Let us assume that for an arbitrary section (Fig. 1.13) the moments of inertia about the coordinate axes z and y are known, and the centrifugal moment of inertia Izy is also known. It is required to establish dependencies for the moments of inertia about the axes 11 zy, rotated by an angle with respect to the original axes z and y (Fig. 1.13). We will consider the angle positive if the rotation of the coordinate system occurs counterclockwise. Let for a given section IzI. yTo solve the problem, let's find the relationship between the coordinates of the area dA in the original and rotated axes. From Fig. 1.13 it follows: From a triangle from a triangle With this in mind, we obtain Similarly for the coordinate y1 we obtain Considering that we finally have ), we determine the moment of inertia relative to the new (rotated) axes z1 and y1: Similarly, the centrifugal moment of inertia I relative to the rotated axes is determined by the dependence . Subtracting (1.27) from (1.26) we obtain Formula (1.30) can serve to calculate the centrifugal moment of inertia about the axes z and y, according to the known moments of inertia about the axes z, y and z1, y1, and formula (1.29) can be used to check the calculations of the moments inertia of complex sections. 1.8. Principal axes and principal moments of inertia of the section With a change in angle (see Fig. 1.13), the moments of inertia also change. For some values ​​of the angle 0, the moments of inertia have extreme values. Axial moments of inertia having maximum and minimum values ​​are called the main axial moments of inertia of the section. The axes with respect to which the axial moments of inertia have maximum and minimum values ​​are the principal axes of inertia. On the other hand, as noted above, the main axes are the axes relative to which the centrifugal moment of inertia of the section is zero. To determine the position of the main axes for sections of arbitrary shape, we take the first derivative with respect to I and equate it to zero: It should be noted that formula (1.31) can be obtained from (1.28) by equating it to zero. If we substitute the values ​​of the angle determined from expression (1.31) into (1. 26) and (1.27), then after transformation we obtain formulas that determine the main axial moments of inertia of the section. In its structure, this formula is similar to formula (4.12), which determines the principal stresses (see Section 4.3). If IzI, then, based on the studies of the second derivative, it follows that the maximum moment of inertia Imax takes place relative to the main axis rotated at an angle with respect to the z-axis, and the minimum moment of inertia - relative to the other main axis located at an angle 0 If II, everything is changing on the contrary. The values ​​of the main moments of inertia Imax and I can also be calculated from the dependences (1.26) and (1.27), if we substitute in them instead of the value. In this case, the question is solved by itself: relative to which main axis is the maximum moment of inertia obtained and relative to which axis is the minimum? It should be noted that if for a section the main central moments of inertia about the z and y axes are equal, then for this section any central axis is the main one and all the main central moments of inertia are the same (circle, square, hexagon, equilateral triangle, etc.). This is easily established from dependences (1.26), (1.27), and (1.28). Indeed, suppose that for some section the z and y axes are the main central axes and, in addition, I. yThen from formulas (1.26) and (1.27) we obtain that Izy , 1a from formula (1.28) we make sure that 11 e. any axes are the main central axes of inertia of such a figure. 1.9. The concept of the radius of gyration The moment of inertia of a section relative to any axis can be represented as the product of the section area by the square of a certain quantity, called the radius of gyration of the section area where iz ─ the radius of inertia relative to the z axis. Then from (1.33) follows: The main central axes of inertia correspond to the main radii of inertia: 1.10. Resistance moments Distinguish between axial and polar moments of resistance. 1. The axial moment of resistance is the ratio of the moment of inertia about a given axis to the distance to the most distant point of the cross section from this axis. Axial moment of resistance relative to the z-axis: and relative to the y-axis: max where ymax and zmax─, respectively, the distances from the main central axes z and y to the points most distant from them. In the calculations, the main central axes of inertia and the main central moments are used, therefore, under Iz and Iy in formulas (1.36) and (1.37) we will understand the main central moments of inertia of the section. Consider the calculation of the moments of resistance of some simple sections. 1. Rectangle (see Fig. 1.2): 2. Circle (see Fig. 1.8): 3. Annular tubular section (Fig. 1.14): . For rolled profiles, the moments of resistance are given in the assortment tables and there is no need to determine them (see appendix 24 - 27). 2. The polar moment of resistance is the ratio of the polar moment of inertia to the distance from the pole to the most distant point of the section max 30. The center of gravity of the section is usually taken as the pole. For example, for a round solid section (Fig. 1.14): For a tubular round section. The axial moments of resistance Wz and Wy characterize purely geometrically the resistance of the rod (beam) to bending deformation, and the polar moment of resistance W characterizes the resistance to torsion.

