{{short description|Gun ballistic calculation}} {{Infobox physical quantity | name = Sectional density | image = frameless|A metal nail has a small cross sectional area compared to its mass, resulting in a high Sectional Density. | caption = A metal nail has a small cross sectional area compared to its mass, resulting in a high sectional density. | unit = kilograms per square meter (kg/m<sup>2</sup>) | dimension = | otherunits = kilograms per square centimeter (kg/cm<sup>2</sup>) <br/>grams per square millimeter (g/mm<sup>2</sup>) <br/>pounds per square inch (lb<sub>m</sub>/in<sup>2</sup>) | symbols = |conserved = }} '''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<ref>[http://derelicta.pagesperso-orange.fr/aubin3.htm Les étranges obus du fort de Neufchâteau {{in lang|fr}}]</ref> and saw very limited use during World War II.
== Formula == In a general physics context, sectional density is defined as:
:<math> SD = \frac{M}{A} </math><ref>[https://books.google.com/books?id=q4jzcfLhBcYC&dq=sectional+density+cross+sectional+area&pg=PA203 Wound Ballistics: Basics and Applications]</ref> * ''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/m<sup>2</sup>). The general formula with units then becomes:
:<math>SD_{\text{kg}/\text{m}^2} = \frac{ m_\text{kg} }{ {A_{\text{m}^2}} }</math> where: * ''SD''<sub>kg/m<sup>2</sup></sub> is the sectional density in kilograms per square meters * ''m''<sub>kg</sub> is the weight of the object ''in kilograms'' * ''A''<sub>m<sup>2</sup></sub> is the cross sectional area of the object ''in meters''
== Units conversion table == {| class="wikitable" |+ {{nowrap|Conversions between units for sectional density}} ! !! kg/m<sup>2</sup> !! kg/cm<sup>2</sup> !! g/mm<sup>2</sup> !! lb<sub>m</sub>/in<sup>2</sup> |- ! 1 kg/m<sup>2</sup> = | '''1''' || '''{{val|0.0001}}''' || '''{{val|0.001}}''' || {{val|0.001422334}} |- ! 1 kg/cm<sup>2</sup> = | '''{{val|10000}}''' || '''1''' || '''{{val|10}}''' || {{val|14.223343307}} |- ! 1 g/mm<sup>2</sup> = | '''{{val|1000}}''' || '''{{val|0.1}}''' || '''1''' || {{val|1.4223343307}} |- ! 1 lb<sub>m</sub>/in<sup>2</sup> = | {{val|703.069579639}} || {{val|0.070306957}} || {{val|0.703069579}} || '''1''' |} (Values in bold face are exact.)<noinclude>
* 1 g/mm<sup>2</sup> equals exactly {{val|1000}} kg/m<sup>2</sup>. * 1 kg/cm<sup>2</sup> equals exactly {{val|10000}} kg/m<sup>2</sup>. * With the pound and inch legally defined as {{val|0.45359237|u=kg}} and 0.0254 m respectively, it follows that the (mass) pounds per square inch is approximately: *: 1 lb{{sub|m}}/in<sup>2</sup> = {{val|0.45359237|u=kg}}/(0.0254 m × 0.0254 m) ≈ {{val|703.06958|u=kg/m<sup>2</sup>}}
== Use in ballistics == {{More citations needed section|date=January 2021}} 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<ref>''Hornady Handbook of Cartridge Reloading: Rifle, Pistol Vol. II'' (1973) Hornady Manufacturing Company, Fourth Printing July 1978, p505</ref>); it yields the projectile's ballistic coefficient.<ref>Bryan Litz. Applied Ballistics for Long Range Shooting.</ref> 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.<ref>{{Cite web |url=http://www.rathcoombe.net/sci-tech/ballistics/wounding.html |title=Shooting Holes in Wounding Theories: The Mechanics of Terminal Ballistics |access-date=2009-07-25 |archive-date=2021-06-24 |archive-url=https://web.archive.org/web/20210624031342/https://rathcoombe.net/sci-tech/ballistics/wounding.html |url-status=dead }}</ref><ref>MacPherson D: Bullet Penetration—Modeling the Dynamics and the Incapacitation Resulting From Wound Trauma. Ballistics Publications, El Segundo, CA, 1994.</ref><ref>{{cite web |url= http://www.gsgroup.co.za/articlesd.html |title= Sectional Density - A Practical Joke? |first= Gerard |last= Schultz |url-status=dead |archive-url= https://web.archive.org/web/20230115024756/http://www.gsgroup.co.za/articlesd.html |archive-date= 2023-01-15 }}</ref>
