Shorter Barrels are more accurate then longer ones, dont take my word for it, take Mr. Lija's.
BY DANIEL LILJA
While there are a number of factors that determine the accuracy of a rifle barrel, one of the more critical elements is its stiffness or rigidity. Obviously the larger in diameter a barrel is, the stiffer it will be. Almost as obviously, as the length of a barrel increases it becomes more limber. So, there is a trade off of sorts if our goal is a stiff barrel, and as a result a potentially more accurate one. If we are limited to a barrel of a certain weight, as we are with a varmint or hunter class rifle, the compromise becomes length verses diameter.
*
Here is a Jay Young railgun with a 2" diameter barrel*
in a barrel-block. *This is a super-stiff barrel.
When a cartridge is fired in a chamber, the barrel undergoes many stresses. It begins to vibrate when the firing pin starts its fall and these vibrations increase dramatically from then on. During recoil and while the bullet is still in the barrel, the barrel will whip vertically. This happens because the thrust axis of the rifle is above the centerline of the stock. During recoil the barrel comes back and up. The muzzle will lag behind the rest of the barrel in this movement and the vertical whipping motion is set up. While these vibrations of the barrel are very small, they do exist. The stiffer a barrel is, the less the muzzle will jump around. This brief description is of course an oversimplification of the dynamics that take place, but they do point out the type of barrel movements occurring and why a stiffer barrel is more accurate.
The stiffness of a barrel can be determined mathematically by knowing a little about the physics involved. A free-floating rifle barrel is a classic example of a cantilevered beam supported at one end and with a load applied to the other, the muzzle end. The basic formula for calculation of muzzle deflection is:
D = (W*l^3)/3*E*Ix
where D is the deflection at the muzzle in inches, W is the force or load applied at the muzzle in pounds, l is the free length of the barrel in inches (not including threads), E is the modulus of elasticity or Young's modulus for the barrel material, and Ix is the moment of inertia for the barrel.
While it is difficult if not impossible to determine the exact force on the muzzle, we can compare the stiffness of one barrel to another by plugging the same load (W) into the formula. In all of the examples shown, this force will be one pound.
The length of a barrel is easily measured. Perhaps surprisingly though, we can see from the formula that the cube of the length is used. This indicates that rigidity decreases greatly as barrel length increases.
The modulus of elasticity is a constant, and in the case of steel it is 30 million PSI. Regardless of the heat treatment of the steel or its alloy, this modulus does not change. So unless we are dealing with a barrel made from some other material, this part of the equation remains the same.
Most gunsmiths are familiar with a somewhat similar example to our rifle barrel, modeled after a beam supported at one end: the boring bar used on a lathe. The farther out the boring bar is slid from its holder, the more easily it deflects during a cut. Increasing its diameter would help greatly, but the diameter of the hole being bored limits bar size. For this reason, the best quality boring bars available are made from solid carbide - that is, the bar itself is carbide. The modulus of elasticity for carbide is about 94 million PSI, or over three times that of steel. As a result, the carbide bars are more than 3 times as still.
With rifle barrels we are limited to steel, but this example shows the importance that the modulus of elasticity plays in overall stiffness.
The moment of inertia of a barrel is the most difficult part to calculate. It is a measure of the cross sectional area of the barrel. A larger diameter barrel will have a higher moment of inertia value and as a result will be stiffer. Calculating the moment for a straight cylinder barrel is relatively easy though. The equation looks like:
Ix = Pi *(D1^4 - D2^4)/64
where; again, Ix is the moment of inertia; Pi is 3.1416; D1 is the outside diameter of the barrel and D2 is the barrel groove diameter. From this equation we can see that the moment of inertia increases with the fourth power of the diameter. As an example of this significance it might be interesting to note that a 2" diameter barrel is 16 times stiffer than a 1" diameter barrel because 2^4 is 16.
Figuring the moment of inertia for a straight tapered barrel, such as a heavy varmint barrel, is much more difficult requiring an integration over its entire length. While I was shooting at the 1991 NBRSA Nationals in Midland, Texas, I had the good fortune of sitting next to my friend Mel Klasi of Rapid City, South Dakota. For those who don't know Mel, he is a professor of Civil Engineering at the South Dakota School of Mines and Technology. Sometime during the week I mentioned to Mel that I was working on a computer program to calculate barrel stiffness. At that time I had the program working for unlimited type barrels but told Mel that I was not getting anywhere fast on the integration aspect for tapered barrels. Mel volunteered to derive the equations for me and a few weeks after getting home I got a nice little floppy disk in the mail that did the calculations. I'm very grateful to Mel for his efforts in this. Without his input, I probably wouldn't be writing this article. Mel's equations are too lengthy and involved to show them here though.
