Supervisor: Mr. Duncan
Word count: 3756
Subject: Physics

Willem Frederik Neeteson International School of Nice
Abstract
The aim of this research paper is to find the optimum structural relationship between length of folds, width and angles of a paper corrugated bridge. The bridges are constructed using A4 sheets of paper that all have the same density and surface area. The method used allows an investigation of force against angle and length, to then be able to study which relationship between length of one fold and angle can withstand the highest force. The main goal is to provide small limitations in the final processing of the data to give a confident and accurate result to where the best suited correlation lies. An apparatus was setup accordingly for the investigation. Nine bridges, each having a different fold length, were constructed and tested between two supporting piers. The angles were varied, on average 6 different reading per bridge, and the effect of having unbalanced forces acting upon the bridge were examined. Graphs of force/angle were plotted and the average angle holding the highest force was found. The accuracy of each relationship was found and compared to the correlation with fold length. The method was accompanied by difficulties that affected the data collected. It wasn’t possible to construct bridges with very short fold lengths, leading to inaccuracies to whether the force/length graph really gave a final peak. In addition, the errors first estimated for the angles did not allow very accurate limits to where the best correlation lies. These problems were taken into account and treated accordingly, leading to more accurate boundaries. The investigation does find an optimum relationship within justifiable limits relative to the paper used and the approach in which the bridges were constructed.
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Willem Frederik Neeteson International School of Nice
Table of Contents
Introduction ....................................................................................................................... 4
Investigation and Aim ....................................................................................................... 6
Hypothesis .......................................................................................................................... 6
The Beam Bridge ............................................................................................................... 8
Truss Design and Strength ............................................................................................... 9
Truss design on bridges ................................................................................................. 10
Method ............................................................................................................................. 11
Limitations in Method ................................................................................................... 13
Processing Data ............................................................................................................... 14
Data Analysis ................................................................................................................... 24
Analysis of limitations .................................................................................................... 25
Conclusion ....................................................................................................................... 29
Bibliography .................................................................................................................... 31
Appendix .......................................................................................................................... 32
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Willem Frederik Neeteson International School of Nice
Introduction 12
Throughout history, bridges have had a major influence on the world.
Society has always relied on transportation to survive. Along with the evolution of transportation, one invention facilitated the development of globalization by
Fig. 1 expanding the reach of travelers, the bridge. They were developed to provide a safe crossing for people, originally on foot but later for mechanized transportation, to travel from one prominent city of industrial strength to another.
A bridge is a structure built to surmount an obstacle or natural barriers such as rivers, lakes or valleys. They range from simple logs thrown across a creek to monuments like the Millau Viaduct in France. Bridges are designed to provide a carriageway for pedestrians, motor vehicles or railroad traffic. They may also be used to transport
Fig. 2 materials such as an oil pipeline or a water aqueduct. The earliest bridges were constructed with no more than felled trees placed to cross rivers or ditches. As communities started to evolve, bridges were required to withstand longer distances. The Roman Arch was the first architectural device to withstand the necessities of their society.
As technology advanced, the principles underlying bridge construction were developed incorporating other materials such as iron, steel and aluminum. A modern bridge must be able to withstand the forces that will act on each component of the completed bridge such as the live load, which are forces that occur as a result of the
Fig. 1 The Millau Viaduct bridge over the River Tarn in the Massif Central
Fig. 2 Roman bridge in Cangas de Onís
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Willem Frederik Neeteson International School of Nice traffic moving across the bridge. However, the dynamic load, which is the force exerted on the bridge as a result of environmental factors such as earthquakes or strong winds, insists that the bridge is very flexible as well as robust.
When constructing a bridge, it is the duty of an engineer to discover and calculate the optimum relationship between the many variables. The main features that control the bridge type are the size of the obstacle, its natural environment and occurrences.
Constructing bridges emphasizes the concepts involved in carrying weight at a distance from the supports 3 . Designs of bridges will vary depending on the function of the bridge and the nature of the terrain where the bridge is to be constructed. The structural concept behind the design of bridges can be classified into building techniques, examples of these are:
Simple Beam Bridge
Pier Span Pier
Arch Bridge
Truss Bridge
Pier
Pier
Span
Span
Pier
Pier
3 http://www.exploratorium.edu/structures/paperbridges.html
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Willem Frederik Neeteson International School of Nice
Until the late 17th century bridges continued to be designed and built largely by architects with a vague idea for engineering. However, such complex and essential work could not rest in the hands of gifted amateurs forever. In 1716 French army engineers took the lead on the rest of the world in bridge building. Today, bridges are truly ubiquitous, a natural part of everyday life. They allow easy travel across major rivers and estuaries, over the new obstacles of motorways and railway lines, and between neighbouring islands. In modern society, international trade and efficient mobility networks depend very much on functioning bridges.
