Slender and highly flexible structures quite often take place in systems designed to meet high to extreme performances. Hence cables, ropes, yarns, hoses and pipelines, which are essential parts of such structures, play a relevant role in practically every engineering field. In mechanical and automotive engineering, large amplitude motions of thin rods can be exploited to design nonlinear vibration absorbers for the reduction of torsional vibrations of drivelines; in assembly and disassembly phases and in system operation, reliable models are needed to predict and analyze the behavior of cables and wiring harnesses, taking also into account effective material properties; accurate structural models of wire ropes are required to study the behavior of rope-ways and cranes on the system level. In aerospace engineering, compact, flexible and slender aerials and booms to be deployed in space are typically used to minimize the room needed to store satellites in launching phases. In textile engineering, complicate interactions among hundreds of yarns have to be controlled to obtain the desired final layout. In biomedical engineering, medical endoscopes characterized by a multilayer structure must be accurately modeled, since they exhibit highly deformed configurations while moving inside narrow curved tubes within the human body. In offshore engineering, floaters, mooring lines, and others structural components of floating wind farms, are subject to structural fatigue and various sources of damping and power cables show complex cross-sectional properties. In civil engineering, estimates of the structural properties from response data coming from non-destructive procedures is critically empowered by a deeper understanding of beam-like structures. However, despite of their ubiquity, slender structures in real operating conditions exhibit responses often too complicated for current modeling tools. In this respect there is a continuous need for reliable models. In this area, this contribution considers a beam model equipped with non-standard constitutive laws and in particular it is aimed at deriving approximate solutions of the equations of motion via asymptotic multiple scale expansion.