Issue 51

A. Chikh, Frattura ed Integrità Strutturale, 51 (2020) 115-126; DOI: 10.3221/IGF-ESIS.51.09 116 by means of both the classical beam theory (CBT), FSDBT and HSDBT Wang et al. [1] given a solution to solve the free vibration, buckling and bending problems of the Timoshenko and Euler-Bernoulli beams based on different models of elastic foundations. There are many areas of application for composite materials (Chikh et al. [2]; Akbaş et al. [3]; Chikh et al. [4]; Fahsi et al. [5]) same the aircraft and aerospace industry. Omidi et al [6] studied the dynamic stability of simple supported FG beams reposing on a linear elastic foundation; with piezoelectric-layers under a periodic axial compression load. Zhong et al. [7] provided an analytical solution for console beams subjected to various types of mechanical loads. Thai et al. [8] studied the free vibration and bending of FG beams by the use of different higher-order beams theories. Zhu, H. [9] established three- dimensional finite element model using finite element software to simulate and compare the stress performance of the strengthening beams with different numbers of CFRP plates. Bouchikhi, A. S et al. [10] investigated the 2D simulation used to calculate the J-integral of the main crack behavior emanating from a semicircular notch and double semicircular notch and its interaction with another crack which may occur in various positions in (TiB/Ti) FGM plate under mode I. Yassine Khalfi et al. [11] developed a refined and simple shear deformation theory for mechanical buckling of composite plate resting on two-parameter Pasternak’s foundations. Meftah Kamel [12] presented a finite element method for analyzing the elasto-plastic plate bending problems. Saidi Hayat [13] presented a new shear deformation theory for free vibration analysis of simply supported rectangular functionally graded plate embedded in an elastic medium. In this paper, a higher-order shear deformation beams theory for bending; buckling and free vibration of FG beams are developed. The present theory differs from other higher-order theories because, in present theory the displacement field which includes undetermined integral terms, which is not considered by the other researchers. The results of the present model are compared with the known data in the literature. V ARIATIONAL FORMULATION AND CINEMATICS onsider an FG beam with length L, width b, and thickness h made of Al/Al 2 O 3 as represents in Fig. 1. The lower part of the FG-beam was totally ceramic and the upper surface was completely made of metal. The beam 0 x L; b / 2 y b / 2; h / 2 z h / 2         in the Cartesian coordinate systems. assumed to be positive in the proposed direction, and the beam is deformed in the x-z plane solely. The x-axis coinciding with the beam inert axis. The beam is supported by Winkler–Pasternak foundations. Figure 1: FGM beam supported by Winkler–Pasternak type elastic foundation. K INEMATICS AND CONSTITUTIVE EQUATIONS n the fundamental of the assumptions expressed in the previous section, the displacement field of present theory can be obtained by:   0 0 1 0 ( , , ) ( , , , ) ( , , ) ( ) , , ( , , , ) ( , , ) w x y t u x y z t u x y t z k f z x y t dx x w x y z t w x y t         (1) where: ( ) ( ) sin , ( ) z f z h f z g z h z             C O