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Encyclopedia > Second moment of area

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 which are directly proportional. This is why beams with higher area moments of inertia, such as I-beams, are so often seen in building construction as opposed to other beams with the same area. It is analogous to the polar moment of inertia, which characterizes an object's ability to resist torsion. Figure 1. ... This article or section does not cite any references or sources. ... I-beams are beams with an I- or H-shaped cross-section. ... Polar moment of inertia is a quantity used to predict an objects ability to resist torsion, in objects (or segments of objects) with an invariant circular cross-section and no significant warping or out-of-plane deformation. ... // Mathmatics In mathematics, the term torsion has several meanings, mostly unrelated to each other. ...


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. Moment of inertia, also called mass moment of inertia and, sometimes, the angular mass, (SI units kg m², Former British units slug ft2), is the rotational analog of mass. ... лInsert non-formatted text hereInsert non-formatted text hereInsert non-formatted text hereInsert non-formatted text hereInsert non-formatted text hereInsert non-formatted text hereInsert non-formatted text hereInsert non-formatted text hereInsert non-formatted text hereInsert non-formatted text hereInsert non-formatted text hereInsert non-formatted text hereInsert non... The former Weights and Measures office in Middlesex, England. ...


See also moment (physics). It has been suggested that this article or section be merged with torque. ...

Contents

Notes on notation

In some circumstances Ix and Ixx 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 Ix, but someone who refers to a section across the x-axis as 'section x-x' will use Ixx for the same variable.


The same applies in the case of Iy and Iyy.


Definition

I_x = int y^2,mathrm dA
  • Ix = 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 farther 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). This article is about the concept of integrals in calculus. ...


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, Ixy, are required.


Unit

The SI unit for second moment of area is metre to the fourth power (m4) Look up si, Si, SI in Wiktionary, the free dictionary. ... This article is about the unit of length. ...


Second moment of area for various cross sections

See list of area moments of inertia for other cross sections. The following is list of area moments of inertia. ...


Rectangular cross section

I_{x}=frac{bh^3}{12}
  • b = width (x-dimension),
  • h = height (y-dimension)
I_{y}=frac{hb^3}{12}
  • 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

I_0 = frac{pi r^4}{4} = frac{pi d^4}{64}
  • r = radius,
  • d = diameter

Hollow Cylindrical Cross Section

I_0 = frac{pi}{64} (D_O ^4 - D_I ^4)
  • DO = Outside Diameter
  • DI = Inside Diameter

Parallel axis theorem

Main article: 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. The parallel axes rule can be used to determine the moment of inertia of a rigid object about any axis, given the moment of inertia of the object about the parallel axis through the objects center of mass and the perpendicular distance between the axes. ... The parallel axes rule can be used to determine the moment of inertia of a rigid object about any axis, given the moment of inertia of the object about the parallel axis through the objects center of mass and the perpendicular distance between the axes. ...

I_z = I_{CG}+Ad^2,
  • Iz = the second moment of area with respect to the z-axis
  • ICG = 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

An axis in the cross section of a beam, shaft or the like along which there are no longitudinal stresses / strains. ...

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}right)
I_{y}= sum left(x^{2}A +I_mathrm{local}right)
  • y = distance from x-axis
  • x = distance from y-axis
  • A = surface area of part
  • Ilocal is the second moment of area for that part of the composite, in the appropriate direction (i.e. Ix or Iy respectively).

"I-beam" cross section

I-beam
I-beam

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. Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ...

  • b = width (x-dimension),
  • h = height (y-dimension)
  • tw = 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).

Ibeam diagram, moment by subtraction
Ibeam diagram, moment by subtraction

Since the I-beam is symmetrical with respect to the y-axis the Ix has no component for the centroid of the blocks removed being offset above or below the x axis. Image File history File links Size of this preview: 398 × 600 pixelsFull resolution (400 × 603 pixel, file size: 11 KB, MIME type: image/jpeg) Diagram for wikipedia article on second moment of area for an I-beam, diagram 2. ... Image File history File links Size of this preview: 398 × 600 pixelsFull resolution (400 × 603 pixel, file size: 11 KB, MIME type: image/jpeg) Diagram for wikipedia article on second moment of area for an I-beam, diagram 2. ... Centroid of a triangle In geometry, the centroid or barycenter of an object in -dimensional space is the intersection of all hyperplanes that divide into two parts of equal moment about the hyperplane. ...

