GEOMETRICAL CHARACTERISTICS OF FLAT SECTIONS.

As experience shows, the resistance of a rod to various deformations depends not only on the cross-sectional dimensions, but also on the shape.

The cross-sectional dimensions and shape are characterized by various geometric characteristics: cross-sectional area, static moments, moments of inertia, moments of resistance, etc.

1. Static moment of area(moment of inertia of the first degree).

Static moment of inertia area relative to any axis is the sum of the products of elementary areas and the distance to this axis, spread over the entire area (Fig. 1)

Fig.1

Properties of static moment of area:

1. The static moment of area is measured in units of length of the third power (for example, cm 3).

2. The static moment can be less than zero, greater than zero and, therefore, equal to zero. The axes about which the static moment is zero pass through the center of gravity of the section and are called central axes.

If x c And y c are the coordinates of the center of gravity, then

3. The static moment of inertia of a complex section relative to any axis is equal to the sum of the static moments of the components of simple sections relative to the same axis.

The concept of static moment of inertia in the science of strength is used to determine the position of the center of gravity of sections, although it must be remembered that in symmetrical sections the center of gravity lies at the intersection of the axes of symmetry.

2. Moment of inertia of flat sections (figures) (moments of inertia of the second degree).

A) axial(equatorial) moment of inertia.

Axial moment of inertia The area of ​​a figure relative to any axis is the sum of the products of elementary areas by the square of the distance to this axis of distribution over the entire area (Fig. 1)

Properties of the axial moment of inertia.

1. The axial moment of inertia of the area is measured in units of length of the fourth power (for example, cm 4).

2. The axial moment of inertia is always greater than zero.

3. The axial moment of inertia of a complex section relative to any axis is equal to the sum of the axial moments of the components of simple sections relative to the same axis:

4. The magnitude of the axial moment of inertia characterizes the ability of a rod (beam) of a certain cross section to resist bending.

b) Polar moment of inertia.

Polar moment of inertia The area of ​​a figure relative to any pole is the sum of the products of elementary areas by the square of the distance to the pole, spread over the entire area (Fig. 1).

Properties of the polar moment of inertia:

1. The polar moment of inertia of an area is measured in units of length of the fourth power (for example, cm 4).

2. The polar moment of inertia is always greater than zero.

3. The polar moment of inertia of a complex section relative to any pole (center) is equal to the sum of the polar moments of the components of simple sections relative to this pole.

4. The polar moment of inertia of a section is equal to the sum of the axial moments of inertia of this section relative to two mutually perpendicular axes passing through the pole.

5. The magnitude of the polar moment of inertia characterizes the ability of a rod (beam) of a certain cross-sectional shape to resist torsion.

c) Centrifugal moment of inertia.

The CENTRIFUGAL MOMENT OF INERTIA of the area of ​​a figure relative to any coordinate system is the sum of the products of elementary areas and coordinates, extended to the entire area (Fig. 1)

Properties of the centrifugal moment of inertia:

1. The centrifugal moment of inertia of an area is measured in units of length of the fourth power (for example, cm 4).

2. The centrifugal moment of inertia can be greater than zero, less than zero, and equal to zero. The axes about which the centrifugal moment of inertia is zero are called the main axes of inertia. Two mutually perpendicular axes, at least one of which is an axis of symmetry, will be the main axes. The principal axes passing through the center of gravity of the area are called the principal central axes, and the axial moments of inertia of the area are called the principal central moments of inertia.

3. The centrifugal moment of inertia of a complex section in any coordinate system is equal to the sum of the centrifugal moments of inertia of the constituent figures in the same coordinate system.

MOMENTS OF INERTIA RELATIVE TO PARALLEL AXES.

Fig.2

Given: axes x, y– central;

those. the axial moment of inertia in a section about an axis parallel to the central one is equal to the axial moment about its central axis plus the product of the area and the square of the distance between the axes. It follows that the axial moment of inertia of the section relative to the central axis has a minimum value in a system of parallel axes.

Having made similar calculations for the centrifugal moment of inertia, we obtain:

J x1y1 =J xy +Aab

those. The centrifugal moment of inertia of the section relative to the axes parallel to the central coordinate system is equal to the centrifugal moment in the central coordinate system plus the product of the area and the distance between the axes.

MOMENTS OF INERTIA IN A ROTATE COORDINATE SYSTEM

those. the sum of the axial moments of inertia of the section is a constant value, does not depend on the angle of rotation of the coordinate axes and is equal to the polar moment of inertia relative to the origin. The centrifugal moment of inertia can change its value and turn to “0”.

The axes about which the centrifugal moment is zero will be the main axes of inertia, and if they pass through the center of gravity, then they are called the main axes of inertia and are designated “ u" and "".

