1.1 History of FEM The term “finite element” was first used by Clough inthe 1960s when he published a paper on plane elasticity problems.
However theideas finite element analysis can be traced much further to the work by A.Hrenikoff (1941) and R. Courant (1942). Inthe 1960s engineers used this method to solve problems regarding stressanalysis, fluid flow, transfer of heat etc.
The key concept of stiffness matrixand assembly of elements came into existence in the late 1950s. In the year1955, Argyris wrote a book on energy theorems and matrix method whichstrengthened the foundation of finite elemental analysis. In the late 1960s afinite element analysis program NASTRAN was developed by NASA. 1.2 Comparison of methods of assembly of stiffness matrix · Direct approachThe direct approach is related to the “directstiffness method” of structural analysis and it is the easiest to understand. However,by direct approach we can solve only simple problemsIt is an extension of matrix displacement approach. · Variational approach This method of assembling stiffness matrix is adoptedfor elasticity problems. In variational approach we use principle of minimumpotential energy.
It is widely used for deriving finite element equations. Abasic knowledge of calculus of variations is required for this approach. Themajor disadvantage is that there are many physical problems.
Variationalapproach is not useful in non-linear problem. The Rayleigh-Ritz method is anapproximate method based on the variational formulation. · Weighted residual method This method of assemblingstiffness matrix is apt for both linear and non-linear differential equation. Itconsists of two steps. In first step we assume the approximate solution whichsatisfies the boundary condition. Since the solution is approximate it givessome error which is called residual. This first step results in a system ofalgebraic equations.
The second step is to solve the equations and getsolution. 1.3 Motivations for development of FEM· In the year 1959, Greenstadt came up with discretisationapproach which involved “cells” other than points.· Greenstadt’s theory is used for the analysis of irregularly shapedstructures with irregularly shaped meshes.· In the year 1965 Zeinkiewicz and Cheung stated that FEM isapplicable to all field problems in variational form. · In 1943 Courant used triangular elements to analyse torsionproblems.· Many physicists and mathematicians became interested in FEMand 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 · Thismethod can handle complex geometry.· Problemscan be solved without prototype.· Itprovides good visualisation of structural behaviour and failures.· Ithas more accuracy.· Itgives better insight into critical design parameters.· Itcalculates the dynamic properties of structure. · Itcan solve non-linear problems.
1.5 Justification of stance taken Stance taken “If Direct Stiffness Method were notinvented, Finite Element Method would not have existed resulting in developmentof other methods of analysis to a greater extent”.Direct stiffness method is the main reason behind thedevelopment of finite element method.The Direct stiffness method was developed to easilyfor the analysis of complicated structures with more number of elements.
Today,most of the computer software’s that are used for structural analysis are basedon the direct stiffness method. 1.6 Conclusions FEM is used tosolve problems regarding stress analysis, fluid flow, transfer of heat etc. inthis method the structure is broken into smaller elements for analysis.
It is an extension of direct stiffnessmethod.FEM is widelyused in software programs for structural analysis.Therefore finiteelement method would not have existed without the invention of direct stiffnessmethod.