Cardboard Bridge design 44

MECHANICAL ENGINEERING DESIGN 1 (MECN2014) Title: BRIDGE DESIGN PROJECT Group Number: 44 Names: Phuti Balty Tjale

Date:

(607911)

Masego Erens

(569332)

Thabo Lepota

(568571)

Humphry Tlou

(681141)

15 September 2015

i

Table of Contents Executive summary .......................................................................................................................1 1.

Introduction...........................................................................................................................2 1.1

Task as given............................................................................................................................. 2

1.2

Literature review ....................................................................................................................... 2

1.3

Material strength ....................................................................................................................... 3

1.4

Material availability .................................................................................................................. 4

1.5

Glue information ....................................................................................................................... 4

1.6

Competition requirements and rules ......................................................................................... 5

2.

Task as understood ................................................................................................................6

3.

Product Requirements and Specifications (PRS) ...................................................................7 3.1

Requirements ............................................................................................................................ 7

3.2

Constraints ................................................................................................................................ 7

3.3

Criteria ...................................................................................................................................... 8

4.

Functional Analysis ...............................................................................................................9

5.

Concept development and analysis ...................................................................................... 10 5.1

Concept 1 ................................................................................................................................ 10

5.2

Concept 2 ................................................................................................................................ 11

5.3

Concept 3 ................................................................................................................................ 12

5.4

Concept 4 ................................................................................................................................ 12

6.

Concept selection ................................................................................................................. 14

7.

Detailed design development ................................................................................................ 16

8.

7.1

Calculation of forces on each member .................................................................................... 16

7.2

Buckling .................................................................................................................................. 18

Design Specifications ........................................................................................................... 21 8.1 Performance Specification ............................................................................................................ 22 8.2 Recommendations ......................................................................................................................... 22

9.

References ........................................................................................................................... 23

10. Appendices .......................................................................................................................... 24

Executive summary This report outlines the design process followed in the construction of a cardboard paper bridge – built from only cardboard paper of 300gsm and glue. The primary aim of the task was to design the bridge that is engineered to complete the task of supporting loadings at its mid-span while flat and when tilted to an angle of 30˚ to the horizontal with a maximum deflection of 5mm. Initially, individual concepts of the bridge were designed separately. Team members brainstormed and integrated ideas for the concepts. New sketches were created from the individual concepts and discussions were held regarding the proposed function of the components and the overall bridge. In total, four concepts were proposed and a selection matrix used to identify which of the four scored the most in terms of the criteria used. Stress, buckling and deflection calculations were incorporated to determine which would withstand the most loads and strength of each. The most suitable design came out as the third concept. It has fewer members than the rest but can withstand the most forces. The deflection thereof was found to be 1.866mm downwards under a load of 5kg and the component through which the axle that will have the rope attached to it will go has very minimal chances of tearing since it is attached to the bottom of the deck of the bridge. The highest stressed member in has a load of 33.04N but with a critical load of 2.89kN, thus the possibility of members failing is minimal. It is manufactured by cutting the specified paper size with a knife cutter and folding until specified thickness. Only triangular and cylindrical members are used, which are easy to fold out. Holes the same size and shape as the cross-section of each member are drilled to the specific member and location they have to be connected to. Glue is smeared onto the joints before they are attached to members. The overall design lacks resemblance to an actual bridge in terms of the small details which make it possible to use a bridge. Hand or side rails could be introduced to avoid passengers of vehicles from falling over. Lane markings on the road to indicate a proper road surface would also enhance aesthetics. Careful attention should be paid to the dimensions used so that one can carefully estimate the mass of the bridge without weighing (which is only done after it is built). 1

1. Introduction 1.1 Task as given

A project has been assigned where it is expected of us to design a cardboard bridge from cardboard paper that has a maximum weight of 300 gsm and restricted dimensions of 841mm x 1189mm (A0). The bridge must be able to support a minimum mid-span load of 3kg without any sign of structural failure with a maximum deflection of 5mm. A truck having the dimensions: 125mm (H) x 75mm (W) must pass through without any obstruction. The bridge will be capable of spanning a river that is 500mm wide while it has a supporting structure which may not extend 20mm below the straight level support line (point where it rests on the testing station). The support structure of the bridge will have a 5mm pin connection to support one side of a bridge while the other bridge support will have a smooth horizontal surface to support the other side of the bridge. In addition, the bridge has to pivot about the pinned connection to an angle of 30° to allow boats to pass under the bridge with sufficient clearance. 1.2 Literature review

