What weight of Martian soil can be supported by the top of an inflatable habitat pressurized at Earth's atmosphere? While habitat's shape is a huge factor, this has 20 meter modules, semi-cylindrical with a 20 meter radius and only the top half is supporting the soil.
a 2,5 meter thick deposit of regolith is sufficient to shield from cosmic radiation. 6 meters would be equivalent to the effect of the Earth´s atmosphere. But the weight from 2,5 meters of deposit on the ceiling would be a fraction of the inner pressure force only . with earth (sea level) atmospheric pressure the force is 100KN/m2 while the weight of the deposit only would be 1500kg/m3 *2.5 *0,38g =14,25KN /m2. If you use lose regolith on the ceiling instead of a mars concrete, then the material would be an abundant resource and the deposit could be thicker, with some other advantages, as heat storage and protection from meteorites. But with a regolith layer of 17,5 meters, the effort for the excavators to shovel all the regolith on the ceiling becomes very high, So i would limit the deposit to 2,5 - 4 meters of thickness.
Since the inner pressure forces always will cause a much higher stress on the material than the weight of the deposit could outbalance , I suggest to reduce the inner pressure to half atmospheric pressure ( 500mbar) which is the given pressure on the peak of a 4000m high mountain and raise the oxygen level to 30%.
the free span of the membrane should not be too much, since the tensile force on the membrane material increases with the free span at the same pressure level
on the following link you see a concept of an inflatable mars habitat with deposit on the ceiling. but transparent fee side walls, that receive visible sunlight without cosmic radiation reflected from a mirror membrane: www.marshabitat.space
You can get a ballpark figure by assuming the roof of the habitat is flat, supported entirely be air pressure and doesn't weigh much in and of itself. Your diagram suggests a full Earth atmosphere inside, which would be nice for the occupants, but means shipping or finding a lot of nitrogen. Probably a lower pressure with a higher oxygen percentage would do. However a full atmosphere is also a nice round number so we'll go with that. So we need a weight of 100 kNewtons on each square meter of roof, which means a mass of about 26500 kg (just divide by Mars surface gravity). https://www.lpi.usra.edu/meetings/LPSC98/pdf/1690.pdf gives a density of about 1500 kg/m^3 for Mars soil, so you'd need a bit under 18m of Mars soil to do the job. Note that Mars soils is pretty porous, so if you compact it, or add a binder of some kind to make "Marscrete" you will need less thickness. As previous replies have suggested this is a lot more that you would need for radiation shielding, however it does have other uses -- thermal insulation is one, and simply holding the roof down against air pressure is another (less structure needed).
That's a difficult question to answer without being flagged for "opinions" because nobody has really designed an inflatable mars habitat yet to be able to give us those exact specs. The best anyone can do is tell you how much radiation is on mars and how much Martian soil is required to provide enough of a halving-distance to get exposure down to around what background levels are on earth.
According to instruments on board the mars probes radiation exposure levels on the martian surface are 11 mSv per year, for reference the regular background level on earth is on average about 1.26 mSv per year (changes with location on our planet but thats about average.) Theoretically one could sustain exposure up to 50 mSv per year but that is really just the maximum dosage one can withstand a year without getting sick or experiencing increased health risks. The reality is that most scientific or physics facilities consider 10 mSV a year to be their actual safe exposure rate with 50 mSv as the maximum tolerable by the human body. So on the Martian surface colonists wouldn't be in immediate danger but would reach the maximum career allowed exposure after about 60 years. Naturally one would rather not reach any kind of limit or ever be exposed to radiation above average earth background at all, so shielding on shelters is highly recommended.
The amount of martian soil required to lower indoor exposure rates down to the same level that we receive here on earth one needs about 4 to 5 meters, but this is assuming your habitat itself does not consist out of various materials that would assist in radiation shielding. Realistically the habitat could be made out of an assortment of materials that would reduce the amount of soil needed for shielding. So, in summary, I'm not trying to give you a vague answer, its just that nobody has really announced a finalized design yet so the closest I can get for you is to say that you need 4 to 5 meters of soil for radiation shielding, and less than that proportional to how much shielding the shelter itself provides.
