Increasing advances in materials engineering and cost reduction in their testing have lead to the study of the stability of vibration of pipes conveying fluid an important problem to deal with. Currently, such analysis is done either by means of simulation with costly specialized software or by making laboratory tests of the selected material. One of the main issues with the last process is that if appears any trouble on the material selection, it is necessary to restart all the process, and it is happening each time there is a mistake on the material selection. In order to avoid such costly tests, a general mathematical description of the dynamic behavior of a clamped-pinned pipeline conveying fluid is presented. The system stability has been studied by means of the eigenvalues of a Hamiltonian linear system associated. From this analysis, characteristic expressions dependent on material constants have been developed as inequalities, which ensure the stability of the material if it matches all expressions. Finally, some specific materials are introduced as study cases to compare the mathematical description proposed with the results obtained from specialized software as ANSYS, in order to validate the results