Distributed strain readings, recorded experimentally, find increased applications in structural health monitoring in recent years. Structural assessment of large span structures like cable-stayed bridges, pipe lines and special structures like tunnels etc find increased applications of such recordings. Mechanics of Fibre Optic Sensors (FOS) utilising Brillouin scattering effect, used for monitoring distributed measurements of temperature and strains, is addressed in this study. FOS are analytically treated in the framework of elastic, Functionally Graded Material Cylinders (FGMCs), under symmetric loading conditions. The achieved results of the studies conducted, characterize the complete set of no-decaying solutions; they present the axial strain in the core-jacket system, not varying with the radius and in close agreement with that of the supporting element These solutions are of particular interest for efficient strain recordings, utilising embedded sensors. Another set of decaying solutions, is also discussed and the rate of decaying of axial strain, between the core and the support, is presented. The proposed solutions shall find advanced applications of structural health monitoring of special and large span structures, by employing FOS.