Abstarct: To achive the best strength and weight distribution in many structures such as wind turbine towers, bridges, buildings, aerospace structures, etc; beams with a variable thickness along the axis, are used. Analyzing these structures when axial force and transverse deflection are coupled is more complicated and more important. One of the applications of the non-linear theory is finding more exact results. In this dissertation, free and forced vibrations in beams with variable thickness with moderately large deflection under transverse dynamic load and axial load will be investigated. The beam is homogeneous and isotropic, in deriving the equations, the displacement field is estimated by the first order shear deformation theory. The problem’s kinematics is baxsed on strain-displacement relation (Von-Karman’s relation) and Hook’s law as a constitutive equation. The governing equation of motion in this problem is a system of non-linear partial differential equations with variable coefficients which are determined by the hamilton principle. Other assumptions that will be considered are as follows: determining response according to space and time in the simply supported beam, determining natural frequencies of the beam for simple and fixed boundary conditions, investigating the amplitude dependency on frequency, investigating geometrical effects on response and frequencies, investigating the accuracy of the first order shear deformation theory in frequency determination, and comparison between the results and numerical solutions.