Issue 29

J. Toti et alii, Frattura ed Integrità Strutturale, 29 (2014) 166-177; DOI: 10.3221/IGF-ESIS.29.15 167 A further alternative is the use of the Finite Element Method (FEM), by adopting a micro-modeling approach [5, 6] or a macro-modeling approach [7, 8]. The computational analysis of these structures subjected to dynamic loadings requires realistic stress-strain material models to satisfactorily reproduce their physical behavior. Research efforts on the static response of the masonry material under cyclic loadings aim at providing an efficient model capable of predicting the hysteretic characteristics of the entire element. Common models used to reproduce the cyclic behavior of cohesive material are based on damage mechanics, plasticity theory and coupling of both. As widely known, if the stress-strain laws with softening are used within the standard continuum theory, the obtained numerical results suffer from pathological sensitivity to the spatial discretization, e.g. to the size of the finite elements. Upon mesh refinement, the energy dissipated by the numerical model decreases and tends to extremely low values, sometimes even to zero. As remedy, regularized models, based on nonlocal continuum approaches, can be adopted [9-12]. Recently, efforts have been made to develop dynamic tests which appear particularly promising for a deeper understanding of the response of masonry arches and vaults. Within this activity structural damage identification has been pursued to extract information on the health of these structures. Indeed, the presence of damage or of other inelastic phenomena modifies the overall structural dynamic response and the damage propagation potentially interacts dynamically with the element vibrations; in particular, changes in its behavior are associated to the decay of the mechanical properties of the system. Based on these considerations, several studies have been devoted to use the variations in the dynamic behavior to detect structural damage. Particular attention has been focused on the use of frequencies only, because of the simplicity of measuring them and, therefore, their experimental reliability. In this framework, dynamic analyses of damaged structures have been performed in [13] with the aim to detect the damage state of an arched masonry structure. In the present paper the damage propagation in a large-scale masonry arch induced by cyclic and dynamic loadings is analyzed. In particular, the masonry arch tested in [13] is considered as case study. A two-dimensional modeling of the masonry arch is proposed. The constitutive material model developed by Toti et al. [12] and extended to the dynamic analyses in [14] is used to reproduce the main hysteretic characteristics of the masonry material under cyclic or dynamic loadings, such as damage, inelastic strains, unilateral phenomena. An inverse procedure is used in order to calibrate the model parameters. The performance of the developed computational tool, in reproducing the experimental response of the arch and the damaging evolution, are investigated in the case of a static cyclic test. The damage propagation and the variations of the dynamic behavior (displacement, frequency contents) with respect to the undamaged condition are examined, when the studied mechanical system is excited by imposed base synchronous harmonic motions. T HE MASONRY ARCH CASE STUDY he masonry arch studied during the experimental campaign presented in [13] is considered as case study. The masonry structure is built with standard clay bricks with size 100×50×25 mm3. The geometrical data of the tested circular arch, schematically reported in Fig. 1, are the following:  internal radius r = 0.77 m;  width w = 0.45 m;  thickness t = 0.05 m;  height h = 0.59 m;  span l = 1.50 m. Fig. 1 shows even the coordinate reference system adopted for the computations. Static tests to induce damage and dynamic identification tests are performed during the experimental campaign [13]. During the static test, the damage of the arch is caused by the application of a point load located with a distance d from the abutment equal to 0.38 m (about a quarter span), where the lowest safety factor for arches is obtained. The point-wise load induces a damage evolution in the structure, which leads a four hinges collapse mechanism. Dynamic tests for modal structural identifications have been conducted on the masonry arch through vibration measurements [13]. The frequencies and the modal shapes are estimated both for the undamaged arch and for damaged arch for different values of the increasing static force. As strains are measured at 11 points of the extrados and intrados of the arch, it was also possible to evaluate the centerline curvature mode shape. Here, the novel two-dimensional model is calibrated in order to reproduce the measured quantities on the experimental apparatus presented in [13], in particular the values of modal frequencies obtained by vibration measurements are compared in Tab. 1 with the same quantities obtained by the proposed numerical model. For sake of brevity, only the modal parameters related to first two in-plane modes of vibration of the arch computed in the undamaged condition are herein considered. T