There are three basic finite element formulations which are used in multibody dynamics. These are the floating frame of reference approach, the incremental method and the large rotation vector approach. In the floating frame of reference and incremental formulations, the slopes are assumed small in order to define infinitesimal rotations that can be treated and transformed as vectors. This description, however, limits the use of some important elements such as beams and plates in a wide range of large displacement applications. As demonstrated in some recent publications, if infinitesimal rotations are used as nodal coordinates, the use of the finite element incremental formulation in the large reference displacement analysis does not lead to exact modeling of the rigid body inertia when the structures rotate as rigid bodies. In this paper, a simple non-incremental finite element procedure that employs the mathematical definition of the slope and uses it to define the element coordinates instead of the infinitesimal and finite rotations is developed for large rotation and deformation problems. By using this description and by defining the element coordinates in the global system, not only the need for performing coordinate transformation is avoided, but also a simple expression for the inertia forces is obtained. The resulting mass matrix is constant and it is the same matrix that appears in linear structural dynamics. It is demonstrated in this paper that this coordinate description leads to exact modeling of the rigid body inertia when the structures rotate as rigid bodies. Nonetheless, the stiffness matrix becomes nonlinear function even in the case of small displacements. The method presented in this paper differs from previous large rotation vector formulations in the sense that the inertia forces, the kinetic energy, and the strain energy are not expressed in terms of any orientation coordinates, and therefore, the method does not require interpolation of finite rotations. While the use of the formulation is demonstrated using a simple planar beam element, the generalization of the method to other element types and to the three dimensional case is straightforward. Using the finite element procedure presented in this paper, beams and plates can be treated as isoparametric elements.

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Abstract:The research aims to understand the relationship between the popularity and attractiveness of commercial buildings, shopping malls and any relationship with visitor satisfaction. This would assist in designing new malls or commercial buildings, predict their degree of popularity, and help achieve both higher revenue resulting from increasing the number of visitors and their satisfaction, comfort and enjoyment of the space. This study will focus on the relationship between shopping mall popularity and wayfinding. Planned and unplanned visits to some specific areas inside the malls will be considered. These areas are: 1-Facilities: prayer rooms, and washrooms, etc.; 2-The largest areas in size and often with the highest number of visitors: food courts, cinemas, play areas; and 3-ATM machines. The objective of this paper is to verify three hypotheses: 1. The popularity of a mall is positively related to visitor satisfaction with wayfinding in the mall; 2. The popularity of a mall is positively related to visitor satisfaction with the location of facilities in the mall; 3. The location of facilities in a mall is positively related to visitor satisfaction with wayfinding in the mall. Surveys were conducted in the city of Abu Dhabi and the Structural Equation Modelling (SEM) technique was used to verify these hypotheses.

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