Here is your answer, Check the attachment:

Mechanical Vibrations, Singiresu S. Rao, Chapter 8:Continuous Systems

This solution is Euler-Bernoulli, but there is no difference in solving your problem bro

Thanks a lot

The tensile force would cause stress stiffening of the beam resulting in an increase in the natural frequencies but if you want to suimulate this in a commercial FEA software like Ansys, prestressed modal analysis has to be performed.

Hi Mohesen,

Even though It is not exactly the same, it might be helpful to think about a string vibrating. For instance, if you look at a guitar string, increasing the tension will result in a higher natural frequency. You cannot really have a compression axial force in a string, but you can decrease the tensile force of a pre-stressed string and see that it will result in a lower natural frequency.

Best,

GP

Hi Mohsen,

Periodic or stochastic axial force can make your beam even dynamically unstable under frequency conditions of parametric resonance (see Ince-Strutt diagram)