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1 Finite Element Method FEM FOR 2D SOLIDS for readers of all backgrounds G. R. Liu and S. S. Quek CHAPTER 7:

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Finite Element Method by G. R. Liu and S. S. Quek 2 CONTENTS INTRODUCTION LINEAR TRIANGULAR ELEMENTS – Field variable interpolation – Shape functions construction – Using area coordinates – Strain matrix – Element matrices LINEAR RECTANGULAR ELEMENTS – Shape functions construction – Strain matrix – Element matrices – Gauss integration – Evaluation of m e

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Finite Element Method by G. R. Liu and S. S. Quek 3 CONTENTS LINEAR QUADRILATERAL ELEMENTS – Coordinate mapping – Strain matrix – Element matrices – Remarks HIGHER ORDER ELEMENTS COMMENTS (GAUSS INTEGRATION)

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Finite Element Method by G. R. Liu and S. S. Quek 4 INTRODUCTION 2D solid elements are applicable for the analysis of plane strain and plane stress problems. A 2D solid element can have a triangular, rectangular or quadrilateral shape with straight or curved edges. A 2D solid element can deform only in the plane of the 2D solid. At any point, there are two components in the x and y directions for the displacement as well as forces.

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Finite Element Method by G. R. Liu and S. S. Quek 5 INTRODUCTION For plane strain problems, the thickness of the element is unit, but for plane stress problems, the actual thickness must be used. In this course, it is assumed that the element has a uniform thickness h. Formulating 2D elements with a given variation of thickness is also straightforward, as the procedure is the same as that for a uniform element.

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Finite Element Method by G. R. Liu and S. S. Quek 6 2D solids – plane stress and plane strain Plane stressPlane strain

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Finite Element Method by G. R. Liu and S. S. Quek 7 LINEAR TRIANGULAR ELEMENTS Less accurate than quadrilateral elements Used by most mesh generators for complex geometry A linear triangular element:

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Finite Element Method by G. R. Liu and S. S. Quek 8 Field variable interpolation where (Shape functions)

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Finite Element Method by G. R. Liu and S. S. Quek 9 Shape functions construction Assume, i= 1, 2, 3 or

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Finite Element Method by G. R. Liu and S. S. Quek 10 Shape functions construction Delta function property: Therefore, Solving,

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Finite Element Method by G. R. Liu and S. S. Quek 11 Shape functions construction Area of triangle Moment matrix Substitute a 1, b 1 and c 1 back into N 1 = a 1 + b 1 x + c 1 y:

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Finite Element Method by G. R. Liu and S. S. Quek 12 Shape functions construction Similarly,

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Finite Element Method by G. R. Liu and S. S. Quek 13 Shape functions construction where i jk i= 1, 2, 3 J, k determined from cyclic permutation i = 1, 2 j = 2, 3 k = 3, 1

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Finite Element Method by G. R. Liu and S. S. Quek 14 Using area coordinates Alternative method of constructing shape functions 2-3-P: Similarly, 3-1-P A2A2 1-2-P A3A3

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Finite Element Method by G. R. Liu and S. S. Quek 15 Using area coordinates Partitions of unity: Delta function property: e.g. L 1 = 0 at if P at nodes 2 or 3 Therefore,

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Finite Element Method by G. R. Liu and S. S. Quek 16 Strain matrix where (constant strain element)

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Finite Element Method by G. R. Liu and S. S. Quek 17 Element matrices Constant matrix

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Finite Element Method by G. R. Liu and S. S. Quek 18 Element matrices For elements with uniform density and thickness, Eisenberg and Malvern (1973):

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Finite Element Method by G. R. Liu and S. S. Quek 19 Element matrices Uniform distributed load:

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Finite Element Method by G. R. Liu and S. S. Quek 20 LINEAR RECTANGULAR ELEMENTS Non-constant strain matrix More accurate representation of stress and strain Regular shape makes formulation easy

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Finite Element Method by G. R. Liu and S. S. Quek 21 Shape functions construction Consider a rectangular element

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Finite Element Method by G. R. Liu and S. S. Quek 22 Shape functions construction where (Interpolation)

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Finite Element Method by G. R. Liu and S. S. Quek 23 Shape functions construction Delta function property Partition of unity

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Finite Element Method by G. R. Liu and S. S. Quek 24 Strain matrix Note: No longer a constant matrix!

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Finite Element Method by G. R. Liu and S. S. Quek 25 Element matrices dxdy = ab d d Therefore,

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Finite Element Method by G. R. Liu and S. S. Quek 26 Element matrices For uniformly distributed load,

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Finite Element Method by G. R. Liu and S. S. Quek 27 Gauss integration For evaluation of integrals in k e and m e (in practice) In 1 direction: m gauss points gives exact solution of polynomial integrand of n = 2m - 1 In 2 directions:

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Finite Element Method by G. R. Liu and S. S. Quek 28 Gauss integration m j wjwj Accuracy n / 3, 1/ 3 1, , 0, 0.6 5/9, 8/9, 5/ , , , , , , , , 0, , , , , , , , , , , , , , , ,

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Finite Element Method by G. R. Liu and S. S. Quek 29 Evaluation of m e

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Finite Element Method by G. R. Liu and S. S. Quek 30 Evaluation of m e E.g. Note: In practice, Gauss integration is often used

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Finite Element Method by G. R. Liu and S. S. Quek 31 LINEAR QUADRILATERAL ELEMENTS Rectangular elements have limited application Quadrilateral elements with unparallel edges are more useful Irregular shape requires coordinate mapping before using Gauss integration

