Thinking outside the box 251 Thinking Outside the Box and Over the Elephant Melissa J. Banister Matthew macauley h j Harvey Mudd College Claremont, cA Advisor: Jon Jacobsen Abstract We present a mathematical model of the collapsing of a box structure, which is to be used to protect a stunt motorcyclist who jumps over an elephant. Reasonable alues of the model's parameters cause it to predict that we should construct the structure out of fifty 6 in x 28 in x 28 in boxes, stacked five high, two wide, and five long. In general, the model predicts that we should use boxes whose height is one-quarter of the harmonic mean of their length and width. We discuss the assumptions, derivation, and limitations of this model Introduction A stunt motorcyclist jumps over an elephant; we use cardboard boxes to cushion the landing. Our goal is to determine how to arrange the boxes to protect the motorcyclist. We determine how many boxes to use ● the size of the boxes the arrangement of the boxes, and any modifications to the boxes In addition, our model must accommodate motorcyclists jumping from dif ferent heights and on motorcycles of different weights. Our goal is to reduce the impulse at landing, thus essentially simulating a much lower jump(of which we assume the rider is capable The UMAP Journal 24(3)(2003)251-262. Copyright 2003 by COMAP, Inc. Allrights reserved Permission to make digital or hard copies of part or all of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial dvantage and that copies bear this notice. Abstracting with credit is permitted, but copyrights for components of this work owned by others than COMAP must be honored. To to republish, to post on servers, or to redistribute to lists requires prior permission from COMAP
Thinking Outside the Box 251 Thinking Outside the Box and Over the Elephant Melissa J. Banister Matthew Macauley Micah J. Smukler Harvey Mudd College Claremont, CA Advisor: Jon Jacobsen Abstract We present a mathematical model of the collapsing of a box structure, which is to be used to protect a stunt motorcyclist who jumps over an elephant. Reasonable values of the model’s parameters cause it to predict that we should construct the structure out of fifty 6 in × 28 in × 28 in boxes, stacked five high, two wide, and five long. In general, the model predicts that we should use boxes whose height is one-quarter of the harmonic mean of their length and width. We discuss the assumptions, derivation, and limitations of this model. Introduction A stunt motorcyclist jumps over an elephant; we use cardboard boxes to cushion the landing. Our goal is to determine how to arrange the boxes to protect the motorcyclist. We determine • how many boxes to use, • the size of the boxes. • the arrangement of the boxes, and • any modifications to the boxes. In addition, our model must accommodate motorcyclists jumping from different heights and on motorcycles of different weights. Our goal is to reduce the impulse at landing, thus essentially simulating a much lower jump (of which we assume the rider is capable). The UMAP Journal 24 (3) (2003) 251–262. c Copyright 2003 by COMAP, Inc. All rights reserved. Permission to make digital or hard copies of part or all of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice. Abstracting with credit is permitted, but copyrights for components of this work owned by others than COMAP must be honored. To copy otherwise, to republish, to post on servers, or to redistribute to lists requires prior permission from COMAP
252 The UMAP Journal 24.3(2003) Figure 1. The landing platform(graphic from Just'In Designs [2001) As the rider breaks through the top layer of boxes, crashing through card- board at a high horizontal speed, it will be difficult to maintain balance. It is too dangerous to rely on the cardboard to stop the horizontal motion of the rider, unless we use such a large pile that keeping it from being visible to the camera would be nearly impossible We are faced with how to cushion the rider's landing without creating merely a pit of boxes. Imagine jumping from a 10-ft roof. If the jumper lands on a large wooden platform resting on a deep foam pit, the risk for injury much less; the foam spreads out the jumper's deceleration over a much longer time,simulating a much lower jumping height Our goal is to create a landing platform for the motorcyclist that behaves uch like the wooden platform on the foam pit. We simulate the foam pi by stacks of boxes. Our"platform"is constructed from boxes unfolded into cardboard flats and placed in a layer on top of the"pit(Figure 1). The idea is that the motorcyclist should never break through this layer of flats but should merely break the boxes underneath it. Safety Considerations Once the motorcycle has landed on the stack of cardboard boxes, its decel- eration should to be as uniform as possible as the structure collapses-the more uniform the deceleration the easier to maintain balance We want the platform to remain as level and rigid as possible. If it is not level, the rider may lose balance; it it is insufficiently rigid, it may bend and collapse into the pile of boxes Terminology The flute type of a gauge of cardboard refers to its corrugated core structure Three sheets of linerboard compose one sheet of corrugated cardboard; tl middle one is shaped into flutes, or waves, by a machine, and then the outer two sheets are glued on either side of it. For example, C-flute corrugated cardboard is the most common form [Mall City Containers n.d. I