Bridge design is constantly revolutionized by the significant demand of bridges by the community or public. While bridges are in demand, many factors which affect the quality of the designs thereof have imposed a competitive innovation of various types of bridges which have different capabilities. The wooden truss bridges were used since 1700s because of their high level of strength. They were also used for railroad bridges mainly because of the heavy loads they can support, however they were not preferred since they are very difficult to construct, requires a high maintenance which is costly and difficult to widen if necessary [1]. Nevertheless, beam bridges came to the rescue since they are simplest to design and build. These beam bridges consist of vertical piers and horizontal beam, while their strength depends on the strength of the roadway so their strength can be increased by additional piers which are mainly a firm combination of concrete and steel. The steel enhances the strength of concrete when stretched under tension [2]. The weight of the beam bridges pushes straight down on the piers 2

[3].Although beam bridges can be quite long, the span or distance between adjacent piers is usually small. However beam bridges have a limited span and do not allow large ships or heavy boat traffic to pass underneath [1]. For that reason, newly innovated suspension bridges were designed and built in 1801 in Pennsylvania. Suspension bridges are strong, have a long span distance and allow large ships and heavy boats traffic to pass underneath the bridge [1]. Since suspension bridges are suspended from the cables, as traffic passes on the road, the weight is carried by the cables which transfer the force of compression to the two towers and the cable also have the constant force of tension which are stretched because the roadway is suspended from them. However they are very expensive because they take a long time to build and require a large amount of materials [1]. It is clear from the existing bridges that various factors which may affect the functionality, strength and criteria of the bridges are considered for the improvement of bridge designs. 1.3 Material strength

Cardboard paper is generally stronger than the average normal paper. In tension it does not snap easily when pulled from both ends with equal forces but components made from it will buckle easily under a compressive loading [6]. The tensile properties of paper are measured by clamping a strip between two grips and applying a tensile load until the strip breaks but it is difficult to obtain the exact value of the tensile strength of cardboard paper. Tensile strength is defined as the breaking force divided by the width of the strip and has the units 𝑁/𝑚 [6]. The modulus of elasticity of cardboard paper is also very difficult to obtain due to paper being very thin and snapping too quickly during testing. Due to the manufacturing process of paper, the elastic modulus 𝐸 thereof is significantly anisotropic, which makes it very different from most materials. The manufacturing process results in paper 𝐸𝑥 > 𝐸𝑦 . This means that the elastic modulus of fibres in the longitudinal direction is larger than that of the fibres in the transverse direction [7]. Experimentally the modulus of elasticity of paper board (100 − 400 gsm) for the 𝑥(MD) and 𝑦(CD) directions have been determined to be 5420𝑀𝑝𝑎 and 1900𝑀𝑝𝑎 respectively. Thus combining the two gives an average of 5724.5𝑀𝑝𝑎. 3

The other factors that influence the strength of cardboard paper are the moisture content of the paper (thus air humidity), paper directionality, folding endurance, stiffness and effects of recycling. The moister the cardboard becomes, the weaker it gets [8]. All types of paper gain or lose moisture due to ambient humidity and the properties of paper change with the moisture content. Paper fibres become weakened through every recycling cycle, thus virgin paper will inherently be stronger than recycled paper. The ability of paper to resist being bent is what stiffness is. When a strip of paper is subjected to continuous folding under tension it will eventually break. The number of folds it can endure before it breaks is the measure of the endurance resistance of paper. Cardboard paper is stiffer when bent and folded across the grain than along the grain (machined direction) [8]. 1.4 Material availability

