Issue 29

L. Facchini et alii, Frattura ed Integrità Strutturale, 29 (2014) 139-149; DOI: 10.3221/IGF-ESIS.29.13 141 analyses are reported in [15]. If for instance the force is harmonic, then the top displacement of the cantilever beam exhibits a slight hysteretic behavior, as shown in Fig. 1. Adopted simplifications From the results of the structural analyses performed by means of ANSYS (Fig.1), it can be inferred that the dynamic behavior of the cantilever beam can be modeled accurately enough by means of its first modal shape. Therefore, a nonlinear single degree of freedom (SDOF) system can be defined whose stiffness is determined by means of the performed nonlinear numerical analyses. The analysis performed with ANSYS applying a horizontal displacement on the top of the masonry cantilever beam, in such a way to obtain the first-loading curve and the successive unloading curve, was hence employed to approximate the system behavior with the BW model, as reported in the following. Approximation of the tower response with Bouc & Wen oscillator Within the first part of the research, the simplified SDOF BW model for the restoring force of the cantilever beam was identified. The behavior of the oscillator is described by an incremental equation of the form:         mx t cx t kg t f t              1 g t x t z t      (1)           n z t x t A z sign x sign z              ⛂⛂ This model represents the restoring g ( t ) function with a linear combination of a linearly elastic force and a history- dependent term: z ( t ). The variable x ( t ) represents the displacement of the top of the cantilever beam. The first set of parameters that must be identified is composed by k ,   , A , n ,  and γ . It can be shown [15] that parameter A in eq. (1) is redundant, and hence in the following it will be considered as unitary: A =1. The tangent stiffness of the oscillator can be obtained by deriving the nonlinear restoring function g ( t ) with respect to the displacement x , obtaining:     1 1 t g z z k k k k x x x                              (2) Being the parameter A assumed unitary without loss of generality, parameter k reduces to the initial stiffness of the system and the initial stiffness can be computed substituting initial conditions z ( 0 )=0 in Eq. (2), obtaining:     0 0 0 1 i t z z k k k k               ⛂ (3) And asymptotically (according Marano and Greco [16] the post-elastic stiffness is given as f k k   ): 0 f t f i k z k k k k          (4) When the maximum displacement is reached, and the unloading process begins, following expression holds:         1/ 1/ γ β β γ α 1 α 1 β γ 1;   ˆ 1 n u z z k k sgn z sgn x                                  (5) and:   1 / β γ β γ 1 α u i u i f k k k k k k         (6) Finally, the elastic limit displacement can be expressed, according to Cunha [17], as: β γ n Y x    . The ratio  /  affects the transitions from the loading curve to the unloading one. Therefore, only one parameter, n , or alternatively  or  , remain undetermined. It is possible to observe that the parameter n influences the transition from elastic to post-elastic behavior and the distance of the unloading path from the first loading.