Structural HP and AC

Interesting houserule, though I'd note that if you do this with the ships in the core book, they become 3.5-4.5x as hard to kill

I did a deep dive into ballistic weapon damage while researching Guns of War, as it was tangential to some other projects I was working on for ACKS post-apocalypse and ACKS cyberpunk.

It turned out to be both robustly studied and yet also impossibly complex to model in a tabletop game. As far as I have been able to gather, there are four basic types of penetration, and each of them has to be modeled differently if one wants to be scientific:

• Fracture due to inertial stress wave Compressive waves propagate into the plate upon impact. If the stress magnitude of this wave exceeds the dynamic yield strength of the target, failure may occur in an unconfined region of the target plate. For a plate target, failure or fracture would occur near the rear target surface. The probability of this type of failure decreases with an increase in target density, hardness or compressive yield or ultimate strength
• Radial fracture behind initial wave front Tensile radial stresses are built up as the compressive wave propagates away from the impact sight. If the target material behavior is tensile and the magnitude of the built up stresses are higher than the ultimate dynamic tensile strength, radial and/or circumferential cracks may occur. The hoop or circumferential stresses will be tensile because of the Poisson’s effect, as the compressive wave propagates outward. Radial cracks are caused by this circumferential tensile stress.
• Spallation The compressive waves reflect from the rear surface as tensile waves. First, the tensile wave cancels the compressive wave. As the compressive wave propagates to the back of the plate, the amplitude of the compressive wave decays, than the net tensile stress may exceed the ultimate dynamic tensile strength of the target material. In this case, a tensile fracture will occur.
• Plugging This type of failure occurs when the projectile pushes a plate plug through the rear surface of the plate. This plug has approximately an equal radius to that of the deformed projectile. As the hardness of the plate is increased (related to the yield strength or the hardness number), the tendency for plugging increases. The reason is that, it becomes harder for the plate material to be pushed radially outward by the projectile. Thus a narrow shear zone builds up in front of the projectile in the periphery region and the plastic flow is confined to this region. Other parameters that effect the formation of plugging are the relative plate thickness and projectile nose shape. Plugging occurs more easily in thinner plates such that even softer plates may fail in this in case that the impact velocity is not sufficiently close to the ballistic limit so that radial momentum transfer causes severe plate bending.

I decided to use "plugging" as the model I'd work from, as it is the most likely outcome for both thick,hard plates and for thin, soft materials.  

In plugging, the material in front of the projectile from the face to the back fails IN ONE PIECE by being force out of the projectile's path out the plate back, either as a cork-like plug or by being torn in the middle and folded back to the sides as thick petals, all as one solid piece of material the total plate thickness deep. This can be approximated by a flat-nosed projectile slicing out a disk of armor like a cookie cutter, but it also occurs in moderately-thick plates hit by pointed projectiles when the resistance is primarily by tearing open in the center over the tip of the point and wedging the hole open by petal-formation, that is, by pushing the plate material forward as it splits open in the center and then having it bend to the sides, since it is still attached to the armor ringing the impact site.

• For the plate thickness T, the plug will not move at all (it is assumed to be incompressible) until the applied force is greater than F_r, the resistance force, and thus allows the entire plug to move as one piece.
• F_r, the resistance force is friction resistance along the sides of the plug, like a cork in the bottle. 
• Assume that, like a cork in the bottle, the friction resistance along sides of the plug to be pushed out is constant everywhere (reasonable enough), so that the resistance force F_r = K_rT for some constant K_r. (The thicker the plug, the  more resistant force it exerts.)
• The Kinetic Energy required to push the plug is the amount of distance moved (its thickness T) against a resistance force (K_rT), so Work = K_rTT = K_rT².
• The KE available is the projectile's motion energy KE = (0.5)(Projectile Mass M)V²
  • To barely penetrate the plate, they are equal: (0.5)MV² = K_rT²
• Since damage is effectively depth of penetration T, then it is proportional to the square root of Kinetic Energy. 

Another similar analysis can be found here - http://forums.sjgames.com/showthread.php?p=135955. I quote:

"It depends on what sort of material is being penetrated. This gets complicated, but the short answer is that

• For materials that respond elastically, the depth of penetration is roughly proportional to the square root of the energy.
• For non-elastic materials that can simply be "crushed" out of the way, penetration is proportional to the energy.
• For materials that primarily interact with a projectile though viscosity, penetration is proportional to the momentum
• For materials that are non-viscous fluids penetration is formally infinite, but the energy and velocity of the projectile decrease exponentially with distance until you reach a velocity regime where non-hydrodynamic forces dominate.

Things used for armor tend to be elastic solids. This means when exposed to a force, they will deform to distribute the strain of the force over a significant part of the volume of the solid. The thicker it is, the more the "layers" behind the first "layer" can buttress the first layer, helping it to resist deformation and failure (there are not really layers, since the material is uniform).

For this reason, the GURPS formulas for damage where damage is roughly proportional to the [square root of the] kinetic energy and DR is roughly proportional to armor thickness tends to reflect real life trends.

Now, things get more complicated when the armor gets thicker than the typical radius of elastic response, or when the projectile is travelling supersonically in the armor medium, so it simply does not have time to respond elastically (in either of these cases, penetration is going to be roughly linear with energy).

Things also get complicated when the projectile itself can deform - in an extreme case you would treat it as a fluid plume interacting hydrodynamically with the armor medium (shaped charge explosive jets fall into this regime)."

No one ever needs to apologize for being pedantic on the Autarch forums!

The amount of powder used was taken straight from the sources. The era I worked from was 100-200 years earlier than the data you are working from and the firearms, the cannon, and the powder were all more primitive.

The kinetic energies were calculated by reverse-engineering from cannon's range and ball weight.