Issue 48

E. Maiorana, Frattura ed Integrità Strutturale, 48 (2019) 459-472; DOI: 10.3221/IGF-ESIS.48.44 460 Although the analytical closed-form solution to determine buckling coefficient is only possible in a few simple cases of geometry and symmetry in loading (Timoshenko and Woinowsky-Krieger 1959 [1]), the Finite Element Method can be used to study many kinds of geometries, boundary conditions, and different configurations of loads, such as patch loading or local symmetrical loads, improving initial imperfections and covering a wide range of constitutive laws for materials in the non-linear range. Figure 1: Bridge web panel girders strengthened by a longitudinal stiffener. The increase in the elastic critical load through optimization of the stiffeners, by means of their shapes and positions, is examined in Xie and Chapman 2004 [2] and Lee et al. 2002 [3] and that in plates subjected to shear loading in Alinia 2005 [4]. The stability of longitudinally stiffened web plates under interactive shear and bending force is described in Alinia and Moosavi 2009 [5] and the influence of longitudinal stiffeners on the elastic stability of girder webs in Maiorana et al. 2011 [6]. An improved design procedure for efficient design of stiffened plates, under combined compression and bending, and in detail the influence of the stiffener location on the stability of the structure, finding the optimum location of it, is proposed in Bedair, 1997 [7]. The ultimate strength of longitudinally stiffened I-girder webs, subjected to combined patch loading and bending, is treated in Graciano and Casanova 2005 [8] and the shear strength of longitudinally stiffened panels with imperfections in Pavlovčič et al. 2007 [9]. The beams examined in this work are reinforced with a longitudinal stiffener and some generalizations were made to improve stiffener design with regard to the rigidity and position, in order to maximize elastic stability. The longitudinal stiffener was connected to the plate at various positions h  / h with respect to the compressed edge, so that the panel of length a at height h was divided into two subpanel: upper plate A with height h  , and the lower plate B with height h  . A Finite Element model was used according to the Strand7 code (G+D Computing 2005 [10]), with a static scheme consisting of a simple hinge bounded plate subjected to combination of bending moment and compression forces (Fig. 2). A sensitivity analysis of the mesh, to define the shape and dimension of the elements was treated deeply in previous papers by the author of this paper. Maiorana and Pellegrino 2011 [11] compares k values obtained with the FE model and Eurocode equations for square and rectangular panels of constant thickness subjected to axial force and bending moment, and Pellegrino et al. 2009 [12] compares the results of the FE model with those of the literature, in the case of perforated steel plates under shear loading. The FE used for the mesh were plates with four nodes (Quad4) and six degrees of freedom per node; the typical size of the elements being h / 20. Boundary conditions were assumed to represent the plate as a web panel, see Fig. 2: sides 1, 2, 3 and 4 displacements restrain u y = 0; side 2 and 4 u z = 0; and displacements u x were symmetric. The applied loads were in equilibrium and directly applied to the nodes on the two lateral vertical edges as a system of conservative forces which do not change direction during deformation, due to symmetry.