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There is a certain sequence, a way to categorize collections of forces that I've been thinking about, but I've never seen a reference to. It's a little bit like the various Legendre Polynomials in it's concept.

It starts with one force acting at the center of mass, we know this gives us $a=\frac{F}{m}$.

If we have two forces that have a net force of zero, then our forces are either in line with the center of mass (where we are either going to compress or stretch an object, perhaps according to Hooke's law) or perpendicular to the line from the force to the center of mass in which case we'll have a torque or some combination of the two.

The next step is where I'm interested. If we have three force that have a zero net force, a zero net torque and a zero net tension/compression, then what is our situation? One of the simplest scenarios is having one large force in the middle (imagine a force perpendicular to the length of a rod or a bridge) and two smaller forces at either end pointing antiparallel to the first. What would be the equation that related the 3 forces to the material properties, the geometry of the rod and the amount the rod bent (a sort of generalization of Hooke's Law)?

Three forces, Zero Net Force, Zero Net Torque, Zero Net Compression/Tension

Or you could have those three forces in line with the rod, one large force in the middle and two smaller forces antiparallel to the first.

Three forces, along the length of the rod.

It seems like there should be more scenarios with three forces with no net force, torque or compression/tension, but I can't see them yet.

If we took this idea further, with four forces we could have a situation like this diagram:

Four forces

That has a zero net force, zero net torque, zero net compression/tension along the length, but has a twisting effect on the material.

I imagine that this concept has been developed, but I the engineer books I've look at have not put things in these terms. I would love to see a reference where this concept has been explored, because it seems like it should be out there somewhere.

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    $\begingroup$ You realize, that there will be compression of the rod that counteracts the forces' tendency to snap it in half? $\endgroup$ – LLlAMnYP Feb 12 '17 at 16:51
  • $\begingroup$ I think the three forces as described wouldn't tend to expand or shorten the entire length of the rod. There would be regions of compression and expansion within the bulk, but that's not what I'm getting at. But I'll clean up the description to "zero net tension/compression". That's not exactly right, but hopefully close enough to tell the story. $\endgroup$ – David Elm Feb 12 '17 at 18:47
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    $\begingroup$ This sounds like the field of beam bending in static mechanics: engineersedge.com/beam_calc_menu.shtml There are many standard scenarious here to cover the different possibilities of where forces are pushing. $\endgroup$ – Steeven Feb 12 '17 at 19:35
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Your first one is an example of a simply supported beam. There is a lot of literature about simply supported beams and similar setups.

In these situations, although there's no net moment or force, there are internal stresses due to the loading. In realistic scenarios these internal stresses create a distortion in the beams depending on how they are loaded. For example, your first beam is given as an example for bending moment.

A topic to look into might be mechanics of materials. Many engineering textbooks cover these topics in great depth.

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