## Introduction

The basic assumption of no-tension masonry model coincides with the hypothesis that the tensile resistance is null. Under this hypothesis, no-tension stress fields are selected by the body through the activation of an additional strain field, the fractures (see Baratta, 1991, Baratta et al. 1981, Baratta & Toscano 1982, Bazant 1996, Heyman 1966). The behavior in compression can be modeled in a number of different ways (elastic linear, elastic non-linear, elastic-plastic; isotropic, anisotropic; etc.), without altering substantially neither the results nor the mathematical treatment of the problem; some convenience exists for practical applications in assuming a isotropic linearly elastic model, in order to keep limited the number of mechanical parameters to be identified for masonry, since increasing the number of data causes increasing uncertainty in the results. Because of these reasons, and being clearly understood that there is no difficulty in introducing more sophisticated models, it is convenient to set up the fundamental theory on the basis of the assumption that the behavior in compression is indefinitely linearly elastic.

Analysis of NRT (Not Resisting Tension) bodies proves that the stress, strain and displacement fields obey extreme principles of the basic energy functionals.

Therefore the behaviour of NRT solids under ordinary loading conditions can be investigated by means of some extensions of basic energy approaches to NRT bodies (Baratta 1984, Baratta & O. Corbi 2005a, 2007, Baratta & al. 1981, Baratta & Toscano 1982).

In details, the solution of the NRT structural problems can be referred to the two main variational approaches:

- the minimum principle of the Potential Energy functional;

- the minimum principle of the Complementary Energy functional.

In the first case the displacements and the fractures are assumed as independent variables; the solution displacement and fracture strain fields are found as the constrained minimum point of the Potential Energy functional, under the constraint that the fracture field is positively semi-definite at any point.

The approach based on the minimization of the Complementary Energy functional assumes the stresses as independent variable. The complementary approach is widely adopted since the existence and uniqueness of the NRT solution are always assured in terms of stress, if some conditions on the compatibility of the loads are satisfied. The stress field can, then, be found as the constrained minimum of the Complementary Energy functional, under the condition that the stress field is in equilibrium with the applied loads and is compressive everywhere in the body.

The solution of both problems can be numerically pursued by means of Operational Research methods (see i.e. Rao 1978) suitably operating a discretization of the analyzed NRT continuum (Baratta & I. Corbi 2004, Baratta & O. Corbi 2003b, 2005a). One should notice that discussion about existence of the solution actually can be led back to some Limit Analysis of the considered NRT continua (Baratta & O. Corbi 2005a).

0,124 mj

0,200 m

0,124 mj

0,200 m

0,775 m 0,750 m Transducer

0 Strain-gauge

Figure 1. (a) Masonry panel geometry, (b) with the applications of a light reinforcement by means of FRP strips.

0,775 m 0,750 m Transducer

0 Strain-gauge m

Figure 1. (a) Masonry panel geometry, (b) with the applications of a light reinforcement by means of FRP strips.

In this regard, a special formulation of Limit Analysis for No-tension structures has been performed, allowing the set up of theorems analogous to the basic kinetic and static theorems of classical Limit Analysis; thus, one can establish efficient procedures to assess structural safety versus the collapse limit state (see e.g. Como & Grimaldi 1983) by specializing and applying fundamental theorems of Limit Analysis to NRT continua (Baratta 1991, Baratta & O. Corbi 2003a, 2005a, Bazant 1996, Como & Grimaldi 1983, Khludnev & Kovtinenko 2000). In details, the individuation of the collapse (live) load multiplier for NRT continua can be referred to the approaches relying on the two main limit analysis theorems:

- the static theorem;

- the kinetic theorem.

Duality tools may also be successfully applied in order to check the relationships between the two theorems of Limit Analysis (Baratta & O. Corbi 2004). In the study of plane mono-dimensional structures featuring a low degree of redundancy, the force/stress approach appears the most convenient to be adopted if compared with the displacement/strain approach, whose number of governing variables is higher and, moreover, increasing with the order of discretization (Baratta & I. Corbi 2003, 2004, 2006).

The following section reports some results showing how the proper implementation of the described theoretical approach for classical masonry structural typologies such as panels, arches and vaults, produces results that are in a very good agreement with experimental data, demonstrating the overall reliability of the mentioned approach, for whose details one should refer to cited references.

As shown in the following, results can also be successfully extended to the case of reinforcements with fiber-reinforced polymers (FRP).