Cardboard Comfortable When It Comes to Crashing 281 Cardboard Comfortable when It Comes to crashing Jeffrey Giansiracusa Ernie esser Simon pai University of washington Seattle, WA Advisor: James Allen Morrow Abstract A scene in an upcoming action movie requires a stunt person on a motorcycle to jump over an elephant; cardboard boxes will be used to cushion the landing We formulate a model for the energy required to crush a box based on size, shape, and material. We also summarize the most readily available boxes on the market. We choose a maximum safe deceleration rate of 5g, based on comparison with airbag rigs used professionally for high-fall stunts Toensure that the stunt person lands on the boxrig, we analyze the uncertainty trajectory and extract the landing point uncertainty We construct a numerical simulation of the impact and motion through the boxes based on our earlier energy calculations. After analyzing the sensitivity and stability of this simulation, we use it to examine the effectiveness of various configurations for the box stack(including different box sizes, types of boxes, and stacking patterns). We find that 200 kg is the most desirable combined mass of the motorcycle and stunt person, and a launch ramp angle of 20 is optimal when considering safety, camera angle, and clearance over the elephant A stack of (30 in)boxes with vertical mattress walls spaced periodically is optimal in terms of construction time, cost, and cushioning capacity. We recom- mend that this stack be 4 m high, 4 m wide, and 24 m long. It will consist of approximately 1, 100 boxes and cost $4, 300 in materials. The stunt persons wages are uncertain but fortunately the elephant works for peanuts Introduction Airbag rigs are commonly used for high-fall stunts [M&M Stunts 20031,but they are designed only to catch humans. The alternative is a cardboard-bc Our objectives are: The UMAP Journal 24(3)(2003)281-298. 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
Cardboard Comfortable When It Comes to Crashing 281 Cardboard Comfortable When It Comes to Crashing Jeffrey Giansiracusa Ernie Esser Simon Pai University of Washington Seattle, WA Advisor: James Allen Morrow Abstract A scene in an upcoming action movie requires a stunt person on a motorcycle to jump over an elephant; cardboard boxes will be used to cushion the landing. We formulate a model for the energy required to crush a box based on size, shape, and material. We also summarize the most readily available boxes on the market. We choose a maximum safe deceleration rate of 5g, based on comparison with airbag rigs used professionally for high-fall stunts. To ensure that the stunt person lands on the box rig, we analyze the uncertainty in trajectory and extract the landing point uncertainty. We construct a numerical simulation of the impact and motion through the boxes based on our earlier energy calculations. After analyzing the sensitivity and stability of this simulation, we use it to examine the effectiveness of various configurations for the box stack (including different box sizes, types of boxes, and stacking patterns). We find that 200 kg is the most desirable combined mass of the motorcycle and stunt person, and a launch ramp angle of 20◦ is optimal when considering safety, camera angle, and clearance over the elephant. A stack of (30 in)3 boxes with vertical mattress walls spaced periodically is optimal in terms of construction time, cost, and cushioning capacity. We recommend that this stack be 4 m high, 4 m wide, and 24 m long. It will consist of approximately 1,100 boxes and cost $4,300 in materials. The stunt person’s wages are uncertain but fortunately the elephant works for peanuts. Introduction Airbag rigs are commonly used for high-fall stunts [M&M Stunts 2003], but they are designed only to catch humans. The alternative is a cardboard-box rig—a stack of boxes that crush and absorb impact. Our objectives are: The UMAP Journal 24 (3) (2003) 281–298. 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
282 The UMAP Journal 24.3 (2003) to catch the stunt person and motorcycle safely, and to minimize the cost and size of the box rig Our approach is We investigate the relationship between the size/shape /material of a box and the work(crush energy) required to crush it . We review the available cardboard boxes with an airbag rig, we estimate the maximum acceptable deceleration that the stunt person can experience during landing We analyze the trajectory of the motorcycle and the uncertainty in its landing location. This determines the proper placement of the box rig and how large an area it must cover Using the crush energy formula, we estimate the number of boxes needed We formulate a numerical simulation of the motorcycle as it enters the box ig. Using this model, we analyze the effectiveness of various types of boxes and stacking arrangements for low, medium, and high jumps As an alternative to catching the stunt person while sitting on the motorcycle we analyze the possibility of having the stunt person bail out in mid-air and land separately from the motorcycle We make recommendations regarding placement, size, construction, and stacking type of the box rig Energy absorbed by Crushing Cardboard We estimate the energy required to crush a box, based on physical consid erations and experimentation. We assume that the primary source of energy absorption is the breakdown of the box walls due to edge compressive forces Commercial cardboard is rated by the edge crush test(ECT), which mea- sures edge compressive force parallel to the flute( the wavy layer between the two wall layers)that the cardboard can withstand before breaking. This can be interpreted as the force against the edge per unit length of crease created [Pflug et al. 1999; McCoy Corporation n d. Once a crease has formed, very little work is required to bend the cardboard further. To understand how the formation of wall creases relates to the crushing a box, we conducted several experiments(Figure 1 ). We found The first wall-creases typically form in the first 15% of the stroke distance These creases extend across two faces of the box a schematic of one such crease is illustrated in Figure 2
