Vibrations of pinned-fixed heterogeneous circular beams pre-loaded by a vertical force at the crown point

This paper deals with the vibrations of isotropic, linearly elastic and heterogeneous circular beams given that a vertical force acts at the crown point. The effect of the loading is taken into account via the axial strain it causes. The material parameters, like Young's modulus, can vary arbitrarily over the symmetric, uniform cross-section. Thus, it is possible to simply model composites (not only multi-layered but also functionally graded material distributions). The main objectives are as follows: (1) to derive the equations of motion, (2) to determine the Green function matrix in closed-form both for a tensile force and for a compressive one; (3) to clarify how the load affects the natural frequencies and (4) to develop a numerical model so that we can obtain how the eigenfrequencies are related to the load. The computational results are presented in graphical format.