If I'm working with standard PLA, and I want to print a box that I can stand on without any risk of it breaking, is there any good way to calculate the appropriate print settings?
I know that structural strength comes from the infill. Knowing this, and knowing the dimensions of the box, the weight of my body, the surface area of my shoes, and the material I'm working with, is there any good way to determine the minimum infill percentage I'd want to use in order to safely bear my weight?
Strictly speaking, it is difficult to do calculations on these materials, but not impossible (I've heard about a few commercial analysis tools that do that). The FDM process (Fused Deposition Modeling) creates a product based of fused slices of material causing an anisotropic material (this means that the properties of the material are different in different dimensions). Basically, your product will be quite strong and similar in the X and Y directions, but fragile in the Z direction (layering direction). You can imagine that every layer may be a seed for cracks to grow when you're pulling at the part.
When applying a compression load on a product like in your example, the walls need to be strong enough to hold the pressure (not all of the load as, based on the type of infill, the infill also can/should take part of the load!) and need to be of sufficiently high percentage, not only to take part of the load, but also support the walls to prevent buckling. I remember that stress calculations for buckling are difficult and require FEA (Finite Element Analysis) for more complex objects other than bars or beams.
I think it is difficult to determine or calculate the infill percentage based on the compression load beforehand as you do not know the exact material properties and the buckling behavior. You do know that a 100% infill will give you enough strength and support, you could try to print at a lower infill, e.g. 75%, and test if that works for you.
I would suggest doing some calibration runs - granted this'll use up a lot of time and filament. But an infill of even 30 to 40%, plus a reasonably thick set of walls and top/bottom layers, should have almost the same strength as a 100% infill. Look at the girders & beams on road bridges, for example. As Oscar wrote, modelling with FEA tools is unreliable, more so because every extrusion printer is a little different.
Try printing a test box, say only 10 by 10 cm, same height, and see if you can stand on that, before printing the full-size item.
A fast way to do this is by using SolidWorks.
You can draw the box in it and run a simulation test with the max load expected.
Here is a link on how to make dynamic load simulations work in SolidWorks, How to apply dynamic load in solidworks simulation ?
The catch in the process is that SolidWorks takes cubes and most objects as complete solids, i.e. 100% infill in 3D printer terms.
You would have to actually design your infill pattern into the cube so as to get the best and most accurate result.