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Finite Element Method by G. R. Liu and S. S. Quek 32 Coordinate mapping Physical coordinatesNatural coordinates (Interpolation of displacements) (Interpolation of coordinates)

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Finite Element Method by G. R. Liu and S. S. Quek 33 Coordinate mapping where,

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Finite Element Method by G. R. Liu and S. S. Quek 34 Coordinate mapping Substitute 1 into or Eliminating,

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Finite Element Method by G. R. Liu and S. S. Quek 35 Strain matrix or where (Jacobian matrix) Since,

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Finite Element Method by G. R. Liu and S. S. Quek 36 Strain matrix Therefore, Replace differentials of N i w.r.t. x and y with differentials of N i w.r.t. and (Relationship between differentials of shape functions w.r.t. physical coordinates and differentials w.r.t. natural coordinates)

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Finite Element Method by G. R. Liu and S. S. Quek 37 Element matrices Murnaghan (1951) : dA=det |J | d d

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Finite Element Method by G. R. Liu and S. S. Quek 38 Remarks Shape functions used for interpolating the coordinates are the same as the shape functions used for interpolation of the displacement field. Therefore, the element is called an isoparametric element. Note that the shape functions for coordinate interpolation and displacement interpolation do not have to be the same. Using the different shape functions for coordinate interpolation and displacement interpolation, respectively, will lead to the development of so-called subparametric or superparametric elements.

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Finite Element Method by G. R. Liu and S. S. Quek 39 HIGHER ORDER ELEMENTS Higher order triangular elements n d = (p+1)(p+2)/2 Node i, Argyris, 1968 :

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Finite Element Method by G. R. Liu and S. S. Quek 40 HIGHER ORDER ELEMENTS Higher order triangular elements (Contd) Cubic element Quadratic element

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Finite Element Method by G. R. Liu and S. S. Quek 41 HIGHER ORDER ELEMENTS Higher order rectangular elements Lagrange type: [Zienkiewicz et al., 2000]

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Finite Element Method by G. R. Liu and S. S. Quek 42 HIGHER ORDER ELEMENTS Higher order rectangular elements (Contd) (nine node quadratic element)

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Finite Element Method by G. R. Liu and S. S. Quek 43 HIGHER ORDER ELEMENTS Higher order rectangular elements (Contd) Serendipity type: (eight node quadratic element)

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Finite Element Method by G. R. Liu and S. S. Quek 44 HIGHER ORDER ELEMENTS Higher order rectangular elements (Contd) (twelve node cubic element)

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Finite Element Method by G. R. Liu and S. S. Quek 45 ELEMENT WITH CURVED EDGES

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Finite Element Method by G. R. Liu and S. S. Quek 46 COMMENTS (GAUSS INTEGRATION) When the Gauss integration scheme is used, one has to decide how many Gauss points should be used. Theoretically, for a one-dimensional integral, using m points can give the exact solution for the integral of a polynomial integrand of up to an order of (2m 1). As a general rule of thumb, more points should be used for a higher order of elements.

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Finite Element Method by G. R. Liu and S. S. Quek 47 COMMENTS (GAUSS INTEGRATION) Using a smaller number of Gauss points tends to counteract the over-stiff behaviour associated with the displacement-based method. Displacement in an element is assumed using shape functions. This implies that the deformation of the element is somehow prescribed in a fashion of the shape function. This prescription gives a constraint to the element. The so- constrained element behaves stiffer than it should. It is often observed that higher order elements are usually softer than lower order ones. This is because using higher order elements gives fewer constraint to the elements.

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Finite Element Method by G. R. Liu and S. S. Quek 48 COMMENTS ON GAUSS INTEGRATION Two Gauss points for linear elements, and two or three points for quadratic elements in each direction should be sufficient for most cases. Most of the explicit FEM codes based on explicit formulation tend to use one-point integration to achieve the best performance in saving CPU time.

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Finite Element Method by G. R. Liu and S. S. Quek 49 CASE STUDY Side drive micro-motor

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Finite Element Method by G. R. Liu and S. S. Quek 50 CASE STUDY Elastic Properties of Polysilicon Youngs Modulus, E169GPa Poissons ratio, Density, 2300kgm -3 10N/m

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Finite Element Method by G. R. Liu and S. S. Quek 51 CASE STUDY Analysis no. 1: Von Mises stress distribution using 24 bilinear quadrilateral elements (41 nodes)

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Finite Element Method by G. R. Liu and S. S. Quek 52 CASE STUDY Analysis no. 2: Von Mises stress distribution using 96 bilinear quadrilateral elements (129 nodes)

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Finite Element Method by G. R. Liu and S. S. Quek 53 CASE STUDY Analysis no. 3: Von Mises stress distribution using 144 bilinear quadrilateral elements (185 nodes)

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Finite Element Method by G. R. Liu and S. S. Quek 54 CASE STUDY Analysis no. 4: Von Mises stress distribution using 24 eight-nodal, quadratic elements (105 nodes)

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Finite Element Method by G. R. Liu and S. S. Quek 55 CASE STUDY Analysis no. 5: Von Mises stress distribution using 192 three-nodal, triangular elements (129 nodes)

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Finite Element Method by G. R. Liu and S. S. Quek 56 CASE STUDY Analysis no. Number / type of elements Total number of nodes in model Maximum Von Mises Stress (GPa) 1 24 bilinear, quadrilateral bilinear, quadrilateral bilinear, quadrilateral quadratic, quadrilateral linear, triangular

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