Axial moments of inertia of the section relative to the axes X And at(see Fig. 32, A) are called definite integrals of the form

When determining axial moments of inertia, in some cases it is necessary to encounter another new geometric characteristic of the section - the centrifugal moment of inertia.

Centrifugal moment of inertia sections relative to two mutually perpendicular axes x y(see Fig. 32, A)

Polar moment of inertia sections relative to the origin ABOUT(see Fig. 32, A) is called a definite integral of the form

Where R- distance from the origin to the elementary site dA.

The axial and polar moments of inertia are always positive, and the centrifugal moment, depending on the choice of axes, can be positive, negative or equal to zero. The units of designation of moments of inertia are cm 4, mm 4.

The following relationship exists between the polar and axial moments of inertia:

According to formula (41), the sum of the axial moments of inertia about two mutually perpendicular axes is equal to the polar moment of inertia about the intersection point of these axes (origin).

Moments of inertia of sections relative to parallel axes, one of which is central (x s,yc)> are determined from the expressions:

Where and Iv- coordinates of the center of gravity C of the section (Fig. 34).

Formulas (42), which have great practical application, read as follows: the moment of inertia of a section about any axis is equal to the moment of inertia about an axis parallel to it and passing through the center of gravity of the section, plus the product of the cross-sectional area and the square of the distance between the axes.

note: coordinates a and c should be substituted into the above formulas (42) taking into account their signs.

Rice. 34.

From formulas (42) it follows that of all the moments of inertia about parallel axes, the smallest moment will be about the axis passing through the center of gravity of the section, i.e., the central moment of inertia.

The formulas for determining the strength and rigidity of a structure include moments of inertia, which are calculated relative to the axes, which are not only central, but also main. In order to determine which axes passing through the center of gravity are the main ones, one must be able to determine the moments of inertia relative to the axes rotated relative to each other at a certain angle.

The relationships between the moments of inertia when rotating the coordinate axes (Fig. 35) have the following form:

Where A- axle rotation angle And And v relative to the axes henna respectively. Angle a is considered positive, if the rotation of the axes And and u happens counterclock-wise.

Rice. 35.

The sum of axial moments of inertia relative to any mutually perpendicular axes does not change when they rotate:

When the axes rotate around the origin of coordinates, the centrifugal moment of inertia changes continuously, therefore, at a certain position of the axes it becomes equal to zero.

Two mutually perpendicular axes about which the centrifugal moment of inertia of the section is equal to zero are called main axes of inertia.

The direction of the main axes of inertia can be determined as follows:

Two angle values ​​obtained from formula (43) A differ from each other by 90° and give the position of the main axes. As we see, the smaller of these angles in absolute value does not exceed l/4. In what follows we will only use the smaller angle. The main axis drawn at this angle will be denoted by the letter And. In Fig. 36 shows some examples of designating the main axes in accordance with this rule. The initial axes are designated by letters hee y.

Rice. 36.

In bending problems, it is important to know the axial moments of inertia of sections relative to those principal axes that pass through the center of gravity of the section.

The main axes passing through the center of gravity of the section are called main central axes. In what follows, as a rule, for brevity, we will simply call these axes main axes, omitting the word “central”.

The axis of symmetry of a flat section is the main central axis of inertia of this section, the second axis is perpendicular to it. In other words, the axis of symmetry and any one perpendicular to it form a system of principal axes.

If a flat section has at least two axes of symmetry that are not perpendicular to each other, then all axes passing through the center of gravity of such a section are its main central axes of inertia. So, in Fig. Figure 37 shows some types of sections (circle, ring, square, regular hexagon, etc.) that have the following property: any axis passing through their center of gravity is the main one.

Rice. 37.

It should be noted that non-central principal axes are of no interest to us.

In the theory of bending, the moments of inertia about the main central axes are of greatest importance.

The main central moments of inertia or main moments of inertia are called moments of inertia about the main central axes. Moreover, relative to one of the main axes, the moment of inertia maximum, relatively different - minimal:

Axial moments of inertia of the sections shown in Fig. 37, calculated relative to the main central axes, are equal to each other: Jy, Then: J u = J x cos 2 a +J y sin a = Jx.

The moments of inertia of a complex section are equal to the sum of the moments of inertia of its parts. Therefore, to determine the moments of inertia of a complex section, we can write:

gd eJ xi , J y „ J xiyi are the moments of inertia of individual parts of the section.

