Issue 29

S. de Miranda et alii, Frattura ed Integrità Strutturale, 29 (2014) 293-301; DOI: 10.3221/IGF-ESIS.29.25 294 by Cocchetti et al. [6]. The authors modelled a tile bonded to a rigid substrate through an elastic adhesive as an eccentrically compressed beam on a Pasternak foundation. The eccentricity of compression is induced by the presence of an out-of-plane workmanship defect leading to a Mode I failure of the adhesive. This approach leads to closed-form estimation of the ultimate strength of tile-substrate adhesive joint. More recently the use of simplified beam model with elastic constraints has been successfully applied also for the estimation of crack opening area in longitudinally cracked pipes [7, 8]. In the present paper, a simple beam model is developed to evaluate the stress state of tiled floors subjected to debonding due to substrate shrinkage. Then, an ad-hoc finite element is developed in order to solve the governing differential equations of the tile-adhesive-substrate system. The presence of inter-tile grouting is modelled through rotational/translational springs. The additional eccentricity induced by workmanship defects is taken into account by means of a partial/eccentric grouting modeled with the same rotational/translational springs collocated in eccentric position. Numerical analyses are performed on different tiled flooring configurations. The robustness and reliability of the proposed model is verified by comparing the model results in terms of normal stresses within the adhesive layer with those obtained with a 2D FE model developed with the commercial software Abaqus. Figure 1 : Tile flooring. M ECHANICAL M ODEL s regards the tile-adhesive-substrate system, reference is made to the mechanical model shown in Fig. 2. Grouting modelling will be described in the next section. The tile, bonded to the flexible substrate by means of an elastic adhesive, is modelled as an Euler-Bernoulli beam on a Pasternak foundation connected to a second layer of vertical springs by means of a shear deformable layer. The adoption of a two-layer system with an interposed shear deformable stratum is inspired by the model developed by Kerr [9] in the contest of geotechnical engineering. Figure 2 : Mechanical Model. In the following, ( ) u x and ( ) v x denote the longitudinal (axial) and the transversal displacement of the tile respectively, and ( ) s v x the transversal displacement of the substrate. Moreover, the Young modulus, the Poisson ratio and the thickness of material i are denoted by i E , i  , i t , respectively. Subscript i can become t , a , or s if the quantity refers to tile, adhesive or substrate, respectively. Plane strain conditions are assumed and, thus, for a material i the plane strain modulus * i i 2 i  E E 1 ν   is used. Indeed, this hypothesis does not consider some specific features related to the bi- dimensional periodicity of the problem, but allows to preserve the main essence of the bi-dimensional problem in its reduction to the one-dimensional beam model and is, in fact, commonly adopted in the literature dealing with the A