Fifty Caliber Barrel Rigidity

By: Daniel Lilja

In this article we will take a look at the rigidity or stiffness of fifty caliber rifle barrels. Barrel stiffness is important because it is one of the most important elements of a barrel that contribute to potential accuracy. Because the centerline of the barrel is above the center of gravity of the typical rifle, the barrel will whip vertically during recoil. The muzzle will lag behind the rest of the rifle and this whipping vibration will be set up. The exact pattern for a particular gun or even individual shot will vary. This can be caused by a number of factors including bedding problems, action face or bolt face misalignment, cartridge case wall variations, and the shooter’s hold on the rifle or its relative position on the sandbags.

We can see then, that a stiffer barrel will whip less and also will be more forgiving of other problems associated with the rifle or ammunition. Two important physical dimensions of a barrel contribute to its stiffness or lack of it. These are barrel length and outside diameter. The size of the bore also has some effect, but as we will explain later, this effect is minimal. Intuitively we can see that a longer barrel will be less stiff and that a larger diameter barrel will be more so.

The relative stiffness of a barrel can be determined mathematically if we are familiar with the physics involved. A free-floating rifle barrel is a classic example of a cantilevered beam. It is supported at one end by the receiver and acts as if a load is applied to the muzzle. The basic formula for deflection of the muzzle is:

D=(W*l^3)/3*E*Ix

Where D in this equation is the deflection at the muzzle in inches, W is the force or load applied to the muzzle in pounds, l is the free length of the barrel (not including the threads), E is the modulus of elasticity (also known as Young’s modulus) for the barrel material, and Ix is the moment of inertia of the barrel.

We’ll take a look now at each element of the formula and explain what might not be clear.

While it would be very difficult to determine the exact load on the muzzle of a barrel, 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 later the load used is one pound.

The length (l) of a barrel is easily measured. It is the free length of the barrel allowed to vibrate and does not include the receiver threads. We can see, though, that the length is raised to the third power in the formula, meaning that rigidity decreases with the cube of its length.

The modulus of elasticity (E) is a constant, and in the case of steel it is 30 million PSI. Perhaps surprisingly, this value does not change with either heat treatment of the steel or the type of alloy being used. It is unchanged for all steels. So unless the barrel is made from some other type of material, this part of the equation is not a variable.

As an example of the importance modulus of elasticity plays in overall stiffness, it might be interesting to compare our rifle barrel to another example of a beam supported at one end that most gunsmiths are familiar with. That is the boring bar used in the lathe. The farther out the bar is slid in its holder, the more easily it deflects during a cut and the tendency for chatter to occur increases too. Increasing the diameter of the bar would help, but bar size is limited by hole diameter. For this reason, the best quality boring bars available are made from carbide not steel. The modulus of elasticity for carbide is about 94 million PSI, or over three times that of steel. As a result, carbide boring bars are more than 3 times as still. The same would be true of carbide versus steel end mills.

With rifle barrels we are limited to steel, but we can see from these examples the role that modulus of elasticity plays in rigidity.

The moment of inertia (Ix) of a barrel is the most difficult part to calculate. It is a sort of measure of the cross sectional area of the barrel integrated over its full length. A larger diameter barrel will have a higher moment of inertia value and will therefore be stiffer. Computing the moment for a straight cylinder barrel is relatively easy. The equation looks like:

Ix = Pi * (D1^4 – D2^4)/64

Where Ix is the moment of inertia; Pi is 3.1416; D1 is the outside diameter of the barrel and D2 is the groove diameter of the barrel. From this formula, we can see that the moment of inertia increases with the fourth power of the diameter of the barrel. As an example of this significance it is interesting to note that a 2″ diameter barrel is 16 times more rigid than a 1″ diameter barrel since 2^4 is 16.

We can now see why the two important physical characteristics of a barrel as related to stiffness are its length and diameter. A barrel loses rigidity with the third power of its length and stiffness increases with the fourth power of its diameter. As we mentioned earlier, the caliber of the barrel does not make a significant contribution. The reason is that in the above formula, the caliber is raised to the fourth power, too, then subtracted from the outside diameter raised to the fourth power. When the inside diameter is a decimal, (as it would be for all small arms, in this case .510″), and it is raised to the fourth power, the number becomes smaller. For a fifty caliber barrel this number becomes .067652″.

