Finite Element Method/ Finite Element Analysis

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This document does not claim any originality and cannot be used as a substitute for prescribed textbooks. I would like to acknowledge various sources like freely available materials from the internet particularly NPTEL/ SWAYAM course material from which the lecture note was prepared. The ownership of the information lies with the respective authors or institutions. Further, this document is not intended to be used for commercial purposes and the BlogSpot owner is not accountable for any issues, legal or otherwise, arising out of the use of this document.

This open resource is a collection of the academic courses of under graduation program for B. Tech (Civil Engineering) and M. Tech as per the syllabus of  Dr. B.A.T University, Lonere, Raigad (m.s), India prepared by Dr. Mohd. Zameeruddin,  Associate Professor, MGM's College of Engineering, Nanded for use in the out-of-class activity. The content covers both theoretical and analytical studies. There are six lessons as part of this document and each deals with an aspect related to Finite Element Analysis and Finite Element Method.

Module 1: Introduction to FEM & Approximate Methods

Module 2: One Dimensional FE Analysis

Module 3: FE Analysis by Direct Approach

Module 4: Two Dimensional FE Analysis

Module 5: Three Dimensional FE Analysis

Module 6: Computer Implementation of FEM

Module 1: Introduction to FEM & Approximate Methods

The finite element method (FEM) is a numerical analysis technique for obtaining approximate solutions to a wide variety of engineering problems [Vishal Jagota et al., 2013]. In this method, all the complexities of problems like varying shape, boundary conditions, and loads are maintained as they are but the solution's obtained are approximate. Because of its diversity and flexibility as an analysis tool, it is receiving much attention in engineering. A number of the popular brands of finite element analysis packages are now available commercially, for example, STAAD PRO, NASTRAN, NISA, and ANSYS.  

    In engineering problems. there are some basic unknowns. If they are found, the behavior of the entire structure can be predicted. The basic unknowns or the field variables which are encountered in the engineering problems are displacements in solid mechanics, velocities in fluid mechanics, electric and magnetic potentials in electrical engineering, and temperatures in heat flow problems. 

     In a continuum, these unknowns are infinite. The finite element procedure reduced such unknowns to a finite number by dividing the solution region into small parts called elements and by expressing the field variables in terms of assumed approximating function within each element. The approximating functions are defined in terms of field variables of a specified point called nodes or nodal points. Thus in finite element analysis, the unknowns are field variables at any point that can be found by interpolation functions.

 After selecting elements and nodal unknown next step in finite element analysis is to assemble element properties for each element. Thus the various steps involved in the finite element analysis are;

1. Select suitable field variables and the elements 

2. Discretize the continua

3. Select the interpolation functions

4. Find the element properties

5. Assemble element properties to get global properties

6. Impose the boundary conditions 

7. Solve the system equations to get the nodal unknowns

8. Make the additional calculations to get the nodal unknowns

The finite element knowledge makes a good engineer better while just user without the knowledge of FEA may produce more dangerous results. To use FEA packages properly the user must know the following points clearly;

1. Which element are to be used for solving problem in hand

2. How to discretize to get good results

3. How to introduce boundary conditions properly

4. How the element properties are developed and what are their limitations

5. To check the ability of available software

 Flowchart [Step-wise procedure of finite element analysis]

Continuum – Discretization of continuum in finite element – Selection of displacement function – Derivation of element stiffness matrix – Assembly of algebraic equation for the overall discrete continuum – Solution for the unknown displacement – Computation of element stress and strain from the nodal displacement

Question: Discuss in step by step finite element process and draw a flow chart for the same

Merits and Demerits of Finite Element Analysis/ Finite Element Method

Compared to other numerical finite element analysis merits are as follows;

1. The modelling of a complex geometries and irregular shapes are made easier by using finite element method.

2. Boundary conditions can be easily incorporated in finite element method.

3. Different types of material properties can be easily assigned in modelling from element to element or within an element.

4. Problems with heterogeneity, anisotropy, non-linearity and time-dependency can be easily dealt with finite element method.

5. The systematic procedure of finite element method makes it a powerful and versatile tool for a wide range of problems. 

6. Finite element method is simple, compact and result oriented and widely popular among engineering community. 

7. Finite element method can be easily coupled with computer aided design (CAD) programs in various disciplines in engineering. 

8. In finite element method it’s relatively easy to control the accuracy by refining the mesh or using higher order elements.

Demerits

1. Closed-form expressions in terms of problem parameters are not available in finite element method.

2. Numerical solution is obtained at only once for a specific problem case.

3. Large amount of data requires as input for mesh

4. Generally, voluminous (lengthy) output data must be analysed and interpreted.

5. Experience, good engineering judgment and understanding of the physical problems are required.

6. Poor selection of element types or discretization may lead to faulty results.

Question:  Explain briefly merits and demerits of finite element method

Application of finite element method

     

 

 

 

Reference Books:

1.     Finite Element Method by S. S. Bhavikati 

2. Finite Element Method with Application inn Engineering by Y. M. Desai, T. I. Eldhoand A.H.Shah