258 The UMAP Journal 24.3 (2003) Numerical results We now return to the question of determining the number of flats needed for the top layer. By Pflug et al. [2000, the flatwise compression test(FCT) result for C-flute board is 1.5x 10 Pa. We expect the motorcyclist to experience an acceleration of approximately 5g upon landing, distributed over a surface area of 3000 mm2=0.003 m. We assume that the motorcycle has mass 100 kg [Kawasaki 2002] and the rider has mass 60 kg. The pressure exerted on the ardboard is P=(160kg)(5)(9.8m/s2) =2.61×106Pa. 0.003m For the cardboard at the bottom of the stack of flats to be bent significantly, enough pressure must be applied to crush most of the cardboard above it. Thus a lower bound on the number of flats is「(2.61×10°Pa)/(1.50×105Pa)= [17.4=18 flats. To be perfectly safe, we double this figure and cross-hatch the flats; that is we want 36 flats in the top platform, the flutes of which alternate in direction Next, we calculate the total mass of cardboard for these flats. We assume that the flats are l m x 3m. From Gilchristet al [1999, we know that the density of C-flute corrugated cardboard is 537 g/m; we obtain a mass of 1.611 kg per flat, or about 60 kg for 36 flats, which is comparable to the weight of a second person. The thickness of a C-flute flat is 4. 4 mm [Mall City Containers n d with 36 flats the height of the stack is 158.4 mm. We now plug some reasonable values into(3 )and get a good approximation of the desired height of the boxes. Let the stacking weight constant k=800 N this is roughly the mean value found in Clean Sweep Supply [2002]. These values, along with g= 5g, give an optimal h of roughly (3m2)0.05m)(800N) ≈ 2(220kg)[98+5(98)m/s2] So the harmonic mean of l and w must be on the order of 4h=0.67 m Converting these values into inches gives h=6.5 in and a value of roughly 26.5 in for the harmonic mean of l and w. The two commercially available box sizes that most closely approximate these valt 6in×26in×26in and 6in x 28 in x 28 in [Uline Shipping Supplies 2002]. Note that we must increase the cross-sectional area beyond what was calculated in order to keep the number of boxes per layer an integer. Doing so increases the total value of B= Akd/m(g+g; thus, we ideally want a somewhat larger box than calculated. Since we cannot increase h(any commercially available box of this rough shape and size has a height of 6 in), we increase l and w. This choice increases the amount by which the cross-sectional area is larger than previously calculated. Since optimum values of l and w only change as a/3,however,the larger box is the closer to the optimum.258 The UMAP Journal 24.3 (2003) Numerical Results We now return to the question of determining the number of flats needed for the top layer. By Pflug et al. [2000], the flatwise compression test (FCT) result for C-flute board is 1.5 × 105 Pa. We expect the motorcyclist to experience an acceleration of approximately 5g upon landing, distributed over a surface area of 3000 mm2 = 0.003 m2. We assume that the motorcycle has mass 100 kg [Kawasaki 2002] and the rider has mass 60 kg. The pressure exerted on the cardboard is P = (160 kg)(5)(9.8 m/s2) 0.003 m2 = 2.61 × 106 Pa. For the cardboard at the bottom of the stack of flats to be bent significantly, enough pressure must be applied to crush most of the cardboard above it. Thus a lower bound on the number of flats is (2.61 × 106 Pa)/(1.50 × 105 Pa) = 17.4 = 18 flats. To be perfectly safe, we double this figure and cross-hatch the flats; that is, we want 36 flats in the top platform, the flutes of which alternate in direction. Next, we calculate the total mass of cardboard for these flats. We assume that the flats are 1 m × 3 m. From Gilchrist et al. [1999], we know that the density of C-flute corrugated cardboard is 537 g/m2; we obtain a mass of 1.611 kg per flat, or about 60 kg for 36 flats, which is comparable to the weight of a second person. The thickness of a C-flute flat is 4.4 mm [Mall City Containers n.d.]; with 36 flats, the height of the stack is 158.4 mm. We now plug some reasonable values into (3) and get a good approximation of the desired height of the boxes. Let the stacking weight constant k = 800 N; this is roughly the mean value found in Clean Sweep Supply [2002]. These values, along with g = 5g, give an optimal h of roughly h = 3  (3 m2)(0.05 m)(800 N) 2(220 kg) 9.8 + 5(9.8) m/s2  ≈ 0.17 m. So the harmonic mean of l and w must be on the order of 4h = 0.67 m. Converting these values into inches gives h= 6.5 in and a value of roughly 26.5 in for the harmonic mean of l and w. The two commercially available box sizes that most closely approximate these values are 6 in × 26 in × 26 in and 6 in × 28 in × 28 in [Uline Shipping Supplies 2002]. Note that we must increase the cross-sectional area beyond what was calculated in order to keep the number of boxes per layer an integer. Doing so increases the total value of B = Akd/m(g + g ); thus, we ideally want a somewhat larger box than calculated. Since we cannot increase h (any commercially available box of this rough shape and size has a height of 6 in), we increase l and w. This choice increases the amount by which the cross-sectional area is larger than previously calculated. Since optimum values of l and w only change as A1/3, however, the larger box is the closer to the optimum.