Cardboard A0 sheets are readily available and do not require any additional work on them. Most printer shops and gift shops (which also have a number of glues) sell these at a price of approximately R50 per sheet. There are plenty of such shops in close vicinity thus it will not be a problem to acquire all the material needed for the construction of the bridge. Literally only two materials are needed for the entire construction; glue and cardboard. 1.5 Glue information

Glues are part of a larger family called adhesives. The two classes are distinguished by the fact that glue comes from organic compounds while adhesives are chemical-based. Adhering materials called epoxies, caulks, or sealants are also chemical compounds that have special additives to give them properties suitable for particular jobs or applications [4]. It is every group’s responsibility to choose their own glue type to use on their construction of the bridge but with the exception of super glue and epoxy. The glue is selected in such a way that it will bond to every component of the bridge; making the joints very strong in order to keep the bridge rigid and stable without causing the cardboard to warp. There’s a quite a variety of glue types to choose from. In order to select glue, certain types of glues had to be investigated to check which product best suits our design. In order to select 4

adhesives, mechanical properties were the simpler way to see which glue best suits our design. The criterion used to select glue is the following: weight distribution, drying time, transparency and colour after drying, strength and temperature resistance [5]. The one glue that fits our criterion is white glue. The fact that it’s friendly to use, not toxic cheap unless ingested, readily available and cleans up with water which is a solvent in them makes them the top contenders for the design. 1.6 Competition requirements and rules

-

The project should be done by groups of a maximum of four members with each member spending at least 50 hours working on it.

-

Strictly only a single sheet of A0 cardboard paper with dimensions 841mmx1189mm can be used for all the parts of the bridge.

-

Any glue other than super glue and epoxy glue are allowed for any joining needed.

-

The scoring equation that will be used to assess the constructed bridge is the following: 0.35 𝑺𝒄𝒐𝒓𝒆 = 0.3 ( ) + 0.4(𝑀𝑎𝑥𝑖𝑚𝑢𝑚 𝐿𝑜𝑎𝑑) + 0.3(𝑆𝑡𝑟𝑢𝑐𝑡𝑢𝑟𝑎𝑙 𝐹𝑎𝑖𝑙𝑢𝑟𝑒) 𝐵𝑟𝑖𝑑𝑔𝑒 𝑚𝑎𝑠𝑠

5

2. Task as understood

A bridge constructed out of cardboard paper has to be designed by groups of four. This should be done from a single A0 sheet of 300gsm cardboard paper and must span a river that is 500mm wide. The height at any location above the deck or road way must be greater than 125mm with a width that is greater than 75mm in order to allow a truck of the dimensions; 125mm (H) x75mm (W) to pass through. It must be capable of supporting a load of at least 3kg at the mid-span without sustaining deflection of more than 5mm below its lowest point. The bridge should have provisions for three 5mm axle pins at the mid-span, one end (below road surface and acting as a pivot for the bridge) and somewhere on the bridge to allow attachment of a cable hoist. It should be able to pivot about a pinned connection to an elevation of 30° to the horizontal for allowance of boats to pass below it. Finally at this tilted position, it must sustain a 0.5kg weight attached to the mid-span for about 10s. The bridge must be designed in such a way that the strength to weight ratio is as high as possible to not compromise on weight even when optimizing its strength.

6

3. Product Requirements and Specifications (PRS)

3.1 Requirements 

Must span a 500mm wide river



A truck with the dimensions (125mm H × 75mm W) must be able to go through the bridge with ease



Must pivot about pinned connection



Must be able to support a load of up to 3kg



Must be able to hold a suspended 0.5 kg mass for at least 10 seconds while being tilted to an angle of 30˚



Provision must be made at the mid-span and one end of the bridge for a 5mm diameter axle pin

3.2 Constraints 

Bridge must be manufactured using only single sheet of A0 cardboard



Weight specification of the cardboard must not exceed 300gsm



The dimensions of the cardboard must be 841 mm × 1189 mm



The design components must be joined together by any type of glue except for super glue and epoxy glue



The bridge must be able to hold a weight without a deflection greater than 5 mm at most



Width of the bridge cannot exceed length of pin (i.e. must be less than 100mm)



Supporting structure may not be extended for more than 20mm below the straight level support line

