Issue 40

N. G. Pnevmatikos et alii, Frattura ed Integrità Strutturale, 40 (2017) 129-136; DOI: 10.3221/IGF-ESIS.40.11 132 n 2 q,e q,e,i i =  F F (3) If F d,max is the maximum control device capacity (maximum possible control force), then the maximum control force that can be applied on the system is: d,max f d,max F  F E (4) where E f is the location matrix for the control devices on the structure. Assuming that one part, up to F d,max , of elastic seismic forces are carried by control devices, the remaining seismic forces which go directly and force the structural elements are: q,e q d,max q d,max q, new q d,max sign( ) if = 0 if        F F F F F F F F (5) F q new is a vector with n forces, where n is the degree of freedom of the system. These forces correspond to a reduced spectral acceleration. From Eq. (2) this new spectral acceleration S d,i,new (T i , ξ i ), corresponding to new seismic forces, can be obtained:   -1 T i q,new d,i,new i i i S (T , ξ ) , i=1,...,n ψ  MΦ F (6) The reduction factor α can be obtained by dividing the new spectral acceleration S d,i,new (T i , ξ i ) by the corresponding initial one: d,i,new i i i e,i i i S (T , ξ ) α , i=1,...,n S (T , ξ )  (7) The elastic spectrum is scaled using the maximum value of α i and the structure is redesigned based on the reduced spectrum. The value of a is: i α max(α )  (8) In order to ensure a linear behavior of the structure, dynamic control analysis is performed for a range of earthquakes (high and low frequency characteristics), with saturation control and time delay. If the response satisfies the elastic criteria, then the value of α is accepted, otherwise it is slightly increased and the above procedure is repeated. The flow chart of this procedure is shown in Fig. 2 with a dashed line. The equation of motion of a controlled structural system with n degrees of freedom subjected to an earthquake excitation ag in the state space approach is: g g f a    X AX B B F  (9) The matrixes X, A, B g , B f are given by new g f 1 1 1 f 2 1 2 1 2 2 2 1 , , , nx nx nx n nx                                      0 I 0 U 0 X A B B U E M K M C M E  (10)