That is, if one drywall anchor supports 50 lb, will two support 100 lb? Four support 200 lb? etc.
This should largely be the case as long as the anchors are relatively far apart.
Think of it this way, two 50lbs. anchors on different walls can each support 50lbs, whether or not the load(s) supported is shared between the two or not.
However, any holes in the drywall weaken it slightly, so while an anchor in drywall may be rated for 50lbs, putting two within a few inches of each other will decrease this rating.
I don't have a good answer for how far apart the anchors need to be, but if you can span anchors to 16" apart, you might as well just mount directly into the studs (assuming 16" OC).
In a perfect world (perfect anchors installed perfectly in perfect locations in perfect drywall), each anchor would add support pretty much in a linear fashion as you describe. However, that's not what happens in reality.
Let's assume the mass-produced anchors are all perfectly uniform, so eliminate that variable.
Your technique is not completely uniform, the drywall isn't completely uniform, and the drywall behaves a little differently depending on things like how close you are to a stud. If the anchors require pre-drilling holes, the holes won't be exactly the same in size, or in the extent to which the surrounding material is degraded. If the anchors make their own holes, local differences in the material and how hard you tighten the anchor will make a difference. So each anchor won't be held with the same strength. Different locations on the wall will be subject to different amounts of vibration, which will affect the anchors differently. So you're starting with the anchors having different strengths.
The loads on each anchor are likely to be different. The weight won't be perfectly distributed, so some anchors will bear more weight than others. If the load has mounting holes, the anchors won't be perfectly aligned with them, so only some will actually be doing most of the supporting.
The forces acting on the anchors may not be just gravity pulling straight down. The weight distribution may make the load want to rotate or tilt away from the wall. Each anchor will be affected differently, bearing different loads and being stressed in different ways.
It isn't really the anchor being affected as much as the drywall around the anchor. Drywall is a brittle, crumbly material, and it can gradually breakdown around the anchor over time under stress and vibration.
Over time, the anchor at the weakest point or under the most stress starts to fail. As it does, its load is borne by the other anchors, which stresses them in different ways, and they may end up supporting more weight. The problem isn't necessarily how much load each anchor supports when you install them, it's a cascade effect, with a sequence of weakest links.
Say you have a 100 lb. load, and it's supported at four corners. You put in four anchors rated at 50 lb. each. But the anchors don't exactly match the mounting holes; they will each be slightly off in different directions relative to the mounting hole pattern.
If the load just rests loosely on the anchors, it will be supported only by the couple of anchors that are highest relative to the pattern. If you tighten the load to the anchors, the anchors end up fighting each other. Tightening all of the mounting screws adds additional load by pulling in the direction that each anchor is off. The anchors that are off in the direction of being slightly higher than the mounting hole pattern in the load end up supporting the weight, and the anchors that are lower may add to the load instead of supporting it.
The result is that N anchors don't give you N times the capacity of each anchor. It isn't because of the math, or even mostly due to the materials. It's mainly because you can't install them perfectly.
No...if one supports 50#, then 2 does not automatically support 100#.
Each material has 1) fiber strength, 2) shear strength, and modulus of elasticity (flexural stress). When checking strength of materials and fasteners, all three stress and strain must be determined/checked for each material...including the fastener.
It’s not linear.