1Analysis of one-dimensional finite elements using the Function Space Approach

The present work demonstrates how computations in finite element method can be interpreted as orthogonal projections of the analytical strains in each element. The Function Space Approach has been adopted to derive a priori estimates of the element strains and errors in finite element analysis. Simple one-dimensional bar and beam elements have been used to illustrate the fundamental principles that guide finite element computations in conservative or self-adjoint problems. Elements with constant and varying cross-sections have been considered. Results predicted are compared with numerical values obtained from standard computer codes. The interesting pathological problem of shear locking in linear Timoshenko beam element has been addressed. An explanation for locking and method to eliminate it have been presented. The rationale behind the practice of using reduced integration for elimination of locking has been revealed explicitly using the rigorous Function Space Approach. Finite element analysis can be regarded as a tool for obtaining approximate solution to differential equations of boundary value problems, using piecewise assumed interpolation functions within the domain of interest. The finite element equations are conventionally derived from the principles o