A Three‐Dimensional Macroelement Formulation for the Biaxial Response of Masonry Walls

ABSTRACT

Equivalent‐frame modeling is a common approach for assessing the seismic response of unreinforced masonry buildings, given its ability to capture the main failure mechanisms of masonry panels without the complexity of micro‐modeling techniques and the associated computational burdens. Typically, the equivalent‐frame strategy assumes that the out‐of‐plane response of walls is negligible, and the analysis focuses on the global building response governed by in‐plane panel mechanisms, reproduced through macroelements corresponding to vertical piers and horizontal spandrels. However, there might be cases where accounting for the coupled in‐plane and out‐of‐plane response is essential for a comprehensive description of the nonlinear behavior of the structure, even when local out‐of‐plane overturning mechanisms are inhibited through adequate structural details. For this reason, this paper presents a two‐node three‐dimensional macroelement that enhances the capabilities of the well‐established bidimensional formulations already implemented in computer programs, effectively and efficiently capturing the bidirectional axial‐flexural response in both elastic and nonlinear regimes. Two sectional integration approaches are proposed for the axial‐flexural response: a stripe formulation that optimizes computational efficiency through bidimensional analytical stress integrations, and a fiber discretization that allows versatility in defining constitutive laws. Three independent shear failure criteria are also implemented. The presentation of the macroelement is completed by an explicit demonstration of the energetic equivalence with a force‐based beam‐column element and by a detailed discussion on the inclusion of second‐order geometrical effects. The capabilities of the proposed macroelement are validated through the simulation of in‐plane and out‐of‐plane quasi‐static cyclic shear‐compression tests on masonry piers characterized by different masonry typologies, aspect ratios, and axial‐load ratios, demonstrating its accuracy and reliability. Moreover, numerical simulations carried out with the proposed formulation demonstrate the opportunity of including in‐ and out‐of‐plane interaction in numerical simulations.