Investigation and Aim
The main objective of this experiment is to find the most suited relationship between variables such as the length of one fold and the angle, to then able to investigate which correlation can hold the largest amount of force possible.
Bridges illustrate the effect of weight or another force at a distance from a pivot or support point (torque), and they also provide experience with beams. Building with naturally weak materials like paper quickly leads to a close consideration of the structural elements and properties of materials.
When one puts weight on a sheet of paper it tends to buckle because it is very thin. It has no strength along the thin direction. By folding or rolling the paper, a "thickness" is created which allows the paper to reinforce itself and not collapse so easily. The bridges will be corrugated using different lengths in folds where the surface area and density of paper remains the same throughout the whole experiment.
Hypothesis
A standard sheet of paper with no modifications in its structure is not able to withstand much force when rested between two supporting piers. In fact, when the supporting piers are at a certain distance from each other, the paper would collapse only
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Willem Frederik Neeteson International School of Nice due to force of gravity. When paper is corrugated, the structure is more rigid and stiff, dissipating the forces along the triangles making the paper much stronger. However, the approach in which the paper is corrugated determines how much force the paper can hold.
The bridges constructed for this experiment resemble closely to a beam – truss bridge, where the principles of truss design and strength are very evident. The definition of a truss is any of various structural frames based on the geometric rigidity of the triangle and composed of straight members subject only to longitudinal compression, tension, or both 4 . The investigation will determine the best relationship between the length of the member, the angle and the width of the bridge.
When examining the fundamentals of truss design, it is known that the optimum relationship between length and angle of the triangle can withstand the largest force.
However, it can not be deduced that bridges with a large amount of folds and very small angles will hold the highest force. The weights will have the more contact with a larger number triangles, however the bridge will not have a wide surface resting on the supporting piers and therefore the force cannot dissipate on a large area. In addition, if the triangle tops are not perfectly aligned, the forces will spread unevenly. Some angles are expected to be pushed towards the outside, producing a weakness along the edge’s length which can cause the bridge to collapse in the middle.
Bridges that have shorter lengths in their folds and larger angles have less contact with the weights and therefore a larger force exerts on a smaller amount of triangles.
With a physically weak material, one single truss cannot endure a large force so it also predictable that a bridge with few triangles cannot withstand a large force. The investigation will study the right balance between the amount of triangles, their length and angle. The bridge needs to able to spread the forces over a large area along the bridge onto the supporting piers.
4 Douglas C. Giancoli, Physics for Scientists and Engineers with Modern Physics, Prentice Hall, 2008 p. 325
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Willem Frederik Neeteson International School of Nice
The Beam Bridge
The bridge with the most suited structural design for this experiment is the beam bridge. The beam bridge consists of a horizontal beam supported at each end by piers or supports. The weight of the beam pushes straight down on the piers. The farther apart its piers, the weaker the beam becomes. A beam bridge is a direct descendant of the log bridge, and in structural terms the simplest of the many bridge types. The main forces acting on the bridge are:
Compression
The force of compression manifests itself on the top side of the beam bridge's deck (or roadway). This causes the upper portion of the deck to shorten.
Tension
The result of the compression on the upper portion of the deck causes tension in the lower portion of the deck. This tension causes the lower portion of the beam to lengthen.
Compression
Pier supports in compression
Pier
Tension
Pier
Pier supports in compression
Any single beam spanning any distance experiences compression and tension. The very top of the beam experiences the most compression, and the very bottom of the beam experiences the most tension. The middle of the beam experiences very little compression or tension. When a standard A4 sheet of paper with no modifications is placed between the support piers (21. 7 +/- 0.1 cm apart), it collapses only due to the force caused by the
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Willem Frederik Neeteson International School of Nice acceleration due to gravity. Paper which is corrugated can hold a much larger force; this is because the truss design distributes the forces of the bridge to the piers, creating a lower force per unit area.
Truss Design and Strength
Load
Compression
Tension
Compression
A truss has the ability to dissipate a load through the truss work. The design of a truss, which is usually a triangle or a variant of a triangle, creates both a very rigid structure and one that transfers the load from a single point to a considerably wider area .
When loads are applied to a truss only at the joints, forces are transmitted only in the direction of each of its sides. That is, the lengths experience tension or compression forces, but not bending forces. If the top and the bottom of the truss represent the beam, while the center of the beam is made up of diagonal sides, one can see that the top and bottom contain more material and for this reason corrugated cardboard is very stiff.
