pile foundation is an essential structural component in civil engineering. the failure of a pile foundation under an earthquake may result in significant economic consequences, such asinterruption of transportation, property damage and failure, or even loss of lives. so, dynamic stability of piles is one of the emerging research topics for civil engineers. due to the excessive use of fractional models in research during recent years, which shows more compatibility of results with experimental models, and a lack of fractional models usage in the field of pile stability, this thesis investigates dynamic stability of piles under periodic earthquake loading, considering the winkler mechanical model with fractional damping for the surrounding soil media. during this research, an approximate theoretical and a numerical method are developed to study dynamic stability behavior and vibration responses of piles with fractional damping foundations under earthquake. solving the equation of motion of a pile loaded by axial periodic load leads to fractional mathieu differential equations. the approximate method is based on the bolotin method and results in two matrices of coefficients. putting determinants of matrices equal to zero results in different orders of approximation for finding instability regions’ boundary. on the other hand, the numerical method is introduced by using block-pulse functions to calculate the vibration response of pile under the periodic load. based on the numerical method, instability regions are generated, and results are used to validate different orders of approximation for each instability region. [...]