>Does the engineer find a large enough weight that deforms the bike, then take a fraction of that as the amount it can support?
Basically, yes.
The materials used by the engineer has known mechanical properties. Those properties are used in design calculations to determine the stress in the structure. Sometimes, this is as easy as F/A, but most of the time it requires several calculations to find all the stress states.
After determining that, a factor of safety may be applied. This just describes how much “extra” load bearing capability the design has. For example, if the rated load for an exercise bike is 250 lb but it has a factor of safety of 2, it can actually hold up to 500 lb before it starts to deform.
This process can start with either determining what the required load capability needs to be, or determining that after the design has been completed. Although usually there are multiple iterations of the design, so you bounce back and forth between these two approaches.
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