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3.3 Criteria 

Mass of the bridge should be at most or preferably 0.35kg (for a good score on the equation)



The design must be for ease of manufacture (e.g. 2D development components that can be cut out and folded easily)



It must be able to be tested with ease



The cost of the bridge should preferably not exceed R100



Number of components should not be too high to avoid needing more material than allowed



The design should be aesthetically pleasing



Load carrying capacity must be greater than 3kg



A high stiffness (strength to weight ratio must be high )



The different 2D components must be easy to assemble



Deflection is minimum

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4. Functional Analysis

START

Place bridge on station for testing

Insert pivot axle through bridge

Load bridge with minimum weight

Stop increasing loads and then remove

Hook rope to axle placed at top corner

No

Is the deflection ≤5mm?

Yes Place the 0.5kg load

Wait 10s

Keep increasing load

No

Pull rope until bridge is tilted 30° to horizontal

Yes Remove weight and bridge from station

Yes

Is the load ≤3kg?

STOP

9

Is there imminent failure?

No

5. Concept development and analysis It is common knowledge in structural engineering that to obtain optimum rigidity in a structure that is under loading it is best to make use of triangulation in it. The concept designs that follow have been developed in light of this fact, thus triangular trusses have been incorporated in all of the concepts as much as possible to take advantage of their rigidity. Cutting triangles from cardboard can also be done fairly easily than most shapes. 5.1 Concept 1

Road level (where cardboard sheet will rest)

One of three beams joining both trapeziums of the bridge together

The two trapeziums are the basic structure

Height of the bridge all throughout is 160mm

3mm diameter axle goes through here and rests on the short member

Hole (±3.5mm) for axle by which bridge pivots

Figure 1

The well-known bridge that inspired this concept is the Warren Bridge. The basic design of this concept is taking two trapeziums and joining them across the width of the bridge at the top joints and midway through the bridge to make a complete bridge. This is done by having one beam defining the height of the bridge that runs from end to end. The advantage with this is that there will be fewer joints at the top of the bridge which suggests that there will be fewer points where failure is most probable to occur. A hole for the 3mm diameter axle pin that functions as a pivot is cut out from the material as part of a component during manufacturing and thus the pin does not rest on any joints. Whereas the axle pin to which the rope will be attached rests on the short member at the top right corner joint. Although the weight of the pin before being pulled rests on a member it will not cause significant buckling as the member onto which it rests is very short in comparison to the others. The joints of the bridge are made more rigid by cutting out gusset plates according to the size and shape of each joint in a particular position on the bridge. All the members forming a specific joint are 10

attached to these cut out gusset plates. These ensure less likelihood of failure at the joints due to loading. When loading is at the top of the bridge, normally trusses would follow the configuration outlined by the dotted lines on the sketch. Since loading is the bottom, they have been placed in such a way that they spread out to the bottom 5.2 Concept 2 The left side and right side are exactly identical. Some members might have not been excluded from the right side for simplicity. The components coloured with black are thick components with a thickness that is a lot more than the other members

6 mm diameter holes through which 5mm diameter axles will be slit

This arm also runs along the right side of the bridge and supports a lot of the downward load

Figure 2

In this concept the main objective was to come up with a bridge that has a strong member that can bare the weight rather that most of it being taken by the lower deck. Another consideration that led to this design was that when it tilts there will be a moment introduced so more mass was moved at the back of the pivot to counteract this moment. The advantage with this one is that where the axles will be positioned is strong and made of thick material unlike being put at the joints. This minimizes stresses at the joints and ensures minimal chances of failure at the axle locations. Another advantage is that while tilting, the problem of having the part of the bridge to the left of the pivot being squashed to the ground is eliminated. The disadvantage is that there are too many joints and thus as far as manufacturing is concerned it might pose as a challenge. One other disadvantage is that the hollow members will be weak, making it easier for failure at the joints.

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5.3 Concept 3 These members are cylindrical The diagonal and upright members are triangular in cross section.