282 The UMAP Journal 24.3 (2003) • to catch the stunt person and motorcycle safely, and • to minimize the cost and size of the box rig. Our approach is: • We investigate the relationship between the size/shape/material of a box and the work (crush energy) required to crush it. • We review the available cardboard boxes. • By comparison with an airbag rig, we estimate the maximum acceptable deceleration that the stunt person can experience during landing. • We analyze the trajectory of the motorcycle and the uncertainty in its landing location. This determines the proper placement of the box rig and how large an area it must cover. • Using the crush energy formula, we estimate the number of boxes needed. • We formulate a numerical simulation of the motorcycle as it enters the box rig. Using this model, we analyze the effectiveness of various types of boxes and stacking arrangements for low, medium, and high jumps. • As an alternative to catching the stunt person while sitting on the motorcycle, we analyze the possibility of having the stunt person bail out in mid-air and land separately from the motorcycle. • We make recommendations regarding placement, size, construction, and stacking type of the box rig. Energy Absorbed by Crushing Cardboard We estimate the energy required to crush a box, based on physical considerations and experimentation. We assume that the primary source of energy absorption is the breakdown of the box walls due to edge compressive forces. Commercial cardboard is rated by the edge crush test (ECT), which measures edge compressive force parallel to the flute (the wavy layer between the two wall layers) that the cardboard can withstand before breaking. This can be interpreted as the force against the edge per unit length of crease created [Pflug et al. 1999; McCoy Corporation n.d.]. Once a crease has formed, very little work is required to bend the cardboard further. To understand how the formation of wall creases relates to the process of crushing a box, we conducted several experiments (Figure 1). We found: • The first wall-creases typically form in the first 15% of the stroke distance. • These creases extend across two faces of the box; a schematic of one such crease is illustrated in Figure 2
Cardboard Comfortable When It Comes to Crashing 283 Figure 1. Experimental apparatus for crushing boxes: We dropped a crush-test dummy (i.e,team member)onto several boxes and observed how the structure( the box, not the dummy )broke down Figure la. Crush-test dummy in action Figure 1b. Crushed box with creases. (Left: Jeff Giansiracusa; right: Simon Pai) (Photos courtesy of Richard Neal. Force Crease forms here Figure 2. The first crease forms in a curve across the side faces as the box is compressed
Cardboard Comfortable When It Comes to Crashing 283 Figure 1. Experimental apparatus for crushing boxes: We dropped a crush-test dummy (i.e., team member) onto several boxes and observed how the structure (the box, not the dummy) broke down. Figure 1a. Crush-test dummy in action. (Left: Jeff Giansiracusa; right: Simon Pai.) Figure 1b. Crushed box with creases. (Photos courtesy of Richard Neal.) Force Crease forms here Figure 2. The first crease forms in a curve across the side faces as the box is compressed
284 The UMAP Journal 24.3 (2003) Once these have formed, the box deforms further with comparatively little resistance, because additional creases are created by torque forces rather than edge compressive forces The primary creases each have length approximately equal to the diagonal length of the face The work done in crushing the box is given by the average force applied times he distance through which it is applied. This and our experimental qualitative results lead us to write the following equation for energy absorbed by a box of dimension lx×ly× l2 crushed in the z-directic E=ECT×2√2+12×l2×0.15 As a reality check, we compute the crush energy for a standard 8.5 in x 17 in x 11 in box with ECT=20 lbs/ in and a C-flute(the type commonly used to store paper). With these numerical values, (1) gives an energy of 187 J. This corresponds roughly to a 140-lb person sitting on the box and nearly flattening it. Crush-test dummy results confirm this estimate Energy can also be absorbed in the process of flattening the flute within the cardboard walls. However, the pressure required to do this is approximately 150 kPa[Pflug et al. 1999] and the surface area involved is more than 1 m2,so a quick calculation shows that the stunt person would decelerate too quickly if the kinetic energy were transferred into flattening boxes. We therefore ignore this additional flattening effect O, any successful box rig configuration must dissipate all of the kinetic energy of the stunt person and motorcycle through box-crushing alone Common Box Types Minimizing cost is important. The cardboard box rig will consist of perhaps unit;so we restrict our attention to commonly available box types Table 1/W hundreds of boxes, and wholesale box prices can range up to $10 or $20 Table 1 Commonly available box types [Paper Mart n d. VeriPack com n d Type Size (in) ECT rating (lbs /in) Price A10×10×10 s0.40 D30×30×30 22822 s5.00 E44×12×12 s175 F80×60×7 $1000
