Barrel Weight Calculations For Contoured Barrels
By: Daniel Lilja
With: Excel File by Jean-Pierre PELLEGRY
In another article on our website we have the formulas and reasoning behind the calculations for determining the weights of straight taper and straight cylinder type barrels. In this article we have the formulas to be used in calculating the weights for contoured barrels, those with a radiused section. These calculations are a little more involved, requiring calculus. But with the use of a computer this is easy to do. And in collaboration with Jean-Pierre PELLEGRY of France we have a downloadable Excel file that will do all of the work. Jean-Pierre’s assistance is greatly appreciated in this endeavor. I e-mailed him the formulas and he worked them into the Excel file for us. My experience with the old GW Basic programming is out-of-step with the newer programs that handle math much easier. The calculations for fluting are not covered in this article but they are in the other article linked to above. But the Excel file does allow for inputing fluting as an option too.
The formulas and calculations below will compute the weight for any contoured barrel at any length, radius through the contoured section, diameter or bore size. To do so the contoured barrel is divided into four basic sections. The same format used in the straight taper barrels is used with these contoured barrels too. First the volume of each geometric section of the barrel is determined. The volumes are added together and then the missing volume from the bore diameter is subtracted. Finally the net volume is multiplied by a weight-per-cubic-inch constant for our steel.
The calculations for the cylinder section, straight tapered section and bore diameter are based on straightforward geometric formulas. The contoured section requires a calculus routine and to make this obvious and simple we have divided this section into 10 equal width discs. This results in an extremely accurate volume with fairly simple math involved.
Lilja Standard Contoured Barrels:
(All dimensions are in inches.)
Formula for determining the volumes and weights for each section:
Volume calculations for contoured section:
Note: the correct answers for each equation are shown at the end of each line in red. The calculations in this example are for a 27” long #7 contour barrel in .308 caliber.
Below is the calculus routine for 10 discs over contoured area based on the equation of a circle: X^2 + Y^2 = R^2. X & Y are the coordinate positions of any given point on the circle. In the case of a contoured rifle barrel our radius is 30″ and it is tangent to the barrel at the B dimension as shown in the drawing above. The math below is similar to the math used and described in more detail in our bullet weight article. There is a drawing in that article that may help visualize the method used here to compute the volume of the contoured section. In the bullet weight article the ogive section of a bullet is very similar to the contoured section of a barrel.
Remark: compute X-axis:
X=(E – D) / 10 Remark: E & D are from the table above
X1=X / 2 Remark: start first disc at ½ of X to arrive at center position .15
Remark: Lilja standard contour radius is 30” for all numbered barrels shown above.
Remark: compute Y-axis:
Y1=R – SQRT(R^2-X1^2) + .5*B .501
Y2=R – SQRT(R^2-X2^2) + .5*B .504
Y3=R – SQRT(R^2-X3^2) + .5*B .510
Y4=R – SQRT(R^2-X4^2) + .5*B .519
Y5=R – SQRT(R^2-X5^2) + .5*B .531
Y6=R – SQRT(R^2-X6^2) + .5*B .546
Y7=R – SQRT(R^2-X7^2) + .5*B .564
Y8=R – SQRT(R^2-X8^2) + .5*B .585
Y9=R – SQRT(R^2-X9^2) + .5*B .609
Y10=R – SQRT(R^2-X10^2) + .5*B .636
Remark: compute volume in cubic inches
V1= Pi * X * Y1^2 .236
V2= Pi * X * Y2^2 .239
V3= Pi * X * Y3^2 .245
V4= Pi * X * Y4^2 .253
V5= Pi * X * Y5^2 .265
V6= Pi * X * Y6^2 .280
V7= Pi * X * Y7^2 .299
V8= Pi * X * Y8^2 .322
V9= Pi * X * Y9^2 .349
V10= Pi * X * Y10^2 .381
Where: Vcontour is the total volume for the contoured section
Volume of cylinder section in cubic inches:
Vcyl = Pi/4 * A^2 * D =4.295
If D < > D then NewD = D + / – NewD
Where: A and D are from the chart above
Volume of tapered section in cubic inches:
Vtapered = Pi / 12 * L (B^2 + B*C + C^2) =.2618 * 20.5 * 2.505 = 13.444
Where: L = length of tapered section and F & E are from the chart above and Vtapered is the total volume for the tapered section.
If F is > or < than 27” then:
NewC = (F – + NewF * .008) + / – C (note standard taper per inch of contours is .008″ per inch)
Volume of bore in cubic inches:
Vbore = grooveDia^2 * Pi/4 * F = .2011
Total Volume in cubic inches:
TotalV = (Vcontour + Vcyl + Vtapered) – Vbore =2.869 + 4.295 + 13.444 – 2.011 = 18.597 cubic inches
Total weight in pounds:
TW = TotalV * .276 = 18.597 * .276 = 5.132 pounds (for a #7 contour 27″ long .30 caliber stainless steel barrel)
Remark: .276 is weight per cubic inch of 416 stainless steel used for barrels and .283 for 4142 chrome-moly.
The Excel program written by Jean-Pierre also allows for the calculations of weights for barrels that don’t fall into our standards. The user can input any diameter, length, taper and radius. The program also allows for the weight removed by fluting. With our CNC contour lathe we can most likely turn whatever contour you come up with.