Calculating Bullet Weights
By: Daniel Lilja
At a gun show recently, my friend Jerry plunked down on a table a very impressive-looking bullet. It was probably the longest bullet I had ever seen, very streamlined and with a boattail. It was a 50 caliber CNC lathe-turned brass bullet that Jerry had designed for long range shooting. It weighed 812 grains. I was impressed and intrigued by the possible bullet designs which lathe turning offered.
Despite the large selection of bullets available to the handloader from the makers of production and custom bullets, there are a number of shooters who, like Jerry, are interested in bullet designs that are not available.
There is a growing demand for special solid-construction bullets used for big game hunting of large and dangerous game. A look through the advertisements in this magazine will show several sources for monolithic solid-type bullets. Some of these are lathe-turned to size, and some are cold-formed from the solid. Usually they are made from a bronze-type alloy.
Target shooters – in particular, benchrest shooters – are continually looking for a new bullet design to give them an accuracy advantage. There have been some recent developments in bullet design in an effort to increase streamlining of the bullet to reduce drag. The goal is to reduce wind drift at a given target range. Most of these bullets are of conventional lead core construction.
Several makers of high quality bullet swage dies for use in handloaders’ presses will make dies that will produce bullets designed by the customer. These dies will make conventional-style, lead core jacketed bullets.
When designing a new bullet, one of the first criteria to consider is the new bullet’s weight. Determining the weight of a bullet prior to its manufacture is necessary if its ballistics and other data are to be predicted.
Before the weight of a proposed bullet can be calculated, the volume of that bullet must first be figured. After the volume is determined, it is multiplied by a density factor for the type of material from which the bullet is constructed.
Bullet Design Types
Spitzer bullets all fall into one of several different design types. A bullet’s geometry can be broken down into symmetrical sections. The ogive, or nose of a bullet, can be one of three varieties, two being common. The ogive can have a tangent radius; that is, the intersection of the body of the bullet and the radius of the ogive blend together at a tangent point. This is the most common type and is typical of most bullets handloaders use. The other common type is the secant ogive. In this case the intersection of the ogive and bullet body does not flow together smoothly. The point of intersection is not tangent, the ogive radius being a secant of the arc circle of the ogive.
The radius of the ogive is generally given as a multiple of the bullet diameter. For example, a 30 caliber bullet might have a 7 caliber tangent ogive. That is to say that the radius, in inches, is seven times .308 or 2.156 inches.
The other nose-type referred to is the cone. Cone-shaped bullets are not common with handloaders, but some experimental work is done with them.
The point of the ogive is called the meplat and can vary in diameter from about .050″ to over an eighth of an inch for conventional spitzer-type bullets.
The body of the bullet is that part of the bullet that comes into full contact with the barrel rifling. It is the cylindrical portion of the bullet of nominal groove diameter for that caliber. It is also referred to as the bearing surface.
The base of the bullet can be either a boattail shape or flat base. If a boattail, its shape is determined by the axial length of the boattail and the angle at which the boattail leaves the body of the bullet.
Volume And Weight Calculated For Geometric Shapes
With a bullet broken down geometrically into these parts, the volume of each can be calculated and these values added together, the sum being the total volume of the bullet. Then, after the volume of the bullet is known, it is multiplied by a density value for the material being used. The result is a calculated bullet weight for a particular design while it is still on the drawing board. Or as is more likely today, still on the computer screen.
The density of a material is known as its specific gravity. The base reference for specific gravity is water, which has a specific gravity of 1 at 62 degrees F. The specific gravity for materials that bullet designers are interested in are available from sources such as physics text books, encyclopedias, the Machinery’s Handbook, etc. For example, lead has a specific gravity of 11.34, copper is 8.89, bronze is 8.78, brass alloys from 8.2 to 8.6, linotype 10.4. Soft point jacketed rifle bullets have a specific gravity of approximately 10.25-10.4. The exact density depends on the ratio of jacket material and lead, and the type of lead alloy being used. The specific gravity of jacketed hollow point bullets measures from about 10 to 10.11, depending on jacket thickness and depth of the hollow point.