Trusses have a high strength to weight ratio and consequently are used in many structures, from bridges, to roof supports, to space stations.
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Willem Frederik Neeteson International School of Nice
Truss design on bridges
F
1
A
B
θ
C
F
3
Length
D
θ F
2
E
The picture represents a free-body diagram of two trusses on a bridge, where each end is supported and exerts upward forces F
1
and F
2
. The diagram assumes that the mass of the bridge acts entirely at the center, on pin C. From symmetry one can see that each of the end supports carries half the weight. At pin A, the torque equation gives:
( F
2
)( l ) – F
3
( l / 2 ) = 0, so F
1
= F
2
= ½ F
3
where l = length
Now consider a heavy load such as a truck being supported by strut BD and the middle.
The strut BD sags under the load, displaying that the strut is under shear stress. The forces that pins B and D exert on the strut BD will have vertical components too and will not only act along the strut BD. The forces are also exerted perpendicular to the strut to balance the weight of the load. The other struts that are not bearing weight remain under pure tension or compression.
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Willem Frederik Neeteson International School of Nice
Method
During this investigation, all experiments are carried out with standard A4 sheets of printing paper. Each sheet of paper has the measurement of 21.0 +/- 0.1 by 29.7 +/- 0.1 cm, obtaining an area 623.7 +/- 0.2 cm². The paper used for the experiment has a density of 100 +/- 1 gm
-2
and the value for the thickness of the paper was obtained by measuring the width of 250 sheets, given to be 3 +/- 0.1 cm, which calculates to be 0.012 +/- 0.004 cm for a single sheet of paper.
There are many variables that have to be taken into account for a corrugated paper bridge. Paper is weak under compression and is somewhat stronger under tension (i.e., it collapses when the ends are pushed together but it doesn't pull apart easily). The sheets of paper will be corrugated to increase the strength of the paper. To corrugate means to draw or bend into folds or alternate furrows and ridges. The corrugations increase the bending strength of the sheet in the direction parallel to the corrugations, but not across them.
The paper was corrugated using a small pin to crease the edges, allowing the paper to fold it itself in a more accurate manner. A ruler was placed on the line so that fold follows equally along the line. At every edge, the procedure was repeated in alternating directions (Fig. 3).
The corrugated paper bridge was placed on the support piers in a manner where it can counteract the tension on the top and compression on the bottom the best possible way so it is in equilibrium when no unbalanced forces act upon the bridge. The edges of the two beams support piers are 21.7 +/- 0.2 cm apart. This means that 4.0 +/- 0.2 centimeters of either side of the paper are supported on the beam (Fig 4). When the bridge is acted upon by a force, it is important that the individual triangles distribute the forces equally along the two sides. To reduce the effect of the angles from expanding, the ends of the bridge are in contact with a putty-like pressure-sensitive adhesive. The weights are distributed across the bridge in a fashion that allows an even spread of forces along the triangles (Fig 5). In addition, the weights are placed along that part of the bridge that has no contact with the support beams. For each procedure, the bridge is tested under the same conditions.
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Willem Frederik Neeteson International School of Nice
Fig. 3
Fig. 3 Construction of the corrugated paper bridges
Fig. 4
Fig.4 The setup of the bridge on the supporting beams
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Willem Frederik Neeteson International School of Nice
Fig. 5
Limitations in Method
Fig.5 The bridge being acted upon by forces
There were certain limitations in the setup that affected the outcome of the results during the experiment. It was difficult to test all the bridges for the same values of width, as some bridges were not able to spread more than 14.0 +/- 0.2 cm. This was because the bridges with large number of folds or very few folds (i.e. bridges 1 and 9) created angles with large errors predicting that the forces equally could not spread equally. Another issue was that the angles were likely to spread out when acted upon by an unbalanced force. This difficulty was taken into account and partly resolved with the use of the puttylike pressure-sensitive adhesive on which the bridge was placed on. When constructing the bridges, the major setback was to have every length completely equal. This was especially evident when building the bridges with large number of folds. An improvement may be to use pre-manufactured corrugated paper; however this would have put limitations on the number of bridges available.
