Convergence problem: Difference between revisions

Revision as of 18:07, 12 March 2013

The following is a list of area moments of inertia. The area moment of inertia or second moment of area has a unit of dimension length⁴, and should not be confused with the mass moment of inertia. If the piece is thin, however, the mass moment of inertia equals the area density times the area moment of inertia. Each is with respect to a horizontal axis through the centroid of the given shape, unless otherwise specified.

Area moments of inertia

Description	Figure	Area moment of inertia	Comment	Reference
a filled circular area of radius r	File:Area moment of inertia of a circle.svg	$I_{0} = \frac{π}{4} r^{4}$		^[1]
an annulus of inner radiusr₁ and outer radiusr₂	File:Area moment of inertia of a circular area.svg	$I_{0} = \frac{π}{4} ({r_{2}}^{4} - {r_{1}}^{4})$	For thin tubes, this is approximately equal to: $π {(\frac{r_{2} + r_{1}}{2})}^{3} (r_{2} - r_{1})$ or $π r^{3} t$ .
a filled circular sector of angle θ in radians and radius r with respect to an axis through the centroid of the sector and the center of the circle	File:Area moment of inertia of a circular sector.svg	$I_{0} = (θ - \sin θ) \frac{r^{4}}{8}$	This formula is valid for only for 0 ≤ $θ$ ≤ $π$
a filled semicircle with radius r with respect to a horizontal line passing through the centroid of the area	File:Area moment of inertia of a semicircle 2.svg	$I_{0} = (\frac{π}{8} - \frac{8}{9 π}) r^{4} \approx 0.1098 r^{4}$		^[2]
a filled semicircle as above but with respect to an axis collinear with the base	File:Area moment of inertia of a semicircle.svg	$I = \frac{π r^{4}}{8}$	This is a consequence of the parallel axis theorem and the fact that the distance between these two axes is $\frac{4 r}{3 π}$	^[2]
a filled semicircle as above but with respect to a vertical axis through the centroid	File:Area moment of inertia of a semicircle 3.svg	$I_{0} = \frac{π r^{4}}{8}$		^[2]
a filled quarter circle with radius r entirely in the 1st quadrant of the Cartesian coordinate system	File:Area moment of inertia of a quartercircle.svg	$I = \frac{π r^{4}}{16}$		^[3]
a filled quarter circle as above but with respect to a horizontal or vertical axis through the centroid	File:Area moment of inertia of a quartercircle 2.svg	$I_{0} = (\frac{π}{16} - \frac{4}{9 π}) r^{4}$	This is a consequence of the parallel axis theorem and the fact that the distance between these two axes is $\frac{4 r}{3 π}$	^[3]
a filled ellipse whose radius along the x-axis is a and whose radius along the y-axis is b	File:Area moment of inertia of an ellipsis.svg	$I_{0} = \frac{π}{4} a b^{3}$
a filled rectangular area with a base width of b and height h	File:Area moment of inertia of a rectangle.svg	$I_{0} = \frac{b h^{3}}{12}$		^[4]
a filled rectangular area as above but with respect to an axis collinear with the base	File:Area moment of inertia of a rectangle 2.svg	$I = \frac{b h^{3}}{3}$	This is a result from the parallel axis theorem	^[4]
a filled rectangular area as above but with respect to an axis collinear, where r is the perpendicular distance from the centroid of the rectangle to the axis of interest		$I = \frac{b h^{3}}{12} + b h r^{2}$	This is a result from the parallel axis theorem	^[4]
a filled triangular area with a base width of b and height h with respect to an axis through the centroid	File:Area moment of inertia of a triangle.svg	$I_{0} = \frac{b h^{3}}{36}$		^[5]
a filled triangular area as above but with respect to an axis collinear with the base	File:Area moment of inertia of a triangle 2.svg	$I = \frac{b h^{3}}{12}$	This is a consequence of the parallel axis theorem	^[5]
a filled regular hexagon with a side length of a	File:Area moment of inertia of a regular hexagon.svg	$I_{0} = \frac{5 \sqrt{3}}{16} a^{4}$	The result is valid for both a horizontal and a vertical axis through the centroid, and therefore is also valid for an axis with arbitrary direction that passes through the origin.
Any plane region with a known area moment of inertia for a parallel axis. (Main Article parallel axis theorem)	File:Parallel Axes Compact.png	$I_{z} = I_{x} + A r^{2}$	This can be used to determine the second moment of area of a rigid body about any axis, given the body's moment of inertia about a parallel axis through the object's center of mass and the perpendicular distance (r) between the axes.

