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Mechanics: Statics





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Second Moment of An Area
  Rectangular Moment of Inertia of an Area
   Elemental Horizontal Strip
   Elemental Vertical Strip
   Elemental Rectangular Strip about the referenc axis perpendicular to the length
  Polar Moment of Inertia of an Area
   Elemental Arc Strip
   Elemental Sector Slice
  Second Moment and First Moment of an Area
  Radius of Gyration of an Area
  Parallel-Axis Theorem

Second Moment of An Area

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By definition, second moment of an area about an axis is equal to the summation of the products of the square of the distance between the elemental area and the reference axis, and the elemental area over an area.

Rectangular Moment of Inertia of an Area

The two most common second moments are the rectangular moments of inertia of an area about the two coordinate axes. Imply

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The elemental differential area is defined as a second order element of dxdy. And a double integration is needed to determine the second moment of an area.

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However, when a group of second order elements can be grouped into a first order element, the elemental differental area can then be expressed as a first order element e.g. a rectangular strip. Only a single integration is needed to determine the second moment of an area.

Elemental Horizontal Strip

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When the second order element areas can be grouped as a first order element, e.g. a horizontal rectangular strip by neglecting the higher order term, only a single integration is needed to determine the second moment of an area. Since the elemental area of a horizontal strip have the same distance from the x-axis. Imply

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Elemental Vertical Strip

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When the second order element areas can be grouped as a first order element, e.g. a vertical rectangular strip by neglecting the higher order term, only a single integration is needed to determine the second moment of an area. Since the elemental area of a vertical strip have the same distance from the y-axis. Imply

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Elemental Rectangular Strip about the referenc axis perpendicular to the length

Because of the distance between the reference axis and elemental area of the rectangular strip is not a constant when the reference axis is not parallel to the lenght of the elemental rectangular strip, the second order differential area element cannot be replaced by the first order differential area element directly. However, the second order differential area element can be replaced by the first order differential area element, if the distance can be expressed as the function along the reference axis or the length of the elemental rectangular strip. For a horizontal elemental rectangular strip, the second moment about an vertical axis is

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For a vertical elemental rectangular strip, the second moment about an horizontal axis is

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Polar Moment of Inertia of an Area

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Besides the rectangular moments of inertia, another important second moment of an area is the polar moment of inertia, which is the second moment of an area about the axis normal to the planar area. And usually the polar moment of inertia of the area is defined with respect to the z axis or the pole at the origin O. Imply

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By considering the same second order differential area element, the polar moment of inertia of the area can be determined from the rectangular moments of inertia, Ix and Iy if the two rectangular moments of inertia are known using the perpendicular axis theorem.   Imply

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If the bounding profile of the area is expressed in form of polar coordinates, the polar moment of inertia about the pole can also be determined by the second moment of an area with the second order differential polar area element. Imply

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Imply

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Or the polar moment of inertia about the pole can be determined by the second moment of an area with the first order differential polar area element.

Elemental Arc Strip

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The polar moment of inertia about the pole can be determined by the second moment of an area with the first order differential elemental arc strip. Since the distance between the differential elemental area and the pole is the same. Imply

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Elemental Sector Slice

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The polar moment of inertia about the pole can be determined by the second moment of an area with the first order differential elemental sector slice. Since the distance between the reference axis and elemental area of the sector slice is not a constant, the second order differential area element cannot be replaced by the first order differential area element directly. However, the second order differential area element can be replaced by the first order differential area element, if the distance can be expressed as the function along the reference pole or the radius of the elemental sector slice. For an elemental sector slice, the second moment about the pole is

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Second Moment and First Moment of an Area

The value of first moment can be positive, negative and zero. But the second moment is always positive because the area element is alway positive, and the square of the distance is also alway positive. And therefore both the rectangular moment of inertia and the polar moment of inertia must be positive regardless of the location of the area element locating on the coordinate system.

Besides, the dimension of second moment of an area is of length to the fourth power, while the first moment of an area had dimensions of length to the third power.

Radius of Gyration of an Area

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For an area, the first moment of the area about an axis or a point can be expressed as product of the imaginary total area of the area at the centroid of the area and the shortest distance between the centroid of the area and the reference axis or the reference point. But for the second moment of the area, there is only an distance, called the radius of gyration, i.e. kx or ky, to describe the distance between  the location of an imaginary total area strip of the area and the reference axis or pole such that an equivalent second moment of the area is created by assuming the imaginary total area of the area is placed at this distance away from the reference axis. Or similarly the polar radius of gyration, kO about the pole. Imply

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Imply

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Parallel-Axis Theorem

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Unlike the first moment about the axis passing through the centroid, i,.e. the centroidal axis, the second moment about the axis, BB' passing through the centroid is not equal to zero. And the second moment of an area, A about the axis passing through its centroid is called the centroidal moment of inertia I of the area. Imply

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Consider the moment of inertia I of the area A about the axis, AA'. Imply

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The distance y in the integral can be expressed in terms of the distances y' and d. Imply

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The integral of the moment of inertia I of the area A about the axis, AA' can be divided into three seperated integrals. The first integral  is the centroidal moment of inertia I of the area. The second integral is the first moment of the area A about the centroidal axis and the value of the first moment of the area about the centroidal axis is equal to zero. The third integral is equal to the total area only. The square of the distance d is alway positive also, therefore the third term is alway positive regardless of the location of the parallel reference axis. Imply

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Therefore the second moment of an area A about an axis AA' is equal to the second moment of an area A about the centroidal axis BB' parallel to the reference axis AA' plus the product of the area A and the square of the distance d between the two parallel axes. This is called the parallel-axis theorem. Similarly, the theorem can also be applied to the polar moment of inertia about a pole O. Imply

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The theorem can also be expressed in terms of the radius of gyration of an area. Imply

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Since the theorem can only be used to relate the second moment of an area about an axis with the second moment of an area about the centroidal axis parallel to the reference axis, two second moments of an area about two parallel axes can not be transferred directly. e.g.

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Imply

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References

  1. I.C. Jong; B.G. rogers, 1991, Engineering Mechanics: Statics and Dynamics, Saunders College Publishing, United States of America
  2. F.P. Beer; E.R. Johnston,Jr.; E.R. Eisenberg, 2004, Vector Mechanics for Engineers: Statics, McGraw-Hill Companies, Inc., New York
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ID: 121000002 Last Updated: 2012/10/16 Revision: 0 Ref:

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