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Willem Frederik Neeteson International School of Nice
Processing Data
The data collected is processed to find the width of single triangle to then be able to obtain the angle. The angle is found by:
B c
Width of 1 triangle / cm
CosA = b
=
2
A a
+ c 2 – a 2 d
Width / Number of triangles
C
2bc
Cos -1
Average Force / N
(F
(Ans) = Angle / °
1
=
+ F
3
2
+F
3
)
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Willem Frederik Neeteson International School of Nice
Bridge 1
Width of 1 triangle / cm
Angle / °
0.5
0.6
0.8
1
1.2
1.4
Error: +/- 0.2
29.0
34.9
47.2
60.0
73.8
88.9
Error: +/- 0.2
Av. Force / N
4.0
4.3
4.2
3.5
2.9
2.5
Error: +/- 0.1
5.0
4.5
4.0
3.5
3.0
y = 0.0004x
2
- 0.0621x + 3.0221
R
2
= 0.8999
2.5
2.0
0 10 20 30 40 50 60 70 80 90 100
Angle / °
All graphs’ trendlines and values for R
2
were found using Microsoft Excel. Horizontal error bars are present, however are too small to be seen.
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Willem Frederik Neeteson International School of Nice
Bridge 2
Width of 1 triangle / cm
Angle / °
0.6
0.7
0.9
1.1
1.3
1.6
Error: +/- 0.2
27.0
32.4
43.5
55.1
67.3
81.1
Error: +/- 0.2
Av. Force / N
5.9
5.9
5.8
5.7
4.0
3.2
Error: +/- 0.1
6.5
6.0
5.5
5.0
4.5
4.0
3.5
3.0
2.5
2.0
y = 0.0004x
R
2
2
- 0.0621x + 3.0221
= 0.8999
0 10 20 30 40 50 60 70 80 90
Angle / °
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3.5
3.0
2.5
2.0
5.0
4.5
4.0
7.0
6.5
6.0
5.5
0
Willem Frederik Neeteson International School of Nice
Bridge 3
Width of 1 triangle / cm
Angle / °
0.6
0.8
1
1.3
1.5
1.8
Error: +/- 0.2
28.0
33.5
45.2
57.5
70.5
84.6
Error: +/- 0.2
Av. Force / N
6.0
6.1
6.3
5.7
4.2
3.6
Error: +/- 0.1
Bridge 3 - Force / Angle y = 0.0004x
2
- 0.0621x + 3.0221
R
2
= 0.8999
10 20 30 40
Angle / °
50 60 70 80 90
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Willem Frederik Neeteson International School of Nice
Bridge 4
Width of 1 triangle / cm
Angle / °
0.9
1.1
1.4
1.7
2.0
Error: +/- 0.2
35.8
48.1
61.4
75.3
91.2
Error: +/- 0.2
Av. Force / N
5.1
5.0
4.0
3.7
3.2
Error: +/- 0.1
5.5
5.0
4.5
y = 9E-05x
2
- 0.0488x + 6.8427
R
2
= 0.95
4.0
3.5
3.0
0 10 20 30 40 50 60
Angle / °
70 80 90 100
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Willem Frederik Neeteson International School of Nice
Bridge 5
Width of 1 triangle / cm
Angle / °
1.0
1.3
1.7
2.0
2.3
Error: +/- 0.2
34.2
46.1
58.9
72.1
86.5
Error: +/- 0.2
Av. Force / N
4.7
5
4
3.8
2.3
Error: +/- 0.1
6.0
5.5
5.0
4.5
4.0
3.5
y = -0.0011x
R
2
2
+ 0.0819x + 3.2193
= 0.9444
3.0
2.5
2.0
0 10 20 30 40 50 60 70 80 90 100
Angle / °
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Willem Frederik Neeteson International School of Nice
Bridge 6
Width of 1 triangle / cm
Angle / °
1.2
1.6
2.0
2.4
2.8
3.2
Error: +/- 0.2
33.2
44.8
56.9
69.7
83.7
99.3
Error: +/- 0.2
Av. Force / N
5.4
5.5
4.2
3.8
2.0
1.0
Error: +/- 0.1
6.0
5.5
5.0
4.5
4.0
3.5
3.0
2.5
2.0
1.5
1.0
0.5
0.0
0 y = -0.0005x
2
- 0.0022x + 6.2202
R
2
= 0.9744
10 20 30 40 50 60
Angle / °
70 80 90 100 110
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Willem Frederik Neeteson International School of Nice
Bridge 7
Width of 1 triangle / cm
Angle / °
1.5
2.0
2.5
3.0
3.5
4.0
Error: +/- 0.2
33.5
45.2
57.5
70.5
84.6
100.6
Error: +/- 0.2
Av. Force / N
4.0
4.1
3.8
3.1
1.5
0.8
Error: +/- 0.1
4.5
4.0
3.5
3.0
2.5
2.0
1.5
1.0
0.5
0.0
y = -0.0007x
R
2
2
+ 0.042x + 3.5426
= 0.9596
0 10 20 30 40 50 60 70 80 90 100 110
Angle / °
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Willem Frederik Neeteson International School of Nice
Bridge 8
Width of 1 triangle / cm
Angle / °
2.0
2.7
3.3
4
4.7
5.3
Error: +/- 0.2
33.2
44.8
56.8
69.7
83.7
99.2
Error: +/- 0.2
Av. Force / N
2.1
2.0
2.1
1.7
1.0
1.0
Error: +/- 0.1
2.5
2.0
1.5
1.0
y = -0.0002x
2
+ 0.0061x + 2.1623
R
2
= 0.8739
0.5
0.0
0 10 20 30 40 50 60 70 80 90 100 110
Angle / °
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Willem Frederik Neeteson International School of Nice
Bridge 9
Width of 1 triangle / cm
Angle / °
3.0
4.0
5.0
6.0
Error: +/- 0.2
38.9
52.8
67.5
83.6
Error: +/- 0.2
Av. Force / N
1.2
0.6
0.6
0.3
Error: +/- 0.1
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0.0
