Axis symmetric
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Transcript of Axis symmetric
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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