The moments of inertia about the principal central axes are called the principal central moments of inertia and are designated , and the main central moments of inertia have extreme values, i.e. one is “min” and the other is “max”.

Let the angle “a 0 ” characterize the position of the main axes, then:

Using this dependence, we determine the position of the main axes. The magnitude of the main moments of inertia after some transformations is determined by the following relationship:

EXAMPLES OF DETERMINING AXIAL MOMENTS OF INERTIA, POLAR MOMENTS OF INERTIA AND MOMENTS OF RESISTANCE OF SIMPLE FIGURES.

1. Rectangular section

Axles x and y - here and in other examples - the main central axes of inertia.

Let us determine the axial moments of resistance:

2. Round solid section. Moments of inertia.

If we draw coordinate axes through point O, then with respect to these axes the centrifugal moments of inertia (or products of inertia) are the quantities defined by the equalities:

where are the masses of points; - their coordinates; it is obvious that, etc.

For solid bodies, formulas (10), by analogy with (5), take the form

Unlike axial ones, centrifugal moments of inertia can be both positive and negative quantities and, in particular, with a certain way of choosing axes, they can become zero.

Main axes of inertia. Let us consider a homogeneous body having an axis of symmetry. Let us draw the coordinate axes Oxyz so that the axis is directed along the axis of symmetry (Fig. 279). Then, due to symmetry, each point of a body with mass mk and coordinates will correspond to a point with a different index, but with the same mass and with coordinates equal to . As a result, we obtain that since in these sums all terms are pairwise identical in magnitude and opposite in sign; from here, taking into account equalities (10), we find:

Thus, symmetry in the distribution of masses relative to the z axis is characterized by the vanishing of two centrifugal moments of inertia. The Oz axis, for which the centrifugal moments of inertia containing the name of this axis in their indices are equal to zero, is called the main axis of inertia of the body for point O.

From the above it follows that if a body has an axis of symmetry, then this axis is the main axis of inertia of the body for any of its points.

The principal axis of inertia is not necessarily the axis of symmetry. Let us consider a homogeneous body that has a plane of symmetry (in Fig. 279 the plane of symmetry of the body is the plane ). Let us draw some axes and an axis perpendicular to them in this plane. Then, due to symmetry, each point with mass and coordinates will correspond to a point with the same mass and coordinates equal to . As a result, as in the previous case, we find that or whence it follows that the axis is the main axis of inertia for point O. Thus, if a body has a plane of symmetry, then any axis perpendicular to this plane will be the main axis of inertia of the body for point O, in which the axis intersects the plane.

Equalities (11) express the conditions that the axis is the main axis of inertia of the body for point O (origin).

Similarly, if then the Oy axis will be the main axis of inertia for point O. Therefore, if all centrifugal moments of inertia are equal to zero, i.e.

then each of the coordinate axes is the main axis of inertia of the body for point O (origin).

For example, in Fig. 279 all three axes are the main axes of inertia for point O (the axis is the axis of symmetry, and the Ox and Oy axes are perpendicular to the planes of symmetry).

The moments of inertia of a body relative to the main axes of inertia are called the main moments of inertia of the body.

The main axes of inertia constructed for the center of mass of the body are called the main central axes of inertia of the body. From what was proved above it follows that if a body has an axis of symmetry, then this axis is one of the main central axes of inertia of the body, since the center of mass lies on this axis. If the body has a plane of symmetry, then the axis perpendicular to this plane and passing through the center of mass of the body will also be one of the main central axes of inertia of the body.

In the examples given, symmetrical bodies were considered, which is sufficient to solve the problems we will encounter. However, it can be proven that through any point of any body it is possible to draw at least three mutually perpendicular axes for which equalities (11) will be satisfied, i.e., which will be the main axes of inertia of the body for this point.

The concept of principal axes of inertia plays an important role in the dynamics of a rigid body. If the coordinate axes Oxyz are directed along them, then all centrifugal moments of inertia turn to zero and the corresponding equations or formulas are significantly simplified (see § 105, 132). This concept is also associated with the solution of problems on the dynamic equation of rotating bodies (see § 136), on the center of impact (see § 157), etc.

Let's look at a few more geometric characteristics of flat figures. One of these characteristics is called axial or equatorial moment of inertia. This characteristic is relative to the axes and
(Fig.4.1) takes the form:

;
. (4.4)

The main property of the axial moment of inertia is that it cannot be less than zero or equal to zero. This moment of inertia is always greater than zero:
;
. The unit of measurement for axial moment of inertia is (length 4).

Connect the origin of coordinates with a straight line segment with infinitesimal area
and denote this segment by the letter (Fig.4.4). The moment of inertia of a figure relative to the pole - the origin - is called the polar moment of inertia:

. (4.5)

This moment of inertia, like the axial one, is always greater than zero (
) and has dimension – (length 4).