y = 0.0004x
R
2
2
- 0.0621x + 3.0221
= 0.8999
0 10 20 30 40 50 60 70 80 90
Angle / °
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Willem Frederik Neeteson International School of Nice
Data Analysis
Examining the results obtained during the experiment provides a good knowledge about which relationship between angle, length and width presents the structure that can resist the highest force. Bridge 1, with the shortest length of 1.0 +/- 0.1 cm and greatest amount of folds 19, in theory should have provided the most rigid structure. However, the bridge did not have a sufficient area in contact with the supporting piers, the forces could not dissipate fully and therefore the bridge did not withstand a force higher than 4.3 +/-
0.1 N. In addition, due to the difficulty in constructing bridges with short lengths, the angles would spread out creating an unequal division of angles and therefore a disproportionate distribution of forces. Bridge 2, with fold length 1.2 +/- 0.1 cm, already portrayed a better balance of angle and length holding more weight than the previous, at most 5.9 +/- 0.1 N. It shows that already a difference in length of 0.2 cm has a significant affect on the rigidness of the structure.
Bridges 4, 5 and 6 all displayed similar outcomes for the amount of force they could resist. The peak force did not overcome 5.4 +/- 0.1 N and as the widths increased, the bridges showed a similar regression in the amount of force they could hold. Bridge 6 obtained unusual results because it doesn’t seem to follow the pattern of the bridges weakening as the length of folds increases. Bridge 4, with a length of 1.4 +/- 0.1 cm for every fold, withstood a force of maximum 5.1 +/- 0.1 N where as bridge 6 held an utmost force of 5.4 +/- 0.1 N. The difference in length of one fold between the two bridges is 0.7 cm, which should portray a significant weakening in the structure. An explanation may be that bridge 6 has longer lengths, therefore a smaller amount of triangles. Bridges with fewer angles make it easier to align the tops evenly as the folding of longer sides is more accurate. The setup must have had a very equal distribution of angles, allowing the forces to dissipate adequately. The obtained results for bridge 6 lead to an important consideration of the accuracy in constructing bridges. It may lead a conclusion that between the setup of bridges 3 and 6 lies the optimum arrangement for paper of density
100 +/- 1 gm
-2
.
Bridge 3 displays the structure that held the highest force of the experiment. With a peak force of 6.3 +/- 0.1 N, it seems to have the best relationship between angle and
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Willem Frederik Neeteson International School of Nice force during this investigation. The bridge was more unproblematic in its setup; the angles were easier to spread evenly, the bridge base had uniform contact with the supporting piers and the weights rested on the bridge completely horizontally.
The best relationship of angle and force on a truss-beam bridge in comparison with its fold length is indicated by the trendlines on the individual graphs. During the investigation, different trend/regression trendlines were examined to display which gives a more accurate value for R 2 . The closer the value is to 1, the more accurate the trendline gives a relationship between angle and force. Primarily, a linear relationship was tested on the obtained data. The values for R
2
were between 0.8565 ≤ R
2
≤ 0.95.
When the graphs were given a polynomial trendlines of order 2, most of the values especially for the bridges with longer folds, gave more accurate values for R
2
. Under the polynomial trendline, the values moved up to 0.8739 ≤ R
2
≤ 0.9744. Bridges 4, 5, 6 and 7 give the most confident values for R
2
. The value peaks at bridge 6, with an accuracy of
0.9744 where as bridge 3 with the peak force has a value of 0.8999. From bridges 1 to 6,
R
2 seems to slowly increase until reaching a peak at 0.9744 where it then starts to decline.
This observation again gives the idea that the optimum relationship may lay within the setups of bridges 3 and 6.
