Static and dynamic analysis of masonry buildings using a new macroelement within the equivalent frame method

In unreinforced masonry buildings, an accurate estimation of the dissipative capacity and of the cyclic behaviour under seismic actions are in general necessary. The most computationally convenient way to model regular and simple masonry buildings is offered from the Equivalent Frame Method, in which piers and spandrels are schematized with beam-type element, and the nodal regions are supposed fully rigid. Within this approach, the non-linear cyclic behaviour of each wall can be reproduced with a lumped plasticity model, e.g. with rotational and shear springs inserted into the beam elements. In this way, the model preparation can take much time, as each masonry panel needs many internal nodes necessary to host the springs and the elastic beams. In this work, a macroelement for the static and dynamic analysis of masonry piers and spandrels is presented. The developed 2-node macroelement has been obtained assembling two flexural springs at both ends and a shear spring in the middle, connected with Euler-Bernoulli beams. The springs characteristics are automatically retrieved by the element on the base of a few mechanical and resisting properties of the masonry, given as input. Each spring has a typical hysteretic law, derived from experimental tests and different for flexural and shear behaviour. Moreover, strength and stiffness degradations law were implemented, and the resisting shear and moments are functions of the compressive load in the panel, which can vary during analysis. The macroelement has been implemented as external library in more than one FE solver, and includes a simple Newton-Raphson algorithm to ensure the force equilibrium. The model has been validated comparing experimental and numerical results on single walls and on an entire building available in literature. It is shown that the developed macroelement is able to adapt the resulting cyclic law on the base of the slenderness and the compressive load of the wall.