Finite Element Method - The Basis (Volume 1)

30 A direct approach to problems in elasticityThe above statement means that for equilibrium to be ensured the total potentialenergy must be stationary for variations of admissible displacements. The ®nite elementequations derived in the previous section [Eqs (2.23)±(2.25)] are simply thestatements of this variation with respect to displacements constrained to a ®nitenumber of parameters a and could be written as8 9@@@a ˆ>=@ ˆ 0@a 2>: . >;.…2:31†It can be shown that in stable elastic situations the total potential energy is not onlystationary but is a minimum. 7 Thus the ®nite element process seeks such a minimumwithin the constraint of an assumed displacement pattern.The greater the degrees of freedom, the more closely will the solution approximatethe true one, ensuring complete equilibrium, providing the true displacement can, inthe limit, be represented. The necessary convergence conditions for the ®nite elementprocess could thus be derived. Discussion of these will, however, be deferred tosubsequent sections.It is of interest to note that if true equilibrium requires an absolute minimum of thetotal potential energy, , a ®nite element solution by the displacement approach willalways provide an approximate greater than the correct one. Thus a bound on thevalue of the total potential energy is always achieved.If the functional could be speci®ed, a priori, then the ®nite element equationscould be derived directly by the di€erentiation speci®ed by Eq. (2.31).The well-known Rayleigh 8 ±Ritz 9 process of approximation frequently used inelastic analysis uses precisely this approach. The total potential energy expressionis formulated and the displacement pattern is assumed to vary with a ®nite set ofundetermined parameters. A set of simultaneous equations minimizing the totalpotential energy with respect to these parameters is set up. Thus the ®nite elementprocess as described so far can be considered to be the Rayleigh±Ritz procedure.The di€erence is only in the manner in which the assumed displacements areprescribed. In the Ritz process traditionally used these are usually given byexpressions valid throughout the whole region, thus leading to simultaneousequations in which no banding occurs and the coe cient matrix is full. In the ®niteelement process this speci®cation is usually piecewise, each nodal parameterin¯uencing only adjacent elements, and thus a sparse and usually banded matrix ofcoe cients is found.By its nature the conventional Ritz process is limited to relatively simple geometricalshapes of the total region while this limitation only occurs in ®nite elementanalysis in the element itself. Thus complex, realistic, con®gurations can be assembledfrom relatively simple element shapes.A further di€erence in kind is in the usual association of the undetermined parameterwith a particular nodal displacement. This allows a simple physical interpretationinvaluable to an engineer. Doubtless much of the popularity of the ®nite elementprocess is due to this fact.