Energies 2021 , 14 , 1095
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In Equation (1), only elastic buckling is took into account. This approach was later modified by Urbanik et al. [9,18] in order to also take into consideration the nonlinear material effects by the following: P f ECT = k P cr ECT un , (2) where n = 1 − r . If ECT > P cr , than u = 1, which corresponds to the situation when elastic buckling is activated first, otherwise, u = 0, which corresponds to the situation when the ultimate load P f depends only on ECT , namely: P f = k ECT . (3) In any case, the critical load for a rectangular orthotropic plate (see Figure 1a) can be assumed as [38–40]: P b cr = k cr π 2 √ D 11 D 22 b 2 , (4) where P b cr is the critical load of the panel of dimensions a × b (height and width, respec- tively), k cr is a dimensionless buckling coefficient, which depends, e.g., on the ratio a / b , boundary conditions applied to the plate edges, material characteristics, the buckling mode, etc.; D 11 is the bending stiffness in the MD, in Nmm, D 22 is the bending stiffness in the CD in Nmm and b is the plate width in mm. Mechanical properties of the corrugated cardboard depend on the direction (CD or MD), which is typical for fibrous materials. Therefore, the buckling coefficient can be assumed as [39]: k cr = D 11 D 22 mb a 2 + 2 ( D 12 + 2 D 33 ) √ D 11 D 22 + D 22 D 11 a mb 2 , (5) where m is the number of half-waves in the direction of loading; a is panel height in mm; D 33 is the torsional stiffness in Nmm; D 12 = ν 21 D 11 = ν 12 D 22 inNmm; ν 12 and ν 21 are the effective in-plane Poisson’s ratios of the panel. 2.2. Buckling—Finite Element Method In our case, the panel may contain various types of perforations with different posi- tions, slopes, etc. Such a situation requires the definition of the buckling coefficient of a panel-by-panel basis, which leads to difficult and time-consuming development of analyti- cal formulas. In such a case, the FEM is much more general because it allows determining the critical load of panels with any shape and boundary conditions, including perforations, even with complex shapes. In this study, the critical load of each panel was calculated using a three-dimensional (3D) finite element (FE) model, where all outer edges are fixed in the out-of-plane direction (see dashed lines in Figure 1a). Moreover, the bottom edge was fixed in the vertical direction. The remaining translations and all rotations were inactive. The critical load in the general case reads [40]: P b cr = λ q , (6) where λ is a critical loading multiplier, i.e., the smallest eigenvalue and q is a distributed load on a panel upper edge [Nmm]. In order to calculate the critical force multiplier, the typical formulation of initial buckling problem is required: [ K 0 + λ K σ ] v = 0, (7) whereK 0 is the global stiffness matrix of the whole panel and K σ is an initial geometrical stress matrix of the whole panel.
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