6mm diameter hole by which the bridge will pivot

Since this design has very fewer members than the others, the members can be made less hollow

Figure 3

The thinking behind this design was to remove as many members as possible, thus as many joints as possible. In doing that it would result in a simple bridge that can be manufactured from 2D rolled sheets of paper instead of having too many members supporting loads. The advantage with this design is that a lot of the members are cylindrical, which is easier to manufacture as one just rolls paper to the desired dimensions. Another advantage is that when tilted the axle will lift the component from the bridge but there is no potential for it to tear from the bridge unlike with the previous design. Disadvantage is that aesthetic wise is not up to scratch, it lacks some creativity 5.4 Concept 4 This curved part is thicker than all the other part on the bridge Road level

The rope will be hooked to axle slit through here

Beam running beneath the bridge

12

The advantage with this design is that below the deck of each bridge runs a beam looking like corrugated cardboard. This will be constructed of two cardboard sheets sandwiching triangular trusses folded from another cardboard paper. This beam together with the trusses above the deck will absorb the load placed at the mid span of the bridge. Another advantage is due to the curved section. It was introduced to allow for easy rotation, thus it is definite that no material after the hole will touch the ground. The disadvantages with this design are that the curve might be difficult to construct physically. There are too many members and therefore many joints, hence there will be too much stresses induced in the hole drilled into the joint to allow an axle for the rope to be put through.

13

6. Concept selection To thoroughly select the best design concept, Matrix selection table is constructed where in all the proposed ideas are judged according to the proposed criteria stated in the PRS. Below is a tabulated data of all the three concepts. Weightings start from 1-5 Scoring

5 = Excellent 4 = Good 3 = Average 2 = poor 1 = very poor

For a concept meeting a certain criteria specified in the requirements outstandingly it will receive a score of 5, and if it meets averagely it will be accredited a score of 4. A concept scoring below 3 is attributed as a poor concept for a particular criterion. Table 1: Matrix Selection Criteria Weighting

Concept

Concept

Concept

1

2

3

Concept 4

Weight

4

3(× 4)

3(× 4)

4(× 4)

4(× 4)

Manufacturability

5

3(× 5)

3(× 5)

4(× 5)

3(× 5)

Effective cost

4

3(× 4)

3(× 4)

4(× 4)

3(× 4)

Max load Support

5

4(× 5)

3(× 5)

4(× 5)

3(× 5)

Deflection

5

3(× 5)

2(× 5)

4(× 5)

2(× 5)

Aesthetics

4

3(× 4)

4(× 4)

3(× 4)

4(× 4)

Stiffness

4

2(× 4)

2(× 4)

3(× 4)

2(× 4)

Material

5

3(× 5)

3(× 5)

2(× 5)

2(× 5)

109

103

126

102

Consumption Total Rating

14

From the matrix above it is observed that all concepts have relatively high total rating with very little differences. All the concepts appear to be meeting the criteria of effective cost very well. Although concept 1 can support relatively huge load, it’s not easy to be manufactured. Thorough analysis revealed that concept 3 of the bridge meets meet most of the criteria listed above in the matrix selection, with zero deflection, average aesthetic and can support at a load of 4 kg which is greater than the one specified in the design requirement, hence this concept is selected as the best among all the proposed designs.

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7. Detailed design development 7.1 Calculation of forces on each member

Assumptions made for force analysis: 

The load placed at the mid-span is 5kg and shared between the two vertical posts equally.



All joints are fixed joints



Thickness of the axles is neglected (thus dealing with point loads)



Area of contact between surface and bridge is negligible, thus reaction is a point load



Thickness of members ignored 𝐶

𝐷

304

193 𝐸

𝐹

𝐵 𝐵𝑥

𝐴

160

260 𝐴𝑦

260 2.5(9.81)

2.5(9.81)

𝐵𝑦

↺ + ∑𝐵 = 0; −𝐴𝑦 (0.68) + 24.53(0.42) + 24.53(0.26) = 0 𝐴𝑦 = 24.53𝑁

↑ + ∑𝐹𝑦 = 0; 𝐵𝑦 + 24.53 − 24.53 − 24.53 = 0 Joint A 𝐹𝐴𝐶 36.59∘ 𝐹𝐴𝐸 𝐴𝑦

16

∴ 𝐵𝑦 = 24.53𝑁

↑ + ∑𝐹𝑦 = 0; 𝐹𝐴𝐶 sin(36.59∘ ) + 24.53 = 0 𝐹𝐴𝐶 = −41.15𝑁 = 41.15(𝐶)

