Issue 29

S. de Miranda et alii, Frattura ed Integrità Strutturale, 29 (2014) 293-301; DOI: 10.3221/IGF-ESIS.29.25 296 * '2 * ''2 T 1 Π 2 t t t L E A u E I v dx        (3) 2 2 ' A , , 0 1 Π 2 2 t a t a l L t k v k u v u dx                    (4)   2 '2 S 1 Π 2 s s s s s L k v v G A v dx         (5) In the above equations, t A , t I , s G and s A  are the tile cross section area, the tile moment of inertia, the substrate shear modulus and the substrate cross section area, respectively. As it can be noted, the potential energy T  Π  of the tile consists of the classical Euler-Bernoulli energy terms, the potential energy of the adhesive A  Π collects all the terms of the Pasternak formulation and the coupling terms and the substrate potential energy S Π collects the elastic energy of the vertical springs and of the shear layer. F INITE E LEMENT DISCRETIZATION Tile-Adhesive-Substrate tandard shape functions are used to represent the axial and transversal displacement fields     ,  u x v x of the tile and the transversal displacement field   s v x of the substrate:       s u v s v s u x v x v x    N u N v N v (6) where u N , v N and s v N collect linear Lagrangian, cubic Hermitian and cubic Lagrangian shape functions, respectively, and u , v , s v are vectors collecting nodal parameters. With these assumptions, following standard procedures, the discrete equilibrium equations are derived:  KU P (7) with: T T 0 0 0 s s s s uu uv u uv vv vv v s vv v v K K u P K K K K U v P P v K K                                    (8) The expressions of the different submatrices of K and of the nodal load vectors u P , v P are given in Appendix. Grouting-Adhesive-Substrate Grouting is modeled by means of a discrete approach based on the use of elastic springs as shown in Fig. 4. The grouting itself is modeled by an axial spring u k , a vertical transversal spring v k and a rotational spring k  . The axial spring can be placed in an eccentric position with respect to tile axis in order to account for partial-eccentric grouting configurations. The attached portion of the adhesive layer is modeled with two vertical springs ad k while the related portion of substrate is modeled with two vertical springs sd k and a transversal spring g k . The definition of each stiffness coefficient is reported in Fig. 4. From the above discrete model, the definition of a grouting-adhesive-substrate stiffness matrix g K is straightforward. The rationale to adopt a discrete approach for modelling inter-tile grouting is twofold: on the one hand the discrete approach allows a simple definition and straightforward implementation of the grouting stiffness matrix g K , on the other hand it allows to preserve all the fundamental stiffness properties of the grouting (axial, transverse and rotational S