Vibrations of Asymmetric Wall-Frame Structures with Varying Mass and Stiffness along the Height

Most of the past studies on the seismic response of asymmetric buildings have concentrated on structures with uniform distribution of mass and stiffness along the building height. Such results clearly do not apply to many modern buildings with a transfer plate system at lower levels. This paper proposes a simple but yet reliable formulation, capable of analyzing the seismic vibrations of irregular tall buildings with varying mass and stiffness along the building height. More specifically, the building is divided into storey. Within each storey, all wall and frame members are replaced by an equivalent flexural beam located at the center of flexure and an equivalent shear beam located at the center of shear, respectively. Different storeys are connected through continuity conditions. Finally, the governing equations of coupled vibrations are established and the results of a simple example are discussed. More detailed analysis will be conducted in the near future. Nevertheless, the present formulation should provide a more reliable way to assess the seismic vulnerability of non-uniform tall buildings in Hong Kong and other regions of moderate seismicity. INTRODUCTION For the majority of real tall buildings, very often asymmetric structural scheming is used for aesthetic reasons or functional needs, or constrained by the shape of the usable land. This results in eccentricity between the centers of mass and stiffness. These asymmetric structures are prone to torsional response even under unidirectional ground shaking. Besides, workmanship and detailing of the structural components as well as the non-structural components of buildings may also be sources of unsymmetrical behavior of symmetric structures. Structural asymmetry has been identified as a cause for the failure or poor performance of multistory buildings during recent earthquakes (Thambiratnam and Corderoy, 1994). Hence, it is important to properly model and analyze the coupled-torsional-translational response of multistory buildings under earthquake loads. Although the effects of torsion on the seismic linear-elastic and nonlinear responses of structures have been investigated extensively in the past few decades, most of the models proposed so far are either too simple, such as those based on idealized single storey models (Chandler and Hutchinson, 1986; Kan and Chopra, 1981), or too tedious and require extensive computational effort, such as the finite element method. At present, some computer packages which employ the finite element method already have the ability to perform three-dimensional linear and/or nonlinear static and/or dynamic _____________ L. X. Wang, PhD student, Department of Civil & Structural Engineering, The Hong Kong Polytechnic University, Kowloon, Hong Kong, China Z. P. Wen, Professor, Institute of Geophysics, China Earthquake Administration (CEA), Beijing, China K. T. Chau, Professor, Department of Civil & Structural Engineering, The Hong Kong Polytechnic University, Kowloon, Hong Kong, China analysis of complicated and irregular structures. However, to set up a suitable input model for realistic multistory structures and to analyze it is very tedious and time-consuming, making this method too complicated to be used in design offices, especially for the preliminary stage of building design. In addition, high uncertainty in the results of the nonlinear response of reinforced concrete members further complicates the use of finite element packages in damage estimation. In view of this, procedures for obtaining the seismic responses and damage of tall buildings with reduced computational effort need to be investigated. Kan and Chopra (1976) developed a simple procedure for the analysis of the elastic seismic response of a particular class of torsionally coupled multistory buildings, in which the centers of mass and resistance all lie on 2 vertical lines and all the floors have the same radius of gyration and the same translational-to-torsional stiffness ratios. The response of such a torsionally coupled building can be determined by analyzing a torsionally uncoupled system and an associated one-story coupled system. Some other efforts (Rutenburg and Heidebrecht, 1975; Bao, 1992) have also been proposed to decouple the torsional coupled differential equations of equilibrium of asymmetric buildings under earthquake loads. More recently, Ng and Kuang (2000) developed a generalized approximate approach for the analysis of triply coupled vibration of asymmetric wall-frame structures by applying the Galerkin method. Those methods can be conducted on microcomputers or even by hand calculations, however, they are only applicable to those buildings with uniform distributions of mass and stiffness along the height. For nonuniform buildings, Wang et al. (1999) developed a methodology for obtaining dynamic responses of frame-shear wall structures having stepped cross sections of shear walls. The wall elements were divided along the vertical direction so that the rigidity of each wall element was constant. However, their formulas are based on planar modeling and contain no torsion components, making it not applicable for the analysis of asymmetric structures. In Hong Kong, one of the most common features of high-rise buildings is the existence of a transfer plate and podium floors, which usually introduce an abrupt change of stiffness (Su et al., 2002). For this kind of structures, the above mentioned approaches seem inadequate. Therefore, to fill in the gap, this study proposes a simple method for analyzing the coupled torsional and translational vibrations of irregular multistory wall-frame buildings with varying mass and stiffness along the building height. The present analysis can be considered as an extension of the earlier work by Wen et al. (2002) who proposed a simple multi-degree-of-freedom model that allows for changing yield shear and storey mass along the building height but only considering equivalent horizontal force approach. In the work, the seismic vulnerability of a typical Hong Kong building found in Mei Foo Sun Chuen was determined for various earthquake intensities and site conditions. The model by Wen et al. (2002) however did not allow for the torsional vibration, and which will be one of the main focuses of the present study. THEORETICAL FORMULATION Consider an asymmetric wall-frame structure with varying mass and stiffness along the height, as shown in Figure 1(a). In this analysis, the whole structure is considered as a multi-degree-of-freedom oscillator depending on the number of storey; and the building properties can change from one storey to another. Basic Assumptions The following basic assumptions are used in the formulation of the theoretical model: (1) The floor slabs are rigid and undeformable in their planes and, therefore, the vibrations of each storey are given by 3 degrees of freedom: ) , ( t x yi and ) , ( t x zi for the translational deflections of the i-th floor in the y and z directions respectively; and ) , ( t x i θ for the rotation of the floor plan about the vertical axis. (2). The concept of continuum method is adopted. That is, at each storey level the discrete set of connecting members can be simplified as equivalent continuous connecting beams along the height and the forces in the connecting members are further smeared into distributed forces. Figure 1. (a). Sketch of the proposed model (where F and S denote the center of flexural rigidity and the center of shear rigidity respectively, Hi is the height of the ith segment); (b) Wall alone can be simulated by a flexural beam; (c) Frame alone can be simulated by a shear beam. (3) All wall members are lumped and replaced by an equivalent flexural beam, as shown in Figure 1(b), which is located at the center of flexural rigidity F and undergoes deformations of both flexure and warping torsion; all frame members are lumped and replaced by an equivalent shear beam (see Figure 1(c)), which is located at the center of shear rigidity S and undergoes deformations of both lateral shear and St. Venant torsion. Hn Hn-1