252 The UMAP Journal 24.3 (2003) Figure 1. The landing platform (graphic from Just’In Designs [2001]). As the rider breaks through the top layer of boxes, crashing through cardboard at a high horizontal speed, it will be difficult to maintain balance. It is too dangerous to rely on the cardboard to stop the horizontal motion of the rider, unless we use such a large pile that keeping it from being visible to the camera would be nearly impossible. We are faced with how to cushion the rider’s landing without creating merely a pit of boxes. Imagine jumping from a 10-ft roof. If the jumper lands on a large wooden platform resting on a deep foam pit, the risk for injury is much less; the foam spreads out the jumper’s deceleration over a much longer time, simulating a much lower jumping height. Our goal is to create a landing platform for the motorcyclist that behaves much like the wooden platform on the foam pit. We simulate the foam pit by stacks of boxes. Our “platform” is constructed from boxes unfolded into cardboard flats and placed in a layer on top of the “pit” (Figure 1). The idea is that the motorcyclist should never break through this layer of flats but should merely break the boxes underneath it. Safety Considerations • Once the motorcycle has landed on the stack of cardboard boxes, its deceleration should to be as uniform as possible as the structure collapses—the more uniform the deceleration, the easier to maintain balance. • We want the platform to remain as level and rigid as possible. If it is not level, the rider may lose balance; it it is insufficiently rigid, it may bend and collapse into the pile of boxes. Terminology • The flute type of a gauge of cardboard refers to its corrugated core structure. Three sheets of linerboard compose one sheet of corrugated cardboard;the middle one is shaped into flutes, or waves, by a machine, and then the outer two sheets are glued on either side of it. For example, C-flute corrugated cardboard is the most common form [Mall City Containers n.d.]
Thinking Outside the box 253 The edge crush test(ECT) value of a gauge of cardboard is the force per unit length that must be applied along the edge before the edge breaks. We make extensive use of the concept of such a value; however, the actual numbers given for gauges of cardboard apply to ideal situations that would not be replicated in the cases that we are considering [Boxland Online 19991 The flatwise compression test(FCT) value of a gauge of cardboard is the pressure that must be applied to collapse it. It does not directly correlate to the stress placed on boxes in practice and therefore is not used as an industr standard [Pflug et al. 2000 The bursting strength of a gauge of cardboard is the amount of air pressure required to break a sample. Because our model is concerned mainly with the strength of a box edge, but the bursting strength more accurately the strength of a face, we do not make use of it [Boxland Online 1999] The stacking weight of a box is the weight that can be applied uniformly to the top of a box without crushing it. In general, the stacking weight of a box is smaller than the ECT or bursting strength, because it takes into account the structural weaknesses of the particular box. We derive most of our numerical values for box strength from manufacturers specified stacking weights[ Bankers Box 2003] Assumptions The force exerted on every layer beneath the top platform is horizontally uniform The force that the motorcycle exerts on the top platform is concentrated where the wheels touch. Ideally, however, this top platform is perfectly flat and rigid so it distributes the force evenly to all lower layers. We approach this ideal by adding additional flats to the top platform The stacking weight of a box is proportional to its perimeter and inversely propor- tional to its height. This assumption is physically reasonable, because weight on a box is supported by the edges and because the material in a taller box This claim can be verified from data Clean Sweep Supply 2002 Orter box on average is farther from the box's points of stability than in a sh Nearly all of the work done to crush a box is used to initially damage its structural integrity. After the structure of a box is damaged, the remaining compression follows much more easily; indeed, we suppose it to be negligible. We denote by d the distance through which this initial work is done and assume for simplicity that the work is done uniformly throughout d. Through rough experiments performed in our workroom, we find that d a 0.03 m. W assume that this value is constant but also discuss the effect of making it function of the size of box and of the speed of the crushing object