→ + ∑𝐹𝑥 = 0; 𝐹𝐴𝐶 cos(36.59∘ ) + 𝐹𝐴𝐸 = 0 −41.15 × cos(36.59∘ ) + 𝐹𝐴𝐸 = 𝐹𝐴𝐸 = 33.04𝑁

Joint C 36.59∘

𝐹𝐶𝐷

𝐶

𝐹𝐴𝐶

𝐹𝐶𝐸

→ + ∑𝐹𝑥 = 0; 𝐹𝐴𝐶 cos(36.59∘ ) + 𝐹𝐶𝐷 = 0 −(−41.15) × cos(36.59∘ ) + 𝐹𝐶𝐷 = 0 𝐹𝐶𝐷 = −33.04𝑁 𝐹𝐶𝐸 = 𝐹𝐶𝐹

𝐹𝐵𝐷 = 𝐹𝐴𝐶

𝐹𝐴𝐸 = 𝐹𝐸𝐹 = 𝐹𝐵𝐹

Joint E 𝐹𝐶𝐸 𝐸 𝐹𝐴𝐸

𝐹𝐸𝐹 24.53𝑁

↑ + ∑𝐹𝑦 = 0; 𝐹𝐶𝐸 − 24.53 = 0 𝐹𝐶𝐸 = 24.53𝑁 17

7.2 Buckling

𝑃𝑐𝑟 =

𝜋 2 𝐸𝐼 (𝐾𝐿)2

Where 𝑃𝑐𝑟 = critical or maximum axial load on the column just before it begins to buckle 𝜎𝑐𝑟 = critical stress, which is an average normal stress in the columns moments before it buckles 𝐸 = modulus of elasticity for the material 𝐼 = least moment of inertia for the column’s cross-sectional area 𝐿 = unsupported length of the column, whose ends are pinned 𝑟 = smallest radius of gyration of the column, determined from 𝑟 = √𝐼 ⁄𝐴 , where I is the least moment of inertia of the column’s cross-sectional area

The members that are under compression are 𝐴𝐶, 𝐶𝐷 and 𝐷𝐵. In the following calculation the maximum axial load on the column before it begins to buckle will be assessed. Since the joints of the members are glued together it is assumed that the ends of the members are fixed, thus the effective length 𝐿𝑒 (𝐾𝐿) is calculated using 𝐾 = 0.5. 𝜋 4 (𝑑 − 𝑑𝑖04 ) 𝐼= 64 0 𝜋 (174 − 94 ) = 3777.77𝑚𝑚4 = 64 = 3.78 × 10−9 𝑚4 𝑟 = √𝐼 ⁄𝐴 = 9𝑚𝑚

3777.77 = 4.81𝑚𝑚 𝜋 (172 − 92 ) 4

17𝑚𝑚

𝜋 2 𝐸𝐼 𝜋 2 (5724.5 × 106 )(3.78 × 10−9 ) 𝑃𝑐𝑟 = = = 33.37𝑘𝑁 (𝐾𝐿)2 (0.5 × 0.16)2 18

𝜎𝑐𝑟 =

𝜋2𝐸 𝐾𝐿 2 (𝑟 )

=

𝜋 2 (5724.5 × 106 ) 2 0.5 (0.00481)

= 5.23 𝑀𝑝𝑎

𝑏ℎ3 32 1 (13.863 )(16) = 1183.33𝑚𝑚4 = 32 𝐼=

= 1.183 × 10−9 𝑚4

60° 13.86𝑚𝑚

𝑟 = √𝐼 ⁄𝐴 =

1183.33 1 (16)(13.86) 2

= 3.27𝑚𝑚

16𝑚𝑚

𝜋 2 𝐸𝐼 𝜋 2 (5724.5 × 106 )(1.183 × 10−9 ) 𝑃𝑐𝑟 = = = 2.89𝑘𝑁 (𝐾𝐿)2 (0.5 × 0.304)2 𝜎𝑐𝑟 =

