Tsinghua Science and Technology, December 2001, 6(5), pp. 446452 Roof Isolation System  A Vibration Absorber for Buildings* TIAN Zhichang , QIAN Jiaru Department of Civil Engineering, Tsinghua University, Beijing 100084, China * Supported by Key Laboratory of Structural Engineering and Vibration of Education Ministry Received: 20000110; revised: 20000831 Code Number: ts01094 Abstract: A roof isolation system is proposed to reduce the dynamic response of buildings to earthquake excitations. In the system, frictional materials are inserted between the roof slab and the beams that support the slab. The roof slab and the beams are connected by springs. The optimum stiffness of the system is determined to minimize the seismic response of the buildings. A comparative study of the responses of an eightstory frame structure with and without the proposed system to ground motions was carried out to assess the system effectiveness. The study showed that the system energy dissipation capacity is nonlinear. The effectiveness of the system is related to the frequency and the acceleration of the ground motion. The system reduces the maximum lateral displacement response and the maximum interstory drift response of the building by as much as 45% except for the roof. Key words: roof isolation system; optimization; building; dynamic response Introduction Experiments and analyses have shown that vibration absorbers, often called passive tuned mass dampers (PTMD), with adequate mass, stiffness and damping, can effectively reduce the dynamic response of buildings[14]. However, vibration absorbers have some shortcomings. First, to be effective, a relatively large mass is required which needs a large space for installation. Secondly, the manufacture of the viscous dampers used in normal PTMD is relatively complex and their damping ratio is difficult to control in practice. Finally, a smooth surface is needed to mount the absorbers to minimize the friction force and to facilitate their free movement. Therefore, although vibration absorbers can potentially protect structures against earthquake effects, these shortcomings must be overcome before they can be accepted by engineers and be widely utilized in buildings. Proposed Roof Isolation System A possible method for overcoming the shortcomings of vibration absorbers is to use a portion of the building as the vibration absorber mass. One possibility is the building roof isolation system as shown in Fig.1. In the system, the frictional materials are inserted between the roof slab and the beams that support the roof slab. The roof slab and the supporting beams are connected by springs which deliver lateral stiffness to the system. 2 Optimization of Lateral Stiffness for the Roof Isolation System Since no general analytic method has been developed to determine the optimal stiffness and the optimal friction of the roof isolation system, a simplified method is proposed here. The objective of a TMD is to absorb vibration and to dissipate energy. The spring stiffness absorbs the vibration while the friction mainly dissipates the energy in the roof isolation system. Numerical analysis shows that the friction hardly influences the vibration absorption, but the stiffness greatly influences the energy dissipation because the dissipating effectiveness depends on how much kinematic energy can be absorbed in the system by the stiffness. The optimization of the roof isolation system is based on the following assumptions (1) The optimal stiffness is independent of the friction. (2) The optimal friction is dependent on the stiffness. Therefore, the optimization process is as follows: (1) Optimizing the stiffness without the friction using the theory of linear random dynamics. (2) Optimizing the friction for the optimal stiffness using the time history integration method. The action of earthquake ground motion on the structures can be explained as an inertial force or energy input. For most engineering practice, the later may be more reasonable. Because the response history of a structure is not similar to the load history, the maximum response and the maximum load do not necessarily occur at the same time and they are not always in the same direction at the same time. So the sum of the strain energy of the critical structural parts is adopted as the target function for the optimization of the system stiffness. Let M, K, and C present the structural mass, stiffness, and damping in matrix form while v(t) and indicate the structural displacement vector and the ground motion acceleration, then the dynamic system can be described by[5] Let w_{n} and f_{n} indicate the natural circular frequency and the modeshape vector of the nth vibration mode of the structure, then Eq. (1a) can be written as[4] here M_{n}, K_{n}, x_{n}, and y_{n} are the generalized mass, generalized stiffness, generalized damping ratio, and generalized displacement time history for mode n, respectively. Suppose that indicates the powerspectral density function of the acceleration time history of ground motion, then the powerspectral density function of the input in the nth mode is given by[5] The response function in the complex frequency domain is Here i is an imaginary number. The powerspectral density function of the displacement response is[5] Suppose that y_{n}(t) is the generalized displacement time history of mode n, then the variance of response y_{n}(t) is [5] Let K^{e} and U^{e} indicate the stiffness matrix and the strain energy of element e, respectively, then According to the orthogonal relation of modes, Eq.(6a) can be simplified as The expected strain energy of element e is Defining as the energy distribution coefficient for the nth mode of element e, then Eq.(7a) is simplified to The weighting sum of the expected strain energy of the important and critical elements is selected as the target function j(k_{sys}, S_{g}), Here a^{e} is the weighting factor of element e. The optimal parameter k_{sys} is the sum of the stiffnesses of the springs in the roof isolation system. It can be calculated according to the minimum value condition, Equations (1)(8) can be easily solved using the finite element method. However, Eq.(9) is not usually used for solving (k_{sys})_{opt} . The optimal value of (k_{sys} )_{opt} is obtained by comparing the values of j for different values of k_{sys} . 