Hi Jerry - Do you know what second moment of inertia is? There are beam calculators on the net that's probably the way to go for a layman. So if you have a beam calculator put 1000N as the centre load and then ratio the number.
https://mechanicalc.com/reference/beam-analysis
See attached. Your current beam in bending is about 8N/um. But its the torsion component that will bite you and that's hard because using hand calcs you have to account for the aspect ratio of the beam etc etc. easier to do it with FE for me anyway. Your CAD may have an FE module? cheers Peter
From analysing a few of these things I look at it like this. 1) Pick a target machine static stiffness say 21N/um. 1/3 of this will be connections, 1/3 will be Z axis and 1/3 will be Z axis. ie 7N/um is the component stiffness target then all of these are inefficient say 50% so you end up at ~10N/um which is a stiff machine, if you get 5N/um for your first your doing well. So the gantry has to be, in this example at least 7N/um but this considers the tool in the bottom position so the gantry is bending and torqueing. Torqueing is the current ???
In several other threads the conclusion for an aluminium cutting machine has been a steel gantry 200x200x10mm thick at least. Not saying you have to do this but here's the calc for an equivalent timber beam. Works out 300x300 solid. You got one direction close. By the way this does not have to be to the ends, it just has to be the middle maybe 75% of the gantry. So maybe just beef up the back of the middle third? More thought bubbles... This is interesting to me because my next machine is going to be plywood....