Analytical solution for multilayer plates using general layerwise plate theory

Abstract

This paper deals with closed-form solution for static analysis of simply supported composite plate, based on generalized laminate plate theory (GLPT). The mathematical model assumes piece-wise linear variation of in-plane displacement components and a constant transverse displacement through the thickness. It also include discrete transverse shear effect into the assumed displacement field, thus providing accurate prediction of transverse shear stresses. Namely, transverse stresses satisfy Hook's law, 3D equilibrium equations and traction free boundary conditions. With assumed displacement field, linear strain-displacement relation, and constitutive equations of the lamina, equilibrium equations are derived using principle of virtual displacements. Navier-type closed form solution of GLPT, is derived for simply supported plate, made of orthotropic laminae, loaded by harmonic and uniform distribution of transverse pressure. Results are compared with 3D elasticity solutions and excellent agreement is found.

U ovom radu prikazano je analitičko rešenje statičke analize slobodno oslonjene kompozitne ploče, zasnovano na opštoj laminatnoj teoriji ploča (GLPT). Matematički model pretpostavlja deo po deo linearnu raspodelu komponenata pomeranja u1,u2 i konstantnu promenu komponente pomeranja u3 po debljini ploče. Pomenuti model u obzir uzima i uticaj smičuće deformacije unutar svakog sloja, čime se dobija realna procena smicanja po debljini ploče. Naime, smičući naponi zadovoljavaju Hook-ov zakon, 3D uslove ravnoteže i granične uslove po naponima. Koristeći pretpostavljeno polje pomeranja linearne veze deformacija i pomeranja i konstitutivne jednačine lamine primenom principa virtualnih pomeranja izvedeni su uslovi ravnoteže. Navier-ovo rešenje GLPT prikazano je za slobodno oslonjenu ploču, sačinjenu od ortotropnih lamina, opterećenu biharmonijskom i konstantnom raspodelom poprečnog opterećenja. Dobijeno je izvrsno slaganje sa 3D rešenjem elastične teorije.