Issue 29

D. De Domenico et alii, Frattura ed Integrità Strutturale, 29 (2014) 209-221; DOI: 10.3221/IGF-ESIS.29.18 211 according to experimental results [18]. Due to the dilatancy of concrete, a non-associated flow rule is postulated for the adopted M–W-type yield surface. Steel is modelled as an isotropic, perfectly plastic material obeying the well-established von Mises yield criterion. For a multi-axial loading scenario the von Mises yield condition is expressed as: ( ) ( ) ij ij y f f       0 (4) where ( ) ij   is the von Mises effective stress and y f is the yield strength. Since in the FE-model the steel reinforcement are modelled by 1D truss elements, a uniaxial stress condition is considered in the following and Eq. (4) applies in the simpler shape r y f   , r  being the stress in the re-bar longitudinal direction. Finally, the FRP strengthening plates are modelled as orthotropic laminates in plane stress conditions obeying a Tsai–Wu- type yield criterion [19]. For a unidirectional lamina in plane stress case the Tsai–Wu polynomial criterion has the following form: F F F F F F              2 2 2 11 1 22 2 66 6 12 1 2 1 1 2 2 2 1 (5) where 1 and 2 denote the principal directions of orthotropy (the fibres are directed along the material axis 1) and 6 12    in the contracted notation. The coefficients i F and ij F ( , 1, 2, 6) i j  entering Eq. (5) are functions of the strength parameters of the unidirectional lamina: : ; : ; : ; : ; : ; : t c t c t c t c F F F F F F F F X X Y Y X X YY S            1 2 11 22 66 12 11 22 2 1 1 1 1 1 1 1 1 2 (6) with: t X , c X the lamina longitudinal tensile and compressive strengths, respectively; t Y , c Y the lamina transverse tensile and compressive strengths, respectively; S the shear strength of the lamina. In the expressions (6) the compressive strengths c X and c Y have to be considered intrinsically negative. Also FRP composite plates are considered as nonstandard material and, therefore, a non-associated flow rule is postulated for their constitutive behaviour. Numerical limit analysis methodology Two distinct limit analysis methods are applied simultaneously. The former, based on the kinematic approach of limit analysis, is able “to build” the collapse mechanism of the analysed structure and to compute an upper bound to the peak load multiplier. The latter, based on the static approach of limit analysis, is instead oriented “to build” a statically and plastically admissible stress field (corresponding to a given load) so giving a lower bound to the peak load multiplier. The reason for computing two bounds arises from the postulated non associativity of concrete and FRP composite material that injects such characteristic on the behaviour of the whole RC-structural element. The two methods, conceived in [20] and [21] with reference to von Mises materials, are known as Linear Matching Method (LMM) and Elastic Compensation Method (ECM), respectively. Both have been rephrased and widely employed by the authors [14], [15]. Their use to bracket the real peak load value of a structure made of a nonstandard material has been also experienced with success [11]–[13]. All the analytical details of LMM and ECM are in the above quoted papers and are here omitted for brevity. The novelty or key-feature of the present study is actually the implementation of LMM and ECM with reference to three different constitutive criteria at the same time. Indeed, the three criteria are those given in the previous section for the three main constituents of the FRP-strengthened RC members here addressed, i.e.: Menétrey–Willam-type for concrete; Tsai–Wu-type for FRP sheets; von Mises for steel bars. For completeness, the two methods are briefly expounded looking only at their geometrical interpretation sketched in Fig. 1 and 2. On taking into account that both methods are performed iteratively, the sketches refer to a current iteration, say ( k -1)th. Looking at the geometrical interpretation of the LMM sketched in Fig. 1, at the current iteration, say at the ( k -1)th FE- analysis, a fictitious structure (i.e. the structure under study with its real geometry, boundary and loading conditions but made of fictitious material) is analysed under loads ( 1) k i P p  , with ( 1) k P  load multiplier and i p assigned reference loads. The fictitious linear solution computed at each Gauss Point (GP) of the FE mesh, can be represented, at the generic GP, by a point ( 1) k L   lying on the complementary dissipation rate equipotential surface referred to the fictitious viscous material, say ( 1) ( 1) ( 1) ( 1) ( , , ) k k k k j I j W D W         , whose geometrical dimensions and centre position depend on the fictitious values ( 1) k I D  and ( 1) k j   fixed at the current GP ( I ranging over the elastic constants entering the considered material; j