Semi-Analytical Finite Element Analysis of Longitudinal-Transverse Bending of Laminated Plates Under Localized Loading Within the Three-Dimensional Theory of Elasticity

Two variants of a semi-analytical finite element method are proposed for solving complex three-dimensional elasticity problems of laminated plates subjected to longitudinal-transverse bending under localized loading. The first variant employs a plate model that accounts for transverse shear and thickness compression. Its distinctive feature is that the unknown functions are defined on the outer surfaces of the layers, which makes it possible to subdivide individual layers into sublayers and thereby improve the accuracy of the numerical results. In contrast, conventional plate models accounting for shear and thickness compression generally use a single reference surface, which makes layer subdivision difficult to implement. In the second variant, the through-thickness distributions of the unknown functions are determined by an exact solution of the corresponding system of differential equations. In both variants, the finite element method is used for approximation along the (x)-axis, whereas trigonometric functions are employed for approximation along the (y)-axis. The development of two independent approaches is motivated by the sensitivity of the considered problems to approximation and numerical errors, which becomes particularly pronounced for layered plates subjected to localized longitudinal-transverse bending when the applied longitudinal load approaches its critical value. The agreement between the results obtained using the two variants provides an independent verification of the proposed approaches. Numerical analyses of the behavior of layered plates under localized longitudinal-transverse bending were performed. The results demonstrate that, for the class of problems considered, polynomial approximation of the unknown functions through the thickness of a layer represented by a single sublayer may produce significant errors, particularly when the longitudinal load is close to its critical value.