beams on elastic foundations have received great attention of researchers for its importance in civil engineering. the present work can be applied, for example, to the study of dynamically loaded finite columns which are embedded in soil and supported by a layer of bedrock, or to the dynamic buckling analysis of longitudinal fibers in a composite elastomer. the dynamic stability of simply supported beam-column under periodic axial loading, and laterally resting on an elastic foundation is investigated. the combined effect of stiffness and damping is exerted on the beam through the foundation. traditionally, the periodical sinusoidal waveforms have been considered as the axial dynamic loading. however, in practical engineering, dynamic loading may exhibit other waveforms. therefore this thesis considers various periodical waveforms as excitations in the derivation of the dynamic stability. the equation of motion for the system is derived. this equation is further processed and transformed into the hill equation. the conditions for dynamic stability regions are developed using pipes matrix method for periodical loadings. the theoretical solutions are provided for various waveforms such as rectangular loading, sawtooth loading, exponential loading and sum of the step loading. in order to conduct numerical simulations and to develop the diagram for dynamic stability region, certain reasonable assumed values are taken for mechanical property of beam and various parameters. using dynamic stability plots, effects of various parameters such as flexural stiffness of beam, damping, and stiffness of foundations are studied and discussed on dynamic stability of a beam. moreover, the first three vibration modes of beam modal analysis are conducted. in order to evaluate the accuracy of the solutions, a comparison is made among the solutions obtained from pipes matrix method, floquet theory, and finite element method. as an example of applications, the study of buckling of rock slope resting on the elastic foundation is carried out and is modeled for blasting vibrations. in addition to that, the factors affecting the buckling of rock slope with blasting are also discussed. the dynamic stability solutions for arbitrary loadings are illustrated using wind sampling data which is obtained.