Research Article Variational Principles for Bending and Vibration of Partially Composite Timoshenko Beams Halil Özer Division of Mechanics, Mechanical Engineering Department, Yıldız Technical University (YTU), Yıldız, Bes ¸iktas ¸, 34349 Istanbul, Turkey Correspondence should be addressed to Halil ¨ Ozer; hozer@yildiz.edu.tr Received 19 April 2014; Accepted 29 June 2014; Published 13 July 2014 Academic Editor: Robertt A. Valente Copyright © 2014 Halil ¨ Ozer. 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. Variational principles are established for the partially composite Timoshenko beam using the semi-inverse method. Te principles are derived directly from governing diferential equations for bending and vibration of the beam considered. It is concluded that the semi-inverse method is a powerful tool for searching for variational principles directly from the governing equations. Comparison between our results and the results reported in literature is given. 1. Introduction Composite beams composed of diferent elastic materials have been widely used in many engineering applications. Te individual beam components of the composite beam are combined by using the shear connectors. Terefore, the overall behavior of the composite beam depends on the stifness of connectors. Connector having infnite stifness eliminates any interlayer shear slip between the individual beam components, which leads to the full interaction con- nection. However, the stifness of connector has a fnite value and the interlayer slip between the individual components occurs. Tis type of connection is called partial-interaction connection. Terefore, analysis of the partial-interaction composite beams requires the consideration of the interlayer slip between the beam components. Te Euler-Bernoulli beam theory has been extensively used in bending, vibration, and buckling analyses. Ecsedi and Baksa [1] analyzed the static behavior of elastic two-layer beams with interlayer slip and developed closed-form solutions for displacements and interlayer slips. Girhammar and Pan [2] presented general solutions for the defection and internal actions for partially composite Euler-Bernoulli beams and beam-columns. Ranzi et al. [3] presented an analytical formulation for the analysis of two-layered composite beams with longitudinal and ver- tical partial-interaction. Teir formulation is based on the principle of virtual work expressed in terms of the vertical and axial displacements of the two layers. Te model was presented in both its weak and its strong forms. Xu and Wu [4] developed a new plane stress model of composite beams with interlayer slips using the one-dimensional theory. Tey concluded that the shear force produced by the shear connectors increases with the increase in rigidity of shear connectors. However, the efect of transverse shear deformation was neglected in the Euler-Bernoulli beam theory. When the beam is thick, the efect of shear deformation becomes signifcant and cannot be neglected for a valid analysis. Te most widely used and fundamentally simpler theory was developed by Timoshenko [5]. Sousa and da Silva [6] studied the behavior of the general case of multilayered composite beams with interlayer slip, under Euler-Bernoulli as well as Timoshenko beam theory (TBT) assumptions. Xu and Wang [7] formulated the principle of virtual work and recipro- cal theorem of work for the partial-interaction composite beams using the kinematic assumptions of Timoshenko’s beam theory. Te variational principles for the frequency of free vibration and critical load of buckling were also Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2014, Article ID 435613, 5 pages http://dx.doi.org/10.1155/2014/435613