How can be determine the soil bearing capacity at a site?

The generally accepted practice is quite a bit different from the mathematical method. But since you say “of a site”, I assume you are talking about a real world application. The standard in North America and indeed most places is to conduct a geotechnical investigation. This is done as follows:

  1. Drilling a sufficient number of boreholes to obtain a reasonable coverage of the site. Ideally if the final design is known, then these boreholes can be placed directly at locations to provide pertinent information (eg. Within the building footprint).
  2. Testing is done while drilling boreholes. Typically this consists of dropping a weight on a rod and counting the number of blows it takes to penetrate one foot (300mm). The test can be conducted with hollow tube ‘spoons’, which collect a soil sample (split spoon tests or SPT’s), or with a capped closed rod (dynamic cone test). The number of blows counted to drive either of these one foot is called an N value. More on this later.
  3. Additional tests may be done depending on the soil type. Shear vane tests can be done in soft clays, and hand penetrometer testing can be done on recovered clayey samples.
  4. Sometimes a well is installed in the open borehole after drilling it. Later, well readings can be taken to determine the groundwater level.
  5. In the lab, recovered samples from the boreholes are usually tested for a variety of properties, including moisture contents, grain size, atterberg limits, etc.
  6. All the lab and field information is combined into borehole logs. Based on these logs and all the information, a geotechnical engineer assigns a bearing value to the soil. Bearing values are often based on depth, so the bearing capacity at 2m depth may be different than at 6m depth, for example. The number of basement levels, the frost penetration in the area of the site, etc. all can determine the founding depth.
  7. Bearing capacities are usually expressed in Europe and Canada as ULS or SLS. The USA is somewhat murkier, but regardless, ULS or ultimate limit state design represents actual failure, while SLS or serviceability limit state represents failure due to exceeding allowable deflections or settlements, i.e. the serviceability of the building is compromised. Typically ULS bearing values are higher for a sight than SLS, though in some cases they can be the same.

As for how an engineer chooses a bearing value, in general we are very conservative. Typical bearing capacities are round numbers, and maybe something like this.

Site is silt and clay / till

N values of 4–8 = firm, low bearing values, suitable for house construction.

N values of 8–15 = stiff soil, bearing of 225 kPa ULS/ 150kPa SLS available. Suitable for low to mid rise buildings.

N values of 16–29= very stiff soil, 300/200 or possibly 400/250 available. Low to Midrise buildings.

N values of 30+ = hard soil, 400–500 ULS and 250–350 SLS available. Some highrise potentially possible.

For very large highrise buildings and higher bearing capacities, foundations would need to be founded on bedrock.

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