I_{x}=frac{{bh^3}-2{{frac{b-t_{w}}{2}}{h_{1}}^3}}{12}

When computing Iy 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.

I_{y}=frac{hb^3}{12}-2left({frac{h_{1}left({frac{b-t_{w}}{2}}right)^3}{12}+Ax^2}right)
  • 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 = h_{1}{frac{b-t_{w}}{2}}
  • x = distance of the centroid of the area contained in the 'C' shape from the y-axis of the beam = {frac{b+t_{w}}{4}}

Doing the same calculation by combining three pieces, the center webbing plus identical contributions for the top and bottom piece:

Ibeam diagram, moment by addition
Ibeam diagram, moment by addition

Since the centroids of all three pieces are on the y-axis Iy can be computed just by adding the moments together. Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ...

I_{y}=frac{h_{1} {t_{w}}^3}{12} + 2 frac{frac{h-h_{1}}{2} b^3}{12}

However, this time the law for composition with offsets must be used for Ix 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=b frac{h-h_{1}}{2}
  • y = offset of the centroid of the top or bottom piece from the centroid of the whole I-beam=frac{h+h_{1}}{4}
I_{x}=frac{t_{w} {h_{1}}^3}{12} + 2 left( frac{b left(frac{h-h_{1}}{2}right)^3}{12} + Ay^2right) =frac{t_{w} {h_{1}}^3}{12} + 2 left( frac{b left(frac{h-h_{1}}{2}right)^3}{12} + b frac{h-h_{1}}{2} left(frac{h+h_{1}}{4}right)^2right)

Product moment of area

The product moment of area, Ixy is defined as

I_{xy} = -int xy,mathrm 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. Stress is a measure of force per unit area within a body. ... Local and global maxima and minima for cos(3Ï€x)/x, 0. ...


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}^* = frac{I_{x} + I_{y}}{2} + frac{I_{x} - I_{y}}{2} cos(2 phi) + I_{xy} sin(2 phi)
{I_y}^* = frac{I_{x} + I_{y}}{2} - frac{I_{x} - I_{y}}{2} cos(2 phi) - I_{xy} sin(2 phi)
{I_{xy}}^* = -frac{I_{x} - I_{y}}{2} sin(2 phi) + I_{xy} cos(2 phi)
  • φ = the angle of rotation
  • Ix, Iy and Ixy = the second moments and the product moment of area in the original coordinate system
  • Ix*, Iy* and Ixy* = the second moments and the product moment of area in the rotated coordinate system.

The value of the angle φ, which will give a product moment of zero, is equal to:

 phi = frac{1}{2} arctan frac{ 2 I_{xy}} {I_x-I_y}

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: This vibrating glass beam may be modeled as a cantilever beam with acceleration, variable linear density, variable section modulus, some kind of dissipation, springy end loading, and possibly a point mass at the free end. ... A statically determinate beam, bending under an evenly distributed load. ...

sigma=-frac{M_y I_x - M_x I_{xy}}{I_x I_y - {I_{xy}}^2 } x - frac{M_x I_y - M_y I_{xy}}{I_x I_y - {I_{xy}}^2} y
  • σ is the bending stress
  • x = the perpendicular distance to the centroidal y-axis
  • y = the perpendicular distance to the centroidal x-axis
  • My = the bending moment about the y-axis
  • Mx = the bending moment about the x-axis
  • Ix = the second moment of area about x-axis
  • Iy = the second moment of area about y-axis
  • Ixy = 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: Stress is a measure of force per unit area within a body. ...

sigma=-frac{M_y}{I_y} x + frac{M_x}{I_x} y

If additionally the beam is only subjected to bending about one axis, the formula simplifies further:

{sigma}= frac{M y}{I_x}

External links

  • http://mywebsite.bigpond.com/npajkic/solid_mechanics/second_moment_of_area/index.html

  Results from FactBites:
 
Reference.com/Encyclopedia/Second moment of area (423 words)
The second moment of area, also known as the area moment of inertia and less precisely as the 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.
Moment of inertia - Wikipedia, the free encyclopedia (806 words)
Moment of inertia should not be confused with the second moment of area or area moment of inertia (SI units: m
For the case where the angular momentum is parallel to the angular velocity, the moment of inertia is simply a scalar.
However, in the general case of an object being rotated about an arbitrary axis, the moment of inertia becomes a tensor, such that the angular momentum need not be parallel to the angular velocity.
  More results at FactBites »

 
 

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