3. Geometric characteristics of cross sections

3.1. First moment of an area

Under some deformations the details strength depends not only on the cross section area but also on its shape. Up to now we studied the deformations which have the stresses depending on the cross section area. To study the deformation of torsion and bending we have to know some other geometric figures plane characteristics.

The first moment of the plane figure about any axis in the plane of the area is given by summation of the elemental areas products and the perpendicular distances between the element and the axis (Fig. 3.1).

Denote by S the first moment of the area with the corresponding axis index:

(3.1)

Fig. 3.1.

The formulas to determine the centroidal coordinates of the figure area are as follows:

(3.2)

Since in these formulas (3.2) by A we can mean dA elemental area, then the expressions of the numerators in the right parts of the formula are the first moment of a figure about the y and x – axes, and is the A area of the all figure. Consequently,

(3.3)

The unit of the first moment of the figure area is the unit of the length.

The first moment of the figure area can be a positive, negative and equal to zero value. Evidently, the first moment of the area about the axis passing through the centroid of the figure area (the central axis) and among the symmetry figure axis is equal to zero.

In the formulas to determine the centroidal coordinate we can imply by the A area the finite figure parts, by x and y – the coordinates of these centroidal parts (i.e. to apply the division method). Hence, to determine the first moment of the area of a compound figure we can also apply the division method, i.e. to determine the first moment of all the figure as an algebraic sum of the first moments of its separate parts:

(3.4)

where Si is the first moment of each figure part area.

3.2. Polar moment of inertia

The Polar moment of inertia of the plane figure about a pole in the plane is called the sum of the products of the elements of the area and the square of the distance between the element and the pole for all the figure area (Fig. 3.2).

Polar moment of inertia is denoted by

. (3.5)

The unit of the polar moment of inertia is the unit of length.

Fig. 3.2.

The Polar moment of inertia is always a positive value and is not equal to zero.

Since we shall need the concept of the polar moment of inertia to study the torsion deformation of the round shaft, we derive formulas to determine the polar moment of inertia of the round solid and circular sections taking the centers of these figures as the pole.

1. The circle has the diameter d (Fig. 3.2). Let us remove the infinite small area dA of the ring shape with the d width lying from the pole ( is the variable value) at the  distance. Thus, Determine the polar moment of inertia:

(3.6)

2. The ring with the dimensions Dxd:

(3.7)

The Polar moment of inertia of the ring section can be determined as the difference of the polar moment of inertia of large and small circles.