PSI - Issue 39

14 Davide Leonetti et al. / Procedia Structural Integrity 39 (2022) 9–19 Author name / Structural Integrity Procedia 00 (2019) 000–000 η ( x ) = P α 2β e - α x ( sin α x + cos α x ) (9) where is the deflection, P is the concentrated force, is is the reaction force per unit length of the beam, α= β ( 4EJ ) ⁄ , E is the Young modulus and J is the second moment of area of the section around its horizontal centroidal axis. The distributed load p(s) from –c ≤ x ≤ c can be interpreted as resulting from the application of the forces P = p∙ds, over an infinitesimal area ds . Starting from Equation 9, the deflection under the distributed load p(s) is given by: η ( x ) = α 2β � p ( s ) e - α ( x-s ) ( sin α ( x-s ) + cos α ( x-s )) c -c ds (10) where all the variables and parameters are depicted in the mechanical schemes of Figures 2 and 3. By differentiation of Equation 10, the internal shear forces and moment along the beam are determined. 3. Results In this section, the results of the model are verified. At first, the through the thickness stress distribution is compared with the results of a finite element model. Successively, the SIFs due to a moving load were obtained with both the stress distribution resulting from the finite element model and the analytical solutions proposed before using both the weight function technique. 3.1. Validation of the stress distribution in an uncracked plate The stress distribution resulting from the superposition of the stresses due to bending moment and contact is compared with the results of a finite element model developed in ANSYS environment with the purpose of validating the results of the analytical model. The considered geometry, consists of a beam on an elastic foundation subjected to a distributed load p(x) acting on the top surface of the beam between - 1≤ x ≤1 mm, therefore c =1 mm. Figure 4 shows the finite element model. Despite the symmetry of the geometry and loads the model is related to the full geometry due to the fact that the prospective crack lines for the SIF calculation are non-symmetrical. The length of the plate is calculated on the basis of the estimated wavelength of the deflection curve. In particular the plate is 10 4 mm long. The elastic foundation has been modeled using SURF154 elements available in ANSYS, in which the real constant number 4 is used to define the stiffness k of the foundation. The geometry has been discretized using PLANE183 elements, an isoparametric element having quadratic displacement behavior. A finer mesh has been applied in the vicinity of the applied load, and a coarser mesh has been used for the rest of the geometry. The transition between these two discretization is made by a conformal structured mesh. The accuracy of the model is ensured by a convergence analysis. 6