When dealing with a weight limit for a rifle barrel, as we usually are, what we want then is the highest moment of inertia for the barrel. As we pointed out, increasing the diameter is the way to achieve this. We are limited by NBRSA rules however to a maximum set of barrel dimensions. This maximum barrel is one that is 1.250" in diameter for a length not greater than 5" at the chamber end with a straight taper to a diameter of .900" at a length of 29". This is a taper of .01458" per inch. A heavy varmint class rifle can easily make weight utilizing this maximum barrel configuration. The amount of weight left from 13.5 pounds for the barrel usually determines the barrel length. Often it is from 22-25 inches long.
When dealing with a light varmint rifle however, limited to 10.5 pounds total weight, usually about half of the weight is in the barrel or about 5 pounds 4 ounces. To utilize a maximum heavy varmint contour barrel and have it weigh 5 and a quarter pounds means the barrel can be no longer than about 18.60". This is shorter than most will accept. The trend today is, from my observation, for a light varmint barrel of around 22". Some like them even longer, up to 24" or so. To get these longer barrels we have several choices; 1) we can cut off some of the chamber end of a heavy varmint barrel and use more of the skinny end; 2) we can use a smaller diameter barrel blank, perhaps having more taper per inch, or; 3) the barrel can be fluted.
If the barrel is fluted a maximum heavy varmint blank is usually used and for good reason. As we saw above, the larger the diameter, the stiffer it is going to be. Using a heavy varmint blank then gives us the maximum diameter allowed. Fluting a barrel removes weight, up to one pound or so depending on flute size. It also lowers a barrel's moment of inertia value but not by very much. Some have the mistaken idea that fluting alone increases the stiffness of a barrel. This is not true. The fluted barrel of a given weight and length will be stiffer than an unfluted barrel of the same weight. The fluted barrel will not be stiffer than the same taper and length barrel that is not fluted though.
If either of the first two options for selecting a 5 pound 4 ounce barrel mentioned above were used, I wondered what barrel contour would be the stiffest. To find out I incorporated the rigidity calculations mentioned already with the barrel weight calculations that I wrote about in the February 1990 issue of NBRSA NEWS. I was then able to easily calculate with the computer, the weights, lengths, diameters, and muzzle deflection for a variety of straight taper barrel configurations all weighing 5 pounds - 4 ounces. The results of this can be seen in the accompanying table. Without the aid of a computer, these calculations would have been so time consuming and lengthy that they never would have been done, at least not by me.
I also made some calculations for maximum heavy varmint dimension barrels of varying lengths. I did the same for different diameter straight cylinder, unlimited type barrels, all of the same length. The results were revealing.
For example, I found that a 24" heavy varmint barrel is 32% stiffer than a 26" version. A 22" HV barrel is about 35% stiffer than that 24" barrel and an 18" HV barrel is 98% stiffer than the 22" barrel. Remember the earlier statement about length being raised to the third power in the deflection equation? The 18" barrel is 3.51 times stiffer than the 26" barrel. Interesting?
I left the lengths of the unlimited barrels the same, 24" with a 1" thread shank, leaving 23" of free barrel. I found that the 1.350" diameter barrel was 36% stiffer than the 1.250" diameter barrel and that the 1.450" diameter barrel was 33% stiffer than the 1.350" example. Because I've made a few barrels from 1 7/8" diameter material, I wondered about them, too. A 1.850" diameter barrel is 2.65 times stiffer than the 1.450" barrel and it is also 4.80 times more still than the 1.250" barrel. Again this emphasizes the importance played by diameter and the resulting higher moment of inertia in rigidity. They play it to the tune of diameter raised to the fourth power.