NB: if the section has a hole, then it is convenient to consider it a section with a negative area.

To perform strength calculations in the future, we will introduce a new geometric characteristic of the strength of a beam subjected to straight bending. This geometric characteristic is called the axial moment of resistance or the moment of resistance during bending.

The ratio of the moment of inertia of a section relative to an axis to the distance from this axis to the most distant point of the section is called axial moment of resistance:

The moment of resistance has dimensions mm 3, cm 3.

The moments of inertia and moments of resistance of the most common simple sections are determined by the formulas given in table. 3.

For rolled steel beams (I-beams, channels, angle beams, etc.), moments of inertia and moments of resistance are given in tables of rolled steel assortments, where, in addition to dimensions, cross-sectional areas, positions of centers of gravity and other characteristics are given.

In conclusion, let us introduce the concept radius of gyration sections relative to coordinate axes X And at - i x And i y respectively, which are determined by the following formulas.

The axes about which the centrifugal moment of inertia is zero are called principal, and the moments of inertia about these axes are called principal moments of inertia.

Let us rewrite formula (2.18) taking into account the known trigonometric relations:

;

in this form

In order to determine the position of the main central axes, we differentiate equality (2.21) with respect to the angle α once and obtain

At a certain value of the angle α=α 0, the centrifugal moment of inertia may turn out to be zero. Therefore, taking into account the derivative ( V), the axial moment of inertia will take an extreme value. Equating

,

we obtain a formula for determining the position of the main axes of inertia in the form:

(2.22)

In formula (2.21) we put cos2 out of brackets α 0 and substitute the value (2.22) there and, taking into account the known trigonometric dependence we get:

After simplification, we finally obtain the formula for determining the values ​​of the main moments of inertia:

(2.23)

Formula (20.1) is used to determine the moments of inertia about the main axes. Formula (2.22) does not give a direct answer to the question: about which axis the moment of inertia will be maximum or minimum. By analogy with the theory for studying a plane stress state, we present more convenient formulas for determining the position of the main axes of inertia:

(2.24)

Here α 1 and α 2 determine the position of the axes about which the moments of inertia are respectively equal J 1 and J 2. It should be borne in mind that the sum of the angle modules α 01 and α 02 should equal π/2:

Condition (2.24) is the condition for the orthogonality of the main axes of inertia of a plane section.

It should be noted that when using formulas (2.22) and (2.24) to determine the position of the main axes of inertia, the following pattern must be observed:

The main axis, relative to which the moment of inertia is maximum, makes the smallest angle with the original axis, relative to which the moment of inertia is greater.

Example 2.2.

Determine the geometric characteristics of flat sections of timber relative to the main central axes:

Solution

The proposed section is asymmetrical. Therefore, the position of the central axes will be determined by two coordinates, the main central axes will be rotated relative to the central axes by a certain angle. This leads to an algorithm for solving the problem of determining the main geometric characteristics.

1. We divide the section into two rectangles with the following areas and moments of inertia relative to their own central axes:

F 1 =12 cm 2, F 2 =18 cm 2;

2. We define a system of auxiliary axes X 0 at 0 starting at point A. The coordinates of the centers of gravity of the rectangles in this axis system are as follows:

X 1 =4 cm; X 2 =1 cm; at 1 =1.5 cm; at 2 =4.5 cm.

3. Determine the coordinates of the center of gravity of the section using formulas (2.4):

We plot the central axes (in red in Fig. 2.9).

4. Calculate the axial and centrifugal moments of inertia relative to the central axes X with and at c according to formulas (2.13) in relation to the composite section:

5. Find the main moments of inertia using formula (2.23)

6. Determine the position of the main central axes of inertia X And at according to formula (2.24):

The main central axes are shown in (Fig. 2.9) in blue.

7. Let's check the calculations performed. To do this, we will carry out the following calculations:

The sum of the axial moments of inertia about the main central and central axes must be the same:

Sum of angle modules α X and α y,, defining the position of the main central axes:

In addition, the provision is fulfilled that the main central axis X, about which the moment of inertia J x has the maximum value, makes a smaller angle with the central axis relative to which the moment of inertia is greater, i.e. with axle X With.