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/mm<sup>2</sup>), the formula then becomes:
:<math> SD_{\text{g}/\text{mm}^2} = \frac{ 4 m_\text{g} }{ { \pi \cdot d _\text{mm}}^2 }</math>
Where: * ''SD''<sub>g/mm<sup>2</sup></sub> is the sectional density in grams per square millimeters * ''m''<sub>g</sub> is the mass of the projectile ''in grams'' * ''d''<sub>mm</sub> is the diameter of the projectile ''in millimeters''
For example, a small arms bullet with a mass of {{convert|10.4|g|gr|sigfig=3}} and having a diameter of {{convert|6.70|mm|in|abbr=on|sigfig=3}} has a sectional density of: : 4 · 10.4 / (π·6.7<sup>2</sup>) = 0.295 g/mm<sup>2</sup>
==== Kilograms per square centimeter ==== Using kilograms per square centimeter (kg/cm<sup>2</sup>), the formula then becomes:
:<math> SD_{\text{kg}/\text{cm}^2} = \frac{ 4 m_\text{kg} }{ {\pi d_{\text{cm}}}^2 }</math>
Where: * ''SD''<sub>kg/cm<sup>2</sup></sub> is the sectional density in kilograms per square centimeter * ''m''<sub>g</sub> is the mass of the projectile ''in grams'' * ''d''<sub>cm</sub> 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 {{convert|154.71|mm|cm}} has a sectional density of: : 4 · 43.2 / (π·154.71<sup>2</sup>) = 0.230 kg/cm<sup>2</sup>
=== 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<sub>m</sub>/in<sup>2</sup>) The formula then becomes:
: <math> SD_{\text{lb}/\text{in}^2} = \frac{ 4 m_\text{lb} }{ {\pi \cdot d_\text{in}}^2} = \frac{ 4 m_\text{gr} }{\pi \cdot 7000 \, {d_\text{in}}^2 }</math><ref>[http://www.chuckhawks.com/sd.htm The Sectional Density of Rifle Bullets By Chuck Hawks]</ref><ref>[http://www.jbmballistics.com/ballistics/topics/secdens.shtml Sectional Density and Ballistic Coefficients]</ref><ref>[http://www.chuckhawks.com/sd_beginners.htm Sectional Density for Beginners By Bob Beers]</ref>
: <math> SD_\mathrm{lbs/sq in} = \frac{ 4 \cdot m_\mathrm{lb} }{ {\pi \cdot d_\mathrm{in}}^2} = \frac{ 4 \cdot m_\mathrm{gr} }{ \pi \cdot 7000 \, {d_\mathrm{in}}^2 }</math>
where: * ''SD'' is the sectional density in (mass) pounds per square inch * the mass of the projectile is: ** ''m''<sub>lb</sub> in pounds ** ''m''<sub>gr</sub> in grains * ''d''<sub>in</sub> is the diameter of the projectile in inches <!-- = \approx {p} * p is pressure --> The sectional density defined this way is usually presented without units. <!-- Units of pressure are kgf/m<sup>2</sup> or lbf/in<sup>2</sup>. -->In Europe the derivative unit g/cm<sup>2</sup> is also used in literature regarding small arms projectiles to get a number in front of the decimal separator.{{citation needed|date=March 2022}}
As an example, a bullet with a mass of {{convert|160|gr|g|sigfig=3}} and a diameter of {{convert|0.264|in|mm|abbr=on|sigfig=2}}, has a sectional density (''SD'') of: : 4·(160 gr/7000) / (π·0.264 in<sup>2</sup>) = 0.418 lb<sub>m</sub>/in<sup>2</sup>
As another example, the M107 projectile mentioned above with a mass of {{convert|43.2|kg|lb|order=flip|sigfig=3}} and having a body diameter of {{convert|154.71|mm|in|order=flip|sigfig=5}} has a sectional density of: : 4 · (95.24) / (π·6.0909<sup>2</sup>) = 3.268 lb<sub>m</sub>/in<sup>2</sup>
== See also == * Ballistic coefficient
==References== {{Reflist}}
Category:Projectiles Category:Aerodynamics Category:Ballistics