Introduction of Finite Element Analysis

General steps of finite element analysis

1. ME8692--FINITE ELEMENT ANALYSIS UNIT I - INTRODUCTION Historical Background – Mathematical Modeling of field problems in Engineering – Governing Equations – Discrete and continuous models – Boundary, Initial and Eigen Value problems– Weighted Residual Methods – Variational Formulation of Boundary Value Problems – Ritz Technique – Basic concepts of the Finite Element Method.

2. Method of Engineering Analysis Experimental methods Analytical methods (or) Theoretical Analysis Numerical methods (or) Approximate methods

3. Experimental methods Prototype can be used Time consuming and costly process Needs man power and material

4. Analytical methods Problems are expressed by mathematical differential equations. It is used only for simple geometries and loading conditions.

5. Numerical methods (or) Approximate methods Problems involving complex material properties and boundary conditions. The following three methods are coming under numerical solutions. Functional Approximation Finite Difference Method (FDM) Finite E lement Method (FEA)

6. Finite Element Method (FEA) Finite element method is a numerical method for solving problems of engineering and mathematical physics. In this method, a body or structure in which the analysis to be carried out is subdivided into smaller elements of finite dimensions called finite elements. Then the body is considered as an assembly of these elements connected at a finite number of joints called Nodes. The properties of each type of finite element is obtained and assembled together and solved as whole to get solution. This method extensively used in the field of structural mechanics, fluid mechanics , heat transfer, mass transfer, electric and magnetic fields problems

7. Based on application, finite element problems are classified as follows Structural Problems Displacement,Stress and Strain in each element can be calculated Non-structural Problems Temperature (or) Fluid pressure at each nodal point is obtained

8. General Steps of the Finite Element Analysis The following two general methods are associated with the FEA. Force Method : Internal forces are considered as the unknowns of the problem Displacement or stiffness method : Displacement of the nodes are considered as the unknowns of the problem.

9. Step 1 : Discretization of structure The art of subdividing a structure into a convenient number of smaller elements is known as discretization. Smaller elements are classified as follows One dimensional elements Two dimensional elements Three dimensional elements Axis symmetric elements

10. One dimensional elements A bar and beam elements are considered as one dimensional elements. The simplest line element also known as linear element has two nodes, one at each end as shown in figure.

11. Two dimensional elements Triangular and rectangular elements are considered as two dimensional elements.

12. Three dimensional elements The most common three dimensional elements are tetrahedral and hexahedral (Brick) elements.

13. Axis symmetric elements The axis symmetric element is developed by rotating a triangle or quadrilateral about a fixed axis located in the plane of the element through 360°.

14. Step 2 : Numbering of Nodes and Elements The nodes and elements should be numbered after discretization process. The numbering process is most important since it decide the size of the stiffness matrix and it leads the reduction of memory requirement.

15. Step 3 : Selection of a Displacement Function or Interpolation Function

16. Step 4 : Define the material behaviour by using Strain-Displacement and Stress-Strain Relationships

17. Step 5 : Derivation of element stiffness matrix and equations

18. Step 6 : Assemble the element equations to obtain the global or total equations

19. Step 7 : Applying Boundary Conditions From the above equation, Global stiffness matrix is a singular matrix. Boundary conditions are applied in that matrix.

20. Step 8 : Solution for the unknown displacements The unknown displacements {u} are calculated by using Gaussian elimination method or Gauss Seidel method.

21. Step 9 : Computation of the element strains and stresses from the nodal displacements

22. Step 10 : Interpret the results Analysis and evaluation of the solution results is referred to as post processing. Post processor computer programs help the user to interpret the results by displaying them in graphical form.

23. Discretization Introduction

24. Discretization The art of subdividing a structure into a convenient number of smaller components is known as Discretization. These smaller components are then put together. The process of uniting the various elements together is called Assemblage. The assemblage of such elements then represents the original body. Discretization can be classified as follows Natural Artificial (Continuum)

25. Natural Discretization In structural analysis, a truss is considered as a natural system. The various members of the truss constitute the elements. These elements are connected at various joints is known as nodes

26. Artificial Discretization Artificial Discretization is generally considered to be a single mass of material as found in a forging, concrete dam, deep beam and plate.

27. Discretization Process Type of elements Size of elements Location of nodes Number of elements