Forces in X,Y and Z

[m_c]

What about the torque reaction from the cutting tool on the Z axis, there must be a twisting motion too there.

HSM Advisor gives tool cutting force and torque etc.

That is your main force that you need to allow for. If you know what torque the cutter needs, you can then calculate the forces required to move the X & Y axis against the cutter, which is the force you'll need for any FEA. However for the axis motion system, you need to add extra to cover any friction/stiction and any inherent losses in the motion system.
It's the Z axis where things get a bit more complicated, but your main force there is going to be while drilling or rotary broaching. Do neither of those, and you're biggest force is going to be from spiral milling cutters continually pulling/pushing the axis down/up.
I've got a couple spreadsheets somewhere for drilling/broaching forces, which I made to work things out for my lathe, however for drilling it won't calculate pull through force, which can potentially pull the Z-axis straight down into the table if the drill grabs as it breaks through the material.

[routercnc]

I believe what the graph shows is the total measured force between the tool and the table (Fx, Fy, Fz) when using a sensor between the part and the bed, so includes the all cutting forces resolved into x,y,z:
Feed force - in direction of cut
Feed normal force - 90 degrees from direction of cut
Feed passive force - feed up/down due to helix)
all of which include any moments caused by cutting

They do talk about another sensor which goes into spindle (RCM) and so torque can be measured directly and extracted out separately.

But in terms of machine design I would have thought you want to know the total force Fx, Fy, Fz on the tool and a target stiffness to aim for. Then use FEA, or hand calcs to work out if you meet the required stiffness when applying the force at the tool tip.

For stiffness targets I've seen figures quoted for commercial machines in the region of 10 - 25 N/um (10,000 - 25,000 N/mm). I think one of my DIY machines was between 1000 - 2400 N/mm and was ~OK but I would aim higher for a good machine and I think it is easily possible.

[PotatoMill]

10-25 N/μm seems reasonable, and a good number to aim for. I searched around for CNC N/μm and found a good preview on work-piece precision https://books.google.no/books?id=0zF...%20cnc&f=false .

Seems like the "hardcore" machine centers have a stiffness around;
X axis: 30-62 N/μm
Y axis: 30-40 N/μm
Z axis: 67-95 N/μm
Radial: 25 N/μm (5000rpm), 20 N/μm (25000rpm)
This is for vertical CNC mill setup. So for a router it would be a bitt less.

According to the link above it seems like the tools are the weakest and the least stiff part of the machine. Where they use the usual deflection calculation for a circular cross section and match it to some real measurements. 0.15 N/μm for 6mm tool.