# Analytical Model for the Structural Behavior of Pipelines During Lowering-In
## 2 Modeling of a Pipeline During Lowering-in
### Mathematical Formulations for Segmental Pipeline Models  
The partitioned points and elements in the vertical plane are schematically shown in Figure 3. Here, SP(Start Point) and EP(End Point) are the starting and ending points of the pipeline, respectively. TDP(Touch Down Point) and SSP(Separated from Skid Point) indicate the points where the pipeline touches the soil and is separated from the skid, respectively. LP1 through LP\(N\) are the lifting points by the pipelayers. Also, IP1 and IP2 are the inflection points of the pipeline. The PCS and PSS elements representing the pipeline contacted with soil and supported by a skid in the vertical plane are formulated by the following equation:  
\[v^{IV}(x)-\frac{T}{EI}v^{\prime\prime}(x)+\frac{k}{EI}v(x)=\frac{q}{EI} \tag{1}\]  
where \(q\) is the constant load per unit length of the pipeline, \(k\) is the soil stiffness in the vertical direction and \(T\) is a constant traction on the laid pipeline. The two parameters of the foundation--\(k\) and \(T\)--are evaluated from the soil properties including the elastic modulus, \(E_{s}\), Poisson's ratio, \(v_{s}\) and vertical deformation parameter within the subsoil, \(\gamma\), defined as \(k=\left[E_{s}\gamma(1-v_{s})\right]/\left[2(1+v_{s})(1-2v_{s})\right]\) and \(T=E_{s}/4\gamma(1+v_{s})\) by Vlasov and Leont'ev formulations [30], respectively. Typically, the range of \(\gamma\) is from 1.0 m\({}^{-1}\) to 2.0 m\({}^{-1}\)[29]. For this reason, the \(\gamma\) was assumed to be 1.0 m\({}^{-1}\) in this study. Upon solving the characteristic equation of Equation (1), there are three combinations of solutions depending on the relationship between \(T\) and \(2\sqrt{kEI}\). Generally, \(T<2\sqrt{kEI}\) is satisfied in most engineering problems [29; 30], so that the behavior of PCS and PSS elements can be defined as follows:  
\[v_{s}=\frac{q}{k}+c_{i}\phi_{1}(x)+c_{i+1}\phi_{2}(x)+c_{i+2}\phi_{3}(x)+c_{i +3}\phi_{4}(x) \tag{2}\]  
where, \(c_{i}\), \(c_{i+1}\), \(c_{i+2}\) and \(c_{i+3}\) are integral constants,  
\(\phi_{1}(x)=\mathrm{e}^{ax}\cos(\beta x)\), \(\phi_{2}(x)=\mathrm{e}^{ax}\sin(\beta x)\),  
\(\phi_{3}(x)=\mathrm{e}^{-ax}\cos(\beta x)\), \(\phi_{4}(x)=\mathrm{e}^{-ax}\sin(\beta x)\),  
\(\alpha=\sqrt{k/4 EI+\sqrt{T/4 EI}}\), \(\beta=\sqrt{k/4 EI-\sqrt{T/4 EI}}\). Figure 3: Segmental pipeline model in the vertical(x-y) plane. As in previous analytical models with two parameters, BOEF assumes that the boundary condition as semi-infinite. This condition provides a convenient calculation but has limitations when considering the separation length increase of the pipeline from the soil or skid. Thus, the semi-infinite boundary condition is invalid to reflect the beginning and final phase of lowering-in due to the small contact surface between the pipeline and the soil or skid. Therefore, a model of the shear layer is necessary to adequately consider the contact problem between the pipeline and the soil or skid beneath the lowering-in phase. The soil pressure applied to the pipeline is defined as \(p(x)=kv(x)-Tv(x)^{\prime\prime}\) in the PCS and PSS elements. The governing equation of the shear layer can be obtained by solving the homogeneous equation of the soil pressure. By substituting boundary conditions at \(x\rightarrow\infty\) (\(\underset{x\rightarrow\infty}{\lim}v_{s}=0\)), the governing equation of the shear layer is as follows:  
\[v_{s}=c_{i+4}\mathrm{e}^{\mu x} \tag{3}\]  
where \(\mu=\sqrt{k/T}\) and \(c_{i+4}\) is an integral constant. The deflected infinitesimal element of the pipeline suspended by pipelayers (PSP element) in the vertical plane is shown in Figure 4. Using the equilibrium equations and the moment-curvature relationship, the PSP element is expressed by the following equation:  
\[\frac{d^{4}v(x)}{dx^{4}}-\frac{T}{EI}\frac{d^{2}v(x)}{dx^{2}}=\frac{q}{EI} \tag{4}\]  
where \(v(x)\) is the vertical displacement of the pipeline, \(T\) is the axial force applied to the pipeline and \(q\) is defined in Equation (1).