Parallel Axis Theorem


Equations
(Eq1)    
Ix =

I
x' + Ad 2
parallel axis theorem


Nomenclature
rdistance from O to the element of area dA


Details

Consider the moment of inertia I of an area A with respect to an axis AA' as shown:



Denoting by y the distance from an element of area dA to AA', it is written:

I y 2 dA

An axis BB' parallel to AA' is drawn through the centroid C of the area; this axis is called a centroidal axis. Denoting by y' the distance from the element dA to BB', the equation y = y' + d follows, where d is the distance between the axis AA' and BB'. Substituting for y in the above integral:

I y 2 dA( y' + d) 2 dA y' 2 dA + 2d y' dA + d 2dA

The first integral represents the moment of inertia of the area with respect to the centroidal axis BB'. The second integral represents the first moment of the area with respect to BB'; since the centroid C of the area is located on that axis, the second integral must be zero. Finally, it is observed that the last integral is equal to the total area A.

In order to calculate the moment of inertia of an area, the area must first be able to be referenced with respect to a defined coordinate system (x and y axes, for example). Calculation of the moment of inertia depends upon how the desired area is positioned with respect to the reference axes and how it is oriented (rotated). The moment of inertia is dependent upon how the desired area is positioned with respect to the reference axes.

In order to calculate the moment of inertia of a composite area (e.g. a complex area that can be subdivided into simple areas), find the moments of inertia of the individual areas separately and add them up.

Click here for many moments of inertia for simple areas

If a desired area is does not lie along the axis, then the parallel-axis theorem must be used.

The parallel-axis theorem is the following:

(Eq1)    
Ix =

I
x' + Ad2

This means that the moment of inertia for the area (Ix) is equal to the moment of inertia of the area with respect to the centroidal x' axis parallel to the x axis (

I
x'), plus the product of the area A and the square of the distance d between the two axes (Ad2).

Eq1 expresses that the moment of inertia I of an area with respect to any given axis AA' is equal to the moment of inertia of the area with respect to the centroidal axis BB' parallel to AA' plus the product of the area A and the square of the distance d between the two axes. This theorem is known as the parallel-axis theorem. Substituting k 2A for I and

k
2A for

I
, the theorem can also be expressed as:

k 2 =

k
2 + d2

A similar theorem can be used to relate the polar moment of inertia JO of an area about a point O to the polar moment of inertia

J
C of the same area about its centroid C. Denoting by d the distance between O and C:

JO =

J
C + Ad2

or:

kO2 =

k
C2 + d2