Finite Element Analysis MEEN 5330 Dustin Grant Kamlesh Borgaonkar Varsha Maddela Rupakkumar Patel Sandeep Yarlagadda

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

Basic Concepts Loads Equilibrium Boundary conditions

Development of Theory Rayleigh-Ritz Method Galerkin’s Method Total potential energy equation Galerkin’s Method

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 Galerkin’s method To solve FEA problems same are modified in the Potential-Energy approach and Galerkin’s approach

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

1D Rod Elements Element strain energy Element stiffness matrix Element -1 Element-2 Element strain energy Element stiffness matrix Load vectors Element body load vector Element traction-force vector

Example 1D Rod Elements Example 1 Problem statement: (Problem 3.1 from Chandrupatla and Belegunda’s 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:

2D Truss 2 DOF Transformations Modified Stiffness Matrix Methods of Solving

2D Truss Transformation Matrix Direction Cosines

2D Truss Element Stiffness Matrix

Methods of Solving Elimination Approach Penalty Approach Eliminate Constraints Penalty Approach Will not discuss Today

Elimination Method Set defection at the constraint to equal zero

Elimination Method Modified Equation DOF’s 1,2,4,7,8 equal to zero

2D Truss Element Stresses Element Reaction Forces

2D Truss Development of Tables Coordinate Table Connectivity Table Direction Cosines Table

2D Truss Coordinate Table

2D Truss Connectivity Table

2D Truss Direction Cosines Table

Example 2D Truss

MATLAB Program TRUSS2D.M

3D Truss Stiffness Matrix 3D Transformation Matrix Direction Cosines

3D Truss Stiffness Matrix 3D Stiffness Matrix

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

References

Homework