When I calculated the deflection for the barrels all weighing 5 pounds - 4 ounces, there were a few surprises. Remember now, these examples all weighed the same but contour and length changed. The stiffest barrel was the already mentioned 18.60" long maximum heavy varmint barrel. The second stiffest barrel was one from what I call a 1.200" heavy varmint blank. It is just like a maximum heavy varmint blank but its diameter is .050" smaller its full length. This barrel is almost 21" long and utilizes all 5" of the 1.200" diameter cylinder. It fared so well because of the full use of the cylinder before the taper. The next stiffest barrel was a maximum heavy varmint blank with four of its five inches of 1.250" diameter cylinder cut off. With a 1" long thread shank (as all the examples have), this left no cylinder in front of the receiver. This barrel was 20.92" long to make the 5 and one quarter pound weight limit. At this length, almost 21", it is long enough that most shooter would accept it. That is probably not true of the short 18.6" long maximum heavy varmint barrel that was the stiffest. This barrel and the previous one are really of the same stiffness for all practical applications, just .000012" of deflection difference between them. The next stiffest barrel surprised me. It was a maximum Hunter class diameter barrel using all four inches of allowable 1.250" diameter shank. It came in at 21.85" in length and has a taper of .02273" per inch. The reason this barrel fared so well has to do with its maximum 1.250" diameter shank at the receiver. Remember the importance of diameter in the moment of inertia calculation. Even over the entire barrel length integration, this maximum diameter, at the breech end, was important. In all honesty, I thought this Hunter barrel would do poorly and used it as an example just to see how badly it would do. Everyone knows how whippy looking those Hunter barrels are.
The worst barrel was the longest - almost 24" long, with a straight taper from 1.200" diameter at the receiver but with no cylinder section. For interest, I also calculated the maximum length 1.250" straight cylinder barrel that would still make weight. It was about 16.5" long and obviously the stiffest of the bunch. It is not legal for use in matches though for two reasons, one, it doesn't meet the diameter requirements of the rulebook, and it is also shorter than the minimum 18" required by the rules. This barrel is the first one listed on the chart under 5 pound - 4 ounce barrels.
Looking at the chart, we can see that as the length of the barrel increases, its stiffness decreases. The stiffest barrel is the shortest and the longest one is the most limber. With just a .050" difference in cylinder diameters in these barrels, the full importance of barrel diameter is not realized as it is with the unlimited barrels.
Another question is, does the caliber of the barrel make any difference in stiffness? The answer: yes it does, but not a great deal. To find out I ran calculations for two 1.250" diameter straight cylinder barrels both 23" long. One of them was a .224 caliber and the other .308 caliber. As might be assumed the 22 was stiffer but not by that much. It would deflect .000112" under the one pound load and the 30 would bend .000132". This means the 22 is about 17% stiffer. In the moment of inertia formula, the caliber is raised to the fourth power and subtracted from the outside diameter raised to the fourth power. When dealing with decimals this results in a fairly small number in the equation from the inside hole diameter.
We can see from these examples that the short and fat barrel is the stiffest. To a degree, fatness is better than shortness (don't forget we're discussing barrels now). The reason goes back to the original equation where length is raised to the third power but diameter is raised to the fourth power in moment of inertia calculations. So, if you need more weight in a barrel and you're wondering whether to make the barrel longer or larger in diameter, especially at the receiver end, I'd suggest going with the increased diameter.
This discussion about length brings me to another point. I think one of the reasons that barrel block unlimited-type rifles shoot so well has to do with the block. Depending on how the barrel is clamped, the free end of the barrel that is left to vibrate is that part of the barrel beyond the block. It is not quite that simple because everything in contact with the barrel is going to vibrate but the block will surely dampen most of the whipping from the clamped portion of the barrel back to the receiver. With a 24" barrel and a 1" thread shank and a 6" block, that leaves about 17" of barrel to vibrate. If we take as an example two 1.450" diameter barrels, one blocked as just mentioned and the other free-floating in front of a conventionally bedded receiver, we are comparing a 17" barrel and a 24" barrel. The difference in rigidity may surprise you; it did me. The shorter barrel is 2.48 times stiffer.
With a long barrel on a long-range rifle or 1000 yard benchrest rifle, the barrel blocks become even more important. With a 30" long barrel, not only is there much unsupported barrel able to vibrate, but that extra length adds up to quite a bit of weight. Hanging 15 pounds of barrel from a short receiver thread shank is a good way to stretch threads and mess up a conventional bedding job too. Perhaps more than any other rifle type, these rifles benefit from a barrel block.
It is probably for this same reason that the short barrels on the Remington XP-100's shoot as well as they do. Phenomenally well in many cases. The little 14-15" tubes are very rigid.
Another advantage to the short barrels is the relocation of the center of gravity towards the shooter. In my opinion a muzzle heavy rifle is hard to control on the sand bags and may cause some vertical stringing in a group. I like the balance point to be as far back as is reasonably possible.