References

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

External links

[1] Template:Cite web

[semicircle-2] 2.0 ^2.1 ^2.2 Template:Cite web

[quartercircle-3] 3.0 ^3.1 Template:Cite web

[rect-4] 4.0 ^4.1 ^4.2 Template:Cite web

[tri-5] 5.0 ^5.1 Template:Cite web

[1]

[2]

[3]

[4]

[5]

@@ Line 1: / Line 1: @@
-Alyson is what my spouse enjoys to contact me but I don't like when individuals use my full title. My wife and I live in Mississippi and I adore each day residing here. My working day occupation is  are psychics real ([http://galab-work.cs.pusan.ac.kr/Sol09B/?document_srl=1489804 galab-work.cs.pusan.ac.kr]) a journey agent. I am truly fond of to go to karaoke but I've been taking on new things lately.<br><br>Also visit my web-site :: telephone [https://www.machlitim.org.il/subdomain/megila/end/node/12300 psychic phone] ([http://www.aseandate.com/index.php?m=member_profile&p=profile&id=13352970 Privacy of Data: This tool is built-with and functions-in Client Side JavaScripting, so only your computer will see or process your data input/output.a cool way to improve])
+The following is a list of [[second moment of area|area moments of inertia]]. The area moment of inertia or [[second moment of area]] has a [[physical unit|unit]] of dimension length<sup>4</sup>, and should not be confused with the [[mass moment of inertia]]. If the piece is thin, however, the mass moment of inertia equals the area density times the area moment of inertia. Each is with respect to a horizontal axis through the [[centroid]] of the given shape, unless otherwise specified.
+==Area moments of inertia==
+{|class="wikitable"
+|-
+! Description || Figure || Area moment of inertia || Comment || Reference
+|-
+| a filled circular area of radius ''r''||[[Image:Area moment of inertia of a circle.svg]]||<math>I_0 = \frac{\pi}{4} r^4</math>|| ||<ref>{{cite web|url=http://www.efunda.com/math/areas/Circle.cfm|title=Circle|accessdate=2006-12-30|publisher=eFunda}}</ref>
+|-
+| an [[annulus (mathematics)|annulus]] of inner radius''r''<sub>1</sub> and outer radius''r''<sub>2</sub>||[[Image:Area moment of inertia of a circular area.svg]]||<math>I_0 = \frac{\pi}{4} \left({r_2}^4-{r_1}^4\right)</math>|| For thin tubes, this is approximately equal to: <math>\pi \left(\frac{{r_2}+{r_1}}{2}\right)^3 \left({r_2}-{r_1}\right)</math>  or <math>\pi {r}^3{t}</math> .||
+|-
+| a filled [[circular sector]] of angle ''θ'' in [[radian]]s and radius ''r'' with respect to an axis through the centroid of the sector and the center of the circle||[[Image:Area moment of inertia of a circular sector.svg]]||<math>I_0 = \left( \theta -\sin \theta \right) \frac{r^{4}}{8}</math>||This formula is valid for only for 0 ≤ <math>\theta</math> ≤ <math>\pi</math>  ||
+|-
+| a filled semicircle with radius ''r'' with respect to a horizontal line passing through the centroid of the area||[[Image:Area moment of inertia of a semicircle 2.svg]]||<math>I_0 = \left(\frac{\pi}{8} - \frac{8}{9\pi}\right)r^4 \approx 0.1098r^4 </math>|| ||<ref name=semicircle>{{cite web|url=http://www.efunda.com/math/areas/CircleHalf.cfm|title=Circular Half|accessdate=2006-12-30|publisher=eFunda}}</ref>
+|-
+| a filled semicircle as above but with respect to an axis collinear with the base||[[Image:Area moment of inertia of a semicircle.svg]]||<math>I = \frac{\pi r^4}{8}</math>||This is a consequence of the [[parallel axis theorem]] and the fact that the distance between these two axes is <math>\frac{4r}{3\pi}</math>||<ref name=semicircle />
+|-
+| a filled semicircle as above but with respect to a vertical axis through the centroid||<center>[[Image:Area moment of inertia of a semicircle 3.svg]]</center>||<math>I_0 = \frac{\pi r^4}{8}</math>|| ||<ref name=semicircle />
+|-
