# 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
There are 34 integral constants and 6 unknown lengths (\(l_{c}\), \(l_{1}\), \(l_{2}\), \(l_{3}\), \(l_{4}\) and \(l_{s}\)) in Equations (11). All unknown parameters can be obtained by applying the following boundary conditions:  
\[\mathrm{SP}\left\{ \begin{array}{l}v_{s}(0)=v_{1}(0)\\ M_{1}(0)=0\\ V_{s}(0)=V_{1}(0)\end{array} \right.\,,\,\,\mathrm{TDP}\left\{ \begin{array}{l}v_ {1}(l_{c})=0\\ v_{2}(0)=0\\ \theta_{1}(l_{c})=\theta_{2}(0)\\ M_{1}(l_{L})=M_{2}(0)\\ V_{1}(l_{c})=V_{2}(0)\end{array} \right.\,,\,\,\mathrm{IP1}\left\{ \begin{array}{l}v _{2}(l_{1})=v_{3}(0)\\ \theta_{2}(l_{1})=\theta_{3}(0)\\ M_{2}(l_{1})=0\\ V_{3}(0)=0\\ V_{2}(l_{1})=V_{3}(0)\end{array} \right.\,,\,\,\mathrm{LP1}\left\{ \begin{array}{l}v _{4}(0)=\delta_{\
u 1}+h\\ v_{3}(l_{2})=\delta_{\
u 1}+h\\ \theta_{3}(l_{2})=\theta_{4}(0)\\ M_{3}(l_{2})=M_{4}(0)\end{array} \right.\,, \tag{12}\] \[\mathrm{LP2}\left\{ \begin{array}{l}v_{5}(0)=\delta_{\
u 2}+h\\ v_{4}(s_{1})=\delta_{\
u 2}+h\\ \theta_{4}(s_{1})=\theta_{5}(0)\\ M_{4}(s_{1})=M_{5}(0)\end{array} \right.\,,\,\,\mathrm{LP3}\left\{ \begin{array}{l}v _{6}(0)=\delta_{\
u 3}+h\\ v_{5}(s_{2})=\delta_{\
u 3}+h\\ \theta_{5}(s_{2})=\theta_{6}(0)\\ M_{5}(s_{2})=M_{6}(0)\end{array} \right.\,,\,\,\mathrm{IP2}\left\{ \begin{array}{l}v _{6}(l_{3})=v_{7}(0)\\ \theta_{6}(l_{3})=\theta_{7}(0)\\ M_{6}(l_{3})=0\\ M_{7}(0)=0\\ V_{6}(l_{3})=V_{7}(0)\end{array} \right.\,,\,\,\mathrm{SP}\left\{ \begin{array}{l}v _{8}(0)=h\\ v_{7}(l_{4})=h\\ \theta_{7}(l_{4})=\theta_{8}(0)\\ M_{7}(l_{4})=M_{8}(0)\\ V_{7}(l_{4})=V_{8}(0)\end{array} \right.\,,\] (13) \[\mathrm{EP}\left\{ \begin{array}{l}v_{8}(l_{s})=h\\ M_{8}(l_{s})=0\\ V_{s}(l_{s})=V_{8}(l_{s})\end{array} \right. \tag{14}\]  
As shown in Figure 6b, the lengths of SP to LP2 and LP2 to EP are \(L_{\text{front}}=l_{c}+l_{1}+l_{2}+s_{1}\) and \(L_{\text{back}}=l_{3}+l_{4}+l_{s}+s_{2}\), respectively. Similar to Model 1, the boundary conditions except for the end displacements of each PSP element, which are \(v_{3}(l_{2})=\delta_{\
u 1}+h\), \(v_{4}(s_{1})=\delta_{\
u 2}+h\), \(v_{5}(s_{2})=\delta_{\
u 3}+h\) and \(v_{7}(l_{4})=h\), are substituted into Equation (11). Then, the integral constants can be defined as a function of the segmental lengths of the elements. Through substituting these integral constants into the excepted boundary conditions, the \(6\times 6\) nonlinear system related to the end displacements of the PSP elements is defined as shown in Equation (13). \[\left\{ \begin{array}{l}f_{1}(l_{c},l_{1},l_{2})=L_{\text{front}}-l_{c}-l_{1}- l_{2}-s_{1}\\ f_{2}(l_{3},l_{4},l_{s})=L_{\text{back}}-l_{3}-l_{4}-l_{c}-s_{2}\\ f_{3}(l_{c},l_{1},l_{2})=-\frac{ql_{1}^{2}}{2T}+C_{9}+C_{10}l_{2}+C_{11} \mathrm{sinh}(\lambda l_{2})+C_{12}\cosh(\lambda l_{2})-\delta_{\
u 1}-h\\ f_{4}(l_{c},l_{1},l_{2})=-\frac{qs_{1}^{2}}{2T_{2}}+C_{13}+C_{14}s_{1}+C_{15} \mathrm{sinh}(\lambda s_{1})+C_{16}\cosh(\lambda s_{1})-\delta_{\
u 2}-h\\ f_{5}(l_{c},l_{1},l_{2})=-\frac{qqs_{2}}{2T}+C_{17}+C_{18}s_{2}+C_{19} \mathrm{sinh}(\lambda s_{2})+C_{20}\cosh(\lambda s_{2})-\delta_{\
u 3}-h\\ f_{6}(l_{3},l_{4},l_{s})=-\frac{ql_{4}^{2}}{2T}+C_{25}+C_{26}l_{4}+C_{27} \mathrm{sinh}(\lambda l_{4})+C_{28}\cosh(\lambda l_{4})-h\end{array} \right. \tag{15}\]  
In this equation, the integral constants \(C_{10}\)-\(C_{13}\) are functions of \(l_{c}\) and \(l_{1}\) and \(C_{14}\)-\(C_{21}\) are functions of \(l_{c}\), \(l_{1}\) and \(l_{2}\). Also, the integral constants \(C_{26}\)-\(C_{29}\) are functions of \(l_{3}\), \(l_{4}\) and \(l_{s}\). This nonlinear system can be solved with the Newton-Raphson method. Model 2 in the lateral plane is assembled with four PSP elements if three pipelayers are applied. Thus, the 4 equations for the horizontal displacements of the elements are shown below. \[\left\{