We can see from these numbers that both length and diameter are very important in determining barrel stiffness and potential accuracy. If the advantages of a short and fat barrel are clear, an obvious question is, are there any disadvantages? The only real disadvantage to the short barrels are a decrease in the muzzle velocity of the bullet and as a result an increase in wind drift.
With a cartridge capacity of a 6PPC's and using a bullet of about 68 grains and a powder with a burning rate close to 322, an inch of change in barrel length is worth just about 25 FPS in velocity. A 20" long barrel then, would produce about 100 FPS less velocity than a 24" long barrel firing the same load. These figures are averages; individual barrels might vary, but the majority of them will fall into this range.
A change in velocity of 25 FPS is worth about .020" of wind drift at 100 yards and about .060" at 200 yards with a 10 MPH crosswind. These figures are true using the velocities expected from a 6PPC shooting a 68 grain flat base bullet with a C1 ballistic coefficient of .265. Wind drift is directly proportional with wind velocity so a 5 MPH wind would have one half of the effect of a 10 MPH breeze.
We have pointed out here the significance of barrel length and diameter in barrel rigidity. The individual shooter must decide if going to a shorter and stiffer barrel is worth the velocity and wind drift penalty. In the unlimited class, where barrel weight is of no concern, I like the barrel block system and as much barrel diameter as possible.
*
STANDARD STRAIGHT TAPER BARRELS
Barrel Number A C D F Approx. Weight at "F" Length
NBRSA HV Taper 1.250" .925" 5.0" 27.0" 6.75 lbs.
1.200 HV Taper 1.200" .875" 5.0" 27.0" 6.25 lbs.
NBRSA Hunter 1.250" .725" 4.0" 27.0" 5.75 lbs.
*
MAXIMUM HEAVY VARMINT BLANK
WEIGHT LENGTH CYL LENGTH CYL DIA MUZZLE DIA DEFLECTION
5 lb 1.8 oz 18" 4" 1.250" 1.060" .000509
5 lb 15.7 oz 22" 4" 1.250" 1.002" .001006
6 lb 6.1 oz 24" 4" 1.250" .973" .001356
6 lb 12.0 oz 26" 4" 1.250" .944" .001789
LIGHT VARMINT BARRELS OF 5 POUNDS - 4 OUNCES
WEIGHT LENGTH CYL LENGTH CYL DIA MUZZLE DIA DEFLECTION
**5 lb 4.0 oz** 16.48" 15.48" 1.250" 1.250" .000344 (CYL)
5 lb 4.0 oz 18.60" 4" 1.250" 1.052" .000569 (HV)
5 lb 4.0 oz 20.82" 4" 1.200" .969" .000988 (1.2 HV)
5 lb 4.0 oz 20.92" 0" 1.250" .959" .000990 (HV)
5 lb 4.0 oz 21.85" 3" 1.250" .844" .001134(HUNTER)
5 lb 4.0 oz 22.52" 1.5" 1.200" .908" .001420 (1.2 HV)
5 lb 4.0 oz 23.88" 0" 1.200" .866" .001879 (1.2 HV)
(**barrel not legal for varmint competition)
STRAIGHT CYLINDER UNLIMITED BARRELS
WEIGHT LENGTH CYL LENGTH CYL DIA MUZZLE DIA DEFLECTION
16 lb 15.6 oz 24" 23" 1.850" 1.850" .000235
10 lb 6.3 oz 24" 23" 1.450" 1.450" .000623
8 lb 15.9 oz 24" 23" 1.350" 1.350" .000830
7 lb 11.2 oz 24" 23" 1.250" 1.250" .001129
Note: The weight of each barrel listed above is expressed in pounds and ounces. For example, the first barrel listed above weighs 5 pounds and 1.8 ounces. In the second column, the length of the barrels is listed as the overall length in inches, including the thread shank length. The length and diameter of the thread shank for each barrel is 1.000" by 1.000". The calculations for deflection are based on the length of barrel extending in front of the receiver. The cylinder length shown in the third column has the 1.000" thread shank length subtracted from it. Listed in parentheses behind the deflection figures, in the group of light varmint barrels, the contour of the barrel blank each was taken from is shown. In some cases, these barrels were trimmed from both ends to get the barrels to weigh the same amount. The deflection shown in the last column is in inches and is for a load of 1 pound, applied at the muzzle perpendicular to the bore. The amount of muzzle deflection is directly proportional with the force applied to the muzzle. That is, doubling the force will double the deflection. All of the barrels in these examples have a .243" groove diameter."
So, I vote SPS Tactical.
Thanks,
P.B.Walsh
