Three-dimensional structural analysis by the integrated force method

The ''integrated force method'', which has been recently developed for analyzing structures, is extended in this paper for three-dimensional structural analysis. A general formulation to generate the stress interpolation matrix in terms of complete polynomials of the required order is developed first. The formulation is based on the definitions of stress tensor components in terms of stress functions. The stress functions are written as complete polynomials and substituted into expressions for stress components. After eliminating dependent coefficients, the expressions for stress components are obtained as complete polynomials, where coefficients are defined as generalized independent forces. Such derived components of the stress tensor identically satisfy Navier's equations of equilibrium. The resulting element matrices are invariant with respect to coordinate transformation and are free of spurious zero energy modes. The formulation provides a rational way to calculate the exact number of independent forces for the required order of approximation with complete polynomials. The reduction in the number of independent forces and its influence on the accuracy of the response are also analyzed. The stress fields derived here are next used to develop a comprehensive finite element library for the three-dimensional structural analysis using the ''integrated force method''. Elements of both tetrahedral and hexahedral shapes, capable of modeling arbitrary geometric configurations, are developed. A number of example problems with available analytical solutions are solved using the present developments and a good agreement of results with the analytical solutions is observed. The responses obtained using the ''integrated force method'' are also compared with those generated with the standard displacement method. A better overall performance of the ''integrated force method'' is observed.