ME300H Introduction to Finite Element Methods

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ME300H Introduction to Finite Element Methods
Finite Element Analysis of Plane Elasticity
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Review of Linear Elasticity
Linear Elasticity A theory to predict mechanical
response of an elastic body under a general
loading condition.
Stress measurement of force intensity
with
2-D
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Review of Linear Elasticity
Traction (surface force)
Equilibrium Newtons Law
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Review of Linear Elasticity
Strain measurement of intensity of deformation
Generalized Hookes Law
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Plane Stress and Plane Strain
Plane Stress - Thin Plate
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Plane Stress and Plane Strain
Plane Strain - Thick Plate
Plane Stress
Plane Strain
Replace E by and by
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Equations of Plane Elasticity
Governing Equations (Static Equilibrium)
Strain-Deformation (Small Deformation)
Constitutive Relation (Linear Elasticity)
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Specification of Boundary Conditions
EBC Specify u(x,y) and/or v(x,y) on G NBC
Specify tx and/or ty on G
where is the
traction on the boundary G at the segment ds.
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Weak Formulation for Plane Elasticity
are components of traction on the boundary G
where
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Finite Element Formulation for Plane Elasticity
Let
where
and
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Constant-Strain Triangular (CST) Element for
Plane Stress Analysis
Let
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Constant-Strain Triangular (CST) Element for
Plane Stress Analysis
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4-Node Rectangular Element for Plane Stress
Analysis
Let
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4-Node Rectangular Element for Plane Stress
Analysis
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Loading Conditions for Plane Stress Analysis
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Evaluation of Applied Nodal Forces
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Evaluation of Applied Nodal Forces
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Element Assembly for Plane Elasticity
5
6
B
3
4
4
3
A
1
2
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Element Assembly for Plane Elasticity
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Comparison of Applied Nodal Forces
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Discussion on Boundary Conditions

• Must have sufficient EBCs to suppress rigid body
• translation and rotation
• For higher order elements, the mid side nodes
  cannot be
• skipped while applying EBCs/NBCs

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Plane Stress Example 2
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Plane Stress Example 3
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Evaluation of Strains
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Evaluation of Stresses
Plane Strain Analysis
Plane Stress Analysis