# Analytical Model for the Structural Behavior of Pipelines During Lowering-In
## 3 Validation of Segmental Pipeline Models
### Model 1 and 2 for the Lowering-in Phase  
Model 1 and Model 2 depending on the configurations of the pipelayer are defined as shown in Figure 6a,b, respectively. Figure 6c shows the lateral plane of Model 2. Model 1 is composed of each single PSS and PSP element and it includes a shear layer in the PSS element. The solutions of Model 1 are presented in Equation (3). \[\left\{ \begin{array}{l}v_{s}(x_{1})=c_{0}\mathrm{e}^{i\pi x_{1}}\\ v_{1}(x_{1})=\frac{q}{k}+c_{1}\phi_{1}(x_{1})+c_{2}\phi_{2}(x_{1})+c_{3}\phi_{ 3}(x_{1})+c_{4}\phi_{4}(x_{1})\\ v_{2}(x_{2})=-\frac{q\pi x_{2}^{2}}{2T}+c_{5}+c_{6}\chi_{2}+c_{7}\mathrm{sinh} (\lambda x_{2})+c_{8}\cosh(\lambda x_{2})\end{array} \right. \tag{8}\]  
where, \(v_{s}(x)\) is the deformation of the shear layer and \(v_{1}(x)\) and \(v_{2}(x)\) are the deformations of the PSS and PSP elements in the vertical plane, respectively. The entire length of the pipeline is known, while the segmental lengths of each element are unknowns. The unknown lengths and integral constants depending on the construction conditions can be obtained by using the following boundary conditions:  
\[\mathrm{SP}\left\{ \begin{array}{l}v_{s}(0)=v_{1}(0)\\ M_{1}(0)=0\\ V_{s}(0)=V_{1}(0)\end{array} \right.,\ \mathrm{TDP}\left\{ \begin{array}{l}v_{1}(l_{s})=0 \\ v_{2}(0)=0\\ \theta_{1}(l_{s})=\theta_{2}(0)\\ M_{1}(l_{s})=M_{2}(0)\\ V_{1}(l_{s})=V_{2}(0)\end{array} \right.,\ \mathrm{LP1}\left\{ \begin{array}{l}M_{2}(l_{1})=0 \\ v_{2}(l_{1})=\delta_{\mathrm{v}}\\ \end{array} \right. \tag{9}\]  
The integral constants are function of the segmental lengths, which are obtained by substituting Equation (9) except for the boundary condition of \(v_{2}(l_{1})=\delta_{\mathrm{v}}\) into Equation (8). Considering the total length of the pipeline, the segmental lengths presented in Figure 6 are determined by substituting the integral constants into the boundary condition of \(v_{2}(l_{1})=\delta_{\mathrm{v}}\). The results are expressed with a \(2\times 2\) system in Equation (10). \[\left\{ \begin{array}{l}f_{1}(l_{s},l_{1})=-\frac{q{l_{1}}^{2}}{2T}+c_{5}(l_{ s},l_{1})+c_{6}(l_{s},l_{1})l_{2}+c_{7}(l_{s},l_{1})\mathrm{sinh}(\lambda l_{1})+c_{8} \cosh(\lambda l_{1})-\delta_{\mathrm{v}}\\ f_{2}(l_{s},l_{1})=L_{t}-l_{s}-l_{1}\end{array} \right. \tag{10}\]  
where, \(l_{s}\) and \(l_{1}\) are the length of each element for Model 1. This nonlinear algebraic \(2\times 2\) system can be solved using the Newton-Raphson method and so forth. Model 2 is defined in Figure 6b,c, which present the behavior of the vertical and lateral planes. If three pipelayers are applied, Model 2 has three discontinuity points of shear forces and two inflection points in the vertical plane and three discontinuity points of shear forces in lateral plane. Therefore, in the vertical plane, Model 2 is assembled with each single PSS and PCS element and six  
Figure 6: Segmental pipeline model during lowering-in (**a**) Model 1; (**b**) Model 2; (**c**) lateral plane of Model 2 (A:SP, B:TDP, C:IP1, D:LP1, E:LP2, F:LP3, G:IP2, H:SSP, I:EP). PSP elements. Thus, 10 equations for the vertical displacements of the elements including the shear layer are applied, as shown in Equation (11). \[\left\{ \begin{array}{l}v_{s}(x_{1})=c_{0}\mathrm{e}^{\mu\chi_{1}}\\ v_{1}(x_{1})=\frac{q}{k}+c_{1}\phi_{1}(x_{1})+c_{2}\phi_{2}(x_{1})+c_{3}\phi_{3} (x_{1})+c_{4}\phi_{4}(x_{1})\\ v_{\mathrm{n}}(x_{\mathrm{n}})=-\frac{q\kappa_{2}}{2T}+c_{4n-3}+c_{4n-2}x_{ \mathrm{n}}+c_{4n-1}\mathrm{sinh}(\lambda x_{\mathrm{n}})+c_{4n}\cosh(\lambda x _{\mathrm{n}})\\ v_{8}(x_{8})=\frac{q}{k}+c_{29}\phi_{1}(x_{8})+c_{30}\phi_{2}(x_{8})+c_{31} \phi_{3}(x_{8})+c_{32}\phi_{4}(x_{8})\\ v_{s}(x_{8})=c_{33}\mathrm{e}^{\mu\chi_{8}}\end{array} \right. \tag{11}\]  
where, \(\mathrm{n}=2,\,3,\cdots,\,6,\,7\) for the case of three pipelayers, \(v_{s}(x_{1})\) and \(v_{s}(x_{8})\) are the deflections of the shear layer, \(v_{1}(x_{1})\) and \(v_{8}(x_{8})\) are the deflections of the PCS and PSS element in the vertical plane, respectively.