# Static Response of Double-Layered Pipes via a Perturbation Approach
## 1 Introduction  
Beams are structural members which are massively used in several application fields. Examples come from civil engineering, e.g., bridges and buildings, from aerospace engineering, e.g., helicopters and arircrafts, from industrial engineering, e.g., turbines and automotive constructions, and many other fields. The requirement for high structural performance of such members led designers to optimize the choice of the constituting materials and their ways of use. Historically, the advent of composite materials [1, 2], for instance, represented a turning point which brought a clear enhancement in the structural efficiency of the members; concurrently, it required a redefinition of the classical beam theories (see [3]), which, however, were mostly developed in linear field and for isotropic materials. In this context, in [4], an in-depth treatment of beam theory is presented, with argumentation addressed to thin-walled beams and composite material applications. In [5], the one-dimensional theory of composite thin-walled beams is consistently derived from a three-dimensional continuum in the framework of elasticity theory. Accounting for local distortion effects in beams--for example, changes in shapes of the cross-sections, such as warping and ovalization--becomes crucial when dealing with thin-walled members. For instance, the contribution of warping in non-uniform torsion is explicitly addressed in [PERSON]'s theory [6], whereas [PERSON]'s theory [7] explains softening effects in bending of pipes as a consequence of ovalization of the cross-section, also in the presence of a soft core [8]. The aforementioned aspects may be addressed in the framework of the generalized beam theory (GBT; see, e.g., [9]) as well. It represents a compelling instrument where, still dealing with one-dimensional continua, the local effects are a-priori determined and described by assumed functions, amplified by parameters which are evaluated through equilibrium conditions. In this context, the choice of the assumed functions, as an outcome of the cross-section analysis, denotes a main step, as discussedin [10; 11; 12], where the use of free-dynamical modes of equivalent frames is proposed, and in case of curvilinear cross-sections [13]. Beam-like structures are also conveniently used as homogeneous counterparts of lattices, when the latter are made of periodic repetitions of a modular cell in one direction. This is the case, for instance, for tall buildings where a shear-shear-torsional beam model can be used for both static [14; 15] and dynamic behavior analysis [16; 17; 18]. In these papers, the kinematic and static (or dynamic) problems relevant to the beam model (referred to as coarse model) are directly defined, while the response law for the homogeneous material, as usual, is deduced through an identification procedure from a refined model. In [19] a beam-like model is proposed to address nonlinear statics and dynamics of a tubular beam; there, as an extension of the classical Euler-Bernoulli beam model, further kinematic descriptors besides the classical ones are introduced in order to account for local distortion, whereas the constitutive law is obtained from a consistent three-dimensional fiber model. Along the same lines, in [20], nonlinear statics of double-layered pipes are addressed using an underlying Timoshenko beam, combined to a local distortion model which draws inspiration from GBT. In particular, the local distortion variables introduced therein represent amplitudes of assumed functions, which describe changes in shape of the annular cross-section, and the obtained coupled equilibrium equations in terms of the kinematic descriptors are solved using numerical tools. In this paper, starting from the model of a double-layered pipe developed in [20], and with the aim of obtaining a closed-form description of the static behavior, the nonlinear equilibrium equations in terms of kinematic descriptors are addressed through a specific perturbation scheme.