See polygon area for calculating area and centroid of the section using similar formulae

Second moment of area

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:
= the angle of rotation(anticlockwise sense):
x * = xcos + ysin
y * = xsin + ycos
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 , 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 in co-ordinate system having origin located at the neutral axis of the beam is (Pilkey 2002, p. 17):
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
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:
If additionally the beam is only subjected to bending about one axis, the formula simplifies further:
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),
Circular cross section
D = diameter
r = radius
This equation is useful in calculating the required strength of masts. Taking the area moment of inertia calculated from the previous formula, and entering it into Euler's formula gives the maximum force that a mast can theoretically withstand.
E is [Young's modulus|Young (elastic) modulus of material]
I is the second moment of area of examined object
l is the length of panel
Hollow Cylindrical Cross Section
DO = outside diameter
DI = inside diameter
rO = outside radius
rI = 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:
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
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)
tw = width of central webbing
h1 = inside distance between flanges (usually referred to as hw, the height of the web)
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).
I-beam 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.
When computing Iy it is necessary to allow for the fact that the pieces being removed are offset from the Y axis, this results in the Ax2 term.
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 =
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:
I-beam 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.
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 =
y = offset of the centroid of the top or bottom piece from the centroid of the whole I-beam =
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:
ai = xiyi + 1 xi + 1yi 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
Pilkey, Walter D. (2002). Analysis and Design of Elastic Beams. John Wiley & Sons, Inc.. ISBN 0-471-38152-7. 
Categories: Structural analysis | Physical quantities