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**S3 : Stadium Roof Design – Cantilever Roof Design Experiment**

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**Cantilever Roof Design**

Liberty Stadium Location : Swansea Home to : Swansea City football and Ospreys Rugby Team Capacity : 20,000 Built in : 2005 Twickenham during the construction of the South Stand. forum.skyscraperpage.com P29

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**Cantilever Roof Design**

Shanghai Sports Centre Stadium Location : China Home to : All sports, but football mainly Capacity : 80,000 Built in : 1997 It’s unique curving roof is the world’s longest cantilevered roof truss structure with fabric canopy, spanning 300metres. eap.ucop.edu P30

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**Cut out and calculate y Roof Depth of Counter Weight, y Counter Weight**

Mast Cut Away Fold to make a stand Cut Cut Away Clockwise Roof Moment = Area x mass per cm² x gravity x lever arm Anticlockwise Counter Weight Moment = Width x y x mass per cm² x gravity x lever arm Equating the two expressions finds a value for y for the roof to balance P31

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**Design roof and balance**

The roof balances when the counter weight is reduced to depth y as the moments about the mast (the pivot) are equal. Cut out a design for the cantilever roof which will hang above spectators. The clockwise roof moment will be reduced, the anticlockwise counter weight moment also needs to be reduced. Progressively reduce it’s depth, checking regularly if it balances. P32

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**Centre of gravity of roof**

Trace the shape onto the piece of card that was removed earlier. How can you find it’s centre of gravity? Punch a hole and hang the roof off a drawing pin. Ensuring that the roof can swing freely, allow it to settle and draw a vertical line from the pin downwards. This line will go through the centre of gravity. If repeated the point where the lines cross will be the centre of gravity. P33

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**Calculate area of new roof**

Again in order to balance the clockwise and anticlockwise moments about the mast must equal. So we can calculate a value for the area of the roof, A. b From measuring, D = 4.1cm b = 5.7 cm Clockwise roof moment = 0.08 x 9.81 x A x ( ) = 5.26A Ncm Anticlockwise moment = 0.08 x 9.81 x 4.1 x 7 x 4.5 = 101 Ncm Equating the two gives A = 101/5.26 = 19.3cm² D P34

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Check area Another way to calculate area is to trace the shape onto square paper and count the squares. This methods gives A = 20cm² What is the percentage difference? Difference = (20 – 19.3) / 19.3 = 3.6 % Does this confirm the earlier calculation? P35

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**More concentrated counter-weight**

The area of the counter weight can be reduced if a more concentrated mass is used, such as coins. This will improve the appearance of the roof structure. The clockwise moment from the roof = 5.26 x A = 5.26 x 19.3 = Ncm Anti-clockwise moment from card counter weight = 0.08 x 9.8 x 3 x 3 x ( ) = 17.6 Ncm The centre of gravity of any coin will be in the middle. If the coins, with mass M, are placed in the middle of a 3cm x 3cm then; Anti-clockwise moment from coins = M x 9.8 x ( ) = 24.5M Equating the clockwise and anticlockwise moments gives M = (101.5 – 17.6) / 24.5 = 3.42 grams P36

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Achieving balance A 1 pence piece weighs 3.56 grams so is the closest to the required weight of 3.42grams. Initially when it is selotaped in placed the structure does not balance as the weight is too big How can this be rectified? The anticlockwise moment is too big so needs to be decreased. Moving the penny closer to the mast will decreased it’s lever arm and it’s moment, allowing it to balance. P37

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Like a Tower crane The system of a overhanging cantilever with a counter balancing weight is exactly the same as how a crane works. The long arm carries the lifting gear whilst the short arm carries the counterweight. How will the mass of the counter weight be decided? The maximum load on the crane will come when the lifting gear is at the very end of the arm. The lever arm for the moment is the greatest here. This counterweight is often made up of very heavy concrete blocks as it must be able to balance this maximum moment. P38

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Tying the roof down A stadium roof very rarely looks like a crane, so where does the counter weight go? By altering the vertical height of the weight, the balancing effect is not compromised, so we can lower the weight until it is out of sight, underground. Punch a hole through the middle of where the coin was located to get the structure to balance. Stick the coins to a piece of string and “tie – down” the roof by tying the string through the hole. The cantilever roof now hangs as if unsupported creating a dramatic looking stadium roof. P39

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