sectional density

Sectional density (often abbreviated SD) is the ratio of an object's mass to its cross sectional area with respect to a given axis. It conveys how well an object's mass is distributed (by its shape) to overcome resistance along that axis.

Sectional density is used in gun ballistics. In this context, it is the ratio of a projectile's weight (often in either kilograms, grams, pounds or grains) to its transverse section (often in either square centimeters, square millimeters or square inches), with respect to the axis of motion. It conveys how well an object's mass is distributed (by its shape) to overcome resistance along that axis. For illustration, a nail can penetrate a target medium with its pointed end first with less force than a coin of the same mass lying flat on the target medium.

During World War II, bunker-busting Röchling shells were developed by German engineer August Coenders, based on the theory of increasing sectional density to improve penetration. Röchling shells were tested in 1942 and 1943 against the Belgian Fort d'Aubin-Neufchâteau and saw very limited use during World War II.

Formula

In a general physics context, sectional density is defined as:

:$SD = \frac$
* ''SD'' is the sectional density
* ''M'' is the mass of the projectile
* ''A'' is the cross-sectional area

The SI derived unit for sectional density is kilograms per square meter (kg/m²). The general formula with units then becomes:

:$SD_ = \frac$
where:
* ''SD''_kg/m² is the sectional density in kilograms per square meters
* ''m''_kg is the weight of the object ''in kilograms''
* ''A''_m² is the cross sectional area of the object ''in meters''

Units conversion table

(Values in bold face are exact.)

* 1 g/mm² equals exactly  kg/m².
* 1 kg/cm² equals exactly  kg/m².
* With the pound and inch legally defined as and 0.0254 m respectively, it follows that the (mass) pounds per square inch is approximately:
*: 1 lb/in² = /(0.0254 m × 0.0254 m) ≈

Use in ballistics

The sectional density of a projectile can be employed in two areas of ballistics. Within external ballistics, when the sectional density of a projectile is divided by its coefficient of form (form factor in commercial small arms jargon); it yields the projectile's ballistic coefficient. Sectional density has the same (implied) units as the ballistic coefficient.

Within terminal ballistics, the sectional density of a projectile is one of the determining factors for projectile penetration. The interaction between projectile (fragments) and target media is however a complex subject. A study regarding hunting bullets shows that besides sectional density several other parameters determine bullet penetration.

If all other factors are equal, the projectile with the greatest amount of sectional density will penetrate the deepest.

Metric units

When working with ballistics using SI units, it is common to use either ''grams per square millimeter'' or ''kilograms per square centimeter''. Their relationship to the base unit ''kilograms per square meter'' is shown in the conversion table above.

Grams per square millimeter

Using grams per square millimeter (g/mm²), the formula then becomes:

:$SD_ = \frac$

Where:
* ''SD''_g/mm² is the sectional density in grams per square millimeters
* ''m''_g is the mass of the projectile ''in grams''
* ''d''_mm is the diameter of the projectile ''in millimeters''

For example, a small arms bullet with a mass of and having a diameter of has a sectional density of:
: 4 · 10.4 / (π·6.7²) = 0.295 g/mm²

Kilograms per square centimeter

Using kilograms per square centimeter (kg/cm²), the formula then becomes:

:$SD_ = \frac$

Where:
* ''SD''_kg/cm² is the sectional density in kilograms per square centimeter
* ''m''_g is the mass of the projectile ''in grams''
* ''d''_cm is the diameter of the projectile ''in centimeters''

For example, an M107 projectile with a mass of 43.2 kg and having a body diameter of has a sectional density of:
: 4 · 43.2 / (π·154.71²) = 0.230 kg/cm²

English units

In older ballistics literature from English speaking countries, and still to this day, the most commonly used unit for sectional density of circular cross-sections is (mass) pounds per square inch (lb_m/in²) The formula then becomes:

: $SD_ = \frac = \frac$Sectional Density for Beginners By Bob Beers
/ref>

: $SD_\mathrm = \frac = \frac$

where:
* ''SD'' is the sectional density in (mass) pounds per square inch
* the mass of the projectile is:
** ''m''_lb in pounds
** ''m''_gr in grains
* ''d''_in is the diameter of the projectile in inches

The sectional density defined this way is usually presented without units.
In Europe the derivative unit g/cm² is also used in literature regarding small arms projectiles to get a number in front of the decimal separator.

As an example, a bullet with a mass of and a diameter of , has a sectional density (''SD'') of:
: 4·(160 gr/7000) / (π·0.264 in²) = 0.418 lb_m/in²

As another example, the M107 projectile mentioned above with a mass of and having a body diameter of has a sectional density of:
: 4 · (95.24) / (π·6.0909²) = 3.268 lb_m/in²

See also

* Ballistic coefficient

References

{{Reflist

Projectiles
Aerodynamics
Ballistics