1.1 History of FEM

The term “finite element” was first used by Clough in

the 1960s when he published a paper on plane elasticity problems. However the concepts

finite element analysis can be traced much further to the work by A. Hrenikoff

(1941) and R. Courant (1942). In the

1960s engineers used this method to solve problems regarding stress analysis,

fluid flow, transfer of heat etc. The key concept of stiffness matrix and

assembly of elements came into existence in the late 1950s. In the year 1955,

Argyris wrote a book on energy theorems and matrix method which strengthened

the foundation of finite elemental analysis. In the late 1960s a finite element

analysis program NASTRAN was developed by NASA.

1.2 Comparison of methods of assembly of stiffness matrix

·

Direct approach

Direct approach

is related to DSM and it is an extension of matrix displacement approach. It is

easy to understand and a simple problem can be solved, whereas it cannot solve

the complex problem.

·

Variational approach

This method of assembling stiffness matrix is adopted

for elasticity problems. In variational approach we use principle of minimum

potential energy. It is widely used for deriving finite element equations. A

basic knowledge of calculus is required for this approach. The major

disadvantage is that there are many physical problems. Variational approach is

not useful in non-linear problem.

·

Weighted residual method

This method of assembling

stiffness matrix is apt for both linear and non-linear differential equation.

It consists of two steps. In first step we assume the approximate solution

which satisfies the boundary condition. Since the solution is approximate it

gives some error which is called residual. This first step results in a system

of algebraic equations. In second step we solve the equations to get solution.

1.3 Motivations for development of FEM

·

In the year 1959, Greenstadt came up with discretisation

approach which involved “cells” other than points.

·

Greenstadt’s theory is used for the analysis of irregularly

shaped structures with irregularly shaped meshes.

·

In the year 1965 Zeinkiewicz and Cheung stated that FEM is

applicable to all field problems in variational form.

·

In 1943 Courant used triangular elements to analyse torsion

problems.

·

Many physicists and mathematicians became interested in FEM

and established a firm background for FEM.

·

In the 1960s FEM got extensive recognition in the field of

engineering.

1.4 Prominence of FEM over other methods

·

This

method can handle complex geometry.

·

Problems

can be solved without prototype.

·

It

provides good visualisation of structural behaviour and failures.

·

It

has more accuracy.

·

It

gives better insight into critical design parameters.

·

It

calculates the dynamic properties of structure.

·

It

can solve non-linear problems.

1.5 Justification of stance taken

Stance taken “If Direct Stiffness Method were not

invented, Finite Element Method would not have existed resulting in development

of other methods of analysis to a greater extent”.

Direct stiffness method is the main reason behind the

development of finite element method.

The Direct stiffness method was developed to easily

for the analysis of complicated structures with more number of elements. Today,

most of the computer software’s that are used for structural analysis are based

on the direct stiffness method.

1.6 Conclusions

FEM is used to

solve problems regarding stress analysis, fluid flow, transfer of heat etc. in

this method the structure is broken into smaller elements for analysis. It is an extension of direct stiffness

method.

FEM is widely

used in software programs for structural analysis.

Therefore finite

element method would not have existed without the invention of direct stiffness

method.