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 can be used to predict the resistance of beams to bending and deflection. The deflection of a beam under load depends not only on the load, but also on the geometry of the beam's cross-section. 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.
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 <tex>I</tex> for both, which may be confusing. Which inertia is meant (accelerational or bending) is usually clear from the context and obvious from the units: second moment of area has units of length to the fourth power whereas moment of inertia has units of mass times area.
See also moment (physics).
Contents
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 <tex>I_{xx}</tex> for the same variable.
The same applies in the case of I_{y} and I_{yy}.
Definition
- <tex>I_x = \int y^2\,\texrm dA</tex>
- I_{x} = the second moment of area about the axis x
- dA = an elemental area
- y = the perpendicular distance from the axis x to the element dA
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).
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.
Product moment of area
The product moment of area, I_{xy} is defined as
- <tex>I_{xy} = \int xy\,\texrm dA</tex>
- 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
- Note that I_{xy} is defined here with positive sign following common practice adopted in structural analysis (see e.g. Pilkey 2002, p. 15). The alternative approach to definition of moments of area could be to define the symmetric tensor for moment of area similarly to the tensor of mass moment of inertia; note that non-diagonal members of this tensor are equal to -I_{xy}:
- <tex> J_{ij} = \int ( r^2 \delta_{ij} - r_i r_j )\, \texrm dA </tex>
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.
Unit
The unit for second moment of area is length to the fourth power (typically mm^{4}, in^{4}, etc.)
Co-ordinate transformations
When calculating moments of the section it is often practical to compute them in one co-ordinate system (typically bound to the section shape) and then transform to another one using co-ordinate transformations.
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 (second moment of area) of the object about the parallel axis through the object's center of mass (or centroid) and the perpendicular distance between the axes.
- <tex>I_x = I_{xCG}+A d^2\,</tex>
- I_{x} = the second moment of area with respect to the x-axis
- I_{xCG} = the second moment of area with respect to an axis parallel to x and passing through the centroid of the shape (coincides with the neutral axis)
- A = area of the shape
- d = the distance between the x-axis and the centroidal axis
Axis rotation
The following formulae can be used to calculate moments of the section in a co-ordinate system rotated relative to the original co-ordinate system:
- <tex>{I_x}^* = \frac{I_{x} + I_{y}}{2} + \frac{I_{x} - I_{y}}{2} \cos(2 \phi) - I_{xy} \sin(2 \phi)</tex>
- <tex>{I_y}^* = \frac{I_{x} + I_{y}}{2} - \frac{I_{x} - I_{y}}{2} \cos(2 \phi) + I_{xy} \sin(2 \phi)</tex>
- <tex>{I_{xy}}^* = \frac{I_{x} - I_{y}}{2} \sin(2 \phi) + I_{xy} \cos(2 \phi) </tex>
- <tex>\phi</tex> = the angle of rotation(anticlockwise sense):
- <tex> x^* = x \cos \phi + y \sin \phi</tex>
- <tex> y^* = -x \sin \phi + y \cos \phi</tex>
- 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 <tex>\phi</tex>, which will give a product moment of zero, is equal to:
- <tex> \phi = -\frac{1}{2} \arctan \frac{ 2 I_{xy}} {I_x-I_y}</tex>
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 in co-ordinate system having origin located at the neutral axis of the beam is (Pilkey 2002, p. 17):
- <tex>\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</tex>
- <tex>{\sigma}</tex> is the normal stress in the beam due to bending
- 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:
- <tex>\sigma=-\frac{M_y}{I_y} x + \frac{M_x}{I_x} y</tex>
If additionally the beam is only subjected to bending about one axis, the formula simplifies further:
- <tex>{\sigma}= \frac{M}{I_x} y</tex>
Second moment of area for various cross sections
See list of area moments of inertia for other cross sections.
