Maximum velocity (rpm) for lead bullets

[Idz]

I have heard stories about bullets spinning apart but never saw any calculations of how it occurs. Not wanting to go through the calculus I cheated and pulled two formulas from "Roarks stress and strain formulas"
stress = 1/8(rho)(w^2 * R^2)(3+nu)
bursting stress = 1/3(rho)(w^2 * R^2)

properties of pure lead
tensile strength = 12 MPa =12e6 N/M^2
rho = density = 11300 kg/m^3
nu = Poisson's ratio = .4 to .45, 0.431 most commonly quoted
some definitions
Tw = rifling twist (in/rev)
V = bullet velocity (fps)
rpm = bullet spin (rev/min) = 720*V/Tw
w = bullet spin (rad/sec) = 2*pi*rpm/60 = 0.1047*rpm = 75.4*V/Tw
Db = caliber (inches)
R = bullet radius (meters) = 0.0127*Db

for spinning failure of a lead bullet set tensile strength = bursting stress
12e6 N/m^2 = 1/3(11300 kg/m^3)(w^2)(R^2)
solve for w
w=1/R sqrt(12e6*3/11300) = 56.4/R == 75.4*Vmax/Tw (from definitions)
re-arrange to solve for maximum velocity
Vmax = 0.748 Tw/R
convert from mks metric to caliber
Vmax = 58.9 Tw/Db

so a few examples:
45 acp = 0.452", Tw=16 yields Vmax = 2084 fps
9mm = 0.356", Tw = 16 yields Vmax = 2647 fps
Ar15 = 0.223", Tw =9 yields Vmax = 2377 fps

Lead alloys have a higher tensile strength strength so can go faster but it looks like its unlikely any pistol round can spin apart. I also worked through the problem of stripping the rifling of a lead bullet but its a bit more complicated because I have to use a QuikLoad calculation of velocity vs time to calculate the torques involved.

Anybody care to check my math? How do your experiences compare with this?

[mrbill2]

The semi-annual argument about bullet RPM is in full swing here: http://castboolits.gunloads.com/showthread.php?225805-Applied-Ballistics-for-Long-Range-Shooting-by-Bryan-Litz
I'm sure they can check your math there. Hope this helps.

[62chevy]

No doubt some of those gurus will be along for a lively discussion on the effects of RPM on boolits shortly. :drinks:

[popper]

Please post you rifling stripping calcs.

[geargnasher]

No, Mrbill, comments like that don't help at all.

Come on guys, don't derail this before it even starts. Idz wasn't even referring to the same thing being discussed on the other thread.

Gear

[303Guy]

There may be a link between the two threads - here we're talking about boolits flying apart, BUT what about boolit distortion due to spin? That would cause the issues being discussed there but here the same calculations could throw light on the extent of shape change due to spin. A bullet/boolit will expand radially by a small amount which causes it to shorten slightly. Is that shape change of any significance? Can the shape change go into the yield point (with greater shape change) without flying apart? Lead has a creep property. Can a boolit slowly change shape as it goes down range? Could that explain non-linear accuracy degradation at increased ranges?

[Idz]

Bullets with intact bases that I've recovered from berms show maybe a few thousandths increase in diameter. At reasonable velocities < 1000 fps I don't think you're getting much if any diameter increase.

[youngda9]

With some Tin and Antimony I'm sure that RPM gets increased....any way to calculate how much with common alloys?

[Idz]

youngda9
Just plug in the higher tensile strength of the alloy instead of the 12 MPa for lead. I don't know if anybody has a convenient table of bullet alloy properties. I suspect the densities of most alloys are close enough to lead that you can use the 11300 kg/m^3.

[geargnasher]

youngda9
Just plug in the higher tensile strength of the alloy instead of the 12 MPa for lead. I don't know if anybody has a convenient table of bullet alloy properties. I suspect the densities of most alloys are close enough to lead that you can use the 11300 kg/m^3.

I wonder how much the compressive strength increase due to heat treatment will affect this.

Gear

[youngda9]

I don't know the tensile strengths of various metal compositions. Perhaps it doesn't increase that much...might help explain why some are able to push the RPM threshold higher using different tensile strength metals. Just thinking out loud.

[Green Lizzard]

check your math? just looking at it makes my hair hurt

[Digital Dan]

Hang tough, you'll get used to it. :-)

[SciFiJim]

I was able to slowly follow the math with difficulty. Not sure of the accuracy, but I could generally tell what was going on.

The telling result though is that for a lead boolit,

Max Velocity = 58.9 x (twist rate / caliber)

Hmm, Looking at a common rifle caliber, say 30-06.

Normal twist rate is 1 in 10

Call the size .308

Max Velocity = 58.9 (10/.308)

Max Velocity = 58.9(32.4675)

Max Velocity before bursting the boolit = ~1900 fps

Is that right? I don't own that caliber, but thought people were driving boolits much faster than that.

Different alloys and heat treating may be the way it is accomplished.

I think it would be difficult to find the edge where the boolit would deform to increase diameter and not burst.

[Idz]

Scifijim
1900 fps for a soft lead bullet sounds fairly reasonable. For your .308 example its spinning at a healthy 137,000 rpm. I doubt anybody would use unalloyed soft lead for that load. I haven't found a table of bullet alloy tensile strength but I know pure tin is about 20 MPa. The fastest 30-06 cast load in my Lyman manual is about 1900 fps using 10:1 alloy, even the harder BHN 15 lyman#2 alloy is only driven up to 1950 fps.

[Old Dawg]

Why should we assume that bullets retrieved intact from target berms a few thousands over have anything to do with bullet rpm.

First bullets will expand in the chamber/ throat due to setback caused by acceleration. This is before they have attained any significant rpm. With any luck they are then sized down by the bore. Upon impact they can be shortened by deceleration cause by impact.

Look at the cast lubed bullets you pick up. If you think they have expanded due to centrifugal force explain how there is still lube in the grooves. Is the lube actually stronger than the cast bullet? Is the bond between lube and bullet stronger than the lead bullet?

[Idz]

I don't find intact soft lead bullets in the berm. I do occasionally find one that hit just right and the base is intact. Lead is not an elastic material (you don't see lead springs) so any distortion caused by spinning the bullet is doesn't go away when it stops spinning. Impact isn't going to reduce the diameter of the base but it often expands it.
So if you find a bullet with an intact base you can be pretty sure that was about the diameter of the bullet in flight.

You are probably correct in your assumption that the lube would let go before the lead. I say probably because the lead is 11 times the density of the lube so the centripetal force is higher on the lead. My recovered bullets usually have lube in the groove.

The formula I am using has both the centripetal force (radial stress) and the tangential stress in it. For a spinning solid you have to consider both stresses to calculate a bursting rpm. Even then it is only a guide and not an absolute answer.