Issue 48

E. Maiorana, Frattura ed Integrità Strutturale, 48 (2019) 459-472; DOI: 10.3221/IGF-ESIS.48.44 462 K (  0 ) = K E + K S (  0 ) (3) The equation governing plate behavior is: d F = [ K E +  K S (  0 )] d u (4) where d u is the vector of displacement variations and d F the vector of load variations. The determinant of the matrix becomes null at buckling, and an increase in displacement without a corresponding increase in load occurs: [ K E +  K S (  0 )] d u = 0 (5) The solution to this is an eigenvalue problem corresponding to the lower eigenvalue  1 related to critical elastic stress, at which buckling occurs:  cr =  1  0 (6) Optimal stiffness and best position On the base of relative bending stiffness, the behavior of the whole web panel-stiffener system can be divided into two categories, defined by the different mechanism of the loss of stability. The flexible stiffening system is characterized by the fact that the web panel and the stiffener reach instability together in a global way, whereas the rigid stiffening system has a local instability mode. In the first case, both web panel and stiffening have the same elastic critical load and the optimal position changes and must be determined numerically for a given relative flexural stiffness ratio for every value of Eqn. 7 (Dubas and Gehri 1986 [14]).  s = ( E I st ) / ( D h ) (7) where D in the flexural rigidity by Eqn. 8 D = ( E t 3 ) / [12 (1 -  2 )] (8) In the second case, stiffening is sufficient against any loss of stability of the web panel, forming a nodal line, and the overall buckling load is the lower value, between that of the subpanels A and B. Both deformation modes, flexible and rigid, occur together and interact closely in the typical buckling shapes of plates with semi-rigid stiffeners, classified as a transient stage. Reaching optimal stiffness, these buckling shapes do not interact (Alinia 2005 [4]) and stiffening remains straight until complete deformation has taken place. The optimal position of the stiffener is reached when the elastic critical load is the same in both subpanels and also depends on stress gradient  : for  < 0 bending dominates compression forces, whereas for  > 0 compression dominates the bending moment. Thus, for a given value of stress gradient  , the distribution of forces changes in subpanels A and B (Fig. 1) and, accordingly, the optimal position changes. Increasing  , the optimal position of the stiffener moves to the center- line of the plate, regardless of relative stiffness (Bedair 1997 [7]). In the design of open beams subject to pure bending forces, the American code (AASHTO 2018 [15]) recommends the longitudinal stiffener to be positioned at an average distance of h  / h = 0.2 from the compressed edge. This position depends on the ratio between rigidities of stiffener and plate, and is not always exactly the same for each situation but changes according to relative flexural stiffness  s . In fact, unlike the case of prevailing compression stresses, when the bending moment predominates, the optimal location for a longitudinal stiffener depends on the geometrical characteristics of the whole plate/stiffener relationship. In conclusion, reaching optimal stiffness with stiffener in its optimum position, maximal stability is reached, ensuring that the stiffener has a minimal cross-sectional area. Effect of initial imperfection Many national steel construction codes prescribe the maximum size/magnitude of the initial imperfection relating to panel height and width. Generally standards take into account the influence of thickness, or refer to slenderness. European Standard (EN 1090-2 2018 [16]) refers to the slenderness, directly with b / t (local) or indirectly with L / 300 (global).