206 L. Pedersen et al. The simplest possible basis for a vibration serviceability check by calculation would be to assume that the floor model is constant in time and is represented by the empty floor dynamic characteristics. This approach would rely on the assumption that the floor occupancy (which could be furniture or passive humans) does not influence the floor dynamic characteristics or at least that the influence of floor occupancy masses is neglected for the evaluation. However, it is a known fact that adding humans or non-structural mass will influence modal properties of a floor [4, 5]. The problem matter is that at the design stage often usage of the floor area is only known by the size of the specified life load. The distribution of the masses later to occupy the floor during its service life is generally unknown. This paper examines different distributions of masses on a floor and how this would influence modal properties of the floor. More specifically, the paper examines the changes occurring in modal properties of a floor when gradually increasing the density of mass added to the floor. Additionally, the paper examines the influence on modal properties occurring when the center of added mass is elevated above the floor midplane. Bookshelves and desks serves as examples of items where the center of mass it elevated above floor midplane. It is assumed that these masses are rigidly attached to the floor. (Some might choose to consider a human mass as a mass rigidly attached to the floor for calculations of floor dynamic behavior. However this might not be appropriate as the human body is not a rigid system [6]). The influences listed above are examined by numerical calculations and it is chosen to monitor changes in natural frequencies and damping ratios of the first nine modes of floor vibration. For the investigations, a case study floor is selected. The floor is described in Sect. 25.2 which also describes the finiteelement (FE) model of the floor. It is a simple floor in terms of geometry and support conditions in order to keep focus on effects of floor occupancy influences. Scenarios considered for usage of the floor is also outlined in Sect. 25.2 along with methods employed for extracting modal parameters of the floor. Section 25.3 presents and discusses the results. Finally, Sect. 25.4 gives the conclusions of the study. 25.2 Methodology 25.2.1 Computational Model of the Floor The case study concerns a floor with the characteristics described below. The floor area is assumed rectangular with side lengths of 8 and 9 m. It is assumed that the floor is pinned along all four sides. The floor is made of reinforced concrete with the material characteristics ED30 GPa (Young’s modulus), andvD0.15 (Poisson’s ratio). Based on a brief review of literature these values are considered to be fairly realistic for reinforced concrete, simplifying the composite material consisting of concrete and rebar into a homogeneous, isotropic and linear elastic material. The thickness of the floor is 180 mm and the mass density is 2400 kg/m3. It has been checked that, with these assumptions, it is possible to meet static ultimate-limit-state requirements as well as static serviceability-limit-state requirements, assuming usage as an office floor area. A FE model of the floor has been constructed using shell elements [7] with five degrees of freedom (d.o.f.) per node, i.e. three displacements and two rotations associated with bending. The element has nine nodes leading to second-order Lagrange interpolation of the displacements and rotations. A small artificial stiffness has been implemented to control the drilling degree of freedom, and selective integration of the stresses has been employed to avoid shear locking. A 12-by-12-element grid has been employed to model the entire floor. Although not shown here, it has been confirmed by calculation that the model of the empty floor has converged, i.e. decreasing the mesh size will not cause significant improvements in the estimates of frequency and damping characteristics of the empty floor for modes up to 70 Hz. A plate model has also been tested for the empty floor and it gave, as expected, minor differences in floor frequencies compared to those extracted using the shell model. Finally, the MATLAB code has been verified by comparison with an ABAQUS [8] model based on Mindlin-Reissner shell elements with eight nodes and reduced integration. However, a shell model is used for the present study, as the attached masses will be elevated above the horizontal mid plane of the floor, thus influencing the inplane displacement d.o.f’s of the shell. The first mode of the floor is found to have a frequency slightly above 8 Hz. Empty floor frequency and damping values for all modes considered in this paper are listed in a later section. In terms of damping characteristics, a Rayleigh damping model is assumed: Cf DaMf CbKf (25.1)
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