Finite Element Analysis

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Finite Element Analysis

• MEEN 5330
• Dustin Grant
• Kamlesh Borgaonkar
• Varsha Maddela
• Rupakkumar Patel
• Sandeep Yarlagadda

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Introduction

• What is finite element analysis, FEA?
• What is FEA used for?
• 1D Rod Elements, 2D Trusses

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Basic Concepts

• Loads
• Equilibrium
• Boundary conditions

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Development of Theory

• Rayleigh-Ritz Method
• Total potential energy equation
• Galerkins Method

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1D Rod Elements

• To understand and solve 2D and 3D problems we
  must understand basic of 1D problems.
• Analysis of 1D rod elements can be done using
  Rayleigh-Ritz and Galerkins method
• To solve FEA problems same are modified in the
  Potential-Energy approach and Galerkins approach

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1D Rod Elements

• Loading consists of three types body force f ,
  traction force T, point load Pi
• Body force distributed force , acting on every
  elemental volume of body i.e. self weight of
  body.
• Traction force distributed force , acting on
  surface of body i.e. frictional resistance,
  viscous drag and surface shear
• Point load a force acting on any single point of
  element

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1D Rod Elements
Element -1 Element-2

• Element strain energy
• Element stiffness matrix
• Load vectors
• Element body load vector
• Element traction-force vector

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Example 1D Rod Elements
Example 1 Problem statement (Problem 3.1 from
Chandrupatla and Belegundas book) Consider the
bar in Fig.1, determine the following by hand
calculation 1) Displacement at point P 2)
Strain and stress 3) Element stiffness matrix 4)
strain energy in element
Given
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2D Truss

• 2 DOF
• Transformations
• Modified Stiffness Matrix
• Methods of Solving

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2D Truss

• Transformation Matrix
• Direction Cosines

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2D Truss

• Element Stiffness Matrix

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Methods of Solving

• Elimination Approach
• Eliminate Constraints
• Penalty Approach
• Will not discuss Today

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Elimination Method

• Set defection at the constraint to equal zero

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Elimination Method

• Modified Equation
• DOFs 1,2,4,7,8 equal to zero

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2D Truss

• Element Stresses
• Element Reaction Forces

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2D Truss

• Development of Tables
• Coordinate Table
• Connectivity Table
• Direction Cosines Table

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2D Truss

• Coordinate Table

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2D Truss

• Connectivity Table

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2D Truss

• Direction Cosines Table

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Example 2D Truss
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MATLAB Program TRUSS2D.M
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3D Truss Stiffness Matrix

• 3D Transformation Matrix
• Direction Cosines

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3D Truss Stiffness Matrix

• 3D Stiffness Matrix

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Conclusion

• Good at Hand Calculations, Powerful when applied
  to computers
• Only limitations are the computer limitations

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References
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Homework