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IN THE PRESENT PAPER, A NEW HYBRID APPROACH BASED ON THE COMBINATION OF THE POWER SERIES EXPANSIONS AND THE RAYLEIGH-RITZ METHOD IS ADOPTED FOR STABILITY AND FREE VIBRATION ANALYSES OF AXIALLY FUNCTIONALLY GRADED NON-UNIFORM BEAMS RESTING ON CONSTANT WINKLER-PASTERNAK ELASTIC FOUNDATION. IN THE PROPOSED NOVEL TECHNIQUE, THE POWER SERIES APPROXIMATION IS FIRST ADOPTED TO SOLVE THE MOTION EQUATION. REGARDING THIS NUMERICAL METHODOLOGY, THE TRANSVERSE DISPLACEMENT AND ALL MECHANICAL PROPERTIES ARE EXPANDED IN TERMS OF POWER SERIES OF A KNOWN DEGREE. BY SOLVING THE EIGENVALUE PROBLEM, ONE CAN ACQUIRE THE FUNDAMENTAL NATURAL FREQUENCIES. ACCORDING TO AFOREMENTIONED METHOD, THE EXPRESSION OF VIBRATIONAL MODE SHAPE IS ALSO DETERMINED. BASED ON THE SIMILARITIES EXISTED BETWEEN THE VIBRATIONAL AND BUCKLING DEFORMATION SHAPES, RAYLEIGH-RITZ METHOD IS FINALLY EMPLOYED TO CONSTRUCT EIGENVALUE PROBLEM FOR OBTAINING THE CRITICAL LOADS. IN ORDER TO ILLUSTRATE THE CORRECTNESS AND CONVERGENCE OF THE METHOD, SEVERAL NUMERICAL EXAMPLES OF AXIALLY NON-HOMOGENEOUS AND HOMOGENEOUS BEAMS. THE OBTAINED OUTCOMES ARE COMPARED TO THE RESULTS OF FINITE ELEMENT ANALYSIS IN TERMS OF ANSYS SOFTWARE AND THOSE OF OTHER AVAILABLE NUMERICAL AND ANALYTICAL SOLUTIONS. THE ACCURACY OF THE METHOD IS THEN REMARKED.
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