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Building under 0.38 g

Forces and motion · level 8, lesson 6 of 6

A question to start

A habitat on Mars is buried under 3 m of soil to shield it from radiation. That is a lot of soil to hold up. Is the roof's main job to stop the soil crushing it, or something else?

The idea

On Earth, buildings mainly hold weight up. On Mars, a habitat must also hold air in. People need a pressure of tens of kilopascals inside (the space station keeps about 101 kPa, a sea-level atmosphere; some designs for Mars use less, around 50 to 70 kPa). Outside, Mars's air presses with only about 0.6 kPa. The difference pushes outwards on every square metre of wall, floor and roof.

Compare the two loads on one square metre of roof, taking an inside pressure of 70 kPa:

On 1 m² of roof Mars Earth (the same building)
Soil above, 3 m at about 1,500 kg/m³ 16.7 kN down 44.1 kN down
Air inside, pushing out 69.4 kN up none (the same air outside)
Net load 52.7 kN up 44.1 kN down

On Mars, the inside air pushes up about four times as hard as the soil pushes down. A buried habitat is not being crushed; it is trying to burst. Its structure works mostly in tension, like a balloon or a pressure tank, and it must be anchored down so it does not lift out of its hole.

soil: 16.7 kN/m²air: 69.4 kN/m²a buried habitat on Mars

For a cylinder of radius \(r\) and wall thickness \(t\), the pressure difference \(P\) makes a hoop stress in the wall

\[\sigma = \frac{P r}{t}.\]

That is what sizes the walls, not gravity.

Worked examples

1. The roof load. Soil: \(3 \times 1500 \times 3.71 = 16{,}700\) N/m². Air: \(70{,}000 - 600 = 69{,}400\) N/m². Net \(52{,}700\) N/m² upwards.

2. A wall. An aluminium cylinder with \(r = 4\) m, \(P = 69.4\) kPa, and an allowed stress of 90 MPa (yield 275 MPa, safety factor about 3): \(t = Pr/\sigma = 69{,}400 \times 4 / 90\times10^6 = 3.1\) mm. The real wall would be thicker, for dents, fatigue, leaks and impacts.

3. How much soil to balance the air? Depth \(d\) with \(1500 \times 3.71 \times d = 69{,}400\): \(d = 12.5\) m. Only under about 12 m of soil would the roof feel no net push.

Try it

  1. Why does a habitat on Mars push out, when the same building on Earth is pushed down?
  2. Find the hoop stress in a steel cylinder of radius 3 m and wall 6 mm, with 70 kPa inside and almost nothing outside.
  3. With 2 m of soil instead of 3, what is the net upward load per square metre?
  4. Why would building at half Earth's pressure inside make the structure lighter?
Answers
  1. Mars's air is so thin that the inside pressure is almost unopposed, and the soil above weighs only 0.38 as much.
  2. \(70{,}000 \times 3 / 0.006 = 35\) MPa.
  3. \(69{,}400 - 2 \times 1500 \times 3.71 = 58{,}300\) N/m², upward.
  4. The hoop stress is proportional to \(P\): half the pressure, half the wall thickness for the same stress.

Common mistake

Thinking of a Mars habitat as a house. A house on Earth stands up against gravity. A pressurised habitat on Mars is first a pressure vessel; gravity is the smaller load.

Where it leads