This paper presents an analytical procedure to predict thebeamsnonlinearnaturalthe beam's nonlinear natural
frequencies using the first order shear deformation theory. The nonlinear kinematics
assumptions include the moderately large deformation of the mid-plane stretching and
transverse deflection which are defined by using the von Kármán relations. The strains are
small and Hooke’s law is used as the constitutive equation. A coupled nonlinear
longitudinal-transverse set of motion equations are derived utilizing Hamilton’s
principle. The equations are solved analytically using the powerful multiple scale method of
the perturbation technique. At first, the linear natural frequencies and mode shapes are
determined. Then the nonlinear frequencies which contain the corrections on the linear
frequencies are calculated. The corrected parts of the nonlinear frequencies are functions of
the axial and transverse amplitudes of vibrations. The influences of the axial load and
aspect ratio on the linear and nonlinear frequencies are studied too. The results for the special
cases are compared with the available results in the literature and the finite element
analysis. The results show the noticeable effect of the axial amplitude as well as the
transverse amplitude of vibrations on the nonlinear frequencies.