Energies 2021 , 14 , 1095
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By solving the eigenvector problem, it is possible to determine the pairs ( λ 1 ,v 1 ) , . . . , ( λ N ,v N ) ,where N is the number of degrees of freedom, λ i is i -th eigenvalue, v i = Δ d i is i -th eigenvector (post-buckling deformation mode). The minimum eigenvalue and the corresponding eigenvector define the critical load multiplier and the buckling mode of the panel. More details are provided in our previous article [12]. In our case, the geometry of each panel does not have to be regular, as well as the position and shape of the perforations can be chosen arbitrarily. Therefore, the triangular shell FE was adopted, see Figure 2a. Due to shear locking that can occur in FE formulation of thick plates, most of the Reissner–Mindlin FE triangular shells were developed using the assumed transverse shear strain techniques [41–43]. Here, the 6-noded quadratic triangle FE with a standard quadratic interpolation for the deflections and rotations, and the assumed linear transverse shear strain field was utilized. More details on the implementation of this FE, including shear locking description, can be found in our previous article [12].
Figure 2. ( a ) Triangular mesh on a single panel with perforation; ( b ) spring connection between two parts separated by a perforation; ( c ) thickness reduction of elements adjacent to perforation.
2.3. Finite Element Model of a Single Panel Used in the Buckling Analysis A new issue, not addressed in our previous article [12], concerns perforations, i.e., crease lines and partial cuts on vertical panels of the box, which is a non-trivial task when they need to be included in numerical calculations. The computational model used in our study to estimate the critical load of each panel is based on the FEM. Therefore, it is required to define the panel geometry, its boundary conditions, and the constitutive model of the corrugated boards, as well as to define the perforation lines behavior. In order to compute the critical load of each vertical panel of the box, we used the 6-node FE described in the previous section and in [12]. An in-house routine, written in MATLAB software, was used for all analyses. Because the eigenvalue buckling prediction belongs to the linear perturbation procedures, we need to define only a linear elastic behavior. To correctly identify whole elastic material properties of corrugated cardboard, it is necessary to perform some laboratory tests, e.g., ECT, 4-point bending test, torsional stiffness test (TST), etc. In this study, we used a box strength estimation (BSE) system, which allows performing four independent tests in two main directions of corrugation and is able to identify all elastic parameters of corrugated board (www.fematsystems.pl/systems/BSE). The following parameters describe the behavior of corrugated board in the elastic range: • D 11 —bending stiffness in the MD; • D 22 —bending stiffness in the CD; • D 66 —twisting bending stiffness; • A 11 —compression stiffness in the MD; • A 22 —compression stiffness in the CD; • A 33 —compression stiffness in the z direction (out of plane); • A 44 —transverse shear stiffness in the 1–3 (x–z) plane; • A 55 —transverse shear stiffness in the 2–3 (y–z) plane.
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