Any permanent or semi-permanent structure on the surface of Mars will need some type of foundation. A foundation anchors a structure to the ground, preferably to the bedrock. This stabilizes the structure against the effects of gravity, wind, and ground movements due to changes in ground moisture (water and carbon dioxide ice) and eventually Marsquakes.
Alternatively, many foundations are of the floating type. This means that after testing the soil, the load bearing capacity is determined and the foundation is built in order to spread the load to less than the load bearing capacity of the soil. That is the main purpose of footings under house walls, for example. In a sense, all building that are not supported directly on the bedrock 'float' on the soil.
Unique Martian Considerations
The environment of Mars presents certain unique factors for foundation design.
The lower gravity means that Martian foundations need not be as strong or as thick as terrestrial counterparts.
Low Air Pressure
Low air pressure means that the Martian winds have a greatly reduced effect upon structures compared to terrestrial winds of the same speeds.
The martian atmosphere has a density of about 2% the one of Earth. As the energy from wind transmitted to a structure is a equation in the form of E=1/2mV2, it is directly proportional to the mass of the atmosphere. So for the same wind velocity the force, and consequently the pressure, on a building will be 50 times less on Mars than on Earth.
The dust in the atmosphere is a tiny fraction of the mass of the gas and exerts negligible pressure on the buildings even in the strongest winds. However, high velocity dust might scratch windows and damages surface finishes or solar cells.
It has been suggested that much of the surface of Mars is a sort of permafrost. Heat generated by structures could cause the water or carbon dioxide in the permafrost to sublimate, resulting in settling of the ground. Any settlement built on this type of ground will need to take measures to protect against this possibility. The loss of CO2 or of water in the ground changes the load bearing capacity of the ground. Permafrost on Mars would be different from permafrost on earth as the frozen material is regolith, and not soil. It is unlikely to be as fluid as melted permafrost and should maintain a fair amount of load bearing capacity.
There is no plant life on Mars, therefore there is no soil. The material on the martian surface is either sand, rock or regolith. Soil has very low load bearing capacity and is generally removed for construction. This would not be an issue on Mars.
A unique characteristic of martian construction is that the pressure inside the building is very high compared to the pressure outside. Depending on how the colony is designed and the interior atmospheric pressure, the pressure on a martian structure should be between 5 and 10 tonnes per meter square of wall or floor surface. So a standard 1000 square foot (100 m2) house on Mars at standard atmospheric pressure would exert 1000 tonnes of force onto its foundations. A large dome, as often illustrated in images of martian settlements will do the same, so a 1000 foot (300m) dome would exert 1502 x π x 10 = 700 000 tonnes of force and require very important foundations.
This is why martian structures are more likely to be pressure vessels; spheres and cylinders with rounded ends, rather than domes or rectangular constructions.
Types of Foundations
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Pressurized habitats and un-pressurized constructions can have different needs as far as foundations go.
Piles can operate as friction elements spreading the load vertically through regolith with low load bearing capacity. Or they can extend down to stable bedrock. Piles reduce the heat transfer to the ground and can be cooled to prevent the melting of permafrost.
A footing is a partial slab under a wall or a column. It reduces the load on the regolith to a safe level below the load bearing capacity of the material.
A slab is a foundation that extends under the entirety of a building, creating a floor and reducing the ground load to the lowest level possible. Slabs on Earth are mostly under compression, however a slab on Mars might serve as part of a pressure vessel and then would be under tension. In such a case it would need to be reinforced with steel bars or other tension resistant materials. Slabs might have high heat transfer to the surrounding ground and might need to be insulated, or actively cooled in permafrost type soils.
Load bearing capacity of martian surface materials
Large areas of Mars appear to be covered with lava flows and rock. The foundations on such materials would be minimal. As there is no liquid water and no soil on Mars, most surface materials should have a high load bearing capacity. Areas with large amounts of water in the regolith might need to be protected from the heat of the settlement to maintain their load bearing capacity. wind blown sand may be difficult to stabilize and build on.
Ice is a plastic material that flows very slowly under pressure. For example, tunnels dug in ice in the american Camp Century base in Greenland, Project Iceworm, closed surprisingly quickly. So construction onto or into a martian glacier (if they exist) would need to be extensively checked before it was started.