Issue 51

M.G. Masciotta et alii, Frattura ed Integrità Strutturale, 51 (2020) 423-441; DOI: 10.3221/IGF-ESIS.51.31 432 N UMERICAL M ODELING ith the aim of reproducing the dynamic behavior identified in the various damage scenarios, the segmental masonry arch described in the previous section was then modeled by a non-commercial finite element (FE) code. In particular, the adopted numerical procedure allowed to take into account the influence of the damage on the structure’s dynamic properties in an automatically way, without resorting to the reduction of the elastic modules of the damaged zone, as usually done in the literature [31, 32]. Constitutive equations and modal analysis of masonry structures Standard modal analysis allows to evaluate the dynamic properties of a structure. This analysis is based on the assumption that the materials constituting the structure are linear elastic, and can be unsuited to be applied to masonry constructions that exhibit a nonlinear behavior and show cracks (either due to permanent and/or accidental loads) which tend to reduce the structural stiffness, hence the frequencies. For these reasons, the dynamic behavior of masonry structures should be evaluated considering the presence of existing cracks. To this end, the FE NOSA-ITACA code [33],[34], developed by the Mechanics of Materials and Structures Laboratory of ISTI-CNR, adopts a numerical method based on linear perturbation and modal analysis, which takes into account the influence of existing damage on the dynamic properties of masonry buildings. The numerical method is widely described in [35, 36]. The NOSA-ITACA code allows to study the nonlinear behavior of masonry structures by adopting the constitutive equation of masonry-like materials. Masonry is modeled as a nonlinear elastic isotropic material with zero or weak tensile strength and infinite or bounded compressive strength [37, 38]. Given the infinitesimal strain tensor E , it is possible to prove the existence of a unique triplet ( T , E e , E f ) of symmetric tensors such that E is the sum of the elastic strain E e and the positive semidefinite fracture strain E f , and the Cauchy stress T , negative semidefinite and orthogonal to E f , depends linearly and isotropically on E e , through the Young’s modulus E and Poisson’s ratio ν [37, 38]. Masonry-like materials are then characterized by the stress function T given by T( E )= T , whose explicit expression is reported in [38], along with its main properties. In particular, T is differentiable on an open dense subset of the set of all strains and the explicit expression of the derivative D E T( E ) of T( E ) with respect to E is reported in [38]. The equation of masonry-like materials was then generalized in order to take into account a weak tensile strength t  ≥ 0 [38]. The procedure implemented in the NOSA-ITACA code to evaluate the dynamic properties of masonry structures in the presence of cracks consists in calculating the numerical solution to the nonlinear equilibrium problem of a masonry structure discretized into finite elements, subjected to given boundary and loading conditions, and then solving the generalized eigenvalues problem: 2 T K M     (1) obtained from the linear equation governing the undamped free vibrations of the structure about the equilibrium solution. K T and M ∈ R n x n are the tangent stiffness and mass matrices of the finite-element assemblage, both symmetric and positive definite,  is a vector of R n and ω a real scalar, with n the number of degrees of freedom of the structure. The counterpart of Eq. (1) for a linear elastic body involves the elastic stiffness matrix K E , replaced in (1) by the tangent stiffness matrix K T , calculated using the solution to the equilibrium problem and then taking into account the presence of cracks in the body. Therefore, assigned a case-study structure discretized into finite elements, and given the mechanical properties of the constituent materials together with the kinematic constraints and loads acting on the structure, the procedure implemented in NOSA-ITACA consists of the following two steps: 1) solution of the nonlinear equilibrium problem of the masonry structure subjected to external loads and calculation of the tangent stiffness matrix K T to be used in the next step; 2) solution of the generalized eigenvalue problem (1), where the matrix K T is employed instead of its elastic counterpart, and the natural frequencies / 2 i i f    and mode shapes i  of the structure are estimated in the presence of cracks [35, 36]. In particular, NOSA-ITACA provides the first k natural frequencies 1 2 ... k f f f    and the corresponding M -orthonormal mode shapes 1 2 , ,..., k    [39]. Reference scenarios model updating NOSA- ITACA is used to model the experimental specimen described in the previous section, employing 53,496 8-node isoparametric brick elements (element 8 of the NOSA-ITACA library [34]) and 64,411 nodes, for a total of 193,233 degrees W