PSI - Issue 8

V.G. Belardi et al. / Procedia Structural Integrity 8 (2018) 368–378

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V.G. Belardi et al. / Structural Integrity Procedia 00 (2017) 000–000

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layers lay-up and ratios of outer to inner radius of the plate were investigated. It is found that the symmetric buckling assumption is not adequate when considering clamped boundary conditions. Rajappa (1963) provided the natural frequencies and mode shapes of a rectilinear orthotropic clamped circular plate applying Galerkin method. Salehi and Sobhani (2004) published a work concerning symmetrically laminated sector plate presenting rectilinear orthotropic characteristics, considering Mindlin plate theory. Two typologies of displacement constraints were ana lyzed (simply supported and clamped edge) meanwhile the applied external load was a constant pressure. The system of equilibrium di ff erential equations of the plate, in both linear and non-linear forms, were discretized through the ap plication of the DR iterative method together with the finite di ff erence discretization technique. Results were presented taking into account various angles of the sector, di ff erent slenderness ratios of the plate and lay-up configurations, the comparison with FE analyses shows a good agreement. With a numerical approach, Kadkhodayan et al. (2012) analyzed square, circular and elliptical rectilinear or thotropic plates under the action of a uniform cross load. The focus is on the combination of the DXDR dynamic relaxation method, that is employed to solve the finite-di ff erence discretized governing plate equations in terms of Cartesian coordinates, and irregular schemes of rectilinear mesh. The study of sector plates has been carried out by Maleki and Tahani (2013) and Andakhshideh et al. (2010). The first analyzed bending and first-order shear deformation with di ff erent lay-ups and displacement boundary con ditions. For the laminate circular plates, both polar and rectilinear orthotropy are considered and the circumferential variation of sti ff ness is taken into account for the latter case. Problem formulation is carried out in the cylindrical coordinate system then, governing equations and related boundary conditions are discretized with the generalized di ff erential quadrature method. Another approach to non-linear static analysis of laminated sector plates is the one of Andakhshideh et al. (2010) where the generalized di ff erential quadrature method is used for geometrically non-linear large deformation of laminated sector plates with any combination of clamped, simply supported and free edges and general laminate lay-ups. Von Ka´rma´n non-linearity is considered in conjunction with first-order shear deformation theory. Moreover, results concerning asymmetric lay-ups are presented. In this frame of much di ff erentiated numerical approaches to the problem of bending of laminates applied to various kind of plates or sector plates and considering various laminate typologies, the present paper outlines an original procedure to achieve the third order di ff erential governing equation for quasi-isotropic rectilinear orthotropic composite circular plates, undergoing external loads transversally disposed with respect to the plate mid-surface, starting from the equilibrium equations and the stress resultants-strains relations and subsequently solved exploiting Galerkin method. This work represents a preliminary study limited to the case of quasi-isotropic stacking sequence for the laminate. This kind of plates are laminates with axisymmetric geometry, made up of unidirectionally reinforced layers with di ff erent orientations; they can be considered circular plates cut-out from a rectangular composite plate. In transver sally loaded circular plates, manufactured with this kind of composite material, the deflected mid-surface is not in dependent from the circumferential coordinate, unlike the circular plates manufactured with an isotropic material. Nevertheless, the quasi-isotropic stacking sequence makes still possible to introduce the hypothesis of axisymmetry for the mid-surface deflection of the rectilinear orthotropic composite circular plate, under transversal load, disregard ing the circumferential variation of the vertical displacement connected to the variable bending sti ff ness. A greater and more general insight of the problem of rectilinear orthotropic composite circular plate transversally loaded can be found in Belardi et al. (2018), where the same authors considered the case of not axisymmetric mid-surface deflection. The Galerkin method is applied to solve the governing third order di ff erential equation in terms of mid-surface deflection, this requires the introduction of specific and appropriate polynomial approximation functions, specifically derived, compliant with the boundary conditions. In particular, fully clamped constraint conditions were considered for the outer diameter of the plate in conjunction with an internal rigid core. The characterization of this model allows to define the sti ff ness matrix terms of a custom composite bolted joint finite element, that is the object of future developments of this work. Results of the original proposed method are presented and compared to those obtained by means of FEA performed with a refined reference model, for quasi-isotropic laminate stacking sequences and di ff erent slenderness and radius ratios; the matching of results is good, so that the analytical procedure proposed and developed here is verified. It