Although computing the moment for a straight cylinder barrel is easy, doing so for a tapered barrel is a real bear. It requires the use of calculus, and without the aid of a computer would be so tedious that it would not likely be done at all. Working with a friend of mine, Mel Klasi from Rapid City, South Dakota, (a professor of Civil Engineering) I developed a computer program that calculates the moment of inertia for tapered barrels, allowing us to compare the relative stiffness of barrels commonly used in benchrest type shooting. The program also computes the weight of the barrel. The moment of inertia formulas for the tapered barrels are much too lengthy and involved to be shown here.

In the table which compares the stiffness of various length barrels, the contour of each is the same. The example barrel has a straight taper starting from a 5″ long cylinder which is 1.850″ in diameter. The taper is .015″ per inch. The thread shank used was 1.5″ long by 1.5″ in diameter. This thread length is not used as part of the free length of the barrel. For example, I started with a 28″ barrel, meaning that 26.5″ of that barrel was free to vibrate. I compared barrels varying in length from the already mentioned 28″ to one 46″ long, increasing the length in 2″ steps. Because the taper was the same for all of the barrels, the muzzle diameter decreased as the barrels got longer. Since the selected taper was .015″ per inch, the muzzle diameter decreased .030″ per each 2″ step.

As an added wrinkle to the comparisons, I included another number to look at as the lengths change. That number is the amount of rearward movement the rifle undergoes from recoil while the bullet is still in the barrel. Editor Eric Williams had an excellent article about recoil and fifty caliber rifles in the third issue of VERY HIGH POWER from 1990. This article dealt with the subject of muzzle brakes and their efficiency. But as Eric pointed out, the gun has started recoiling long before the powder gasses which make muzzle brakes work so well ever reach the brake. Because fifties fire such a heavy bullet for a shoulder-fired rifle, using a large powder charge to do so, their recoil while the bullet is in the barrel is significantly more than for smaller arms. Eric compared a fifty to a 6MM PPC benchrest rifle, and the movement was about four times as much for the fifty. I won’t go into detail concerning the formula for this calculation, but anyone interested can look up the article. The highlights were that overall gun weight decreases movement, and barrel length increases it. On the downside, it becomes more difficult to consistently shoot good groups with a rifle that moves a lot while the bullet is still under its influence.

For the examples shown, I assumed that the rifle weighed 25 pounds without the barrel. I then added the weight for each sample barrel (again calculated by the computer) to the 25 pounds to arrive at a total gun weight for the recoil distance calculations. So as the barrel length increased the overall weight of the rifle did, too. This gives us a fair comparison of recoil as barrel length increases. I assumed that in each case the bullet weighed 700 grains and that the powder charge weight was 235 grains – typical fifty caliber loads.

From the chart we can make some interesting comparisons. For example, the 28″ barrel is 573% stiffer at the muzzle than the 46″ version is. The 28″ barrel is also 27% stiffer than its 30″ neighbor. Other comparisons can be made by dividing the deflection figure for the longer barrel by the deflection for the shorter one.

We can clearly see that the shorter barrels are stiffer. A stiffer barrel is potentially more accurate. The obvious question, then, is what are the advantages to a longer barrel? One benefit is that the expansion ratio increases with length and results in higher velocity. There is also some indication that the muzzle velocities for bullets will be more uniform for longer barrels. The obvious advantage of higher velocity is reduced wind drift over the course of fire, and to a lesser extent, a flatter bullet trajectory.

When target shooting at 1000 yards, reduced wind drift is a real advantage. So the next question: how much velocity increase can we expect from each inch of barrel length, and how much will it reduce wind drift at 1000 yards?

The best way to learn how much an inch of barrel contributes to velocity would be to start out with a long barrel and cut off each inch, chronographing as we cut. I have not done that with a fifty but I have with a 338/378 Weatherby Mag firing the 300 grain Sierra MK HP bullet and using IMR 5010 powder. With that cartridge and load, I found that each inch was worth about 12 fps from 46″ down to 30″. Below 30″ each inch was worth about 25 fps.

