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.

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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.

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