In this work, the free vibration analysis of Timoshenko microbeams made of the Functionally Graded Material (FGM) on the Winkler–Paternak elastic foundation based on the Modified Coupled Stress Theory (MCST) is investigated. Material characteristics of the beam vary throughout the thickness according to the power distribution and are estimated though Mori–Tanaka, Hashin–Shtrikman and Voigt homogenization techniques. | Vietnam Journal of Mechanics Vol. 46 No. 1 2024 pp. 31 43 DOI https 0866-7136 20579 VIBRATION ANALYSIS OF TIMOSHENKO MICROBEAMS MADE OF FUNCTIONALLY GRADED MATERIALS ON A WINKLER PASTERNAK ELASTIC FOUNDATION Tran Van Lien1 Le Thi Ha2 1 Hanoi University of Civil Engineering Hanoi Vietnam 2 University of Transport and Communications Hanoi Vietnam E-mail lethiha@ Received 21 January 2024 Revised 19 March 2024 Accepted 22 March 2024 Published online 31 March 2024 Abstract. In this work the free vibration analysis of Timoshenko microbeams made of the Functionally Graded Material FGM on the Winkler Paternak elastic foundation based on the Modified Coupled Stress Theory MCST is investigated. Material characteristics of the beam vary throughout the thickness according to the power distribution and are es- timated though Mori Tanaka Hashin Shtrikman and Voigt homogenization techniques. The Timoshenko microbeam model considering the length scale parameter is applied. The free vibration differential equations of FGM microbeams are established based on the Fi- nite Element Method FEM and Kosmatka s shape functions. The influences of the size- effect foundation material and geometry parameters on the vibration frequency are then analyzed. It is shown that the study can be applied to other FGMs as well as more complex beam structures. Keywords FGM microbeam nondimensional frequency MCST. 1. INTRODUCTION Functionally Graded Materials FGMs are inhomogeneous composites which have attracted considerable attention due to their novel thermo-mechanical properties that en- able them to be used in a wide range of applications in many industries such as aircrafts biomedical products space vehicles. Micro Electro-Mechanical Systems MEMS are the new field in which FGMs have been utilized to achieve the desired performance. Micro-sized structures as plates sheets beams and framed structures are widely used in MEMS devices for example electrically actuated micro .