The specific gravity of existing bullets can be determined easily with the use of an accurate powder scale and a glass of water. First, hang the bullet from the scale with a thin wire and weigh it. Second, weigh the bullet while it is hanging in the glass of water. Subtract the wet weight from the dry weight and divide the difference into the dry weight. The result is the specific gravity. It is best to use a scale that reads to two decimal places if possible, as a small error in the weight can change the specific gravity by more than is desired.
Weight In Grains
Handloaders are interested in the weight of bullets in grains. There are 252.7 grains in one cubic inch of water. To calculate the total bullet weight in grains, the bullet volume, in cubic inches, is multiplied by the specific gravity value, and then by 252.7. The product is bullet weight in grains.
Calculating Volume Of Cylinder And Boattail
Calculating bullet volume is a fairly straightforward math problem when broken down into the geometric parts that were described earlier. If the bullet is a flat-base type, the volume of a cylinder is calculated. The diameter of this cylinder is the bullet’s diameter and its length is the axial length of the bearing surface. The formula for the volume of a cylinder is .7854 x bullet diameter squared x bearing length. The result is the volume in cubic inches. (See Figure One).
If the bullet is to have a boattail base, the formula to use is the one for calculating the volume of a frustum of a cone. It is .2618 x boattail length x (bullet dia.^2 + bullet dia. x boattail end dia. x boattail end dia.^2). If only the boattail length and angle are known, the end diameter can be calculated using a bit of trigonometry. The formula is end dia. = bullet dia.- 2 x (boattail length x tangent of the boattail angle). The boattail volume is then added to the volume of the remaining bearing surface. (See Figure One).
Computing Ogive Volume
The remaining volume is that of the ogive. As mentioned above, there are two types of ogives: tangent and secant. To calculate the volume of the ogive, a calculus technique will be used, broken down into an algebraic series of equations.
First the location of the ogive radius center must be determined in relation to the bullet. For convenience its location will be referenced from the long axis of the bullet, or its center line, and from the base of the bullet. The distance axially, from the radius center to the bullet base, is known as its X coordinate. Similarly the distance from the bullet axis and the radius center vertically is its Y axis. (See Figure Two).
For a tangent radius ogive, it is quite simple to determine these two measurements. The Y axis is the ogive radius in inches minus one half bullet diameter. In the case of the 7 caliber, 308 diameter bullet, it would be 7 x .308 – (.308/2) = 2.002 inches.
The X coordinate of a tangent ogive is determined by first calculating the ogive length axially. It is ogive length = SQR(R x 2 x C – C^2) where R is the ogive radius in inches and C=(bullet dia. – meplat dia.)/2. The X coordinate is then bullet length minus ogive length.
Determining the X-Y coordinates of the radius center for a secant ogive is more complicated. There are two methods that can be used. One is to derive the location using point slope equations, and the other uses trigonometry. In this article the trigonometry method will be described, as it is probably easier for someone not familiar with the coordinate system to visualize the location of the radius center using right triangles. Again the center will be referenced from the bullet base and long axis. (See Figure Two).
The length for a secant ogive is left up to the bullet designer. It is not determined mathematically, as it is with a tangent ogive. Streamlined bullets often have a secant ogive radius of 10 to 15 calibers and a length of approximately that of a tangent ogive of half the radius in calibers. For example, a 12 caliber secant ogive might be the same length as a 6 caliber tangent ogive. The method for calculating the length for a tangent ogive was described above.
After the X-Y coordinates of the ogive radius center are calculated, the actual volume is computed, based on calculating the individual volumes of discs superimposed in the ogive. (See Figure Three). It has been found that using ten discs gives a very accurate volume. After changing the number of discs in the accompanying computer program from one to 1000 discs, it was found that when using 5 discs or more, the volume was virtually the same for a sample of ogives. Obviously, the greater the number of discs the more accurate the total volume will be. When rounded to the nearest grain, the weight will remain the same with at least five discs.