𝜋2𝐸 𝐾𝐿 2 (𝑟 )

=

𝜋 2 (5724.5 × 106 ) 2 0.5 (0.00327)

= 2.42 𝑀𝑝𝑎

7.3 Deflection Assumptions Moment of inertia of the beam is 4 times that of each cylinder it comprises of and load acts directly at the mid-span of the beam: 49.05𝑁

24.525𝑁

680 𝑚𝑚 24.525𝑁

19

↑ + ∑𝐹𝑦 = 0; −𝑉 + 24.525 = 0 ∴ 𝑉 = 24.525𝑁 ↺ + ∑𝐴 = 0; −24.525(𝑥) + 𝑀 = 0

𝑥

24.525𝑁

𝑉

∴ 𝑀(𝑥) = 24.525𝑥

𝑀

𝐸𝐼

𝐸𝐼

𝑑2𝑣 𝑑𝑥2

𝑑𝑣 24.525 2 = 𝑥 + 𝐶1 𝑑𝑥 2

= 𝑀(𝑥) = 24.525𝑥

⟹ 𝐸𝐼𝑣 =

24.525 3 𝑥 + 𝐶1 𝑥 + 𝐶2 6

𝑎𝑡 𝑥 = 0, 𝑣 = 0 𝑎𝑛𝑑 𝑎𝑡 𝑥 = 0.68, 𝑣 = 0 ∴ 0 = 0 + 𝐶2 ⟹ 𝐶2 = 0 0=

24.525 (0.68)3 + 𝐶1 (0.68) ⟹ 𝐶1 = −1.89006 6

Due to the symmetry of arrangement, it can easily be seen that 𝑣𝑚𝑎𝑥 occurs at 𝑥 = 0.68⁄2: ∴ 𝐸𝐼𝑣 = ( 𝐼=

24.525 3 ) 𝑥 − 1.89006𝑥 6

𝜋 (234 − 154 ) = 11251.61𝑚4 = 1.125(10−8 )𝑚𝑚4 ⟹ 𝐼𝑡𝑜𝑡 = 4.50(10−8 )𝑚𝑚4 64 𝑣𝑚𝑎𝑥 =

1 24.525 [( ) (0.34)3 − 1.89006(0.34)] = −1.87𝑚𝑚 𝐸𝐼𝑡𝑜𝑡 6

From the appendix, the displacement of the joint F was calculated to be −3.55(10−3 )𝑚𝑚. The negative sign indicates that it is displaced upwards. It makes complete sense since the member DF is in tension and thus pulls the joint upwards. Since the joint F is identical to joint E the same calculation applies. The resultant deflection of the entire bridge should be approximately 1.866𝑚𝑚 downwards.

20

8. Design Specifications The following is a selection matrix on the best material to use for the building of the Bridge. Since the Constrain in the PRS require the bridge to be constructed out of a cardboard of maximum size of 300gsm, it was necessary to determine the best cardboard size that will be suitable for the construction of the components for the selected design. The cardboard material was selected based on the ability to be folded in triangular or round shapes because these were easier to construct as compared to rectangular or square shapes. Table 2: Material Selection Matrix Material Size

240gsm

300gsm

Components Road surface

 

Vertical triangular beam



Diagonal triangular beam Horizontal shafts with



protrusions Hook



21

8.1 Performance Specification

Components

Specification

Road surface

Allow smooth mobility of transport and persons

Hook

Provide allowance for lifting the bridge

Vertical triangular

Prevent the bridge from failing in compression, when subjected to a

column

downward load.

Diagonal triangular

Provide a tensile upward force, thus resist the bridge from failing at

column

the ends.

Top Protruded

Ensures that the bridge becomes more rigid and stable

horizontal column Bottom Horizontal

Prevent the bridge from failing in the middle and minimizes

column

deflection

8.2 Recommendations

The bridge built, performed very well, and indeed it was well designed with very ingenious ideas. Modification that can be applied to improve the bridge is to include side or hand rails for mobility guidance along the road surface. Including diagonal members between the two horizontal members on both sides of the bridge will increase the rigidity of the bridge especially at the top. To Prevent buckling or deflection of the bridge, it would be necessary to constrain the horizontal columns by introducing flat sheets at both ends.