Analysis of limitations
The investigation and setup obtain rather large gaps in the data due to large errors that have not been taken into account. When studying the force/angle graphs, it is clearly seen that the trendline does not seem to cross the majority of the points and their error bars. Calculating the amount of points that are crossed by the trendline over the total amount of points (23 / 50); one obtains a standard deviation of 46% within the mean.
This is significantly smaller than an approximately normal distribution of 68% within 1 standard deviation of the mean. This suggests that the error given to the angles is too small. With further analysis and a better consideration of the limitations in the method, the width of one triangle had about an error of +/- 0.2 cm. This has a significant impact of the error of the angles, calculated to be about 10% of the given value (calculation example shown below using the data obtained for bridge 6).
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Willem Frederik Neeteson International School of Nice
3.5
3.0
2.5
2.0
1.5
1.0
0.5
6.0
5.5
5.0
4.5
4.0
0.0
0
With these obtained error values for the angles, far more points are crossed by the trendlines of the force/angle graphs (example for bridge 6).
A = Cos -1 (b 2 + c 2 – a 2) / (2bc)
A = Cos
-1
(2.1
2
+ 2.1
2
– 2.0
2
) / (2*2.1*2.1) = 56.9°
Using an error of +/- 0.2 cm for the width
A = Cos
-1
(2.1
2
+ 2.1
2
– 1.8
2
) / (2*2.1*2.1) = 50.6°
A = Cos
-1
(2.1
2
+ 2.1
2
– 2.2
2
) / (2*2.1*2.1) = 63.2°
Percentage error
56.9 / 63.2 = 0.90
56.9 / 50.6 = 1.12
Average error of 10 %
y = -0.0005x
2
- 0.0022x + 6.2202
R
2
= 0.9744
10 20 30 40 50 60
Angle / °
70 80 90 100 110
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Willem Frederik Neeteson International School of Nice
Bridge 6 - Force / Angle
3.0
2.5
2.0
1.5
1.0
0.5
0.0
6.0
5.5
5.0
4.5
4.0
3.5
y = -0.0005x
2
- 0.0022x + 6.2202
R
2
= 0.9744
0 10 20 30 40 50 60 70 80 90 100 110 120
Angle / °
The force/angle graphs were modified so that error bars represented 10% of the given value for the angle. If these changes applied, the data gives a standard deviation of
72%, which is very close to the normal distribution of 68%. The results now give a better idea of which angle provides the most rigid structure in comparison with the length of one side. For further analysis, the relationship between length and force was examined.
Bridge
8
9
6
7
4
5
2
3
1
Angle of highest force / ° Force / N
34.9
29.7
45.2
35.8
34.2
39
45.2
45
38.9
Error: +/- 0.2
5.3
5.4
4.1
2.1
4.3
5.9
6.3
5.1
1.2
Error: +/- 0.1
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Willem Frederik Neeteson International School of Nice
The table above displays the angles that held the highest force for every different bridge. The new angle error (+/- 10%) must be taken into account meaning that all the values for angle must be rounded to the nearest whole number. When taking all the angles of the bridges that held the highest force and finding their average, one obtains an angle of 39° +/- 10%, or 35° ≤ A ≤ 43° (If one bridge had two values for greatest force, an average was taken for that bridge).
34.9 + 29.7 + 45.2 + 35.8 + 34.2 + 39 + 45.2 + 45 + 38.9
9
= 39° +/- 10%
This angle holds the highest force for any bridge with different lengths. The following graph displays the correlation between length in cm and force in Newton at the angle that held the highest force.
Length / Force
5.0
4.0
3.0
2.0
7.0
6.0
1.0
0.0
0.0
B.1
B.2
B.3
B.4
B.5
B.6
y = -0.3066x
2
+ 0.3454x + 5.4642
R
2
= 0.8431
B.7
B.8
0.5
1.0
1.5
2.0
2.5
Length / cm
3.0
The labels indicate the bridge number
3.5
4.0
B.9
4.5
5.0
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Willem Frederik Neeteson International School of Nice
The graph displays a peak at 1.3 +/- 0.1 for a force of 6.3 +/- 0.1 N. In relation to the data obtained, the graphs indentify that the optimum relationship for the bridges tested during this investigation is a length of 1.3 +/- 0.1 cm against an angle of 38° +/-
10%.
Conclusion
The aim of this investigation was to find the optimum relationship between length and angle for a corrugated paper bridge. The experiment was carried using a variety of bridges that all had different fold lengths. The bridges were tested under the same conditions, placed between two supporting piers with a distance of 21.7 +/- 0.2 cm apart.
The angles were varied to examine which angle can withstand the highest force. An average for the best angle was found and compared to the best fold length. With the data available, limitations were discovered in which the optimum relationship lies for paper of density 100 +/- 1 gm
-2
.