3 Comparative Study The effectiveness of the proposed roof isolation system was evaluated by analyzing an eightstory frame structure. Without the proposed roof isolation system, the structure was modeled as a lumped parameter model shown in Fig.2, which is referred to as Model1. The structure is assumed to be in elasticity. The same structure with the proposed roof isolation system is called Model2. For Model2 the frictional coefficient between the roof slab and the beams supporting the slab is taken as 0.2. The friction force contributes the systematic damping for Model2. The structural damping ratio is assumed to be 5% for the fundamental modes of both Model1 and Model2. The structural damping matrices of both models are proportional to their mass and stiffness matrices. 3.1 Lateral stiffness of the roof isolation system The optimal lateral stiffness of the roof isolation system was calculated as described in Sec.3. The powerspectral density function of the acceleration time history S_{g}(2pf) was assumed to be constant for f Î(0.0, 6.4). This range covers the first three natural frequencies of Model1 and the first four natural frequencies of Model2. When S_{g}= 1.0 , j(k_{sys}, S_{g}) varies only to k_{sys} , and j(k_{sys} ,1,0) means only the first to fourth stories were considered as part of the target function, because they can be more easily damaged than the others. Equations (1)(8) were used to calculate j as a function of k_{sys}. jk_{sys} curve was drawn in Fig.3. The results showed that j reached the minimum for k_{sys} = 47.5 MN/m . 3.2 Response to ground motions The input ground acceleration time history was modeled as a harmonic process with a sweep frequency, i.e., =Asin(2pft) (10) Here f is the frequency of the harmonic ground motion which varied from 0.0 through 6.4 Hz. A is the peak acceleration that was selected to be 0.5, 1.0, and 2.0 m/s^{2} in the comparative study. The second type of ground motion considered in the study was an artificial earthquake wave, the Shanghai 1  4 wave. The peak acceleration of this ground motion was scaled to 1.0, 2.0, and 4.0 m/s^{2}. The responses of Model1 and Model2 to the ground motions were calculated numerically. The results are given in Figs. 4, 5 and Tables 1, 2, 3, 4. The figures show the peak response and the time history response of both lateral displacement and interstory drift. The tables list the peak displacements and the peak interstory drifts. The tables also show the percentage reductions in each case. The analytical results are as follows: (1) For the harmonic ground motion with sweep frequency, the proposed roof isolation system considerably reduces the peak displacement response and the peak interstory drift response of the structure in the resonant regime. Outside of the resonant regime, the peak responses of Model2 and Model1 are about the same. Therefore, the roof isolation system may significantly reduce the dynamic response of a structure to the ground motions if the ground motions have a dominant frequency close to the fundamental frequency of the structure. Otherwise the effectiveness of the roof isolation system is not significant (Fig.4). Similar results were found in previous studies[6] in which earthquake records were used as the ground excitation. Therefore, the roof isolation system provides protection against sharp resonant ground motions that can cause significant structural damage. (2) The roof isolation system does not effectively reduce the seismic response if the peak acceleration of the ground motion is not sufficientlylarge. The effectiveness of the roof isolation system increases with the increase of the peak acceleration. The effectiveness is not proportional to the peak acceleration of the ground motion. For example, the peak lateral displacements and the peak interstory drifts are reduced, on average, by 5%, 32% and 41% when peak acceleration of the Shanghai 14 earthquake wave is scaled to 1.0, 2.0, and 4.0 m/s^{2}, and by 38% and 58% when the peak acceleration of the harmonic ground motionis 0.5 and 1.0 m/s^{2}. This means that the roof isolation system is fixed on the top of the structure if the ground motion does not cause the roof to move. The friction between the roof and the supporting beams begins to dissipate energy only when the inertial force acting on the roof is large enough. It also means that the energy dissipation capacity of the proposed roof isolation system is nonlinear. (3) The earthquake response reduction of the roof isolation system is limited. For example, the reduction of the peak displacement or the peak interstory drift is about the same for the peak accelerations of the harmonic ground motion of 1.0 and 2.0 m/s^{2} because the energy dissipation capacity of the roof isolation system depends on its mass and friction coefficient. Therefore, the energy dissipation capacity is limited for a roof isolation system with a specific mass and a specific friction force. (4) The peak displacement and the peak interstory drift are reduced by the proposed roof isolation system from the first story to the seventh story of the structure, but they increase at the roof due to the sliding of the roof on the supporting beams. However, this sliding would not damage the beams and columns of the eighth story. 4 Conclusions A roof isolation system is proposed to reduce the seismic response of buildings. The results of a comparative study of an eightstory frame structure with and without the roof isolation system show that: (1) The proposed roof isolation system significantly reduces the seismic response of the building structures if the ground motion has a dominant frequency close to the fundamental frequency of the building. (2) The energy dissipation capacity of the proposed roof isolation system is nonlinear. In addition, the roof isolation system has an upper limit for reducing the seismic response of the building structures. The proposed roof isolation system has potential to be a practical and effective way to reduce earthquake damage of buildings. Further studies are needed to find solutions to overcome some of the practical problems associated with it, such as how to assess the amount of friction coefficient in a simple way, whether the natural frequency is changed by the friction coefficient, etc. References
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