A Three-Variable Geometrically Nonlinear New First-Order Shear Deformation Theory for Isotropic Plates: Formulation and
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A Three‑Variable Geometrically Nonlinear New First‑Order Shear Deformation Theory for Isotropic Plates: Formulation and Buckling Analysis Rameshchandra P. Shimpi1 · P. J. Guruprasad1 · Kedar S. Pakhare1 Received: 23 August 2019 / Accepted: 20 March 2020 © Shiraz University 2020
Abstract This paper presents a displacement-based geometrically nonlinear first-order shear deformation theory for the analysis of shear deformable isotropic plates. Nonlinear strain–displacement relations as utilized by the von Kármán plate theory and linear stress–strain constitutive relations are used to formulate this theory. Governing equations of this theory are derived by utilizing equilibrium equations for an infinitesimal plate element. Commonly occurring plate edge boundary conditions are described based on the physical understanding of the plate deformation. As against other geometrically nonlinear first-order shear deformation plate theories reported in the literature, main contributions of this theory are that (1) it incorporates the rotation-free shear deformation plate kinematics of the first order; (2) this theory involves only three governing equations involving only three unknown functions; (3) expressions of governing equations of this theory have a striking resemblance to corresponding expressions of the von Kármán plate theory; (4) this theory describes two unique, physically meaningful plate clamped edge boundary conditions. Illustrative examples pertaining to the buckling of shear deformable isotropic Lévytype plates and the comparison of the results obtained with the corresponding results reported in the literature demonstrate the efficacy of the proposed theory. Keywords First-order shear deformation plate theory · Geometrically nonlinear plate theory · von Kármán plate theory · Plate buckling
1 Introduction The von Kármán plate theory is a displacement-based theory which deals with deformations of thin plates wherein the plate transverse displacement is of the order of the plate thickness and the plate in-plane displacements are small as compared to the plate thickness. It neglects transverse shear deformation effects present through the plate thickness in its displacement field. The von Kármán plate theory,
* Kedar S. Pakhare [email protected] Rameshchandra P. Shimpi [email protected] P. J. Guruprasad [email protected] 1
Department of Aerospace Engineering, Indian Institute of Technology Bombay, Mumbai, India
as discussed by Shames and Dym (2016), involves three governing differential equations involving three unknown functions. The importance of transverse shear deformations with respect to plate theories has been brought out by Reissner (1945) and Mindlin (1951). It should be noted that Reissner’s plate theory (Reissner 1945) is a stress-based first-order shear deformation plate theory (FSDT), whereas Mindlin’s plate theory Mindlin (1951) is a displacement-based FSDT. Displacement-based FSDTs assume displacement fields which give rise to constant transverse shear strains t
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