A discrete‐element approach accounting for P‐Delta effects

Abstract

This paper presents a discrete macro‐element accounting for P‐Delta effects to describe the rocking response of masonry walls subjected to out‐of‐plane (OOP) loadings. Both constitutive and geometric nonlinearities are considered within the discrete macro‐element method (DMEM), which is a modeling approach characterized by a very low computational cost compared to traditional distinct element method (DEM) and detailed finite element (FEM) strategies. OOP failure mechanisms are the main cause of severe damage for unreinforced masonry (URM) buildings, subjected to seismic actions. These mechanisms are generally activated by low seismic excitation and displacements. However, after their activation, they can evolve towards large displacements related to rigid‐block‐like kinematics that strongly affects the nonlinear mechanical response. Therefore, geometric nonlinearities, often ignored, should be included in the analyses. A new simplified, still accurate P‐Delta formulation is presented, according to which the global equilibrium is imposed by referring to the undeformed system configuration, avoiding assembling the geometric stiffness matrix. Namely, the system load vector is updated at each step of the analysis, accounting for the additional moments generated by the in‐plane compression forces acting on the macro‐elements in the deformed configuration. The proposed model is validated against closed‐form analytical solutions of rigid‐block benchmarks in large displacements and the results of experimental tests already available in the literature. In addition, extensive parametric analyses are performed to investigate the role of different mechanical and geometric parameters characterizing the ultimate non‐linear response of masonry walls subjected to horizontal forces. The results show how the proposed model, including P‐Delta effects, accurately predicts the non‐linear rocking response of masonry walls until the attainment of the unstable configuration.

Out-of-Plane Response of Masonry Church Facades Including P-Delta Effects

“…According to the P-Delta formulation proposed in [15], P-Delta effects are considered by updating the current positions of the external and along-interface internal forces applied to the macro-elements. This simplified procedure avoids assembling and updating the geometric stiffness matrix according to the current system configuration, ensuring a good efficiency of the model.…”

“…This simplified procedure avoids assembling and updating the geometric stiffness matrix according to the current system configuration, ensuring a good efficiency of the model. More details on the model formulation and validation can be found in [15].…”

“…Belonging to relatively simplified approaches is also the discrete macro-element method (DMEM), recently upgraded to include P-Delta effects [15]. The DMEM was proposed initially for the in-plane response of masonry walls [16] and subsequently extended to 3D kinematics to account for the coupled in-plane and out-of-plane responses of masonry walls when subjected to earthquake loadings [17].…”

“…In this spatial formulation, the model has been adopted to assess monumental structures subjected to a non-box behaviour, and thus vulnerable against out-of-plane mechanisms ( [19], [20]). The model has been implemented within the HiStrA (Historical Structure Analysis) software [21] and has been validated in [15] against analytical rigid-body solutions and quasi-static experimental tests.…”