position predicted by the numerical model based on the strain measurements;
* the relative error percentage calculated as \((\mathrm{Y_{predicted}-Y_{measured}})/\mathrm{Y_{measured}}\times 100\).  
In addition, the deformed configuration in the plane X-Y is also illustrated in Figures 13 and 14 for the two tests respectively. We can observe a satisfactory estimate of the displacement trend provided by the proposed finite element model coupled with the strain acquisition system. A mean error in the displacement of about 18% was obtained with practically just three effectively monitored sections (A, B, and F) in the pipe stretch of 18 m mainly affected by the landslide. This demonstrates the feasibility of using the proposal as an effective alarm indicator. The error is mainly attributed to the very few discrete strain measurements, which can only approximate the actual complex strain field along the pipeline, in addition to instrumental errors in the point-wise strain. Another possible approximation consists in neglecting the tangential strains in the measurements, although their effects are unlikely to be significant given the slenderness of the problem.  
The finite element analysis also predicts a displacement along Z of about 1/4 of the Y component showing the necessity of a 3D model. Furthermore, a linear analysis gives 343 mm and 404 mm as maximum displacement in Y for the two tests respectively, proving the necessity of a geometrically nonlinear model that takes into account the cable-like effect of the axial stiffness in reducing the transversal displacement.