Keywords: masonry, fibre reinforced polymer, non-linear model, homogenization, finite element method.

The foreseen inadequate performance of masonry under earthquakes is a common issue in many countries and arises from the weakness of mortar joints. Conventional retrofitting, e.g. reinforcement with steel plates, has proven to be impractical, expensive and add considerable mass to the structure. In this context, the utilization of externally bonded fibre reinforced polymer (FRP) strips [1] seems an interesting solution because of the limited invasiveness and good performance at failure.

An efficient analysis of FRP strengthened large scale masonry structures in the non-linear range would require a micro-macro or macro-computational approach [2,3], allowing a fast evaluation of (a) collapse loads, (b) displacements near collapse, (c) failure mechanism, and (d) post peak behaviour of the structures.

In this paper, a two-step micro-macro model is used to analyse efficiently masonry FRP strengthened structures. In the first step, masonry is substituted by a macroscopic equivalent material through the application of a simplified averaging procedure, in which a representative element of volume (REV) constituted by a central brick interconnected with its six neighbours with zero thickness joints is meshed with rigid wedges and non-linear softening interfaces. The approach allows estimating, in an approximate way, masonry macroscopic non-linear behaviour subject to in- and out-of-plane loads.

In the second step, full masonry structures are analysed in the non-linear range through a non-commercial finite element (FE) code constituted by six-node rigid wedges, with elastic and inelastic deformation allowed only at the interfaces between adjoining elements. FRP strips are modelled by means of triangular rigid elements. Masonry and FRP layers interact by means of interfacial tangential actions exhibiting an elasto-damaging relationship, in agreement with codes of practice formulas dealing with delamination [1].

In order to circumvent some drawbacks of standard FEs when dealing with softening materials, a sequential quadratic programming approach (SQP) is adopted to solve the global non-linear problem [4] at a structural level. The algorithm is finally tested on a medium size deep beam reinforced with diagonal and horizontal FRP strips.

1

CNR-DT 200-2004, "Guide for the design and construction of externally bonded FRP systems for strengthening existing structures", C.N.R. National Research Council, Italy, 2004.

2

T. Massart, R.H.J., Peerlings, M.G.D. Geers, "Mesoscopic modeling of failure and damage-induced anisotropy in brick masonry", Eur J Mech A/Solids, 23, 719-735, 2004. doi:10.1016/j.euromechsol.2004.05.003

3

P. Pegon, A. Anthoine, "Numerical strategies for solving continuum damage problems with softening: application to the homogenisation of masonry", Computers & Structures, 64(1-4), 623-642, 1997.

4

O. De Donato, A. Franchi, "A modified gradient method for finite element elastoplastic analysis by quadratic programming", Computer Methods in Applied Mechanics and Engineering, 2(2), 107-131, 1973. doi:10.1016/0045-7825(73)90010-8

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