Presentation on theme: "Finite Element Method CHAPTER 8: FEM FOR PLATES & SHELLS"— Presentation transcript:
1 Finite Element Method CHAPTER 8: FEM FOR PLATES & SHELLS for readers of all backgroundsG. R. Liu and S. S. QuekCHAPTER 8:FEM FOR PLATES & SHELLS
2 CONTENTS INTRODUCTION PLATE ELEMENTS SHELL ELEMENTS Shape functions Element matricesSHELL ELEMENTSElements in local coordinate systemElements in global coordinate systemRemarks
3 INTRODUCTIONFE 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.
4 PLATE 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.
12 Element matrices Substitute into Recall that: where (Can be evaluated analytically but in practice, use Gauss integration)
13 Element matrices Substitute into potential energy function from which we obtainNote:
14 Element matrices(me can be solved analytically but practically solved using Gauss integration)For uniformly distributed load,
15 SHELL 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.
16 Elements in local coordinate system Consider a flat shell element
17 Elements in local coordinate system Membrane stiffness (2D solid element):(2x2)Bending stiffness (plate element):(3x3)
18 Elements in local coordinate system Components related to the DOF qz, are zeros in local coordinate system.(24x24)
19 Elements in local coordinate system Membrane mass matrix (2D solid element):Bending mass matrix (plate element):
20 Elements in local coordinate system Components related to the DOF qz, are zeros in local coordinate system.(24x24)
22 RemarksThe 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.
24 CASE 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