Presentation on theme: "Finite Element Method CHAPTER 8: FEM FOR PLATES & SHELLS"— Presentation transcript:
1Finite Element Method CHAPTER 8: FEM FOR PLATES & SHELLS for readers of all backgroundsG. R. Liu and S. S. QuekCHAPTER 8:FEM FOR PLATES & SHELLS
2CONTENTS INTRODUCTION PLATE ELEMENTS SHELL ELEMENTS Shape functions Element matricesSHELL ELEMENTSElements in local coordinate systemElements in global coordinate systemRemarks
3INTRODUCTIONFE equations based on Mindlin plate theory will be developed.FE equations of shells will be formulated by superimposing matrices of plates and those of 2D solids.Computationally tedious due to more DOFs.
4PLATE ELEMENTSGeometrically similar to 2D plane stress solids except that it carries only transverse loads. Leads to bending.2D equilvalent of the beam element.Rectangular plate elements based on Mindlin plate theory will be developed – conforming element.Much software like ABAQUS does not offer plate elements, only the general shell element.
5(Mindlin plate theory) PLATE ELEMENTSConsider a plate structure:(Mindlin plate theory)
6PLATE ELEMENTS Mindlin plate theory: In-plane strain: where (Curvature)
12Element matrices Substitute into Recall that: where (Can be evaluated analytically but in practice, use Gauss integration)
13Element matrices Substitute into potential energy function from which we obtainNote:
14Element matrices(me can be solved analytically but practically solved using Gauss integration)For uniformly distributed load,
15SHELL ELEMENTS Loads in all directions Bending, twisting and in-plane deformationCombination of 2D solid elements (membrane effects) and plate elements (bending effect).Common to use shell elements to model plate structures in commercial software packages.
16Elements in local coordinate system Consider a flat shell element
17Elements in local coordinate system Membrane stiffness (2D solid element):(2x2)Bending stiffness (plate element):(3x3)
18Elements in local coordinate system Components related to the DOF qz, are zeros in local coordinate system.(24x24)
19Elements in local coordinate system Membrane mass matrix (2D solid element):Bending mass matrix (plate element):
20Elements in local coordinate system Components related to the DOF qz, are zeros in local coordinate system.(24x24)
22RemarksThe membrane effects are assumed to be uncoupled with the bending effects in the element level.This implies that the membrane forces will not result in any bending deformation, and vice versa.For shell structure in space, membrane and bending effects are actually coupled (especially for large curvature), therefore finer element mesh may have to be used.
24CASE STUDY Mode Natural Frequencies (MHz) 768 triangular elements with 480 nodes384 quadrilateral elements with 480 nodes1280 quadrilateral elements with 1472 nodes17.675.084.86237.877.447.41410.588.528.305613.8411.6911.447814.8612.4512.17CASE STUDY