Rectangular cross section
- <tex>I_{x}=\frac{bh^3}{12}</tex>
- b = width (x-dimension),
- h = height (y-dimension)
- <tex>I_{y}=\frac{hb^3}{12}</tex>
- b = width (x-dimension),
- h = height (y-dimension)
Circular cross section
- <tex>I_0 = \frac{\pi}{64} D^4 = \frac{\pi}{4} r ^4</tex>
- D = diameter
- r = radius
this equation is very useful in most calculations. Having given moment of inertia calculated previously and entering it to this equation, gives you rapid approximation of required mast diameter
Hollow Cylindrical Cross Section
- <tex>I_0 = \frac{\pi}{64} (D_O ^4 - D_I ^4) = \frac{\pi}{4} (r_O ^4 - r_I ^4)</tex>
- <tex>D_O</tex> = outside diameter
- <tex>D_I</tex> = inside diameter
- <tex>r_O</tex> = outside radius
- <tex>r_I</tex> = inside radius
Composite cross section
When it is easier to compute the moment for an item as a combination of pieces, the second moment of area is calculated by applying the parallel axis theorem to each piece and adding the terms:
- <tex>I_{x}= \sum \left(I_\texrm{local}+y^{2}A\right)</tex>
- <tex>I_{y}= \sum \left(I_\texrm{local}+x^{2}A\right)</tex>
- 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. This section only covers doubly symmetric I-beams, meaning the shape has two planes of symmetry.
- b = width (x-dimension),
- h = height (y-dimension)
- t_{w} = width of central webbing
- h_{1} = 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 <tex>I_{x}</tex> has no component for the centroid of the blocks removed being offset above or below the x axis.
- <tex>I_{x}=\frac{{bh^3}-2{{\frac{b-t_{w}}{2}}{h_{1}}^3}}{12}</tex>
When computing I_{y} it is necessary to allow for the fact that the pieces being removed are offset from the Y axis, this results in the Ax^{2} term.
- <tex>I_{y}=\frac{hb^3}{12}-2\left({\frac{h_{1}\left({\frac{b-t_{w}}{2}}\right)^3}{12}+Ax^2}\right)</tex>
- A = Area contained within the middle of one of the 'C' shapes of created by two flanges and the webbing on one side of the cross section = <tex>h_{1}{\frac{b-t_{w}}{2}}</tex>
- x = distance of the centroid of the area contained in the 'C' shape from the y-axis of the beam = <tex>{\frac{b+t_{w}}{4}}</tex>
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.
- <tex>I_{y}=\frac{h_{1} {t_{w}}^3}{12} + 2 \frac{\frac{h-h_{1}}{2} b^3}{12}</tex>
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=<tex>b \frac{h-h_{1}}{2}</tex>
- y = offset of the centroid of the top or bottom piece from the centroid of the whole I-beam=<tex>\frac{h+h_{1}}{4}</tex>
- <tex>I_{x}=\frac{t_{w} {h_{1}}^3}{12} + 2 \left( \frac{b \left(\frac{h-h_{1}}{2}\right)^3}{12} + Ay^2\right) =\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)^2\right)</tex>
Any cross section defined as polygon
The second moments of area for any cross section defined as a simple polygon on XY plane can be computed in a generic way by summing contributions from each segment of a polygon.
For each segment defined by two consecutive points of the polygon, consider a triangle with two corners at these points and third corner at the origin of the coordinates. Integration by the area of that triangle and summing by the polygon segments yields:
- <tex>I_x = \frac{1}{12} \sum_{i = 1}^{n} ( y_i^2 + y_i y_{i+1} + y_{i+1}^2 ) a_i \,</tex>
- <tex>I_y = \frac{1}{12} \sum_{i = 1}^{n} ( x_i^2 + x_i x_{i+1} + x_{i+1}^2 ) a_i \,</tex>
- <tex>I_{xy} = \frac{1}{24} \sum_{i = 1}^{n} ( x_i y_{i+1} + 2 x_i y_i + 2 x_{i+1} y_{i+1} + x_{i+1} y_i ) a_i \,</tex>
- <tex>a_i = x_i y_{i + 1} - x_{i + 1} y_i</tex> is twice the (signed) area of the elementary triangle,
- index i passes over all n points in the polygon, which is considered closed, i.e. point n+1 is point 1
These formulae imply that points defining the polygon are ordered in anticlockwise manner; for clockwisely defined polygons it will give negative values. See polygon area for calculating area and centroid of the section using similar formulae.
See also
- List of area moments of inertia
- List of moments of inertia
- Moment of inertia
- Polar moment of inertia
- Parallel axis theorem
- Perpendicular axis theorem
References
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