Research Article Free Vibration Behavior of a Gradient Elastic Beam with Varying Cross Section Mustafa Özgür Yayli Department of Civil Engineering, Faculty of Engineering, Bilecik S ¸eyh Edebali University G¨ ul¨ umbe Kamp¨ us¨ u, 11210 Bilecik, Turkey Correspondence should be addressed to Mustafa ¨ Ozg¨ ur Yayli; mozgur.yayli@bilecik.edu.tr Received 12 July 2013; Accepted 3 December 2013; Published 17 February 2014 Academic Editor: Nuno Maia Copyright © 2014 Mustafa ¨ Ozg¨ ur Yayli. Tis is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Based on strain gradient elasticity theory, a fnite 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 frst as in common classical fnite element procedure in terms of various kinds of boundary conditions. Gradient elastic shape functions are used for interpolating defection inside a fnite element. Stifness 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. Te results are given in a series of fgures and compared with their classical counterparts. Te efect 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. 1. Introduction Microbeams have held wide applications in micro-electronic- mechanical systems (MEMS) such as those in actuators [1], microswitches [2], microresonators [3], Atomic Force Microscopes [4], and sensors [5] in which thicknesses and lengths of microbeams are typical on the order of microns and submicrons; therefore the small scale efects in their static and dynamic behavior are considerable. In the past decades, some researchers have tried to establish experimental and theoretical models for vibration of microbeams. However, most of these studies are based on the classical continuum mechanics. Te classical model is doubt- ful whether it can describe the static or dynamic behaviour of elastic materials with microstructure, since it is associated with the concepts of locality of the stress. Te experimen- tal studies have also revealed that the classical continuum mechanics is unable to consider size-dependent static and dynamic behaviours in microscaled structures [6, 7]. Lam et al. [8] observed experimentally that the normalized bending stifness increases by about 2.4 times when the thickness was reduced from 115 to 20 m; Stolken and Evans [9] examined that the plastic work hardening shows a great increase as the microbeam thickness decreases from 50 to 12.5 m. Tese results demonstrate that the size dependence is signifcant to certain materials. During past years, some nonclassical elasticity theories such as the strain gradient, nonlocal, and couple stress theories have been introduced and employed to investigate the microbeams. Nonclassical continuum theories can be classifed to couple stress and strain gradient approaches. Te couple stress theory includes higher-order stresses, known as the couple stresses. Te classical couple stress theory was originated by Mindlin [10] and others including Toupin [11] in 1960s. Some related research works have been performed to model the static and dynamic problems based on the classical couple stress theory [12, 13]. A new modifcation to couple stress theory has been proposed by Mindlin [14] in which a new equilibrium equation of high order is considered in addition to the classical equilibrium equations of forces and moments of forces. So far Park and Gao [15] have developed the static bending; Kong et al. [16] have developed free vibration problems of a Euler-Bernoulli beam. Ma et al. [17] have developed the static bending and free vibration problems of a Timoshenko beam. dos Santos and Reddy [18] have analyzed classical and nonclassical frequency ratios for Hindawi Publishing Corporation Shock and Vibration Volume 2014, Article ID 801696, 11 pages http://dx.doi.org/10.1155/2014/801696