Let us calculate the moments of inertia of an arbitrary shape about the axes rotated about the given axes and
on the corner (Fig.4.14)

Let the moments of inertia about the axes
And
known. Choose an arbitrary site
and express its coordinates in the system of axes
And
through the coordinates in the old axes
And
:

Let's find the axial and centrifugal moments of inertia of the figure relative to the rotated axes
And
:

Taking into account that

;
And
,

Install in the same way:

The centrifugal moment of inertia takes the form:

. (4.30)

We express the axial moments in terms of the sine and cosine of the double angle. To do this, we introduce the following functions:

. (4.31)

Substituting (4.31) into formulas (4.27) and (4.28), we obtain:

If we add the expressions for the axial moments of inertia (4.32) and (4.33), we get:

Condition (4.34) represents the condition of invariance of the sum of axial moments of inertia with respect to two mutually perpendicular axes, i.e. the sum of the axial moments of inertia about two mutually perpendicular axes does not depend on the angle of rotation of the axes and is a constant value. Previously, this condition was obtained on the basis that the sum of the axial moments of inertia about two mutually perpendicular axes was equal to the value of the polar moment of inertia about the point of intersection of these axes.

We investigate the equation for the moment of inertia to the extremum and find such an angle value , at which the moment of inertia reaches an extreme value. To do this, we take the first derivative of the moment of inertia by angle (expression (4.32)) and equate the result to zero. At the same time, we put
.

(4.35)

The expression in brackets is the centrifugal moment of inertia about axes inclined to the axis
at an angle . With respect to these axes, the centrifugal moment of inertia is zero:

, (4.36)

which means that the new axes are the main axes.

It was previously determined that the principal axes of inertia are the axes with respect to which the centrifugal moment of inertia is zero. Now this definition can be expanded - these are the axes, relative to which the axial moments of inertia have extreme values. The moments of inertia about these axes are called main moments of inertia.

Find the position of the main axes of inertia. From expression (4.36) one can get:

. (4.37)

The resulting formula gives for the angle two meanings: And
.

Consequently, there are two mutually perpendicular axes with respect to which the moments of inertia have extreme values. As noted above, such axes are called the principal axes of inertia. It remains to establish which of the axes the moment of inertia reaches the maximum value, and relative to which - the minimum value. This problem can be solved by studying the second derivative of the expression (4.32) with respect to the angle . Substituting in the expression for the second derivative the value of the angle or
and examining the sign of the second derivative, one can judge which of the angles corresponds to the maximum moment of inertia, which to the minimum. Below are formulas that will give an unambiguous value of the angle .

Find the extreme values ​​for the moments of inertia. To do this, we transform the expression (4.32) , taking out of the brackets
:

We use the function known from trigonometry and substitute expression (4.37) into it, we get:

. (4.39)

Substituting expression (4.39) into formula (4.38) and making the necessary calculations, we obtain two expressions for extreme moments of inertia, which do not include the angle of inclination of the axes :

; (4.40)

. (4.41)

From formulas (4.40) and (4.41) it can be seen that the values ​​of the main moments of inertia are determined directly through the moments of inertia about the axes
And
. Therefore, they can be determined without knowing the position of the principal axes themselves.

Knowing the extreme values ​​of the moments of inertia
And
in addition to formula (4.37), it is possible to determine the position of the main axes of inertia.

We give formulas without derivation that allow us to find angles And between axle
and main axes:

;
(4.42)

Corner determines the position of the axis, relative to which the moment of inertia reaches its maximum value (
), corner determines the position of the axis, relative to which the moment of inertia reaches a minimum value (
).

We introduce another geometric characteristic, which is called the radius of gyration of the section. This characteristic is denoted by the letter and can be calculated with respect to the axes
And
in the following way:

;
(4.43)

The radius of gyration is widely used in problems of strength of materials and its application will be discussed in the following sections of the course.

Let us consider several examples of structural calculations taking into account the rotation of the axes and using the radius of gyration of the section.

Example 4.7. The moments of inertia of a rectangular section about the main axes are, respectively,
cm 4,
cm 4. When rotated by 45 0, the moments of inertia relative to the new axes turned out to be the same. What is their value?

To solve the problem, we use expression (4.28), taking into account the fact that the centrifugal moment of inertia about the main axes is zero:

Substitute in the formula (a) the numerical values ​​for the moments of inertia and the angle of rotation of the axes:

Example 4.8. Which of the figures (Fig. 4.15), having the same area, has a radius of inertia relative to the axis , will be the largest? Determine the largest radius of gyration of the section relative to the axis .

1. Find the area of ​​each of the figures and the dimensions of the sections. The area of ​​\u200b\u200bthe figures is equal to cm 2 for the third figure.

We find the diameter of the first section from the expression:

cm.

Square side size:

Triangle base:

cm.

2. We find the moments and radii of inertia of each of the sections relative to the central axis .

For a circular section:

cm 4;
cm.

For a square section:

cm 4;
cm.

For a rectangular section:

;

For a triangular section:

cm 4;
cm.

