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Presentation on Shape Function of Axisymmetric Element PSG COLLEGE OF TECHNOLOGY COIMBATORE-641005 Presented by, GOWSICK C S (16MI34) KARTHIKEYAN K (16MI06) 1 st year ME-CIM Department Of Mechanical Engineering PSG College of Technology

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Page 1: Axis symmetric

Presentation on Shape Function of Axisymmetric

Element

PSG COLLEGE OF TECHNOLOGYCOIMBATORE-641005

Presented by,GOWSICK C S (16MI34)

KARTHIKEYAN K (16MI06)1st year ME-CIM

Department Of Mechanical EngineeringPSG College of Technology

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Introduction

• Axisymmetric element is an two-dimensional element with 3 nodes and 6 DOF.

• When element is symmetry with respect to geometry and loading exists about an axis of the body

Application:

• Soil masses subjected thick-walled pressure vessels.

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Introduction

Advantages

– Smaller models (3D to 2D)

– Faster execution

– Easier post processing (FEA software)

To model This ?

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How to model ?

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How to model ?

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Axisymmetric Element

• In Triangular tori,each element is symmetric with respect to geometry and loading about z axis. z axis is called the axis of symmetry or the axis of revolution.

• Nodal points are I,j,m.

• r, Φ, and z indicate the radial, circumferential, and longitudinal direction.

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Examples

• Domed pressure vessel

• Engine valve stem

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Derivation of the Stiffness Matrix

N,M-Mid side nodes

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z axial stress

, Φ hoops stress

r radial stress

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Derivation of the Stiffness Matrix

• The normal strain in the radial direction is then given by

• The tangential strain is then given by

• The longitudinal normal strain given by

• Shear strain in the r-z plane given by

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Properties

• Isotropic E≡G≡K≡v (uniform) in x,y,z

E.g All metals except mercury

• Orthotropic- E≡G≡K≡v varies orthogonal wrtx,y

E.g composite fibre, plywood

• Anisotropic- E≡G≡K≡v varies non uniformly in x,y,z

E.g Rocks

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• Isotropic stress/strain relationship

• Step 1-Select Element Type

o The element has three nodes with two degrees of freedom per node(that is, ui, wi at node i )

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• Step 2 Select Displacement Functions

o The element displacement functions are taken to be

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• The nodal displacements are

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• The general displacement function is then expressed in matrix

• Substituting the coordinates of the nodal points

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• Performing the inversion operations

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Shape Function

• Interpolation function w.r.t fixed nodes

• Input – nodal position

• Output - deformation

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• Shape functions

• General displacement function

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• Step 3 Define the Strain/Displacement and Stress/Strain Relationships

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Strain Stress

Step 4 Derive the Element Stiffness Matrix and Equations

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• Centroid point of element

• Surface Forces

• Body force

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EXAMPLE

Bulb

Drilling platform

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Problem

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Global martrix

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PROBLEM 2

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PROBLEM 2

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PROBLEM 2

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PROBLEM 2

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Reference

• Daryl L. Logan, "A first course in finite element method”

•“Introduction to finite element in engineering” by D.Belegundu

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Thank you