Thinking Outside the Box 253 • The edge crush test (ECT) value of a gauge of cardboard is the force per unit length that must be applied along the edge before the edge breaks. We make extensive use of the concept of such a value; however, the actual numbers given for gauges of cardboard apply to ideal situations that would not be replicated in the cases that we are considering [Boxland Online 1999]. • The flatwise compression test (FCT) value of a gauge of cardboard is the pressure that must be applied to collapse it. It does not directly correlate to the stress placed on boxes in practice and therefore is not used as an industry standard [Pflug et al. 2000]. • The bursting strength of a gauge of cardboard is the amount of air pressure required to break a sample. Because our model is concerned mainly with the strength of a box edge, but the bursting strength more accurately measures the strength of a face, we do not make use of it [Boxland Online 1999]. • The stacking weight of a box is the weight that can be applied uniformly to the top of a box without crushing it. In general, the stacking weight of a box is smaller than the ECT or bursting strength, because it takes into account the structural weaknesses of the particular box. We derive most of our numerical values for box strength from manufacturers’ specified stacking weights [Bankers Box 2003]. Assumptions • The force exerted on every layer beneath the top platform is horizontally uniform. The force that the motorcycle exerts on the top platform is concentrated where the wheels touch. Ideally, however, this top platform is perfectly flat and rigid, so it distributes the force evenly to all lower layers. We approach this ideal by adding additional flats to the top platform. • The stacking weight of a box is proportional to its perimeter and inversely proportional to its height. This assumption is physically reasonable, because weight on a box is supported by the edges and because the material in a taller box on average is farther from the box’s points of stability than in a shorter box. This claim can be verified from data [Clean Sweep Supply 2002]. • Nearly all of the work done to crush a box is used to initially damage its structural integrity. After the structure of a box is damaged, the remaining compression follows much more easily; indeed, we suppose it to be negligible. We denote by d the distance through which this initial work is done and assume for simplicity that the work is done uniformly throughout d. Through rough experiments performed in our workroom, we find that d ≈ 0.03 m. We assume that this value is constant but also discuss the effect of making it a function of the size of box and of the speed of the crushing object
254 The UMAP Journal 24.3(2003) Figure 2. Tire before and after landing, acting as a shock absorber When the motorcycle lands, we ignore the effects of any shock absorbers and assume that the motorcyclist does not shift position to cushion the fall. This is a worst case scenario. To calculate how much force the tires experience per unit area, we consider a standard 19-inch tire of height 90 mm and width 120 mm [Kawasaki 2002]. It compresses no less than 50 mm(Figure 2). A simple geometry calculation then tells us that the surface area of the tire touching the platform is approximately 3000 mm. We assume that the force exerted on the motorcycle on landing is uniformly distributed over this area The pressure required to compress a stack of cardboard flats completely is the sun of the pressures required to compress each individual flat In a uniformly layered stack of boxes, each layer collapses completely before the layer beneath it begins to collapse. This is probably an oversimplification; however, it is reasonable to suppose that the motorcycle is falling nearly as fast as the force that it is transmitting The motorcyclist can easily land a jump 0. 25 m high The model Crushing an Individual Box For a cardboard box of height h, width w, and length l, by the assumptions ade above the stacking weight s is S(h, l, w) k(l+w) where k is a constant(with units of force) Once a box is compressed by a small amount, its spine breaks and very little additional force is required to flatten it. Thus most of the force that the box exerts on the bike is done over the distance d h, and we assume that the work is done uniformly over this distance; this work is W=(force)(distance) k(l +w)d
254 The UMAP Journal 24.3 (2003) Figure 2. Tire before and after landing, acting as a shock absorber. • When the motorcycle lands, we ignore the effects of any shock absorbers and assume that the motorcyclist does not shift position to cushion the fall. This is a worstcase scenario. To calculate how much force the tires experience per unit area, we consider a standard 19-inch tire of height 90 mm and width 120 mm [Kawasaki 2002]. It compresses no less than 50 mm (Figure 2). A simple geometry calculation then tells us that the surface area of the tire touching the platform is approximately 3000 mm2. We assume that the force exerted on the motorcycle on landing is uniformly distributed over this area. • The pressure required to compress a stack of cardboard flats completely is the sum of the pressures required to compress each individual flat. • In a uniformly layered stack of boxes, each layer collapses completely before the layer beneath it begins to collapse. This is probably an oversimplification; however, it is reasonable to suppose that the motorcycle is falling nearly as fast as the force that it is transmitting. • The motorcyclist can easily land a jump 0.25 m high. The Model Crushing an Individual Box For a cardboard box of height h, width w, and length l, by the assumptions made above, the stacking weight S is S(h, l, w) = k(l + w) h , where k is a constant (with units of force). Once a box is compressed by a small amount, its spine breaks and very little additional force is required to flatten it. Thus, most of the force that the box exerts on the bike is done over the distance d h, and we assume that the work is done uniformly over this distance; this work is W = (force)(distance) = k(l + w)d h .
