Energies 2021 , 14 , 1095
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of the perforation on the calculation results can be observed both in the decrease of the critical force, as well as in the compression force/reaction. No specific mesh study was performed here, as buckling analysis is a linear perturba- tion procedure, where the sensitivity of the critical load concerning mesh size is low. In all computations, we used the seed value 1/10 of the mean edge length of the analyzed panel (see Figure 2a). 2.4. Box Compression Strength—McKee’s Formula In order to compute failure load of a single panel, we substitute the buckling force derived from Equation (4) into Equation (1), after [3] we have:
1 − r
r D
11 D 22
Z − 2 ( 1 − r ) ,
(20)
P f = k 1 ECT
2 ( 1 − r ) ∼
where k 1 = 1.33 k ( 4 π ) = 2.028, while r = 0.746 and k = 0.4215. To obtain the failure load for the whole corrugated box, BCT , it is sufficient to multiply the Equation (20) by the box perimeter Z . This is related to the Mckee et al. assumption [3], details can be found in [11,12]. Thus, the ultimate compressive strength, also known as the long McKee formula, reads:
1 − r
r D
11 D 22
Z 2 r − 1 .
(21)
BCT MK 1 = k 1 ECT
A further simplification, due to the empirical observations of McKee et al., leads to a final form of well-known short McKee’s formula: BCT MK 2 = 5.874 ECT h 0.508 Z 0.492 . (22) It is the most widely used formula. However, its accuracy is only acceptable for the simplest boxes. If the ratio of length to width or height to length of the box is relatively high, the McKee formula is no longer valid [11]. The same limitations apply if the box contains any perforations. To overcome this limitation, it is sufficient to derive in Equation (1) the critical load calculated by numerical methods instead of analytical formulas. Both will be shown in the following sections. 2.5. Box Compression Strength—General Case In order to obtain the compressive strength of the box, BCT , the ultimate loads for each panel must be summed up, whereas the compressive load of each panel (see Figure 1a) can be calculated by multiplying the value of the force P f described by the Equation (1) by the width of the i -thpanel b or c . Thus, in a general case, we get the following: BCT = k ECT r 2 ∑ i = 1 P b cr 1 − r i γ b p i γ b b + 2 ∑ i = 1 ( P c cr ) 1 − r i γ b p i γ c c , (23) where ( · ) i indicates the i -th panel of width b or c ( i = 1, . . . ,2). P b cr and P c cr are the critical forces of panels of width b and c , respectively (see Figure 1a). γ b p and γ c p are the reduction factors taking into account the ratio of the compressive strength of the plate with and without the perforations for plate width b and c , respectively. γ b and γ c are the reduction coefficients due to the in-plate aspect ratio of the box dimensions, defined as: γ b = √ b / c , γ c = 1 i f b ≤ c , (24) γ c = √ c / b , γ b = 1 i f b > c . (25)
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