Although the experiment did not give a definite idea of which relationship works, the limitations discovered are rather confident and do satisfy the theoretical concept behind truss design and strength. As mentioned in the hypothesis, the bridge with most angles could not withstand the highest due to the lack of contact with the supporting piers. The peak force was at 6.3 +/- 0.1 N, where the angle was 45° +/- 10% and fold length 1.3 +/- 0.1 cm. As expected, bridges with very large fold lengths withstood the least force of the whole investigation, at most about 1.2 +/- 0.1 N. The cluster of bridges in the force/length graph indicates the setups with a larger holding force giving a visual idea of the limitations. The peak of curve was calculated to be 0.56 cm when using differentiation: y = -0.6132x
2
+ 0.3454x = 0, because gradient at peak is 0 dy/dx = -0.6132x + 0.3454 = 0 x = -0.3454 / -0.6132 = 0.56
Page 29 of 40

Willem Frederik Neeteson International School of Nice
This suggests that error for the fold length is greater than 0.1 cm. However, it is difficult to determine whether the graph is at a peak because there is not sufficient data to the left of the graph (folds lengths smaller than 1.0 +/- 0.1 cm). The conclusion of the data seems to rely on half of a quadratic parabola. This is due to the difficulties that are encountered when building bridges with very small fold lengths. An interesting continuation of this experiment would be to construct bridges with the help of paper corrugating machines, allowing the investigation of very small fold lengths and therefore a large number of triangles. This could a give a more accurate peak where the limitations of the optimum relationship can be reduced.
In conclusion, the best suited correlation between length and angle for a corrugated paper can be found when taking into account the manner in which the bridges were built and the paper used. Further investigation can be done applying new technologies that allow more accurate and symmetrical constructions of trusses as well as more realistic simulations of forces acting upon a paper corrugated bridge.
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Willem Frederik Neeteson International School of Nice
Bibliography
Figure 1
<http://www.wayfaring.info/wp-content/uploads/2007/06/bridge-2.jpg>
Figure 2
<http://upload.wikimedia.org/wikipedia/commons/4/46/Roman_bridge_near_Cov adonga_Spain.jpg>
K.A. Tsokos, Physics, A text for the International Baccalaureate Program, Fifth Edition,
2008
<http://www.faculty.fairfield.edu/jmac/rs/bridges.htm>
<http://www.architecture.about.com/od/bridgegallery/Bridge_Construction_and_Enginee ring.html>
<http://www.pbs.org/wgbh/buildingbig/lab/loads.html>
<http://www.design-technology.org/beambridges.html>
<http://www.scienceclarified.com/Bi-Ca/Bridges.html>
<http://www.makingthemodernworld.org.uk/learning_modules/maths/02.TU.03.hmtl>
<http://www.123helpme.com/preview.asp?id=62636>
<http://42explore.com/bridge.html>
<http://science.howstuffworks.com/bridge4.html>
<http://www.jhu.edu/virtlab/bridge/bridge.html>
Page 31 of 40

Willem Frederik Neeteson
Appendix
10
12
14
5
6
Bridge 1
Length of folds / cm
Number of folds
Triangles along the base
Density / gm -2
Width / cm Force
1
/ N Force
2
/ N
8
3.8
4.2
4.1
3.9
4.4
4.2
3.2
2.9
3.7
2.9
2.3
+/- 0.1
2.5
+/- 0.1 +/- 0.2
International School of Nice
1.0 +/- 0.1
Force
3
/ N
4.2
4.3
4.3
3.6
2.8
2.6
+/- 0.1
19
10
100 +/- 1