Thinking Outside the box 255 Crushing a layer of boxes To ensure that a layer of boxes collapses uniformly, we build it out of n identical boxes. The total amount of work required to crush such a layer is k(l+ w)d T=n Once the structure starts to collapse, we want the rider to maintain a roughly constant average deceleration g over each layer. It follows that the layer should do total work m(g +h, so nkd(l+ w) h m(g+g)h Define A= nlw to be the cross-sectional area of a layer of boxes rearrangil (1) produces adk how B (g+9)l The constant B gives a necessary relationship among the dimensions of the box if we wish to maintain constant deceleration throughout the collision Finally, we would like to minimize the total amount of material, subj the above constraint. To do so, consider the eficiency of a layer with a given composition of boxes to be the ratio of amount of work done to amount of material used. If the motorcyclist peaks at a height ho, we must do work mgho to stop the motorcycle. We minimize the total material needed by maximizing the efficiency of each layer The amount of material in a box is roughly proportional to its surface area, S=2(hl+ lw+ wh). Thus the amount of material used by the layer is pro- portional to nS=2n(hl+ lw wh). It follows that the efficiency E of a layer composed of boxes of dimensions h×l×wis nkd(l+w) kd(l+ w) Eo nS 2nh(hl+lw+wh) 2h(hl +lw+ hw) We maximize E for each layer, subject to the constraint(2). The calculation tre easier if we minimize 1/E. Neglecting constant factors, we minimize f(h,,)=;(h+h+lu) subject to the constraint holy B where B is the constant defined in(2). However, as long as we are obeying this constraint(that each layer does the same total work), we can write hlu B
Thinking Outside the Box 255 Crushing a Layer of Boxes To ensure that a layer of boxes collapses uniformly, we build it out of n identical boxes. The total amount of work required to crush such a layer is WT = n k(l + w)d h . Once the structure starts to collapse, we want the rider to maintain a roughly constant average deceleration g over each layer. It follows that the layer should do total work m(g + g )h, so WT = nkd(l + w) h = m(g + g )h . (1) Define A = nlw to be the cross-sectional area of a layer of boxes; rearranging (1) produces B ≡ Adk m(g + g ) = h2lw l + w . (2) The constant B gives a necessary relationship among the dimensions of the box if we wish to maintain constant deceleration throughout the collision. Finally, we would like to minimize the total amount of material, subject to the above constraint. To do so, consider the efficiency of a layer with a given composition of boxes to be the ratio of amount of work done to amount of material used. If the motorcyclist peaks at a height h0, we must do work mgh0 to stop the motorcycle. We minimize the total material needed by maximizing the efficiency of each layer. The amount of material in a box is roughly proportional to its surface area, S = 2(hl + lw + wh). Thus the amount of material used by the layer is proportional to nS = 2n(hl + lw + wh). It follows that the efficiency E of a layer composed of boxes of dimensions h × l × w is E ∝ WT nS = nkd(l + w) 2nh(hl + lw + wh) = kd(l + w) 2h(hl + lw + hw) . We maximize E for each layer, subject to the constraint (2). The calculations are easier if we minimize 1/E. Neglecting constant factors, we minimize f(h, l, w) = h l + w (hl + hw + lw) subject to the constraint h2lw l + w = B, where B is the constant defined in (2). However, as long as we are obeying this constraint (that each layer does the same total work), we can write f(h, l, w) = h2 + hlw l + w = h2 + B h ,