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:''This article is about the moment of inertia as related to the 'bending of a plane'. For the moment of inertia dealing with rotation of an object, see Moment of inertia.''
The 'second moment of area', also known as the 'area moment of inertia' or 'second moment of inertia', is a property of a shape that is used to predict its resistance to bending and deflection. It is analogous to the polar moment of inertia, which characterizes an object's ability to resist torsion.
The ''second moment of area'' is not the same thing as the moment of inertia, which is used to calculate angular acceleration. Many engineers refer to the ''second moment of area'' as the ''moment of inertia'' and use the same symbol $I$ for both, which may be confusing. Which inertia is meant (accelerational or bending) is usually clear from the context and obvious from the units.
See also moment (physics).

Contents

Notes on notation

Definition

Unit

Second moment of area for various cross sections

Rectangular cross section

"I-beam" cross section

Circular cross section

Cylindrical Cross Section

Parallel axis theorem

Composite cross section

"I-beam" cross section

Product moment of area

Stress in a beam

★ ''Mechanics of solids and structures'', Benham, P.P. ISBN: 0273361910

Notes on notation

In some circumstances ''I_x'' and ''I_xx'' mean the same thing depending on what notation the person conducting the calculations is using. For example some people referring to a section across the x-axis will just use ''I_x'', but someone who refers to a section across the x-axis as 'section x-x' will use $I\_\{xx\}$ for the same variable.
The same applies in the case of ''I_y'' and ''I_yy''.

Definition

: $I\_x\; =\; int\; y^2,\; dA$

★ ''I_x'' = the moment of inertia about the axis ''x''

★ ''dA'' = an elemental area

★ ''y'' = the perpendicular distance to the element ''dA'' from the axis ''x''
Intuitively, the second moment of area, measuring the resistance to bending, can be likened to a person's attempt to stop a force from turning a lever: the further a person places a hand from the pivot the more leverage is obtained and the easier it is to resist the turning force. In the above formula, the hand's resisting turning are replaced with the sum of small sections of the object (infinitesimally small in the limit); the leverage is proportional to the square of the distance from the 'pivot'. Each small section adds its own contribution depending on its position and proportional to how big it is in cross section; each piece can be split into smaller pieces being summed up until the infinitesimal size is reached and the result is accurate (i.e. the limit of the integral).
The above can only be used on its own, when sections are symmetrical about the ''x''-axis.
When this is not the case, the second moment of area about both the ''x''- and the ''y''-axis and the product moment of area, ''I_xy'', are required.

Unit

The SI unit for second moment of area is metre to the fourth power (''m''⁴)

Second moment of area for various cross sections

See list of area moments of inertia for other cross sections.

Rectangular cross section

:

★ ''b'' = width (''x''-dimension),

★ ''h'' = height (''y''-dimension)
:

★ ''b'' = width (''x''-dimension),

★ ''h'' = height (''y''-dimension)

"I-beam" cross section

:''See below, as an example of applying the formula for a composite cross section.''

Circular cross section

:

★ ''r'' = radius,

★ ''d'' = diameter

Cylindrical Cross Section

:

★ ''$D\_O$'' = Outside Diameter

★ ''$D\_I$'' = Inside Diameter

Parallel axis theorem

Main articles: parallel axis theorem

The parallel axis theorem can be used to determine the moment of an object about any axis, given the moment of inertia of the object about the parallel axis through the object's center of mass and the perpendicular distance between the axes.
: $I\_z\; =\; I\_\{CG\}+Ad^2,$

★ ''I_z'' = the second moment of area with respect to the ''z''-axis

★ ''I''_CG = the second moment of area with respect to an axis parallel to z and passing through the centroid of the shape (coincides with the neutral axis)

★ ''A'' = area of the shape

★ ''d'' = the distance between the ''z''-axis and the centroidal axis

Composite cross section

When it is easier to compute the moment for an item as a combination of pieces the ''composite cross sections'' the second moment of area is calculated by applying the parallel axis theorem to each piece and adding the terms:
: $I\_\{x\}=\; sum\; left(y^\{2\}A\; +I\_mathrm\{local\}$
ight)
: $I\_\{y\}=\; sum\; left(x^\{2\}A\; +I\_mathrm\{local\}$
ight)

★ ''y'' = distance from ''x''-axis

★ ''x'' = distance from ''y''-axis

★ ''A'' = surface area of part

★ ''I''_local is the second moment of area for that part of the composite, in the appropriate direction (i.e. ''I_x'' or ''I_y'' respectively).

