# Effects of New Tunnelling on a Buried Pipeline with Joints and Its Detachment
## 2 Jointed Pipeline Subjected to Tunneling  
### Calculation Model of Subgrade Reaction Analysis  
In the traditional methods, the pipeline is regarded as an elastic beam resting on an elastic foundation. The interaction between the soil and pipeline is taken into account by a series of independent springs. A shear layer connecting these independent springs is applied to consider the shear interaction between the springs. The tunnelling-induced ground movement adds pressure on the elastic beam. After considering the arching effect, the profile of the ground movement is described by the Gaussian distribution function. Since most buried pipelines have joints in practice, the traditional deformation of buried jointed pipelines cannot be calculated using the beam with equivalent bending stiffness. If the joints are in the least helpful position, the mechanical behaviors of the buried pipeline will be significantly affected. Different deformation mechanisms depend on the number of joints and the relative position between the joints and the new tunnel [29]. Figure 1 shows the deformation characteristics of the jointed pipelines subjected to new tunnelling. Figure 2 indicates the calculation model of the subgrade reaction analysis. Some assumptions required in the calculation model are detailed as follows: (1) The soil material is modeled by the Pasternak foundation model; (2) The settlement of the foundation is represented by the Gaussian function; (3) The pipe material is linearly elastic, homogeneous and isotropic; (4) The pipeline is regarded as the Bernoulli-Euler beam model because the deflections of the pipelines are small; (5) The jointed pipeline is considered as a continuous pipeline with additional moments at the joints; (6) Axial forces in the pipeline and friction on the contact surface are not taken into account; (7) Detachment is allowed between the soil and the pipeline, as shown in Figure 3. Due to the existence of the last hypothesis, both the linear and nonlinear deformation of the pipeline can be considered. Figure 1: Typical deformation of buried jointed pipeline subjected to tunnelling-induced ground settlement while the joints are in the least helpful positions. Figure 2: Calculation model of subgrade reaction analysis. The Pasternak foundation model assumes the existence of shear interactions between the spring elements. It connects the ends of the springs to a shear layer comprising incompressible vertical elements and deforms only by transverse shear, as shown in Figure 2.  
The differential equation of Bernoulli-Euler beam on the Pasternak foundation is  
\[EI\frac{d^{4}w}{dx^{4}}=q(x)-p(x) \tag{1}\]  
where \(EI\) is the flexural rigidity of the beam; \(w\) is the deflection of the beam taking downside positive; \(q(x)\) is the external load, composed of the self-weight of pipelines and covered soil and additional moment at joints; and \(p(x)\) is the reaction force of the elastic foundation. To derive the constitutive relationship between the load and deflection of the foundation, the vertical equilibrium of an element in the shear layer is considered, as shown in Figure 4.  
Assuming that the foundation material is homogeneous and isotropic in the \(x-y\) plane, we have \(G_{x}=G_{y}=G\), then  
\[\tau_{xz}=G_{x}\gamma_{xz}=G\frac{\partial(w-S)}{\partial x} \tag{2a}\]  
Figure 4: An element of shear layer in Pasternak foundation. Figure 3: Detachment between underlying soil and pipeline. \[\tau_{yz}=G_{y}\gamma_{yz}=G\frac{\partial(w-S)}{\partial y} \tag{2b}\]  
Noting that \(S\) is the tunnelling-induced ground movement described as a function of soil deflection. The deformation of the contact area is \((w-S)\). The shear forces per unit length of the shear layer are  
Figure 1:0 Figure 1:1  
Figure 5 shows an element in the shear layer.