Equivalent frame models are an effective tool for the seismic assessment of existing masonry structures. Due to their simplicity, these models can be used to perform multiple nonlinear dynamic analyses, accounting explicitly for different sources of modeling and input uncertainty. In the past, equivalent frame models have been used to effectively estimate the global response of buildings whose behavior is dominated by in-plane failure modes of piers and spandrels. The recent development of a three-dimensional macroelement formulation for modeling both the in-plane and out-of-plane response extends the use of equivalent frame models to the additional study of local out-of-plane mechanisms of a building. This work applies the newly developed formulation, implemented in OpenSEES (McKenna et al., 2000), to the modeling of two shaking table tests on a stone masonry building and on a modern mixed concrete-unreinforced masonry structure. Since the approach explicitly accounts for the quality of connections in the building (i.e., wall-to-wall and floor-to-wall connections), specific elements and material models were developed for modeling these connections in an equivalent frame idealization of the three-dimensional structure. Through comparison with the experimental results, the performance of the modeling approach is discussed, and the sensitivity of the response to the major sources of modeling uncertainty (quality of connections, damping model) is assessed. The comparisons show that these new equivalent frame models can capture the onset of out-of-plane failure for historical structures with poor floor-to-wall connections and for modern URM buildings with stiff RC slabs, where the slab can uplift from the URM wall, which leads to changing static and kinematic boundary conditions of the out-of-plane loaded wall. The results further show that 1–2% of damping leads to good agreements with the experimental results if initial stiffness proportional Rayleigh damping is used.

Though flexible diaphragms play a role in the seismic behaviour of unreinforced masonry buildings, the effect of the connections between floors and walls is rarely discussed or explicitly modelled when simulating the response of such buildings. These flexible diaphragms are most commonly timber floors made of planks and beams, which are supported on recesses in the masonry walls and can slide when the friction resistance is reached. Using equivalent frame models, we capture the effects of both the diaphragm stiffness and the finite strength of wall-to-diaphragm connections on the seismic behaviour of unreinforced masonry buildings. To do this, we use a newly developed macro-element able to simulate both in-plane and out-of-plane behaviour of the masonry walls and non-linear springs to simulate wall-to-wall and wall-to-diaphragm connections. As an unretrofitted case study, we model a building on a shake table, which developed large in-plane and out-of-plane displacements. We then simulate three retrofit interventions: Retrofitted diaphragms, connections, and diaphragms and connections. We show that strengthening the diaphragm alone is ineffective when the friction capacity of the wall-to-diaphragm connection is exceeded. This also means that modelling an unstrengthened wall-to-diaphragm connection as having infinite stiffness and strength leads to unrealistic box-type behaviour. This is particularly important if the equivalent frame model should capture both global in-plane and local out-of-plane failure modes.

Existing masonry buildings can be strengthened to reduce their seismic vulnerability. While the effect of strengthening on the structural elements of a building has been well researched, there are few computationally efficient numerical modelling approaches that capture the effect of strengthening interventions. This paper shows how a recently developed macro-element for unreinforced masonry walls, which can capture both in-plane and out-of-plane failure modes, can model the effect of fibre-reinforced polymer (FRP) strengthening on masonry walls. Practically, FRP strengthening increases the flexural in-plane and out-of-plane strength through longitudinal FRP strips placed against the masonry. In the macro-element, this is modelled through adding fibres to the section model that represent the longitudinal FRP strips. Transversal FRP strips affect the shear strength and in the macro-element this is accounted for by increasing the cohesion in the equation for the shear strength of the macro-element. This paper models quasi-static and dynamic in-plane shear–compression tests as well as out-of-plane bending tests on walls with and without FRP strengthening and compares the numerical simulations to experimental results taken from the literature. The proposed modelling approach for FRP-strengthened masonry walls accurately predicts their in-plane and out-of-plane response, indicating that equivalent frame models can be used to predict the response of masonry buildings with FRP-strengthened walls. The application of the approach is illustrated through hypothetical retrofit configurations of a masonry building that was tested on a shake table in its unstrengthened state.