Let's write it down invariance condition the sum of equatorial moments of inertia about two mutually perpendicular axes. From Fig. 4.4 it is clear that
.

Substituting this expression into formula (4.5), we obtain:

The invariance condition is formulated as follows: the sum of axial moments of inertia relative to any two mutually perpedicular axes is a constant value and equal to the polar moment of inertia relative to the intersection point of these axes.

The moment of inertia of a flat figure relative to two simultaneously perpendicular axes is called biaxial or centrifugal moment of inertia. The centrifugal moment of inertia has the following form:

. (4.7)

The centrifugal moment of inertia has the dimension – (length 4). It can be positive, negative or zero. Axes about which the centrifugal moment of inertia is zero are called main axes of inertia. Let us prove that the axis of symmetry of a plane figure is the main axis.

Consider the flat figure shown in Fig. 4.5.

Select left and right from the axis of symmetry two elements with infinitesimal area
. The center of gravity of the entire figure is at point C. Let’s place the origin of coordinates at point C and denote the vertical coordinates of the selected elements with the letter “ ”, horizontally – for the left element “
”, for the right element “ " Let us calculate the sum of centrifugal moments of inertia for selected elements with an infinitesimal area relative to the axes And :

If we integrate expression (4.8) from the left and right, we get:

, (4.9)

because if the axis is an axis of symmetry, then for any point lying to the left of this axis there is always a point symmetric to it.

Analyzing the obtained solution, we come to the conclusion that the axis of symmetry is the main axis of inertia. Central axis is also the main axis, although it is not an axis of symmetry, since the centrifugal moment of inertia was calculated simultaneously for two axes And and turned out to be zero.

DEFINITION

Axial (or equatorial) moment of inertia section relative to the axis is called a quantity that is defined as:

Expression (1) means that to calculate the axial moment of inertia, the sum of the products of infinitesimal areas () multiplied by the squares of the distances from them to the axis of rotation is taken over the entire area S:

The sum of the axial moments of inertia of the section relative to mutually perpendicular axes (for example, relative to the X and Y axes in the Cartesian coordinate system) gives the polar moment of inertia () relative to the intersection point of these axes:

DEFINITION

Polar moment inertia is called the moment of inertia section with respect to some point.

Axial moments of inertia are always greater than zero, since in their definitions (1) under the integral sign there is the value of the area of ​​the elementary area (), always positive, and the square of the distance from this area to the axis.

If we are dealing with a section of complex shape, then often in calculations we use the fact that the axial moment of inertia of a complex section relative to the axis is equal to the sum of the axial moments of inertia of the parts of this section relative to the same axis. However, it should be remembered that it is impossible to sum up the moments of inertia that are found relative to different axes and points.

The axial moment of inertia relative to the axis passing through the center of gravity of the section has the smallest value of all moments relative to the axes parallel to it. The moment of inertia about any axis () provided that it is parallel to the axis passing through the center of gravity is equal to:

where is the moment of inertia of the section relative to the axis passing through the center of gravity of the section; - cross-sectional area; - distance between axes.

Examples of problem solving

EXAMPLE 1

Exercise	What is the axial moment of inertia of an isosceles triangular cross-section relative to the Z axis passing through the center of gravity () of the triangle, parallel to its base? The height of the triangle is .
Solution	Let us select a rectangular elementary area on a triangular section (see Fig. 1). It is located at a distance from the axis of rotation, the length of one side is , the other side is . From Fig. 1 it follows that: The area of ​​the selected rectangle, taking into account (1.1), is equal to: To find the axial moment of inertia, we use its definition in the form:
Answer

EXAMPLE 2

Exercise	Find the axial moments of inertia relative to the perpendicular axes X and Y (Fig. 2) of a section in the form of a circle whose diameter is equal to d.
Solution	To solve the problem, it is more convenient to start by finding the polar moment relative to the center of the section (). Let us divide the entire section into infinitely thin rings of thickness , the radius of which will be denoted by . Then we find the elementary area as:

product of inertia, one of the quantities characterizing the distribution of masses in a body (mechanical system). C. m. and. are calculated as sums of products of masses m to points of the body (system) to two of the coordinates x k, y k, z k these points:

Values ​​of C. m. and. depend on the directions of the coordinate axes. In this case, for each point of the body there are at least three such mutually perpendicular axes, called the main axes of inertia, for which the centrifugal mass and. are equal to zero.

The concept of C. m. and. plays an important role in the study of rotational motion of bodies. From the values ​​of C. m. and. depend on the magnitude of the pressure forces on the bearings in which the axis of the rotating body is fixed. These pressures will be the smallest (equal to static) if the axis of rotation is the main axis of inertia passing through the center of mass of the body.

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