Free Vibration Behavior of a Gradient Elastic Beam with Varying Cross Section

Abstract

Based on strain gradient elasticity theory, a finite element procedure is proposed for computation of natural frequencies for the microbeams of constant width and linear varying depth. Weak form formulation of the equation of motion is obtained first as in common classical finite element procedure in terms of various kinds of boundary conditions. Gradient elastic shape functions are used for interpolating deflection inside a finite element. Stiffness and mass matrices are then calculated to solve the microbeam eigen value problem. A solution for natural frequencies is obtained using characteristic equation of microbeam in gradient elasticity. The results are given in a series of figures and compared with their classical counterparts. The effect of various slope values on the natural frequencies are examined in some numerical examples. Comparison with the classical elasticity theory is also performed to verify the present study.