256 The UMAP Journal 24.3 (2003) and thus f depends only on h. The function f is minimized at adk (3) 2 2m(g+g′ At this value of h, the constraint reduces to B 1+=n=4B This implies that the harmonic mean of l and w should be 2l=2y4B=4h H=I+u So, in the optimal situation, the box should be roughly four times as long and wide as it is tall. However, there are other considerations For commercially available boxes, we must choose among some discrete set of realizable box dimensions That the number n of boxes in a layer must be an integer affects the possible values of many of the parameters on which h is based(most notably A, the cross-sectional area of the layer). We select the box among the potential candidates which most nearly compensates for this change in parameters The entire structure We need to determine the gross parameters of the entire structure. We determine the cross-sectional area of the structure by considering how much space the motorcyclist needs to land safely. The motorcycle has length about 1 m; we should leave at least this much space perpendicular to the direction of travel. We need substantially more in the direction of travel, to ensure that the motorcyclist can land safely with a reasonable margin of error. So we let the structure be 1 m wide and 3 m long, for a cross-sectional area of 3 m How many corrugated cardboard flats should we use? We do not want them all to crease under the weight of the motorcycle and rider, for then the motorcyclist could be thrown off balance. So we must determine how much we can expect the flats to bend as a result of the force exerted by the motorcycle To calculate the number of flats required, we use the flatwise compression test(FCT) data in Pflug et al. [2000] for C-flute cardboard. Though our goal is to prevent substantial creasing of the entire layer of flats, we note that if v have a reasonable number of flats, creasing the bottom flat requires completely crushing a substantial area along most of the e remainins ng layers. Less than 20% of this pressure is required to dimple a sheet of cardboard to the point where it may be creased. Since we assume that the pressure required to crush the flats scales linearly with the number of flats, we find the maximum pressure that256 The UMAP Journal 24.3 (2003) and thus f depends only on h. The function f is minimized at h = 3 B 2 = 3  Adk 2m(g + g ) . (3) At this value of h, the constraint reduces to lw l + w = B h2 = √3 4B. This implies that the harmonic mean of l and w should be H ≡ 2lw l + w = 2√3 4B = 4h. So, in the optimal situation, the box should be roughly four times as long and wide as it is tall. However, there are other considerations. • For commercially available boxes, we must choose among some discrete set of realizable box dimensions. • That the number n of boxes in a layer must be an integer affects the possible values of many of the parameters on which h is based (most notably A, the cross-sectional area of the layer). We select the box among the potential candidates which most nearly compensates for this change in parameters. The Entire Structure We need to determine the gross parameters of the entire structure. We determine the cross-sectional area of the structure by considering how much space the motorcyclist needs to land safely. The motorcycle has length about 1 m; we should leave at least this much space perpendicular to the direction of travel. We need substantially more in the direction of travel, to ensure that the motorcyclist can land safely with a reasonable margin of error. So we let the structure be1m wide and3mlong, for a cross-sectional area of 3 m2. How many corrugated cardboard flats should we use? We do not want them all to crease under the weight of the motorcycle and rider, for then the motorcyclist could be thrown off balance. So we must determine how much we can expect the flats to bend as a result of the force exerted by the motorcycle. To calculate the number of flats required, we use the flatwise compression test (FCT) data in Pflug et al. [2000] for C-flute cardboard. Though our goal is to prevent substantial creasing of the entire layer of flats, we note that if we have a reasonable number of flats, creasing the bottom flat requires completely crushing a substantial area along most of the remaining layers. Less than 20% of this pressure is required to dimple a sheet of cardboard to the point where it may be creased. Since we assume that the pressure required to crush the flats scales linearly with the number of flats, we find the maximum pressure that