"I-beam" cross section

The I-beam can be analyzed as either three pieces added together or as a large piece with two pieces removed from it. Either of these methods will require use of the formula for composite cross section.

★ ''b'' = width (''x''-dimension),

★ ''h'' = height (''y''-dimension)

★ ''t_w'' = width of central webbing

★ ''h₁'' = inside distance between flanges
This formula uses the method of a block with two pieces removed. (While this may not be the easiest way to do this calculation, it is instructive in demonstrating how to subtract moments).

Since the I-beam is symmetrical with respect to the y-axis the $I\_\{x\}$ has is no component for the centroid of the blocks removed being offset above or below the x axis.
:
When computing ''I_y'' it is necessary to allow for the fact that the pieces being removed are offset from the X axis, this results in the ''Ax²'' term.
:
ight)^3}{12}+Ax^2}
ight)

★ ''A'' = Area contained with in the middle of one of the 'C' shapes of created by two flanges and the webbing on one side of the cross section =

★ ''x'' = distance of the centroid of the area contained in the 'C' shape from the y-axis of the beam =
Doing the same calculation by combining three pieces, the center webbing plus identical contributions for the top and bottom piece:

Since the centroids of all three pieces are on the y-axis ''I_y'' can be computed just by adding the moments together.
:
However, this time the law for composition with offsets must be used for ''I_x'' because the centroids of the top and bottom are offset from the centroid of the whole I-beam.

★ ''A'' = Area of the top or bottom piece=

★ ''y'' = offset of the centroid of the top or bottom piece from the centroid of the whole I-beam=
:
ight)^3}{12} + Ay^2
ight) =rac{t_{w} {h_{1}}^3}{12} + 2 left( rac{b left(rac{h-h_{1}}{2}
ight)^3}{12} + b rac{h-h_{1}}{2} left(rac{h+h_{1}}{4}
ight)^2
ight)

Product moment of area

The product moment of area, ''I_xy'' is defined as
: $I\_\{xy\}\; =\; -int\; xy,\; dA$

★ ''dA'' = an elemental area

★ ''x'' = the perpendicular distance to the element ''dA'' from the axis ''y''

★ ''y'' = the perpendicular distance to the element ''dA'' from the axis ''x''
The product moment of area is significant for determining the bending stress in an asymmetric cross section. Unlike the second moments of area, the product moment may give both negative and positive values. A coordinate system, in which the product moment is zero, is referred to as a set of principal axes, and the second moments of area calculated with respect to the principal axes will assume their maxima and minima. A coordinate system with origin in the centroid of the cross section and with both axes being axes of symmetry are always principal axes.
Additionally, the product moment may be used to calculate the second moments of area for a coordinate system rotated relative to the original coordinate system.
:$\{I\_x\}^$
★ = rac{I_{x} + I_{y}}{2} + rac{I_{x} - I_{y}}{2} cos(2 phi) + I_{xy} sin(2 phi)
:$\{I\_y\}^$
★ = rac{I_{x} + I_{y}}{2} - rac{I_{x} - I_{y}}{2} cos(2 phi) - I_{xy} sin(2 phi)
:$\{I\_\{xy\}\}^$
★ = -rac{I_{x} - I_{y}}{2} sin(2 phi) + I_{xy} cos(2 phi)

★ $phi$ = the angle of rotation

★ ''I_x'', ''I_y'' and ''I_xy'' = the second moments and the product moment of area in the original coordinate system

★ ''I_x
^★ '', ''I_y
^★ '' and ''I_xy
^★ '' = the second moments and the product moment of area in the rotated coordinate system.
The value of the angle $phi$, which will give a product moment of zero, is equal to:
:
This angle is the angle between the axes of the original coordinate system and the principal axes of the cross section.

Stress in a beam

The general form of the classic bending formula for a beam is:
:

★ $\{sigma\}$ is the bending stress

★ ''x'' = the perpendicular distance to the centroidal ''y''-axis

★ ''y'' = the perpendicular distance to the centroidal ''x''-axis

★ ''M_y'' = the bending moment about the ''y''-axis

★ ''M_x'' = the bending moment about the ''x''-axis

★ ''I_x'' = the second moment of area about ''x''-axis

★ ''I_y'' = the second moment of area about ''y''-axis

★ ''I_xy'' = the product moment of area
If the coordinate system is chosen to give a product moment of area equal to zero, the formula simplifies to:
:
If additionally the beam is only subjected to bending about one axis, the formula simplifies further:
:

References

★ ''Mechanics of solids and structures'', Benham, P.P. ISBN: 0273361910

External links==

★ Illustrations and examples (de.wikipedia.org)