284 The UMAP Journal 24.3 (2003) • Once these have formed, the box deforms further with comparatively little resistance, because additional creases are created by torque forces rather than edge compressive forces. • The primary creases each have length approximately equal to the diagonal length of the face. The work done in crushing the box is given by the average force applied times the distance through which it is applied. This and our experimental qualitative results lead us to write the following equation for energy absorbed by a box of dimension lx × ly × lz crushed in the z-direction: E = ECT × 2 l2 x + l2 y × lz × 0.15 (1) As a reality check, we compute the crush energy for a standard 8.5 in × 17 in × 11 in box with ECT = 20 lbs/in and a C-flute (the type commonly used to store paper). With these numerical values, (1) gives an energy of 187 J. This corresponds roughly to a 140-lb person sitting on the box and nearly flattening it. Crush-test dummy results confirm this estimate. Energy can also be absorbed in the process of flattening the flute within the cardboard walls. However, the pressure required to do this is approximately 150 kPa [Pflug et al. 1999] and the surface area involved is more than 1 m2, so a quick calculation shows that the stunt person would decelerate too quickly if the kinetic energy were transferred into flattening boxes. We therefore ignore this additional flattening effect. So, any successful box rig configuration must dissipate all of the kinetic energy of the stunt person and motorcycle through box-crushing alone. Common Box Types Minimizing cost is important. The cardboard box rig will consist of perhaps hundreds of boxes, and wholesale box prices can range up to $10 or $20 per unit; so we restrict our attention to commonly available box types (Table 1). Table 1. Commonly available box types [Paper Mart n.d.; VeriPack.com n.d.] Type Size (in) ECT rating (lbs/in) Price A 10 × 10 × 10 32 $0.40 B 20 × 20 × 20 32 $1.50 C 20 × 20 × 20 48 $3.50 D 30 × 30 × 30 32 $5.00 E 44 × 12 × 12 32 $1.75 F 80 × 60 × 7 32 $10.00
Cardboard Comfortable When It Comes to Crashing 285 Some quick estimates Maximum Safe acceleration To determine acceptable forces and accelerations for the stunt person, we compare the box rig with other cushioning devices. In the stunt rigging busi ness, it is common practice to use an air bag for high falls of up to 30 m; such airbags are approximately 4 m deep Assume that a stunt person falls from 30 m above the airbag. Gravity accelerates the performer from rest to speed u when the performer strikes the airbag and is decelerated completely, so we have V2gdall-v2abaghbag, where dfall is the fall distance, abag is the deceleration rate the stunt person experiences in the airbag, hbag is the height of the airbag, and g is the acceleration due to gravity. Thus, dfall 30 m 7.5 4 We therefore conclude When using an airbag the stunt person experiences an average acceleration of at most 7 5g. This provides an upper bound on the maximum acceleration that a person can safely withstand With the airbag the stunt person is able to land in a position that distributes forces evenly across the body. In our stunt, however, the stunt person lands in the box rig while still on the motorcycle, with greater chance for injury under high deceleration We choose 5g as our maximum safe deceleration Displacement and Energy Estimates If the deceleration is constant through the boxes, then we can estimate the distance required to bring them to rest. Since any deviation from constant acceleration increases either the stopping distance or the peak deceleration, this will give us a lower bound on the stopping distance and hence on the required dimensions of the box rig Suppose that the stunt person enters the rig at time t=0 with speed vo and experiences a constant deceleration a until brought to rest at time t=tf. The persons speed is u(t)=vo-at Since the stunt person is at rest at time tf,we have
Cardboard Comfortable When It Comes to Crashing 285 Some Quick Estimates Maximum Safe Acceleration To determine acceptable forces and accelerations for the stunt person, we compare the box rig with other cushioning devices. In the stunt rigging business, it is common practice to use an air bag for high falls of up to 30 m; such airbags are approximately 4 m deep. Assume that a stunt person falls from 30 m above the airbag. Gravity accelerates the performer from rest to speed v when the performer strikes the airbag and is decelerated completely, so we have 2gdfall = 2abaghbag, where dfall is the fall distance, abag is the deceleration rate the stunt person experiences in the airbag, hbag is the height of the airbag, andg is the acceleration due to gravity. Thus, abag = dfall hbag g = 30 m 4 m g = 7.5g. We therefore conclude: • When using an airbag, the stunt person experiences an average acceleration of at most 7.5g. This provides an upper bound on the maximum acceleration that a person can safely withstand. • With the airbag, the stunt person is able to land in a position that distributes forces evenly across the body. In our stunt, however, the stunt person lands in the box rig while still on the motorcycle, with greater chance for injury under high deceleration. • We choose 5g as our maximum safe deceleration. Displacement and Energy Estimates If the deceleration is constant through the boxes, then we can estimate the distance required to bring them to rest. Since any deviation from constant acceleration increases either the stopping distance or the peak deceleration, this will give us a lower bound on the stopping distance and hence on the required dimensions of the box rig. Suppose that the stunt person enters the rig at time t = 0 with speed v0 and experiences a constant deceleration a until brought to rest at time t = tf . The person’s speed is v(t) = v0 − at. Since the stunt person is at rest at time tf , we have tf = v0/a