Moment of inertia about an axis parallel to the central one (Steiner's theorem)

PREFACE

Lecture No. 1 “Geometric characteristics

Preface…………………………………………………………………….4

flat sections"……………………………………………………………….5

2. Lecture No. 2 “Principal axes and principal moments of inertia”..………………………………………….…………………………...13

3. Lecture No. 3 “Torsion. Calculations for strength and torsional rigidity"………………………………………………………………………16

4. Lecture No. 4 “Shear and crushing. Strength calculations"…….………………………………………………………………..32

5. Questions to check the material covered...……………………..36

6. References…………………………………………………………37

Part 2 of the lecture notes contains the basic theoretical principles and calculation formulas on the following topics: Geometric characteristics of plane sections, Torsion, Shear and crushing.

The purpose of the lecture notes is to assist students in studying the subject, in solving and defending computational and graphic works on the strength of materials.

Lecture No. 1 “Geometric characteristics of plane sections”

The geometric characteristics of flat sections include:

· cross-sectional area F,

· static moments of area S x , S y ,

axial moments of inertia J x , J y ,

· centrifugal moment of inertia J xy,

polar moment of inertia Jρ ,

moment of resistance to torsion W ρ,

· moment of resistance to bending W x

1.1. Static moments of area S x , S y

The static moment of the cross-sectional area relative to a given axis is equal to the sum of the products of elementary areas and the distance to the corresponding axis.

Units S x And S y : [cm 3 ], [mm 3 ]. The sign “+” or “-” depends on the location of the axes.

Property: Static moments of the cross-sectional area are equal to zero (S x =0 and S y =0) if the point of intersection of the coordinate axes coincides with the center of gravity of the section. The axis about which the static moment is equal is called the central axis. The point of intersection of the central axes is called the center of gravity of the section.

Where F is the total cross-sectional area.

Example 1:

Determine the position of the center of gravity of a flat section consisting of two rectangles with a cutout.

Negative area is subtracted.

1.2. Axial moments of inertia J x ; Jy

The axial moment of inertia is equal to the sum of the products of the elementary areas and the square of the distance to the corresponding axis.

The sign is always "+".

Cannot be equal to 0.

Property: Takes a minimum value when the intersection point of the coordinate axes coincides with the center of gravity of the section.

The axial moment of inertia of a section is used in calculations of strength, rigidity and stability.

1.3. Polar moment of inertia of the section J ρ

Relationship between polar and axial moments of inertia:

The polar moment of inertia of the section is equal to the sum of the axial moments.

Property:

When the axes are rotated in any direction, one of the axial moments of inertia increases and the other decreases (and vice versa). The sum of the axial moments of inertia remains constant.

1.4. Centrifugal moment of inertia of the section J xy

The centrifugal moment of inertia of the section is equal to the sum of the products of the elementary areas and the distances to both axes

Unit of measurement [cm 4 ], [mm 4 ].

Sign "+" or "-".

If the coordinate axes are axes of symmetry (example - I-beam, rectangle, circle), or one of the coordinate axes coincides with the axis of symmetry (example - channel).

Thus, for symmetrical figures the centrifugal moment of inertia is 0.

Coordinate axes u And v , passing through the center of gravity of the section, about which the centrifugal moment is equal to zero, are called the main central axes of inertia of the section. They are called main because the centrifugal moment relative to them is zero, and central because they pass through the center of gravity of the section.

For sections that are not symmetrical about the axes x or y , for example at a corner, will not be equal to zero. For these sections, the position of the axes is determined u And v by calculating the rotation angle of the axes x And y

Centrifugal moment about the axes u And v -

Formula for determining axial moments of inertia about the principal central axes u And v :

where are the axial moments of inertia relative to the central axes,

Centrifugal moment of inertia about the central axes.

Steiner's theorem:

The moment of inertia about an axis parallel to the central one is equal to the central axial moment of inertia plus the product of the area of ​​the entire figure and the square of the distance between the axes.

Proof of Steiner's theorem.

According to Fig. 5 distance at to the elementary site dF

Substituting the value at into the formula, we get:

The term since point C is the center of gravity of the section (see the property of static moments of the sectional area relative to the central axes).

For a rectangle with heighth and widthb :

Axial moment of inertia:

Bending moment:

the moment of resistance to bending is equal to the ratio of the moment of inertia to the distance of the most distant fiber from the neutral line:

For a circle:

Polar moment of inertia:

Axial moment of inertia:

Torsional moment:

Bending moment:

Example 2. Determine the moment of inertia of a rectangular cross-section about the central axis Cx .

