Issue 29

D. De Domenico et alii, Frattura ed Integrità Strutturale, 29 (2014) 209-221; DOI: 10.3221/IGF-ESIS.29.18 219 The numerical methodology also gives some hints on the state of specimens at incipient collapse by pointing out the plastic zones (collapse mechanism) built by the LMM at the last converged solution. Fig. 6a and 7a show the strain rate components at collapse, c xx   , of concrete FEs in the deformed configuration for beam F- SB and S6-PRE3, respectively. The plastic zones arise at the mid-span of the element, while the remaining portions of the beam rotate rigidly around a sort of plastic hinge as observed in the experimental flexural collapse mechanism. The predicted plastic zones appear sufficiently confined and reasonably close to the damaged zones experimentally detected, see [4] and [16]. The beams fail due to concrete crushing near the loading point in the compression zone as observed in the experimental test. The collapse mechanism is also described by the strain rates at collapse of FRP FEs in the fibre direction, i.e. 1 c   , reported in Figs. 6b and 7b in the un-deformed configurations for the two analysed beams. The most critical FRP zones are highlighted and, obviously, these zones are those where the FRP sheets, to a greater extent, bear the load and act compositely with concrete in the global collapse mechanism. Finally, it is worth noting that for both the predicted collapse mechanisms the stresses numerically obtained in the steel FEs at the mid-span (where a plastic hinge develops) are just yielded as observed in the experimental outcomes. Other types of FRP-strengthened RC-elements have been analysed within the same research programme, obtaining encouraging confirmations on the predictive performance of the proposed approach, see e.g. [26]. a) b) Figure 6 : Collapse mechanism of the specimen F-SB: a) contour plot of the strain rate components   c xx of concrete FEs reported in the deformed configuration of the beam; b) contour plot of the strain rates 1   c of FRP FEs in the fibre direction a) b) Figure 7 : Collapse mechanism of the specimen S6-PRE3: a) contour plot of the strain rate components   c xx of concrete FEs reported in the deformed configuration of the beam; b) contour plot of the strain rates 1   c of FRP FEs in the fibre direction C ONCLUDING REMARKS numerical limit analysis methodology has been presented to analyse RC members strengthened with externally bonded FRP plates. A multi-yield-criteria formulation has been proposed to appropriately describe the behaviour, at a state of incipient collapse, of the three main constituent materials: concrete, steel-bars and FRP laminates. The latter formulation is essential to deal with concrete crushing, steel bars yielding and FRP rupture that may occur at ultimate limit states. The lack of associativity postulated for concrete and FRP composite laminates has resulted in adopting a nonstandard limit analysis approach which underlies the use of two numerical methods for limit analysis, the LMM and the ECM, to search for an upper and a lower bound to the actual peak load multiplier. Operationally, as compared to previous results presented in [12] or to alternative numerical approaches e.g. [16, 27, 28], the multi-yield-criteria formulation here proposed does not entail any significant computational cost: simple FE analyses (performable with any commercial FE-code) have to be solved. The more accurate and consistent 3D modelling that accounts for three materials through three distinct yield criteria seems to give good results. The reliability and effectiveness of the proposed methodology have been proved by analysing full-scale laboratory tests on RC beams strengthened with externally bonded FRP sheets. The obtained numerical results, in terms of peak load A