Issue 22

S. Bennati et alii, Frattura ed Integrità Strutturale, 22 (2012) 39-55; DOI: 10.3221/IGF-ESIS.22.06 40 mainly by the stresses that develop on the interface between the bonded elements. In order to account for these possible effects, various theoretical models have been developed that enable evaluating such stresses. Moreover, a number of special laboratory tests have been studied to evaluate the strength of the bonding under different loading conditions. One very commonly adopted test procedure is to glue an FRP strip to a block of concrete and then subject it to tension until it detaches completely [5, 6]. Various theoretical models have been proposed for this test and differ mainly in the constitutive law assumed for the interface between the concrete and FRP. In general, the different models assume an initially linear elastic response, in which the stresses tangential to the interface are proportional to the corresponding relative displacements. Subsequently, the models assume a softening response, which is meant to represent the progressive damage occurring prior to complete detachment of the reinforcement. Cottone and Giambanco [7] have formulated a linear decreasing law for the softening phase, while Cornetti and Carpinteri [8] propose an exponential decreasing one. In the present work we address the problem of a simply supported beam, strengthened with an FRP strip bonded to its intrados and subjected to bending couples applied to its end sections. A similar problem, though with the concentrated load applied to the mid-span section, has been considered by Carpinteri et al. [9] and by De Lorenzis and Zavarise [10]. In the mechanical model proposed here, the beam and reinforcement strip are bonded by a zero-thickness interface representing the adhesive layer and the superficial layers of the bonded elements. The interface transfers tangential stresses only from one element to the other. Furthermore, the following simplifying hypotheses are adopted: a) the beam is flexible, but inextensible, and exhibits indefinitely elastic behaviour; b) the reinforcement strip is extensible, but completely lacking flexural rigidity, hence it is subject to solely axial stresses; c) the tangential stresses transferred by the interface are functions of the relative displacements between the beam and the strip through a piecewise linear function defined over three intervals (elastic response − softening response − debonding). The mechanical model is described by a set of differential equations, supplemented by appropriate boundary conditions. The resulting mathematical problem is solved analytically and explicit expressions for the main quantities of interest ( displacements, internal forces, interface stresses) are determined. The model predicts an overall non-linear mechanical response for the FRP-strengthened beam, which passes through various distinct, recognisable stages of behaviour as the intensity of the applied bending couples is increased. In the first stage, the entire system exhibits a linearly elastic response and the beam can be divided into two parts: the portion initially lacking reinforcement and that connected to the reinforcement through the interface, which is still completely in the elastic field. The first stage of behaviour ends when the tangential interfacial stresses at the extremities of the reinforcement reach the elastic limit. In the second stage, the interface portions near the reinforcement extremities enter the softening response domain and the beam is thus divided into three parts: the portion initially lacking reinforcement and those portions connected to the reinforcement through the interface, which is now partly damaged and partly still in the elastic field. The second stage of behaviour ends when the relative displacements at the extremities of the reinforcement reach their ultimate value, corresponding to interface debonding. In the third stage, the reinforcement strip progressively detaches from the beam, beginning at the extremities, and the beam can be thought of as still divided into three parts: an unreinforced portion made up of the area initially lacking reinforcement plus the area with the now detached strip, a portion whose reinforcement is connected via the damaged interface and the portion with reinforcement still connected by an elastic interface. As the loads increase, the damaged and debonded interface portions extend to involve the entire length of the beam. Upon complete detachment, the overall response of the system reduces to that of the beam lacking any reinforcement. It is interesting to note that this last stage is however only reached asymptotically with infinite growth of the couples applied to the beam. M ECHANICAL MODEL et us consider a beam AB of length 2  L l , simply supported at its ends, where it is loaded by two equal and opposite couples of magnitude M ( Fig. 1). A strip of FRP reinforcement is bonded along the centre of the beam intrados for a length 0 2 b . We indicate H as the height of the cross section of the beam and h the distance of the section’s centre from its intrados (for a rectangular cross section, / 2  h H ). Thanks to the symmetry of the problem, it is possible to limit the study to the left half alone by introducing appropriate constraints in correspondence to the axis of symmetry (Fig. 2). The positions of the beam and strip cross sections are given by an abscissa s , whose origin is placed at the left-hand extremity of the active (not detached) portion of the reinforcement strip. This choice helps obtaining simpler expressions for the integration constants characterising the analytical solution. With 0 0   a l b the length of the initially unreinforced segment of the beam, we indicate by ( ) v s and L