# Stone Bridges, Integrals, SHP.

So, I have this stone bridge, and I’d like to be able to state how to apply SHP damage at it to bring it down.

This quickly got out of hand.

It’s a bridge 80 ft long, 10 ft wide, a walkway 3 ft thick (I’m going by a module text here, don’t judge me. Dwarves did it.). It has a stone “handrail”, in clear violation of scifi/fantasy Anti-OSHA, which I’ll treat as battlement.

The artwork for it seems to set it as a very long 80’ single arch over this 150’ deep canyon rather than just a flat stone plank over the canyon, which seems a bit more realistic, even for dwarves.

If I spindle and mutilate the entry for “Wall, stone castle, 20’ high, 100’ long, 10’ thick” I can say that stone walls have 1 SHP per 13.33 cu ft. (20,000 cu ft div by 1500 SHP)

I’m not real sure how to judge a “breach” in a horizontal structure, espc. given construction type. This bridge is “ridiculously long arch” style; but I don’t think you’d see anything other than arch bridges or simple beam bridges in these-olden DND times.

What would be a more logical SHP for a stone arch bridge, even one that’s a bit fantastical at a single arch over 80’?

The bridge is bookended by two small towers on each side, 30’x20’. Assume the abutments for the bridge extend down the sides of the cliff the same height - 30’. That gives us a parabola that is 80’ wide by 30’ high, that we build the inverse of with stone and slap a 3ft thick walkway on top.

The standard form of the parabolic equation is y = ax^2 + bx.

(math happens)

y = -0.01875x^2 + 1.5x is the parabola we’re working with.

Wolfram Alpha is cool, by the way, type that in to see the archway we’re constructing.

What I’d like to get is an approximation of the volume of stone under different sections of the bridge.

So, I then integrate for the first time in 16 years (using Wolfram Alpha “integrate -0.01875 x^2+1.5 x from 0 to 10” - why can I have this in 2014 and TI calculators are still stuck in the mid-90s?)

I may have done this wrong, but:

The area under the first 10ft of the span is 68.76 ft.
10ft to 20ft: 181.25.
20ft to 30ft: 256.25
30ft to 40ft: 293.75

The middle 10 ft of the bridge, the thinnest part, is 298.438.

SO! The area of a rectangle of stone that is 33 ft high (to add the 3ft walkway thickness) and 10ft wide is 330 sq ft. Subtracting, then, the area of our integral for each 10 ft section then applying width (10ft wide bridge), we can say:

First 10 ft: 261.24 sq ft - 2612.4 cu ft - 196 SHP
Second 10 ft: 148.74 sq ft - 1487.5 cu ft - 112 SHP
Third 10 ft: 73.75 sq ft - 737.5 cu ft - 55 SHP
Fourth 10 ft: 36.25 sq ft - 362.5 cu ft - 27 SHP

And then the second 40 ft span reverses those counts. The bridge as a whole has ~780 SHP; adding the battlements, that’s 940 SHP.

If we go as far as to say the bridge will collapse at 2/3rd SHP, that’s still ~333 SHP in damage and will take a while with artillery, counting misses, the bridge is strong stone so it’s AC 6.

A mage with Disintegrate can take out 125 SHP of bridge, which is most of the middle - the middle 10 ft of the bridge at it’s thinnest point is 43 SHP - 23 for the bridge and 20 for the battlements on either side.

Stone to Flesh would be a disgustingly entertaining way to do it. Wolfram tells me the “volume of a man” is 2.345 cu ft, which, sadly, is not even 1 SHP, so we’ve only succeeded in being gross.

Transmute Rock to Mud would make short work of one of the abutments, slipping it from it’s supporting natural stone, and at 5th level is probably the most efficient way - 625 SHP definitely kills the span.

So, as long as I can accept this bridge can be built this way at all, I at least know how to take it down.

Assuming I didn’t frak this all up, that is. I welcome any corrections, cause I’m pretty sure Diff Eq killed that part of my brain, so this was a lot of flailing to get to this point.

I’m having trouble visualizing the bridge. Which of the following does it resemble?

For what it’s worth, ACKS assumes that 1 cubic foot of stone weighs 160lbs and that each ton of stone (2,000lbs) has approximately 1 SHP.

It’s closest to the second link; assume that where the arch touches the ground, sheer cliffs rise up instead of those primary pylons.

Also, it’s solid through the middle, rather than the pylons.

If you type “y = -0.01875x^2 + 1.5x” into Wolfram, it’ll draw the curve of the bottom.

Alternatively, I’ve taken that shot from Wolfram and drawn an incredibly crude side view: