30 H.G. Kim and R. Wiebe Fig. 4.1 Geometry and coordinate system for a rectangular plate different types of laminated composite plate theories have been proposed or discussed in the literature such as Classical laminated plate theory (CLPT), First-order shear deformation theories (FSDT), Higher-order shear deformation theories (HOSDTs), and Layerwise theory (LT) [1]. Despite the challenges from other theories, CLPT is still preferred to model behaviors of thin composite plates and shows relatively accurate predictions on balanced symmetric laminates under pure bending or tension [1]. In this paper, CLPT is used to establish the theoretical model of free vibration of quasi-isotropic plates. The experiment results on free vibration of isotropic plates under fully clamped (CCCC) and cantilever (CFFF) boundary conditions are used to prove the validity of the model. In addition, the nonlinear dynamic behaviors of initially flat, and post-buckled quasi-isotropic plates are analyzed. 4.2 Theoretical Model for Linear Free Vibrations The derivation of the equation of motion for linear free vibrations of anisotropic plates are well explained many books such as [9] and [10]. In this section, the derivation of the theoretical model will be presented based on the Reddy’s work in [9]. Based on Hamilton’s principle, the dynamic version of the principle of virtual work is 0 D Z T 0 .ıUCıV ıK/dt (4.1) where the virtual strain energyıU, virtual work done by applied force ıV, and the virtual kinetic energyıKare given by ıUD Z 0 Z h 2 h 2 . xxı xx C yyı yy C2 xyı xy/dzdxdy (4.2) ıV D Z 0 hqb.x; y/ıw x; y; h 2 C qt.x; y/ıw x; y; h 2 idxdy Z Z h 2 h 2 ŒO nnıun C O nsıus C O nzıw dzds (4.3) ıKD Z 0 Z h 2 h 2 0h Pu0 z @Pw0 @x ıPu0 z @ıPw0 @x C Pv0 z @Pw0 @y ıPv0 z @ıPw0 @y C P w0ıPw0idzdxdy (4.4) where qb and qt are the distributed forces at the bottom and at the top, respectively, and O nn, O ns, and O nz are the specified stress components on the portion of the boundary .
RkJQdWJsaXNoZXIy MTMzNzEzMQ==