+| a filled quarter circle with radius ''r'' entirely in the 1st quadrant of the [[Cartesian coordinate system]]||[[Image:Area moment of inertia of a quartercircle.svg]]||<math>I = \frac{\pi r^4}{16}</math>|| ||<ref name=quartercircle>{{cite web|url=http://www.efunda.com/math/areas/CircleQuarter.cfm|title=Quarter Circle|accessdate=2006-12-30|publisher=eFunda}}</ref>
+|-
+| a filled quarter circle as above but with respect to a horizontal or vertical axis through the centroid||[[Image:Area moment of inertia of a quartercircle 2.svg]]||<math>I_0 = \left(\frac{\pi}{16}-\frac{4}{9\pi}\right)r^4</math>||This is a consequence of the [[parallel axis theorem]] and the fact that the distance between these two axes is <math>\frac{4r}{3\pi}</math> ||<ref name=quartercircle />
+|-
+| a filled [[ellipse]] whose radius along the ''x''-axis is ''a'' and whose radius along the ''y''-axis is ''b''||[[Image:Area moment of inertia of an ellipsis.svg]]</td><td><math>I_0 = \frac{\pi}{4} ab^3</math>|| ||
+|-
+| a filled rectangular area with a base width of ''b'' and height ''h''||[[Image:Area moment of inertia of a rectangle.svg]]||<math>I_0 = \frac{bh^3}{12}</math>|| ||<ref name=rect>{{cite web|url=http://www.efunda.com/math/areas/rectangle.cfm|title=Rectangular area|accessdate=2006-12-30|publisher=eFunda}}</ref>
+|-
+| a filled rectangular area as above but with respect to an axis collinear with the base||[[Image:Area moment of inertia of a rectangle 2.svg]]||<math>I = \frac{bh^3}{3}</math>||This is a result from the [[parallel axis theorem]]||<ref name=rect />
+|-
+| a filled rectangular area as above but with respect to an axis collinear, where ''r'' is the perpendicular distance from the centroid of the rectangle to the axis of interest|| ||<math>I = \frac{bh^3}{12}+bhr^2</math>||This is a result from the [[parallel axis theorem]]||<ref name=rect />
+|-
+| a filled triangular area with a base width of ''b'' and height ''h'' with respect to an axis through the centroid||[[Image:Area moment of inertia of a triangle.svg]]||<math>I_0 = \frac{bh^3}{36}</math>|| ||<ref name=tri>{{cite web|url=http://www.efunda.com/math/areas/triangle.cfm|title=Triangular area|accessdate=2006-12-30|publisher=eFunda}}</ref>
+|-
+| a filled triangular area as above but with respect to an axis collinear with the base||[[Image:Area moment of inertia of a triangle 2.svg]]||<math>I = \frac{bh^3}{12}</math>||This is a consequence of the [[parallel axis theorem]] ||<ref name=tri />
+|-
+| a filled [[regular hexagon]] with a side length of ''a''||[[Image:Area moment of inertia of a regular hexagon.svg]]||<math>I_0 = \frac{5\sqrt{3}}{16}a^4</math>||The result is valid for both a horizontal and a vertical axis through the centroid, and therefore is also valid for an axis with arbitrary direction that passes through the origin.||
+|
+|-
+| Any plane region with a known area moment of inertia for a parallel axis. (Main Article [[parallel axis theorem]]) ||[[Image:Parallel Axes Compact.png]]||<math>I_z = I_x + Ar^2</math> || This can be used to determine the second moment of area of a rigid body about any axis, given the body's moment of inertia about a parallel axis through the object's center of mass and the perpendicular distance (r) between the axes.
+|}
+==See also==
+* [[List of moments of inertia]]
+==References==
+{{reflist}}
+==External links==
+* [http://www.amesweb.info/SectionalPropertiesTabs/SectionalProperties.aspx Online Sectional Properties Calculator]
+* [http://civilengineer.webinfolist.com/str/micalc.htm Online Calculator for Area Moment of Inertia ]
+[[Category:Mechanics|Area moment of inertia]]
+[[Category:Physics-related lists|Area moments of inertia]]

Convergence problem: Difference between revisions

en>Xnn

Revision as of 18:07, 12 March 2013

Contents

Area moments of inertia

See also

References

External links

Navigation menu

Convergence problem: Difference between revisions

en>Xnn

Revision as of 18:07, 12 March 2013

Area moments of inertia

See also

References

External links

Navigation menu

Search