4.2
3.5
2.9
Av. Force / N
4.0
4.3
2.5
+/- 0.1
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Willem Frederik Neeteson International School of Nice
10
12
14
+/- 0.2
5
6
8
Bridge 2
Length of folds / cm
Number of folds
Triangles along the base
Density / gm -2
Width / cm Force
1
/ N Force
2
/ N
6.0
6.0
5.7
5.9
6.0
5.7
5.6
3.9
3.2
+/- 0.1
5.4
4.0
3.1
+/- 0.1
1.2 +/- 0.1
17
9
100 +/- 1
5.9
4.0
3.2
+/- 0.1
Force
3
/ N
5.9
5.7
6.0
5.7
4.0
3.2
+/- 0.1
Av. Force / N
5.9
5.9
5.8
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Willem Frederik Neeteson
Bridge 3
Length of folds / cm
Number of folds
Triangles along the base
Density / gm -2
International School of Nice
1.3 +/- 0.1
15
8
100 +/- 1
Width / cm
5
6
8
10
12
14
+/- 0.2
Force
1
/ N
6.0
6.0
6.4
5.6
4.0
3.5
+/- 0.1
Force
2
/ N
6.0
6.1
6.3
5.6
4.2
3.6
+/- 0.1
Force
3
/ N
6.0
6.1
6.3
5.8
4.3
3.6
+/- 0.1
Av. Force / N
6.0
6.1
6.3
5.7
4.2
3.6
+/- 0.1
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Willem Frederik Neeteson International School of Nice
12
14
16
+/- 0.2
Width / cm
6
8
10
Bridge 4
Length of folds / cm
Number of folds
Triangles along the base
Density / gm -2
Force
5.0
4.7
4.0
3.5
3.0
N/A
1
+/- 0.1
/ N
3.6
3.4
N/A
+/- 0.1
Force
2
/ N
5.1
5.3
4.1
Page 35 of 40
3.9
3.2
N/A
+/- 0.1
Force
3
/ N
5.1
5.0
4.0
1.4 +/- 0.1
13
7
100 +/- 1
3.7
3.2
N/A
+/- 0.1
Av. Force / N
5.0
5.0
4.0

12
14
16
+/- 0.2
6
8
10
Willem Frederik Neeteson
Bridge 5
Length of folds / cm
Number of folds
Triangles along the base
Density / gm -2
Width / cm Force
1
/ N Force
2
/ N
4.0
4.7
3.9
5.3
5.0
4.1
3.4
2.8
N/A
+/- 0.1
4.1
2.0
N/A
+/- 0.1
Page 36 of 40
International School of Nice
1.7 +/- 0.1
11
6
100 +/- 1
3.9
2.3
N/A
+/- 0.1
Force
3
/ N
4.7
5.3
4.0
3.8
2.4
N/A
+/- 0.1
Av. Force / N
4.7
5.0
4.0

Willem Frederik Neeteson
6
8
Bridge 6
Length of folds / cm
Width / cm
Number of folds
Triangles along the base
Density / gm -2
Force
1
/ N Force
2
/ N
10
5.8
5.6
4.2
5.0
5.3
4.1
12
14
3.5
2.1
4.0
2.0
16
+/- 0.2
0.8
+/- 0.1
1.2
+/- 0.1
International School of Nice
2.1 +/- 0.1
Force
3
/ N
5.4
5.5
4.2
3.9
2.0
1.1
+/- 0.1
5
9
100 +/- 1
Av. Force / N
5.4
5.5
4.2
3.8
2.0
1.0
+/- 0.1
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Willem Frederik Neeteson International School of Nice
10
12
14
16
+/- 0.2
6
8
Bridge 7
Length of folds / cm
Number of folds
Triangles along the base
Density / gm -2
Width / cm Force
1
/ N Force
2
/ N
3.8
4.3
4.1
4.2
3.8
2.9
1.6
0.7
+/- 0.1
3.8
3.4
1.4
0.9
+/- 0.1
Force
3
/ N
4.3
3.9
3.7
3.1
1.6
0.8
+/- 0.1
2.6 +/- 0.1
4
7
100 +/- 1
Av. Force / N
4.0
4.1
3.8
3.1
1.5
0.8
+/- 0.1
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Willem Frederik Neeteson International School of Nice
10
12
14
16
+/- 0.2
6
8
Bridge 8
Length of folds / cm
Number of folds
Triangles along the base
Density / gm -2
Width / cm Force
1
/ N Force
2
/ N
2.2
1.9
2.3
1.9
2.1
1.6
0.8
1.1
+/- 0.1
2.1
1.7
1.2
1.0
+/- 0.1
Force
3
/ N
1.9
2.1
2.0
1.9
1.0
1.0
+/- 0.1
3.5 +/- 0.1
3
5
100 +/- 1
Av. Force / N
2.1
2.0
2.1
1.7
1.0
1.0
+/- 0.1
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Willem Frederik Neeteson International School of Nice
10
12
14
16
+/- 0.2
6
8
Bridge 9
Length of folds / cm
Number of folds
Triangles along the base
Density / gm -2
Width / cm Force
1
/ N Force
2
/ N
1.3
0.6
1.1
0.6
0.7
0.2
N/A
N/A
+/- 0.1
0.7
0.5
N/A
N/A
+/- 0.1
Force
3
/ N
1.1
0.7
0.6
0.3
N/A
N/A
+/- 0.1
4.5 +/- 0.1
2
3
100 +/- 1
Av. Force / N
1.2
0.6
0.7
0.3
N/A
N/A
+/- 0.1
Page 40 of 40