Solution. Let's divide the area of ​​the rectangle into elementary rectangles with dimensions b (width) and dy (height). Then the area of ​​such a rectangle (shaded in Fig. 6) is equal to dF=bdy. Let's calculate the value of the axial moment of inertia J x

By analogy we write

Axial moment of inertia of the section relative to the central

Centrifugal moment of inertia

Since the axes Cx and C y are axes of symmetry.

Example 3. Determine the polar moment of inertia of a circular cross-section.

Solution. Let us divide the circle into infinitely thin rings of thickness with radius, the area of ​​such a ring is . Substituting the value into the expression for the polar moment of inertia and integrating, we get

Taking into account the equality of the axial moments of the circular section and

We get

The axial moments of inertia for the ring are equal

With– the ratio of the cutout diameter to the outer diameter of the shaft.

Let's consider how the moments of inertia change when the coordinate axes are rotated. Let us assume that the moments of inertia of a certain section relative to the 0 axes are given X, 0at(not necessarily central) -, - axial moments of inertia of the section. It is required to determine - axial moments about the axes u, v, rotated relative to the first system by an angle (Fig. 8)

Since the projection of the broken line OABC is equal to the projection of the trailing line, we find:

Let us exclude u and v in the expressions for the moments of inertia:

Let's consider the first two equations. Adding them term by term, we get

Thus, the sum of the axial moments of inertia about two mutually perpendicular axes does not depend on the angle and remains constant when the axes are rotated. Let us note at the same time that

Where is the distance from the origin of coordinates to the elementary area (see Fig. 5). Thus, using the angle and equating the derivative to zero, we find

At this angle value, one of the axial moments will be the largest, and the other will be the smallest. At the same time, the centrifugal moment of inertia becomes zero, which can be easily verified by equating the formula for the centrifugal moment of inertia to zero .

Axes about which the centrifugal moment of inertia is zero and the axial moments take extreme values ​​are called main axes. If they are also central (the point of origin coincides with the center of gravity of the section), then they are called main central axes (u; v). Axial moments of inertia about the principal axes are called main moments of inertia - And

And their value is determined by the following formula:

The plus sign corresponds to the maximum moment of inertia, the minus sign to the minimum.

There is another geometric characteristic - radius of gyration of the section. This value is often used in theoretical conclusions and practical calculations.

The radius of gyration of the section relative to a certain axis, for example 0x, is called the quantity , determined from equality

F– cross-sectional area,

Axial moment of inertia of the section,

From the definition it follows that the radius of gyration is equal to the distance from the axis 0 X to the point at which the cross-sectional area F should be concentrated (conditionally) so that the moment of inertia of this one point is equal to the moment of inertia of the entire section. Knowing the moment of inertia of the section and its area, you can find the radius of gyration relative to the 0 axis X:

The radii of gyration corresponding to the main axes are called main radii of inertia and are determined by the formulas

AXIS OF INERTIA

AXIS OF INERTIA

The main, three mutually perpendicular axes drawn through the k.-l. point of the body and having the property that if they are taken as coordinate axes, then the centrifugal inertia of the body relative to these axes will be equal to zero. If TV a body fixed at one point is put into rotation around an axis, which at a given point is manifested. main O. and., then the body in the absence of external. forces will continue to rotate around this axis, as if around a stationary one. The concept of the main O. and. plays an important role in the dynamics of TV. bodies.

Physical encyclopedic dictionary. - M.: Soviet Encyclopedia. . 1983 .

AXIS OF INERTIA

The main ones are three mutually perpendicular axes drawn through the k.n. point of the body, coinciding with the axes of the ellipsoid of inertia of the body at this point. Main O. and. have the property that if they are taken as coordinate axes, then the centrifugal moments of inertia of the body relative to these axes will be equal to zero. If one of the coordinate axes, for example. axis Oh, is for the point ABOUT main O. and., centrifugal moments of inertia, the indices of which include the name of the axis, i.e. I xy And I xz, are equal to zero. If a solid body, fixed at one point, is brought into rotation around an axis, which at a given point is the main O. and., then the body in the absence of external. forces will continue to rotate around this axis, as if around a stationary one.

Physical encyclopedia. In 5 volumes. - M.: Soviet Encyclopedia. Editor-in-chief A. M. Prokhorov. 1988 .