22

9. References

[1] Meter, N. V. (2004). Cities/layout L. Bridge Basics, 14-17. Retrieved May 10 2015: http://www.nbm.org/assets/pdfs/youth-education/bridges_erpacket.pdf [2] Beam Bridges. (2015). Retrieved May 14, 2015, from Design-technology.org: http://www.design-technology.org/beambridges.htm [3] Beam. (2015). Retrieved May 14, 2015, from Warwickallen.com: http://www.warwickallen.com/bridges/BeamBridges.htm [4] Mechanical properties of adhesives. (2015). Retrieved May 14, 2015, From Adhesiveandglue.com: http://www.adhesiveandglue.com/mechanical-properties-adhesives.html [5] Wood Glue uses and information . (2015). Retrieved May 14, 2015, From Naturalhandyman.com: http://www.naturalhandyman.com/iip/infadh/infadhe.html [6] Karlson, H. (2010). Strength Properties of Paper produced from Softwood Kraft Pulp. Karlstads University: Faculty of Technology and Science. Retrieved 14 May 2015, From: http://www.diva-portal.org/smash/get/diva2:317178/fulltext01.pdf [7] Sekulić, B. (2013). Structural Cardboard: Feasibility Study of Cardboard As A Long-Term Structural Material In Architecture, University of Politècnica De Catalunya. Retrieved 24 June 2015, From: http://upcommons.upc.edu/pfc/bitstream/2099.1/21603/1/BrankoSekulic_TFM.pdf. [8] Guyana. (2010). Paper and Board packaging: Properties, specifications and sourcing. Retrieved 24 June 2015, From: http://www.iica.int/Eng/regiones/caribe/guyana/IICA%20Office%20Documents/tfo_packaging_ workshop/Guyana%20TFO%20Pkg%20W'shp%20Session%203%20%20Paper%20and%20Board.pdf

23

10. Appendices

24

6

5

4

3

2

1

160,00

D

40,00

33,50

25,50

36,8 2

221,00

D

50,00 680,00

C

C PARTS LIST 6

5

ITEM 1

QTY 1

PART NUMBER cover page

2

4

bottom_column

3 4

2 4

5

2

bottom beam vertical triangular column horizontal shaft with protrusions

6

2

across column

7

4

8

1

diagonal triangular beam hook

7 B

1

A

Designed by

Checked by

Date

Approved by

Humphry Tlou - 681141

2

8

3

Date

4 5

4

3

2

B

A

2015-09-14

full assembly 6

DESCRIPTION Cut from 240 gsm,0.09724 metre squared paper 240gsm paper hollow rolled tubes.Use ponal wood glue for every joint. 300gsm folded paper 300gsm paper folded in triangular shapes 240gsm rolled solid paper tubes with two steps 240gsm rolled hollow paper,cuttings made with a cutting knife 300gsm paper folded in triangular shapes 240gsm paper folded

Edition

Sheet

1/1 1

6

5

4

3

2

1

Part 6

Hole drill to suit part 5 150,00

9,00

15,00

D

25,00

D

Part 5

190,00 C

C

17,00

9,00

160,00

Part 3

155,00

16,00 B

8,00

B

Hole cut to fit vertical triangular column

25,00

A

60,00

50,00

16

,00

,00 16

Designed by

Checked by

Approved by

Date

Humphry Tlou - 681141

6

5

4

3

Date

A

2015-09-14

acros column 2

Edition

Sheet

1/1 1

6

5

4

3

2

1

Part 1

23,00

D 92,00

25,50

680,00 D

97,00 Part 2

23,00

15,00

680,00

C

C

Bottom Support

B

B

A

Designed by

Checked by

Date

Approved by

Humphry Tlou - 681141

5

4

3

A

2015-09-14

cover page 6

Date

2

Edition

Sheet

1/1 1