The largest radius of gyration turned out to be at a rectangular section and is equal to
cm.

Principal axes and principal moments of inertia

When the coordinate axes are rotated, the centrifugal moment of inertia changes sign, and therefore, there is such a position of the axes at which the centrifugal moment is equal to zero.

The axes about which the centrifugal moment of inertia of the section vanishes are called main axes , and the main axes passing through the center of gravity of the section -main central axes of inertia of the section.

The moments of inertia about the main axes of inertia of the section are calledmain moments of inertia of the sectionand are denoted by I1 and I2 with I1>I2 . Usually, speaking of the main moments, they mean axial moments of inertia about the main central axes of inertia.

Let's assume the axes u and v are principal. Then

From here

.

(6.32)

Equation (6.32) determines the position of the main axes of inertia of the section at a given point relative to the original coordinate axes. When the coordinate axes are rotated, the axial moments of inertia also change. Let us find the position of the axes, relative to which the axial moments of inertia reach extreme values. To do this, we take the first derivative of Iu by α and equate it to zero:

from here

.

The condition dIv / dα. Comparing the last expression with formula (6.32), we come to the conclusion that the principal axes of inertia are the axes with respect to which the axial moments of inertia of the section reach extreme values.

To simplify the calculation of the main moments of inertia, formulas (6.29) - (6.31) are transformed, excluding trigonometric functions from them using relation (6.32):

.

(6.33)

The plus sign in front of the radical corresponds to the larger I1 , and the minus sign to the smaller I2 from the moments of inertia of the section.

Let us point out one important property of sections in which the axial moments of inertia about the principal axes are the same. Let's assume the axes y and z are principal (Iyz =0), and Iy = Iz . Then according to equalities (6.29) - (6.31) for any angle of rotation of the axesα centrifugal moment of inertia Iuv =0, and axial Iu=Iv.

So, if the moments of inertia of the section about the main axes are the same, then all the axes passing through the same point of the section are the main ones and the axial moments of inertia about all these axes are the same: Iu=Iv=Iy=Iz. This property is possessed, for example, by square, round, annular sections.

Formula (6.33) is similar to formulas (3.25) for principal stresses. Consequently, the main moments of inertia can also be determined graphically by the Mohr method.

Changing the moments of inertia when rotating the coordinate axes

Let us assume that the system of coordinate axes is given and the moments of inertia are known Iz, Iy and Izy figures about these axes. Let's rotate the coordinate axes by some angleα counterclockwise and determine the moments of inertia of the same figure relative to the new coordinate axes u and v.

Rice. 6.8.

From fig. 6.8 it follows that the coordinates of any point in both coordinate systems are interconnected by the relations

Moment of inertia

Hence,

	(6.29)
	(6.30)

centrifugal moment of inertia

.

(6.31)

It can be seen from the obtained equations that

,

i.e., the sum of the axial moments of inertia remains constant when the coordinate axes are rotated. Therefore, if relative to any axis the moment of inertia reaches a maximum, then relative to an axis perpendicular to it, it has a minimum value.

Calculate the moments of inertia J u , J v and J uv:

Adding the first two formulas (3.14), we obtain J u + J v= Jz+ Jy, i.e. for any rotation of mutually perpendicular axes, the sum of the axial moments of inertia remains constant (invariant).

Principal axes and principal moments of inertia

Exploring the function J u(a) to the extreme. To do this, we equate the derivative to zero J u(a) by a.

We obtain the same formula by equating to zero the centrifugal moment of inertia

.

Principal axes are called axes, relative to which the axial moments of inertia take on extreme values, and the centrifugal moment of inertia is zero.

An infinite number of principal axes of inertia can be drawn by taking any point on the plane as the origin. To solve the problems of resistance of materials, we are only interested in main central axis of inertia. Principal central axes of inertia pass through the center of gravity of the section.

Formula (3.17) gives two solutions that differ by 90°, i.e. allows you to determine two values ​​of the angle of inclination of the main axes of inertia relative to the original axes. With respect to which of the axes is the maximum axial moment of inertia J 1 = J max , and relative to which one - the minimum J 2 = J min , will have to be solved according to the meaning of the problem.

More convenient are other formulas that uniquely determine the position of the principal axes 1 and 2 (given without derivation). In this case, the positive angle is measured from the axis Oz counterclock-wise.

In formula (3.19), the sign "+" corresponds to the maximum moment of inertia, and the sign "-" to the minimum.

Comment . If the section has at least one axis of symmetry, then relative to this axis and any other perpendicular to it, the centrifugal moment of inertia is equal to zero. In accordance with the definition of the principal axes of inertia, we can conclude that these axes are the principal axes of inertia, i.e. the axis of symmetry is always the main central axis.

For symmetrical profiles presented in the assortment, channel or I-beam, the main central axes of inertia will be the vertical and horizontal axes intersecting at half the height of the profile.