The computation of the fundamental mode in structural moment frames provides valuable insight into the physical response of the frame to dynamic or time-varying loads. In standard practice, it is not necessary to solve for all n mode shapes in a structural system; it is therefore practical to limit the system to some determined number of r significant mode shapes. Current building codes, such as the American Society of Civil Engineers (ASCE), require certain class of structures to obtain 90% effective mass participation as a way to estimate the accuracy of a solution for base shear motion. A parametric study was performed from the collected data obtained by the analysis of a large number of framed structures. The purpose of this study was the development of rules for the required number of r significant modes to meet the ASCE code requirements. The study was based on the implementation of an algorithm and a computer program developed in the past. The algorithm is based on Householders Transformations, QR Factorization, and Inverse Iteration and it extracts a requested s (s<< n) number of predominate mode shapes and periods. Only the first r (r < s) of these modes are accurate. To verify the accuracy of the algorithm a variety of building frames have been analyzed using the commercially available structural software (RISA 3D) as a benchmark. The salient features of the algorithm are presented briefly in this study.

Current trends in the Computer Aided Engineering (CAE) involve the integration of legacy mesh-based finite element software with newer solid-modeling kernels or full CAD systems in order to simplify laborious or highly specialized tasks in engineering analysis. In particular, mesh generation is becoming increasingly automated. In addition, emphasis is increasingly placed on full assembly (multi-part) models, which in turn necessitates an automated approach to contact analysis. This task is challenging due to increases in algebraic system size, as well as increases in the number of distorted elements - both of which necessitate manual intervention to maintain accuracy and conserve computer resources. In this investigation, it is demonstrated that the use of a mesh-free B-Spline finite element basis for structural contact problems results in significantly smaller algebraic systems than mesh-based approaches for similar grid spacings. The relative error in calculated contact pressure is evaluated for simple two dimensional smooth domains at discrete points within the contact zone and compared to the analytical Hertz solution, as well as traditional mesh-based finite element solutions for similar grid spacings. For smooth curved domains, the relative error in contact pressure is shown to be less than for bi-quadratic Serendipity elements. The finite element formulation draws on some recent innovations, in which the domain to be analyzed is integrated with the use of transformed Gauss points within the domain, and boundary conditions are applied via distance functions (R-functions). However, the basis is stabilized through a novel selective normalization procedure. In addition, a novel contact algorithm is presented in which the B-Spline support grid is re-used for contact detection. The algorithm is demonstrated for two simple 2-dimensional assemblies. Finally, a modified Penalty Method is demonstrated for connecting elements with incompatible bases.