Three-Dimensional Closed Form Solutions and ∞³Theories for Orthotropic Plates

Abstract

A Navier-type method for finding the exact three-dimensional solution for orthotropic rectangular plates is presented. The present procedure is an extension of a previous author's work (which was valid for isotropic plates) and uses the Mixed Form of Hooke's Law (MFHL) which leads to the formulation of the boundary conditions on the top and bottom surfaces of the plate directly in terms of transverse stresses. The paper also extends the Generalized Unified Formulation (GUF) to the case of orthotropic materials. ∞ 3 theories are presented. The unknown displacements are expanded along the thickness of the plate by using a powerful compact formulation which leads to the writing, with a single formal theory, of a class of ∞ 3 theories. This formal technique allows the user to treat each unknown independently and, therefore, different numerical approaches can be used in the FEM codes based on this Generalized Unified Formulation. This technique can be very useful in multifield problems such as thermoelastic applications and multilayered plates embedding piezo-layers.The paper analyzes the effects of the different orders of expansion for the displacement variables for various material properties. It is shown that when the anisotropy of the mechanical properties is strong the results are more sensitive to the order used to approximate the displacements.Thirty different theories are assessed for both thick and thin plates and the results are compared against the present exact solution.

Analytic solution of angle-ply laminated plates under extension, bending, and torsion

“…The applied pure tension force (P), bending moment or torsion may cause the coupling of the tension strain ("), bending curvature (b 1 ), and twisting curvature (#). Therefore, to find the stress field due to the applied tension force (P), bending moment (M 1 ), and torsion (M t ), as shown in equations (8) to (10), the following relationships should be used.…”

“…Pan 7 derived exact solutions for 3D magneto-electro-elastic plates under static loadings. Demasi 8 developed a Navier-type method to obtain an exact 3D solution for simply supported rectangular plates. Yang et al.…”

“…Pan 7 derived exact solutions for 3D magneto-electro-elastic plates under static loadings. Demasi 8 developed a Navier-type method to obtain an exact 3D solution for simply supported rectangular plates. Yang et al 9 used pseudo-Stroh formalism to derive an exact solution for simply supported plates under surface loadings.…”

Variational analysis of free-edge stress and displacement fields in general un-symmetric and thin-ply laminates under in-plane, bending and thermal loading

“…It is worth mentioning that unlike the current approach based on an assumed form for the through-thickness variations of the in-plane stress fields, some authors have tried to solve exactly the three-dimensional differential equations of elasticity based on the assumption of arbitrary shape functions for the displacement components through both the in-plane and out-of-plane coordinates. However, these approaches are still restricted to cross-ply laminates with free-edges [50] and general laminates with simply-supported boundary conditions [51,52] without considering the effects of residual stresses. The future works necessitate the extension of these works to deal with general laminates containing straight free edges, ply cracks and delaminations considering the effects of residual stresses.…”

Static Analysis Of Orthotropic Plates By Double Fourier Series

“…A., (2015)). Demasi, daha önceden izotrop plaklar için yapmış olduğu çalışmayı geliştirerek ortotrop dikdörtgen plakların üç boyutlu kesin çözümünü Navier metodu ile hesaplamıştır (Demasi, L., (2010)). Kabir, yaptığı çalışmada basit mesnetli tabakalı dikdörtgen plak için bir analitik çözüm sunmuştur.…”