Using only a hand calculator and ten discs, the bullet designer can compute an extremely accurate volume for an ogive. Surprisingly, using only two discs will yield an accurate volume. In the example of the bronze 50 caliber bullet, the weight was calculated using two discs and found to be just 3 grains different compared to using ten or more discs.
The underlying formula for determining the X and Y positions and resulting dimensions of the ten discs is the basic equation of a circle from its origin. That equation is X^2 + Y^2 = radius^2. Using algebra to change the formula, the Y coordinate of the ogive at any given X coordinate is Y = SQR(R^2 – X^2) – YC where R is the ogive radius in inches, X is the distance on the X axis from the radius center, and YC is the Y coordinate of the radius center as calculated above. The disc thickness, represented by an increasing X value, is one tenth of the ogive length.
The volume of each disc is calculated as volume = Pi x disc thickness x Y^2 where Pi is 3.1416 and Y is the Y coordinate as calculated above.
Total ogive volume is simply the volume of the ten individual discs added together.
Total bullet volume is, as mentioned above, the sum of the boattail volume if present, the bearing surface volume, and the ogive volume. Weight is determined by multiplying the volume in cubic inches by the specific gravity for the bullet material being used, and then multiplying by the constant 252.7 for a bullet weight in grains.
If the point of the bullet is not an ogive but a cone, the formula for calculating the volume is .2618 x bullet diameter^2 x cone length. As in the above examples, the volume of the nose-cone is added to the volume of the bearing length cylinder and boattail if present.
Computing bullet volumes and weights prior to actual manufacture of bullets is necessary if the bullet is to have the desired characteristics for its intended purpose. With the aid of a personal computer, the labor and time involved in these calculations is greatly reduced. Included is a listing of a simple program in BASIC format. However, with just the aid of a pocket calculator, the bullet designer can easily determine the weight of yet-to-be-made bullets.
|JACKETED SOFT POINT||10.25-10.4|
|JACKETED HOLLOW POINT||10.0-10.11|
For those readers with an IBM compatible personal computer, a listing of a short computer program written in GWBASIC is given below. The user will be asked to input the bullet diameter, the overall length of the bullet, and the meplat diameter. He will be asked if the bullet has a boattail, and if it does, its length and angle or end diameter. The program will need to know if the bullet has a secant or tangent ogive and its radius. If the bullet has a secant ogive the program will ask for the ogive length, and the specific gravity of the bullet.
From the above information the computer will perform all of the calculations described in the accompanying article. These include the calculation of the volumes of a boattail, bearing cylinder, and location of the ogive radius center and the resulting ogive volume. The total bullet weight, in grains, is then calculated from the specific gravity value determined by the user. The volume of the ogive is calculated using 100 discs in this program.
In April of 2003 we added the Bullet Design code from another article in this section as a Free Download in an Excel file. The bullet weight calculation code listed below is included as a part of that Excel file.