In the 4th edition of VERY HIGH POWER from 1993, Dennis Chapman wrote an article about Bull Pup rifles. He mentioned that he compared the velocities from one of the 26″ long barreled Bull Pups to his own 36″ long barreled bench rest rifle. He found that the velocity difference amounted to 13 fps per inch firing the same ammunition in each rifle.

I also have an internal ballistics program that calculates velocities and expansion ratios, and I ran some numbers through it changing the barrel length. In the past this program has always come quite close in predicting actual velocities. In this case it indicated that we could expect a change of 12-16 fps per inch, with the greater change coming with the shorter barrels.

So, it would seem that each additional inch of barrel in a fifty would produce at least 12 fps and possibly as much as 16 fps.

It is an easy matter now to calculate how much wind drift penalty there is for using a short barrel at 1000 yards. If we assume that we can drive a 700 grain bullet with a C1 ballistic coefficient of .80 at 2650 fps from a 28″ barrel, we can run those numbers through an external ballistics program that calculates wind drift. In this case I used the program TRAG1Q from Tioga Engineering. I found that with the above mentioned bullet and velocity, a 10 MPH direct cross wind would deflect the bullet 52.98 inches from its course under standard sea level conditions. Increasing the velocity by 16 fps to simulate the velocity expected from a 1″ longer barrel showed that the bullet would drift about 52.48 inches or .50 inch less. If we carry this line of reasoning along a little farther and add 160 fps to simulate a 38″ long barrel the drift is 48.17 inches or 4.81 inches less. Actually, according to the program the velocity gain for a 38″ barrel would not be quite this great but we will give it the benefit of the doubt. As mentioned these drift figures are for a 10 mph wind. For a 20 mph gust the drift would be doubled and likewise for a 5 mph push it would be cut in half.

It can be seen, then, that there are some advantages for longer barrels as far as velocity improvement is concerned. It is possible to some degree to have both a long barrel and one that is stiff. The barrel block bedding system holds the barrel, and the block is bedded into the stock. The action and remaining free length of the barrel both float. The length of the barrel block is not critical but with smaller caliber 1000 yard type guns they are often in the 8″ long range. Some blocks clamp the barrel and others are epoxied onto the barrel. Using an 8″ long block with 1″ of barrel between the block and action would effectively reduce the amount of free barrel by 9″. Referring to the chart we can see that this would make a marked decrease in muzzle deflection. Also blocks are able to support a long and heavy barrel better than some actions.

We can now see some of the advantages and disadvantages to long and short barrels. To summarize, barrels decrease in stiffness with the third power of their length, but increase in rigidity with the fourth power of their outside diameter. Increasing barrel length increases the expansion ratio of the barrel and results in higher velocities. Higher velocities translate into less bullet wind drift and potentially better groups on the target. After all is said and done, when making any changes to a rifle in search of greater accuracy, the results have to be seen on the target to make them worthwhile.

BARREL LENGTH MUZZLE DIA WEIGHT RECOIL DISTANCE DEFLECTION
28″ 1.505″ 15.83# .071″ .000431″
30″ 1.465″ 16.68# .075″ .000546″
32″ 1.445″ 17.49# .079″ .000682″
34″ 1.415″ 18.26# .083″ .000842″
36″ 1.385″ 19.00# .087″ .001029″
38″ 1.355″ 19.70# .091″ .001244″
40″ 1.325″ 20.37# .094″ .001493″
42″ 1.295″ 21.00# .098″ .001777″
44″ 1.265″ 21.59# .102″ .002101″
46″ 1.235″ 22.16# .106″ .002470″

Notes about the chart: The barrel length shown is for the overall barrel length. The free length of the barrel in the calculations is the overall length minus the thread shank length used which is 1.5″ The muzzle diameter decreases as length increases because the barrel is a straight tapered one with a taper of .015″ per inch. The barrel also has a cylinder length of 5″ and cylinder diameter of 1.850″. The weight is calculated for the particular barrel indicated. Recoil distance is the amount the rifle will move rearwards while the bullet is still in the barrel. The deflection value is the amount the muzzle would deflect if a one pound load were applied at the muzzle perpendicular to the bore axis.

As of January, 2000 we have added the computer code, used to compute the above values, to the end of the article entitled “A Look At the Rigidty of Bench Rest Barrels“.