In most practical applications, Euler-Bernoulli beam theory is often sufficient to provide useful information about the relation between axial loading and free vibrations. However, there are a number of instances in which the axial loading, or some related effect, results in relatively highly deflected states of the system, especially when the structure under consideration is very slender. For example, a pipeline or cable is characterized by having one of its dimensions very much greater than the other two, and the loads to which it is subject may often result in large deflections, even in cases in which self-weight is the only appreciable loading. Elastic bending stiffness does not necessarily dominate the effects of gravity, for example. In these cases, a more sophisticated description of the geometry is needed, and it is these types of flexible structures that form the basis for this chapter. In Chapter 7, we saw how initial postbuckling could be handled by retaining extra terms in the various energy expressions. But now, we allow (static) deflections to become large by using an arc-length description of the geometry and then consider small-amplitude oscillations about these nonlinear equilibrium configurations. In the final section, a FE solution is also shown for a specific case (essentially with the same approach as used toward the end of Chapter 9). It also turns out that experimental verification is relatively easy, especially if thermoplastics like polycarbonate are used.