This study investigates the dynamical attitude of a nonlinear elastic circular rod’s longitudinal oscillation with lateral inertia and finite radius. This model was derived in 1986 by Wei and Gui-tong with a fourth-order nonlinear mixed derivative. The axial symmetry of this model has been thought through by using cylindrical coordinates. Furthermore, the strain and kinetic energy in the length unit of the rod have been determined. Two recent computational (extended Fan-expansion (EFE) and generalized rational (GR)) techniques are employed to construct some novel solitary wave solutions. The soliton wave solutions are obtained using Mathematica 13 software and are given with the distinct physical properties of trigonometric, hyperbolic and rational solution species. The stability of the investigated model and the obtained solutions through the suggested two analytical schemes are tested. Putting different values of the parameters explains these solutions through some numerical simulations in two-dimensional, three-dimensional and contour plots.