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

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.

Introduction

Advantages

– Smaller models (3D to 2D)

– Faster execution

– Easier post processing (FEA software)

To model This ?

How to model ?

How to model ?

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.

Examples

• Domed pressure vessel

• Engine valve stem

Derivation of the Stiffness Matrix

N,M-Mid side nodes

z axial stress

, Φ hoops stress

r radial stress

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

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

• 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 )

• Step 2 Select Displacement Functions

o The element displacement functions are taken to be

• The nodal displacements are

• The general displacement function is then expressed in matrix

• Substituting the coordinates of the nodal points

• Performing the inversion operations

Shape Function

• Interpolation function w.r.t fixed nodes

• Input – nodal position

• Output - deformation

• Shape functions

• General displacement function

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

Strain Stress

Step 4 Derive the Element Stiffness Matrix and Equations

• Centroid point of element

• Surface Forces

• Body force

EXAMPLE

Bulb

Drilling platform

Problem

Global martrix

PROBLEM 2

PROBLEM 2

PROBLEM 2

PROBLEM 2

Reference

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

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

Thank you