10 CLEAR: CLS: PRINT CODE ‘BULLETWT”WRITTEN IN GWBASIC BY DANIEL
20 PRINT “THIS PROGRAM WILL CALCULATE THE WEIGHTS OF BULLETS
FROM PHYSICAL DIMENSIONS”
30 PRINT “AND A SPECIFIC GRAVITY VALUE. THE WEIGHT WILL BE
RETURNED IN GRAINS.”:PRINT””
40 INPUT “ENTER BULLET DIAMETER (INCHES):”; BD
50 INPUT “ENTER BULLET OVERALL LENGTH (INCHES):”;BL
60 INPUT “ENTER MEPLAT DIAMETER (INCHES):”;MD
70 INPUT “DOES THIS BULLET HAVE A BOATTAIL (1 = YES 2 = NO):”;QBT
80 IF QBT>1 THEN BTL = 0:ED = 0: GOTO 130
90 INPUT “ENTER BOATTAIL LENGTH (INCHES):”;BTL
100 INPUT “ENTER (1 = BOATTAIL ANGLE 2 = END DIAMETER):”;QD
110 IF QD=1 THEN INPUT “ENTER BOATTAIL ANGLE DEGREES) :”;BTA
120 IF QD=2 THEN INPUT “ENTER BOATTAIL END DIAMETER (INCHES):”:ED
130 INPUT “WHAT TYPE OF OGIVE DOES THIS BULLET HAVE (1=TANGENT
140 IF QO=2 THEN INPUT “ENTER THE LENGTH OF THE OGIVE (INCHES):”;NL
150 INPUT “ENTER THE OGIVE RADIUS (CALIBERS):”;RC
160 INPUT “ENTER THE SPECIFIC GRAVITY OF THE BULLET”;SG
180 R=RC*BD:REM OGIVE RADIUS IN INCHES
190 IF QO=2 THEN GOTO 250
200 REM COMPUTE X-Y COORDINATES FOR TANGENT OGIVE
210 NL=SQR(R*2*C-C^2):REM FIND NOSE LENGTH
220 XC=BL-NL: REM FIND X COORD OF OGIVE RADIUS CENTER FROM BASE
230 YC=R-.5*BD:REM FIND Y COORD OF OGIVE RADIUS CENTER FROM AXIS
240 GOTO 340
250 REM COMPUTE X-Y COORD FOR SECANT OGIVE
260 HYP=SQR(NL^2+C^2):REM HYPOTENUSE OF TRIANGLE IN OGIVE
270 H=R-.5*SQR(4*R^2-HYP^2):REM HEIGHT OF CIRCULAR SEGMENT
280 A=ATN(C/NL):REM FIND ANGLE OF OGIVE SLOPE
290 HX=H/COS(A):REM HYPOTENUSE OF RIGHT TRIANGLE IN CIRCULAR
300 X=R*SIN(A):REM FIND AXIAL LENGTH OF X COORD FROM MIDPOINT OF
310 B=SQR(R^2-X^2):REM FIND LENGTH OF Y COORD
320 YC=(B-HX-(BD+MD)/4):REM FIND Y COORD FOR OGIVE RADIUS CENTER
330 XC=BL-(NL/2+X): REM FIND X COORD FOR OGIVE RADIUS CENTER
340 REM:CALCULATE VOLUMES AND WEIGHT
350 LB=BL-NL-BTL:REM CALC BEARING LENGTH
360 LBV=.7854 *BD^2*LB:REM CALCULATE VOLUME OF BEARING LENGTH
370 BTV=.2618*BTL*(BD^2+BD*ED+ED^2):REM CALCULATE BOATTAIL
380 REM CALCULATE VOLUME OF OGIVE
390 IF QO=1 THEN XSEC=0:GOTO 410
400 IF QO=2 THEN XSEC=BL-NL-XC:REM DISTANCE FROM OGIVE ORIGIN TO
JUNCTION OF OGIVE AND BEARING CYLINDER ON X AXIS FOR SECANT
420 XX=NL/200+XSEC:REM START FIRST DISC AT ONE HALF THICKNESS
430 DIM Y(100),XX(100),OGV(100),V(100)
440 FOR D=1 TO 100:REM 100 DISCS
450 Y(D)=SQR(R^2-XX^2)-YC:REM EQUATION OF CIRCLE = X^2 + Y^2 = R^2
460 V(D)=PI*NL/100*Y(D)^2:REM VOLUME EACH DISC
470 OGV=V(D)+OGV:REM OGIVE VOLUME
490 NEXT D
500 V=LBV+BTV+OGV:REM ADD VOLUMES OF BEARING LENGTH,
BOATTAIL, AND OGIVE
510 W=(V*SG*252.7):REM CALC WEIGHT IN GRAINS (252.7 GRAINS IN CUBIC
520 PRINT “””” :PRINT”THE WEIGHT IN GRAINS IS:”;:PRINT USING”###”;W