# Analytical Model for the Structural Behavior of Pipelines During Lowering-In
## Appendix A  
As shown in Figure 11, an example for segmental pipeline model is a lowering phase applied three pipelayers. Each of spacing between pipelayers is 20 m (LP1 to LP2) and 15 m (LP2 to LP3), respectively. The lifting heights at LP1, LP2 and LP3 measured from the trench bottom are assumed to be 2 m, 2.5 m and 2.4 m, respectively. Also, the height from the trench bottom to the skid is assumed to be 2 m.  
The equations for each element of the segmental pipeline model are summarized in Table 11. The material and geometric properties of the pipeline and trench are the same as in Table 1 in the Section 3.3. Also the variables in the equations for each element such as \(q\), \(T\), \(\lambda\) and \(\mu\) are calculated according to the assumptions and model settings of this paper. In order to analyze behavior of pipeline in the vertical plane, the integral constants and unknown lengths of the equations for each element should be obtained firstly. The boundary conditions at the partitioning points to obtain the unknown variables are represented in Equation (A1). \[\text{SP}\left\{ \begin{array}{l}v_{s}(0)=v_{1}(0)\\ M_{1}(0)=0\\ V_{s}(0)=V_{1}(0)\end{array} \right.,\ \text{TPP}\left\{ \begin{array}{l}v_{1}(l_{s})=0\\ v_{2}(0)=0\\ \theta_{1}(l_{c})=\theta_{2}(0)\\ M_{1}(l_{c})=M_{2}(0)\\ V_{1}(l_{c})=V_{2}(0)\end{array} \right.,\ \text{TPP}\left\{ \begin{array}{l}v_{2}(l_{1})= \theta_{3}(0)\\ M_{2}(l_{1})=0\\ M_{3}(0)=0\\ V_{2}(l_{1})=V_{3}(0)\end{array} \right.,\ \text{LP1}\left\{ \begin{array}{l}v_{4}(0)=2 \\ \theta_{3}(l_{2})=\theta_{4}(0)\\ M_{3}(l_{2})=M_{4}(0)\end{array} \right.,\ \text{LP1}\left\{ \begin{array}{l}v_{ 4}(0)=2\\ \theta_{3}(l_{2})=\theta_{4}(0)\\ M_{3}(l_{2})=M_{4}(0)\end{array} \right.,\ \text{LP1}\left\{ \begin{array}{l}v_{ 4}(0)=2\\ \theta_{3}(l_{2})=\theta_{4}(0)\\ M_{3}(l_{2})=M_{4}(0)\end{array} \right.,\ \text{LP1}\left\{ \begin{array}{l}v_{ 4}(0)=2\\ \theta_{3}(l_{2})=\theta_{4}(0)\\ M_{3}(l_{2})=M_{4}(0)\end{array} \right.,\ \text{LP1}\left\{ \begin{array}{l}v_{ 4}(0)=2\\ \theta_{3}(l_{2})=\theta_{4}(0)\\ M_{3}(l_{2})=M_{4}(0)\end{array} \right.,\ \text{LP1}\left\{ \begin{array}{l}v_{ 4}(0)=2\\ \theta_{3}(l_{2})=\theta_{4}(0)\\ M_{3}(l_{2})=M_{4}(0)\end{array} \right.,\ \text{LP1}\left\{ \begin{array}{l}v_{ 4}(0)=2\\ \theta_{3}(l_{2})=\theta_{4}(0)\\ M_{3}(l_{2})=M_{4}(0)\end{array} \right.,\] \[\text{EP}\left\{ \begin{array}{l}v_{ 8}(l_{s})=2\\ M_{8}(l_{s})=0\\ V_{s}(l_{s})=V_{8}(l_{s})\end{array} \right.\] (A1)  
By approaching the boundary value problem using equations for elements, summarized in Table A1 and Equation (A1), all integral constants can be obtained as a function of elemental lengths. The displacement of the PSP elements of LP1, LP2, LP3 and SSP to estimate the unknown variables can be obtained as follows. \[v_{3}(l_{2})=2,\ v_{4}(20)=2.5,\ v_{5}(15)=2.4,\ v_{7}(l_{4})=2\] (A2)  
Substituting the integral constants, which are the function of the elemental length, into Equation (A2) then the following nonlinear system is obtained:  
\[\left\{\begin{array}{l}f_{3}(l_{c},l_{1},l_{2})=-\frac{\theta l_{2}^{2}}{2T} +C_{9}+C_{10}l_{2}+C_{11}{\sinh(0.0102l_{2})}+C_{12}\cosh(0.0102l_{2})-2.0\\ f_{4}(l_{c},l_{1},l_{2})=-\frac{\theta(20)^{2}}{2T}+C_{13}+C_{14}(20)+C_{15}{ \sinh(0.204)}+C_{16}\cosh(0.204)-2.5\\ f_{5}(l_{c},l_{1},l_{2})=-\frac{\theta(15)^{2}}{2T}+C_{17}+C_{18}(15)+C_{19}{ \sinh(0.153)}+C_{20}\cosh(0.153)-2.4\\ f_{6}  
In the lateral plane, the pipeline contacted with the soil and supported by the skid is assumed to be a rigid beam. Also, since the pipeline does not have inflection point and is only partitioned at the lifting point of the pipelayer. Therefore, the pipeline in the lateral plane can be described as shown in Figure A2. The locations of TDP, LP1, LP2, LP3 and SSP were assumed to be the same as both the vertical and lateral plane. Therefore, the elemental length of the model in the lateral plane can be calculated as the sum of \(l_{1}\) and \(l_{2}\), the sum of \(l_{3}\) and \(l_{4}\), respectively.