# Static Response of Double-Layered Pipes via a Perturbation Approach
## 2 Model Description  
The object of this study is a double-layered beam with a thin annular cross-section (see Figure 1). The beam length is \(l\), the average radius of the annular cross-section is \(R\) and the two plies have thicknesses \(h_{i}\) and \(h_{e}\), where the subscripts \(i\) and \(e\) stand for internal and external, respectively \((h_{i},h_{e}\ll R)\). A thinner inter-layer is present as well, constituted by the adhesive, and its thickness is \(h_{a}\ll h_{i},h_{e}\). A beam-like model is used here, with the aim of roughly describing the in-plane static behavior of the aforementioned pipe, under the action of transversal forces. The model was taken from the literature (see [20] for details), and it is briefly recalled here for the sake of completeness. It is constituted by an initially straight axis-line, whose direction coincides with the unitary vector \(\bar{\mathbf{a}}_{x}\), and which is spanned by the abscissa \(s\in[0,l]\). Cross-sections are initially transverse to the axis-line and are connected to it at their center of gravity. With reference to Figure 2, they initially lie on the plane spanned by the unitary vectors \(\bar{\mathbf{a}}_{y}\) and \(\bar{\mathbf{a}}_{z}\), where \((\bar{\mathbf{a}}_{x},\bar{\mathbf{a}}_{y},\bar{\mathbf{a}}_{z})\) is the canonical basis. The points of the axis-line perform displacements \(\mathbf{u}(s)=u(s)\bar{\mathbf{a}}_{x}+v(s)\bar{\mathbf{a}}_{y}\), i.e., limited to the \((\bar{\mathbf{a}}_{x},\bar{\mathbf{a}}_{y})\)-plane, and the cross-sections perform independent rotations about axis \(\bar{\mathbf{a}}_{z}\) of amplitude \(\vartheta(s)\), as in a planar Timoshenko beam. Besides \((u(s),v(s),\vartheta(s))\), further scalar kinematic descriptors are defined in the model, in order to account for local distortions of the beam, i.e., changes in shape for the cross-sections: indicated as \((a_{p}(s),a_{w}(s),a_{g}(s))\), the physical interpretation of the distortion variables is given in [20] and comes from the expression of the displacement of a generic point of the beam, seen as a corresponding three-dimensional body and functionally introduced for homogenization purposes. Specifically, as in the spirit of generalized beam theory (GBT--see, e.g., [9]), each of them represents the amplitude of a trial function, defined in the cross-section domain and describes an elementary distortion, namely, ovalization \((a_{p}(s))\) and warping \((a_{w}(s))\) of the cross-section itself, and longitudinal sliding of the plies \((a_{g}(s))\) due to local opposite bending, as shown in Figure 3.  
Figure 1: The double-layered beam with an annular cross-section: (**a**) complete view; (**b**) details of the cross-section. Figure 3: Physical meaning of the distortion variables: (**a**) assumed trial function for ovalization and amplitude \(a_{p}\); (**b**) assumed trial function for warping and amplitude \(a_{w}\); (**c**) assumed trial function for longitudinal sliding of the layers under opposite bending and amplitude \(a_{g}\). Figure 2: Initial configuration of the beam. The strain-displacement relationship is consistently introduced: classical strain components for Timoshenko beam are defined, namely, longitudinal strain \(\epsilon_{0}\), shear strain \(\gamma_{0}\) and bending curvature \(\kappa_{0}\), plus strain components relevant to each distortion variables, namely, \(\alpha_{j}\), which are equal to the distortion variable themselves, and \(\beta_{j}\), which are their gradient (for \(j=p,w,g\)). The strain-displacement relationship follows:  
\[ \begin{split}\epsilon_{0}:=&(1+u^{\prime})\cos \vartheta+v^{\prime}\sin\vartheta-1\\ \gamma_{0}:=&-(1+u^{\prime})\sin\vartheta+v^{\prime }\cos\vartheta\\ \kappa_{0}:=&\vartheta^{\prime}\\ \alpha_{j}:=& a_{j}\\ \beta_{j}:=& a_{j}^{\prime}\end{split}  \tag{1}\]  
where prime stands for \(s\)-derivative. Equation (1) is combined to geometrical boundary conditions, evaluated at the restrained cross-section \(\tilde{s}=0\) and/or \(\tilde{s}=l\). \[