Issue 40

N. G. Pnevmatikos et alii, Frattura ed Integrità Strutturale, 40 (2017) 129-136; DOI: 10.3221/IGF-ESIS.40.11 133 where M and C denote the mass and damping matrices of the structure, respectively, K new is the new stiffness matrix of the redesigned structure, and F is the control force matrix. The control force F is determined by linear state feedback as follows:   1 2 1 2              U F G U G U G G GX U   (11) G is the gain matrix, which will be calculated by pole assignment method and according to the desired poles of the controlled system. If the response obtained for the controlled system satisfies the design criteria, then the reduction by q or by a scale factor, α, is accepted. In this work a representative design criterion was used, that the story drift does not exceed h/300 (where h is the story height). This value does not cause member yielding. In a similar way, additional design criteria concerning the rotation and strength of structural members can be used. The above procedure was tested for a number of numerical simulations, and some representative examples are presented next. R ESULTS AND D ISCUSSION he proposed approach is demonstrated by means of numerical example where an eight-story building, described in the work of Yang et al, 1995 [7], is analyzed. Initially the elastic and design spectra are calculated based on Eurocode 8 (EC8) seismic code. Based on those spectra and on dynamic characteristics of building the seismic forces F q,i for each eigenmode and their combination are calculated for both elastic and design spectrum. The seismic forces which are obtained from elastic and design spectrum and their differences are shown in Fig. 3(a). Assuming that the control devices are installed on each floor and the maximum capacity is 1000kN, following the proposed procedure the scale factor α is calculated to be equal to 0.49 or the equivalent reduction from the elastic spectrum 1-α which is equal to 51%. The elastic and design spectra and the reduced spectrum by 51% from the elastic spectrum, for which the structure will be redesigned, are illustrated in Fig. 3(b). In order to ensure that the structure remains in the elastic range after redesigning, dynamic time control analysis history, with saturation control and time delay, for a wide range of earthquakes should be performed. The numerical simulations were performed in Simulink toolbox of Matlab software. The numerical simulation of the control scheme is described in Fig. 3(c). The response (displacement and acceleration) of the system subjected to Athens earthquake 1999 were calculated. From the numerical results it was seen that full compensation of the displacements was achieved. According to the work of Yang et al. (2003) when one control force corresponds for each degree of freedom then complete compensation of the response can be achieved and the response state vector can be reduced to zero. Another reason that the relative displacements are near to zero is that the elastic response spectrum of the Athens earthquake are lower than the elastic spectrum that was used initially for the design procedure. The acceleration is equal to the external signal and the building behaves like executing a rigid body motion. The control forces are identical, with maximum value at 917 kN and rms value at 134 kN, because the mass of each story is the same. The storey drift between the floors was not exceeded the limit value h/300=10 mm. Time history of displacement and the acceleration from 8th floor for the controlled and uncontrolled structure is shown in Fig. 4. S UMMARY AND CONCLUSIONS procedure to design a structure equipped with control devices is described. The structure is designed based on a reduced spectrum. A scale factor α which multiplies the elastic spectrum and produces a reduced spectrum is proposed. The design philosophy is that one part of seismic forces are taken by control devices and to the rest of earthquake forces taken up from the structure. The numerical results indicate that reduction of the spectrum can be achieved using control devices. The cost of repairing the post-earthquake damages of an uncontrolled structure which was design based on ductility demand can be considered as a motivation to install a control system which will keep the structure in the elastic range. The control system is acceptable if the results obtained from the dynamic control analysis T A