Assuming a Cartesian printer with a belt and smooth rod design in which one axis moves another (i.e. the X-axis rails 'sit' on the Y-axis rails), what are the main considerations in sizing the rods and belts? For example, given a base design using 8 mm diameter rods and 6 mm belts (assume these are the limiting factors of the printer and that the frame, etc. can handle whatever you throw at it), what is roughly the maximum load, print speed and build size that this should be expected to support? If you were to increase the rod diameters to 10 mm or even 12 mm on one or both axes (assume the steppers could handle the increased load), what would the increased rigidity buy you in terms of maximum speed and/or build size and would 6 mm belts still be appropriate? Ballpark calculations or rules of thumb are fine as I understand the variables are likely not trivial and am looking more for a rough range of guidance to understand the trade-offs involved.
8 mm rods and 6 mm GT2 belts are generally accepted as a good tradeoff between price and performance, an exact calculation is possible but might not be very relevant if another part is flexing. Also, generally speaking, the smaller the part the sooner it will wear out of specification. Thus your service interval might be higher compared to an over-engineered printer.
In short, it depends on what your goal is, if you desire low maintenance and accurate machine, you might be better off with heavier gauge parts. Obviously, this will also affect the speed of printing. A 6 mm GT2 belt might have a higher stretch factor compared to a 10 mm belt, but can be mitigated by adjusting the acceleration. In addition, a 10 mm belt has a larger pulley reducing the number of steps per mm, lowering precision. As such you might be better of using two 6 mm belts.
Increased rod size for the print bed will not affect printing speed much but might help with accuracy since the bending modulus is lower. Play around with the calculators below to get an idea of the force your beam will have to withstand. That said, there are a lot of other factors that will flex under load, for example, the bed leveling springs. You can replace them with solid spacers, but that might warp the bed when it heats up.
To conclude, I would use the calculators to figure out if the 8 mm rods are within tolerance for the intended speeds and load, but don't forget to look at the overall picture. The quality of parts you choose is one such thing.
The 3D printing revolution started out with the idea/community project to build self-replicating machines:
RepRap was the first of the low-cost 3D printers, and the RepRap Project started the open-source 3D printer revolution. It has become the most widely-used 3D printer among the global members of the Maker Community. (From RepRap.org)
The main aspects for this project was to build self-replicating machines from cheap and "simple" available materials and making them freely available for the benefit of everyone. The rod solution is a simple and affordable solution for linear motion with fair tolerances.
As far as the rigidity of the X carriage, rods aren't the best solution, increasing rod diameter will surely increase the stiffness, but it will be smaller than a design that uses a (quality) linear rail, these are much stiffer. Note that instead of steel, carbon rods can be used, these are stiff and light and reduce the weight of the carriage, allowing for higher acceleration and speed. The solution of using rods is mainly based on availability and being a cheap solution.
Do note that rods flex much more than linear rails, even when you increase the diameter, this will limit the length of the X-axis.
Any decent rail will have far less give. In terms of 3D printing, this leads to greatly reduced backlash and ringing artifacts, yielding cleaner print surfaces and edges.
Belts allow the rotational motion to be transferred into a linear motion. Belt width depends on the force (carriage weight) you need to move. Same as for rods, an increased height will allow for a larger load to be driven/moved, but for a fixed load, the increasing height allows for less stretch and thus less printing defects. Note the different types of belts are available, all having their own stiffness and load bearing capacity.