PROPERTIES OF AREA
Certain mathematical expressions of the properties of sections are used in design calculations for various design shapes and loading conditions. These properties include the moment of inertia, cross sectional area, neutral axis, section modulus, and radius of gyration of the design shape in question. These properties are described below.
1. Moment of inertia. The moment of inertia I of the cross section is defined as the sum of the products of the differential areas into which the section may be divided, multiplied by the squares of their distances from the neutral axis of the section (Figure 3.1).
If the section is subjected to a bending moment about the X-X axis of the cross section, the moment of inertia about X-X is denoted by Ixx,
n = total number of differential areas A і = area of element i Yi = distance between element i and X-X axis
If the member is subjected to a bending moment about axis Y-Y of the cross section, we denote the moment of inertia associated with it as 1%,,
I, = 1 A, X)
m = total number of elementary areas
Aj = area of element j
X, = distance between element j and Y-Y axis
2. Cross sectional area. This is the area of a section taken through the member, perpendicular to its longitudinal axis.
3. Neutral axis. The neutral axis is a line through the cross section of the member along which the fibers sustain neither tension nor compression when subjected to a loading.
4. Section modulus. Denoted as S, this is the moment of inertia divided by the distance between the neutral axis and the extreme fibers (maximum stressed fibers) of the cross section.
If c is the distance from the neutral axis to the extreme
fibers in inches, one can write:
Radius of gyration. This property, denoted as r, is the square root of the quantity of the moment of inertia divided by the area of the cross section.
Here rx and ry are the radii of gyration about X-X and Y-Y axes, respectively.