See what "AXIS OF INERTIA" is in other dictionaries:

The main three mutually perpendicular axes, which can be drawn through any point of a solid body, differ in that if a body fixed at this point is brought into rotation around one of them, then in the absence of external forces it will... ... Big Encyclopedic Dictionary

Main, three mutually perpendicular axes that can be drawn through any point of a solid body, characterized in that if a body fixed at this point is brought into rotation around one of them, then in the absence of external forces it will... ... encyclopedic Dictionary

The main, three mutually perpendicular axes drawn through some point of the body, having the property that, if they are taken as coordinate axes, then the centrifugal moments of inertia (See Moment of inertia) of the body relative to these axes ... ... Great Soviet Encyclopedia

The main, three mutually perpendicular axes, which can be drawn through any point on the TV. bodies, characterized in that if a body fixed at this point is brought into rotation around one of them, then in the absence of external strength it will continue... ... Natural science. encyclopedic Dictionary

main axes of inertia- Three mutually perpendicular axes drawn through the center of gravity of the body, having the property that if they are taken as coordinate axes, then the centrifugal moments of inertia of the body relative to these axes will be equal to zero.... ... Technical Translator's Guide

main axes of inertia- three mutually perpendicular axes drawn through the center of gravity of the body, having the property that if they are taken as coordinate axes, then the centrifugal moments of inertia of the body relative to these axes will be equal to zero.... ...

- ... Wikipedia

Main axes- : See also: main axes of inertia, main axes (tensor) of deformation... Encyclopedic Dictionary of Metallurgy

Dimension L2M SI units kg m² SGS ... Wikipedia

The moment of inertia is a scalar physical quantity that characterizes the distribution of masses in a body, equal to the sum of the products of elementary masses by the square of their distances to the base set (point, line or plane). SI unit: kg m².… … Wikipedia

Books

• Thoretic physics. Part 3. Mechanics of solids (2nd edition), A.A. Eichenwald. The third part of this course in theoretical physics is a natural continuation of part II: the basic principles of mechanics are applied here to a solid body, i.e. to a system...

Task 5.3.1: For the section, the axial moments of inertia of the section relative to the axes are known x1, y1, x2: , . Axial moment of inertia about the axis y2 equal...

1) 1000 cm4; 2) 2000 cm4; 3) 2500 cm4; 4) 3000 cm4.

Solution: The correct answer is 3). The sum of the axial moments of inertia of the section relative to two mutually perpendicular axes when the axes are rotated through a certain angle remains constant, that is

After substituting the given values, we get:

Task 5.3.2: Of the indicated central axes of the section of an equal angle angle, the main ones are...

1) x3; 2) everything; 3) x1; 4) x2.

Solution: The correct answer is 4). For symmetrical sections, the axes of symmetry are the main axes of inertia.

Task 5.3.3: Main axes of inertia...

• 1) can only be drawn through points lying on the axis of symmetry;
• 2) can only be drawn through the center of gravity of a flat figure;
• 3) these are the axes about which the moments of inertia of a flat figure are equal to zero;
• 4) can be drawn through any point of a flat figure.

Solution: The correct answer is 4). The figure shows an arbitrary flat figure. Through the point WITH two mutually perpendicular axes are drawn U And V.

In the course on strength of materials it is proven that if these axes are rotated, then their position can be determined in which the centrifugal moment of inertia of the area becomes zero, and the moments of inertia about these axes take extreme values. Such axes are called main axes.

Task 5.3.4: Of the indicated central axes, the main section axes are...

1) everything; 2) x1 And x3; 3) x2 And x3; 4)x2 And x4.

Solution: The correct answer is 1). For symmetrical sections, the axes of symmetry are the main axes of inertia.

Task 5.3.5: Axes about which the centrifugal moment of inertia is zero and the axial moments take extreme values ​​are called...

• 1) central axes; 2) axes of symmetry;
• 3) main central axes; 4) main axes.

Solution: The correct answer is 4). When the coordinate axes are rotated by an angle b, the moments of inertia of the section change.

Let the moments of inertia of the section relative to the coordinate axes be given x, y. Then the moments of inertia of the section in the system of coordinate axes u, v, rotated at a certain angle relative to the axes x, y, are equal

At a certain value of the angle, the centrifugal moment of inertia of the section becomes zero, and the axial moments of inertia take extreme values. These axes are called main axes.

Task 5.3.6: Moment of inertia of the section about the main central axis xC equal...

1); 2) ; 3) ; 4) .

Solution: The correct answer is 2)

To calculate we use the formula