Non-Linear Seismic Analysis of Masonry Buildings

Non-linear analysis is the most viable tool to get accurate predictions of the actual response of masonry structures under earthquake loading. Analytical methods based on the idealisation of masonry walls with openings as systems of macro-elements allow not only to capture the main failure modes observed after past earthquakes, but also to ensure a limited computational demand in engineering practice. The present thesis deals with non-linear incremental static (pushover) analysis on masonry buildings modelled through evolutionary spread plasticity macro-elements. In the first part of the work, fundamentals of structural analysis, seismic risk, and performance-based seismic design are reviewed along with non-linear static procedures and macro-element methods for seismic analysis of masonry buildings. The second part of the work deals with theoretical advances in non-linear seismic analysis of masonry buildings. In particular, an evolutionary spread plasticity macro-element has been developed and static pushover procedures for individual masonry walls, as well as entire buildings, are presented and discussed in detail. The proposed macro-element has been defined as ‘evolutionary’ because its inner reacting domain changes as the lateral drift demand increases. Such an evolution is caused by the spreading of cracking and yielding within the masonry. Namely, tensile cracking of masonry induces significant reductions in the effective width of cross-sections, which then depends on the magnitude of the given lateral drift (geometrical non-linearity). Yielding of masonry develops near the extreme parts of the macro-element, which are subjected to maximum bending moment. Mechanical non-linearity of masonry in compression is also taken into account through a deformation-based approach. In fact, the mechanical behaviour of the macro-element is characterised for different constitutive laws of masonry, by means of (1) two- and three-dimensional flexural strength domains (to be coupled with classical strength domains), (2) moment-curvature relationships, and (3) force-displacement diagrams. Two-dimensional strength domains derived from strength degrading constitutive models allow to account for more real characteristics of masonry. The comparison between flexural strength domains corresponding to different constitutive models have shown that current simplified formulas lead to higher values of ultimate shear force and bending moment, if the given axial force is not significantly higher than one-half of the allowable axial force. The implementation of full non-linear stress-strain relationships (which have been obtained by recent uniaxial compression tests in the direction orthogonal to mortar bed joints of masonry) has been found to provide more conservative estimations of ultimate shear force corresponding to flexural failure of macro-elements (i.e., toe crushing). Flexural strength domains have been defined at cracking, elastic, and ultimate limit states of masonry cross-sections and panels, in both cracked and uncracked conditions. The explicit consideration of strain ductility of masonry has allowed to assess the evolution in strength domains. In this regard, less significant variations in both ultimate shear force and bending moment have been detected for a given axial force lower than one-fourth of the allowable axial force. Finally, flexural strength of macro-elements has been also investigated for any boundary condition through the development of three-dimensional strength domains corresponding to elastic and ultimate limit states, in both cracked and uncracked conditions. Moment-curvature relationships have been defined for rectangular unreinforced masonry cross-sections by means of an incremental iterative procedure. Their development has allowed to assess key parameters of sectional behaviour, such as flexural overstrength, strength degradation due to strain softening of masonry, yielding and ultimate curvatures, and curvature ductility. Such parameters have been estimated for a number of constitutive laws and the relationship between curvature and strain ductilities has been also investigated. The implementation of empirical stress-strain relationships presented in the first appendix of this thesis has let to derive moment-curvature relationships where both yielding and ultimate strains of masonry, as well as strain softening, are explicitly considered. It has been found that, if the applied axial force does not exceed one-half of the allowable axial force, the ratio between curvature ductility and strain ductility is greater than unity and does not depend on the magnitude of the applied axial force. In order to define force-displacement diagrams of macro-elements, a specific incremental iterative procedure has been developed; it is based on the monitoring of the maximum axial strain over the cross-section(s) subjected to the maximum bending moment. It has been shown that force-controlled procedures can lead to significant underestimations of displacement capacity of masonry panels and to underestimations of both lateral stiffness and maximum resisting shear force. Force-based pushover procedures in response control have been developed for individual walls with openings and entire masonry buildings, separately, in order to predict also their non-linear softened response. Such procedures have been implemented in a novel computer program named RAN CODE, which is specifically devoted to structural analysis of masonry buildings. Amongst several numerical applications aimed at validating the developed procedures, the outcomes of a series of global pushover analyses on a masonry building designed in compliance with Eurocode 8 and Italian building code are discussed. The pushover procedure developed for single masonry walls with openings could be employed in the case of existing buildings (either single buildings or building units within aggregates), which have often flexible floor diaphragms and lacking, or poor, connections between diaphragms and walls, as well as between orthogonal walls. The procedure developed for global pushover analysis of masonry buildings accounts for torsional effects due to both inherent (i.e., structural) and accidental eccentricities between centres of mass and centres of stiffness. The use of spread plasticity macro-elements which change with the given deformation state allows to relate the ‘local’ response of masonry panels to the ‘global’ response of the structure. Finally, three appendices include results and empirical models obtained through experimental programs aimed at supporting non-linear modelling and analysis of masonry buildings. Appendix A deals with mechanical characterisation of masonry under uniaxial compression along directions parallel and orthogonal to mortar bed joints. Such a characterisation is consisted in the definition of mechanical parameters and constitutive models able to simulate non-linear behaviour of masonry up to large inelastic strains. Appendix B deals with mechanical characterisation of masonry in sliding shear along mortar bed joints. Also in this case, both classical and advanced mechanical parameters have been defined and empirical models have been derived. Such models include shear stress versus shear strain relationships and a shear response surface, which allows to simulate non-linear shear behaviour of unit-mortar interfaces over the whole range of allowable strains (that is, from elastic to inelastic range). Appendix C summarises the main results of three quasi-static lateral loading tests on a full-scale masonry wall with a opening and no tensile-resistant elements (e.g., reinforced concrete bond beams, steel ties), which is the typical case of existing masonry buildings. Namely, the first monotonic test allowed to investigate non-linear behaviour of the wall up to the first significant damage to the spandrel panel above the opening. The second test was carried out on the pre-damaged wall under cyclic displacements, in order to assess residual properties and to reach some hints on seismic performance of masonry walls subjected to earthquake sequences. The last test was performed under cyclic displacements on the wall after repairing and upgrading of the spandrel panel with an inorganic matrix-grid composite system. The aim of that test was to assess the effectiveness of the strengthening system for seismic retrofit of masonry structures and rapid remedial works during seismic emergency scenarios. Data processing for all lateral loading tests has shown that the damage to the spandrel panel affected both load-carrying capacity and strength degradation of the wall, whereas rocking behaviour of piers produced large displacement capacity and low residual drifts (that is, high re-centring capacity)