Designing a space elevator
R is then 42,167,533 metres from the centre of the Earth so the height above the Earth's surface at which centrifugal force equals gravity is about 36,000 km. From here, construction would begin in space, where a satellite would lower a tether to connect somewhere at the equator. However, a counterweight attached to the cable extending into space would be needed to balance the mass of the cable extending towards Earth. From here, you could attach one end of the cable to a base station at the equator and have lifts run up the tether. Firstly, the challenge of building a tether on which the elevator would climb is a massive problem towards building a functional elevator, as the materials we have at this point in time are inadequate. The reason that the materials we currently have are inadequate is because of the maximum tension in the tether, which is created by one
half of the tether stretching into space and the other half being attracted by the Earth's gravitational field. The maximum tension is reached in geostationary orbit. It is possible to calculate the mass tension in the cable using the formula for the uniform cross-section:
𝑅 2 2(𝑅 𝑔 )
𝑇(𝑅 𝑔 )= 𝐺𝑀𝜌 [ 1 𝑅
3 2𝑅 𝑔
−
+
]
3
𝐺 −𝐺𝑟𝑎𝑣𝑖𝑡𝑎𝑡𝑖𝑜𝑛𝑎𝑙 𝐶𝑜𝑛𝑠𝑡𝑎𝑛𝑡
𝑅 − 𝑅𝑎𝑑𝑖𝑢𝑠 𝑜𝑓 𝑡ℎ𝑒 𝐸𝑎𝑟𝑡ℎ
𝑀 − 𝑀𝑎𝑠𝑠 𝑜𝑓 𝑡ℎ𝑒 𝐸𝑎𝑟𝑡ℎ 𝜌−𝑀𝑎𝑡𝑒𝑟𝑖𝑎𝑙 𝐷𝑒𝑛𝑠𝑖𝑡𝑦
𝑅 𝑔 −𝑅𝑎𝑑𝑖𝑢𝑠 𝑜𝑓 𝑔𝑒𝑜𝑠𝑡𝑎𝑖𝑜𝑛𝑎𝑟𝑦 𝑜𝑟𝑏𝑖𝑡
Here you enter the density of the material of your choice, for example, steel. Steel at 7900kg/m 3 is a very dense material; the maximum tensile strength of the cable at a point would be 382GPa, which is 240 times the maximum strength of steel. There is an improved design for the shape of the cable called a tapered tower: the design was developed to economize; maximum stress is at geostationary orbit and the least stress is at both ends of the cable, the base and the counterweight. Therefore, it would not be necessary to have a uniform cross-section as there is no need for such a strong cable where the stress is close to zero. With this equation you can now calculate the taper ratio for different materials:
3
𝐴 𝑔 𝐴 𝑠
𝑅 2𝐿 𝑐
𝑅 𝑅 𝐺
𝑅 𝑅 𝐺
=exp(
((
)
−3(
)+2))
𝑅 −𝑅𝑎𝑑𝑖𝑢𝑠 𝑜𝑓 𝑡ℎ𝑒 𝐸𝑎𝑟𝑡ℎ
𝐿 𝑐 − 𝐿𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑡ℎ𝑒 𝐶𝑎𝑏𝑙𝑒
𝑅 𝐺 −𝑅𝑎𝑑𝑖𝑠𝑠 𝑜𝑓 𝐺𝑟𝑎𝑣𝑖𝑡𝑎𝑡𝑖𝑜𝑛𝑎𝑙 𝑂𝑟𝑏𝑖𝑡
𝐴 𝑠 − 𝐶𝑟𝑜𝑠𝑠 −𝑠𝑒𝑐𝑡𝑖𝑜𝑛𝑎𝑙 𝑎𝑟𝑒𝑎 𝑎𝑡 𝑡ℎ𝑒 𝑏𝑎𝑠𝑒
exp − Exponential Function
𝐴 𝑔 −𝐶𝑟𝑜𝑠𝑠 −𝑠𝑒𝑐𝑡𝑖𝑜𝑛𝑎𝑙 𝑎𝑟𝑒𝑎 𝑎𝑡 𝑡ℎ𝑒 𝑤𝑖𝑑𝑒𝑠𝑡 𝑝𝑜𝑖𝑛𝑡
Material
Taper Ratio (A g /A s )
Diameter g (m)
4.97 x 10 113
3.53 x 10 54
Steel
2.64 x 10 8
Kevlar
81.3
6.37 x 10 -3 8.26 x 10 42
Carbon Nanotubes
1.62
2.73 x 10 90
Ti-6Al-7Nb
1.17 x 10 9
Zoltex PX35 Carbon Fibre
170.7
313
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