Structural considerations are going to revolve around mass, stiffness & degrees of freedom (motion) requirements relative to your stool's functions.
Degrees of FreedomWe discussed DOF* in how we want the base to move relative to the world. Still we may wish to leave a DOF free or only partially constrained within the structure for instance elevation and yaw (swivel). You may want a height adjustment on your stool so pneumatic cylinders may be an option, also threaded wooden rods (piano stools). Cylinders allow some cushioning potential as well in that direction. Either one usually also means you will have a swivel, which may be desired. Some sort of 4 bar linkage could be created to allow 2 elevations or folding away, but can't seem to find any examples for that beyond the convertible kitchen stool/step combinations.
MassMass is going to be a consideration if your stool is to be mobile and especially if you intend on picking it up to move it (as opposed to rolling) Woodworkers tend to overbuild everything, which means too much mass. To illustrate,
Another mass factor to consider is where the Center of Gravity will be, especially relative to any handhold (should be vertically in line with CG for minimum frustration) or if there will be tipping forces present.
StiffnessStiffness is related to mass, as whatever material we use has a given density and modulus of elasticity. Still, design geometry plays the more significant roll. Consider standing carefully on an empty aluminum can vs an equivalent mass of aluminum foil arranged as you wish. Likewise a structure that keeps wire members in tension is very stiff relative to mass (suspension bridge.) or the Eames Molded Chair frame
Ideally, you could simulate all expected load cases on your design with some sort of finite element analysis (FEA) model optimizing it so none of your materials go beyond their elastic limits. In reality, this would be a ton of work and a bit of unreliability is involved given the inconsistency of sawn wood as a material for razor edge designs (if you can even really understand all the potential use cases; is a 300lb teenager rocking on the back legs a valid design criteria?), So most are going to just overbuild the geometry a little and if it flexes, then they'll add a brace etc. The problem with this is every structure flexes and acts like a spring to some extent, it's just a question of how much, and adding that brace can cause flexing in other directions/locations...
Flexing components and especially joinery past their limits will cause catastrophic failures, but flex-body design can be used to good effect when kept within reason. Consider the difference between the structure of 4 leg pedestal base table vs. 4 leg table with aprons. A pedestal base is very stiff and so it wobbles on an uneven floor. The apron table typically might touch on diagonals initially (setting up for the wobble) but if the top and aprons together aren't too stiff (likely given their size), it will flex like a hinge a small amount (staying within the elastic range and allowing it to bounce back when moved) and make contact on all four legs under it's own weight. A similar thing happens with the typical 5 legged rolling office chair, note how the legs are on arms (that sounds weird) when on an un-flat floor (within reason) the "high point" arm will flex a bit until the other arms start to take some of the flex until finally most or all will bear the combined weight (and stay within the elastic limit). If the 5 legs were more like tripod legs (going straight toward the swivel) then in addition to the legs getting in the way of curling your legs under you, I suspect it would be too rigid and you'd get the dreaded wobble. You could also use the flexing of a springy wood to give a slight suspension effect mimicking the spring pole lathe or bow's principles.
Hopefully I've given you some things to think about in designing your shop stool and beyond. In my next post, I'll discuss how I plan on combining these variables (along with the short build time-line) into my own design.
*If you've enjoyed this sort of technical engineering stuff, I recommend this short design reference, it's excellent:
Exact Constraint: Machine Design Using Kinematic Processing
Douglass L. Blanding