3.3. Axial moment of inertia

The axial moment of inertia about any axis in the plane of the figure is called the sum of the products of the elemental area and the square of the distance from the element and axis (Fig. 3.1).

Let us denote the moment of inertia by I with the index corresponding to the axis:

(3.8)

It is evident that the axial and the polar moments are expressed in the same units –the unit of length.

The axial moment is always positive and it is not equal to zero.

Fig. 3.3.

The sum of the axial moments about two mutually perpendicular axes is equal to the polar moment of inertia about the origin:

(3.9)

As the sum integral is equal to the integrals sum, then we can determine the inertia moment of complicated figures as the sum of simple figures into which they can be divided. The concept of the axial moments of inertia will be necessary in studying the theory of bending. Determine the axial moments of inertia of some simple figures.

1. The rectangle with sides b and h (Fig. 3.3).

Select the infinite small area dA in the form of a stripe b with the height dy, then dA=bdy.

(3.10)

For the square with the a side .

2. The circle with the b diameter about x and y axes. By symmetry we have As

then

(3.11)

3. The ring with dimensions of D х d about x and y axes:

(3.12)

3.4. The moment of inertia at parallel displacement of axis

The axes passing through the figure centroid are called central. The inertia moment about the central axis is called the central moment of inertia.

The moment of inertia of an area is equal to the moment of inertia about a parallel axis through the centroid of the area plus the product of the area and square of the distance between the axes.

Let us take an arbitrary plane figure having the area A, the centroid at the point С and the moment of inertia through the x axis is I_x. Determine the inertia moment of the figure about the axis which is parallel to the centroidal axes and located at the distance a (Fig. 3.4):

(3.12)

The first item on the right part is the inertia moment of the figure about x axis i.e. I_x; the second item contains the first moment of the area about x axis and it is equal to zero, because x axis is central; the third item is equal to а²А.

Thus, we get

(3.13)

One should keep in mind the fact that we can use the last formula only when one of parallel axes is central.

Analyzing the derived formula we can do the conclusion that the moment of inertia of the number of parallel axes will be the minimum one about the centroid axis.

Fig. 3.4.

Derive the formula of the inertia moment calculation of a rectangular about х₁ axis passing through its base:

(3.14)

3.5. Principal axes and principal moment of inertia

Let us imagine the plane figure that has the moment of inertia about the coordinate axes I_x and and the polar moment of inertia about the initial coordinates.

As it was determined earlier

The polar moment of inertia will be constant but the axial moments of inertia will change if we rotate the initial coordinates in its plane and what is more

.

If the sum of two variable values remains constant, one of them decreases but the other one increases. Hence, under some axis position one of the second moments reaches the maximum value and the other  the minimum one.

The axes about which the inertia moments have both maximum and minimum values are called the principal axes of inertia.

The moment of inertia about the principal axis is called the principal moment of inertia.

If the principal axis passes throw the figure centroid, it is called the principal central axis and the moment of inertia about this axis is called the principal central moment of inertia.

The following circumstance is extremely important: if the figure has the axis of symmetry, the axis will always be one of the principal central axis.

Let us introduce another geometric characteristics of the plane section.

The product of inertia of the plane figure is called the sum of the products of the area elements and the perpendicular distance between the elements and the axes taken over the figure area:

, (3.15)

where x, y are the distances between the elemental area dA and x and y axes. The product of inertia may be positive, negative and in a special case zero.

If the mutually perpendicular axes x and y or one of them are the axis of the symmetry, the product of inertia is equal to zero about these axes.

The product of inertia makes part of the formulas to determine the position of the principal axes of asymmetrical sections.