S3 : StadiumS3 : Stadium Roof Roof Design – Cantilever Roof Design – Cantilever Roof

Design ExperimentDesign Experiment

Cantilever Roof Design

www.liberty-stadium.com/

forum.skyscraperpage.com

Twickenham during the construction of the South Stand.

Liberty Stadium

Location : Swansea

Home to : Swansea City football and Ospreys Rugby Team

Capacity : 20,000

Built in : 2005

P29

Cantilever Roof Design

eap.ucop.edu

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.P30

Cut out and calculate y

Cut

Fold to make a stand

Depth of Counter Weight, y

Cut Away

Cut Away

Roof

Counter Weight

Mast

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 balanceP31

Design roof and balanceThe 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

Centre of gravity of roofTrace 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

Calculate area of new roof

bAgain 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.

From measuring,

D = 4.1cm

b = 5.7 cm

Clockwise roof moment

= 0.08 x 9.81 x A x (5.7 + 1) = 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

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

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 = 101.5 Ncm

Anti-clockwise moment from card counter weight

= 0.08 x 9.8 x 3 x 3 x (1.5 + 1) = 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 (1.5 + 1) = 24.5M

Equating the clockwise and anticlockwise moments gives

M = (101.5 – 17.6) / 24.5 = 3.42 gramsP36

Achieving balanceA 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

Like a Tower craneThe 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

Tying the roof downA 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.

1. Punch a hole through the middle of where the coin was located to get the structure to balance.

2. 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