§ 4.5. Calculation of moments of inertia sections of a simple form

As indicated in § 1.5, the geometrical characteristics of complex sections are determined by dismembering them to a number of simple figures, the geometric characteristics of which can be calculated by the corresponding formulas or determine by special tables. These formulas are obtained as a result of direct integration of expressions (8.5) - (10.5). Receptions of their preparation are discussed below on the examples of the rectangle, triangle and circle.

Rectangular cross section

We define the axial moment of the inertia of the rectangle height H and width B relative to the axis passing through its base (Fig. 11.5, a). We highlight from the rectangle with lines parallel to the axis elementary strip of height and width b.

The area of \u200b\u200bthis strip distance from the strip to the axis is equal to them. Substitute these values \u200b\u200bin the expression of the moment of inertia (8.5):

Similarly, for the moment of inertia, there can be an expression on the axis

To determine the centrifugal moment of inertia, select the lines parallel to the axes from the rectangle (Fig.

11.5, b), elementary area of \u200b\u200bmagnitude. We first determine the centrifugal moment of the inertia of not the entire rectangle, but only a vertical strip of height H and the width located at a distance from the axis

The work is made per sign of the integral, as for all sites belonging to the vertical strip under consideration, it is constantly.

Integrating the expression within the limits from to

We will define the axial moments of the inertia of the rectangle relative to the axes of y and passing through the center of gravity in parallel to the sides of the rectangle (Fig. 12.5). For this case, the integration limits will be from

The centrifugal moment of the inertia of the rectangle relative to the axes (Fig. 12.5) is zero, since these axes coincide with its axes of symmetry.

Triangular cross section

We define the axial moments of the inertia of the triangle relative to three parallel axes passing through its base (Fig. 13.5, a), the center of gravity (Fig. 13.5, b) and the vertex (Fig. 13.5, e).

For the case when the axis passes through the base of the triangle (Fig. 13.5, a),

For the case when the axis passes through the center of gravity of the triangle parallel to its base (Fig. 13.5, b),

In the case where the axis passes through the vertex of the triangle in parallel its base (Fig. 13.5, B),

The moment of inertia is much larger (three times) than the moment of inertia as the main part of the triangle area is more removed from the axis than from the axis

Expressions (17.5) - (19.5) were obtained for an equifiable triangle. However, they are true for unequal triangles. Comparing, for example, the triangles shown in Fig. 13.5, a and 13.5, g, of which the first is a preceded, and the second is unequal, we establish that the size of the platform and the limits in which it changes from (from 0 to) for both triangles is the same. Consequently, the moments of inertia are also the same. Similarly, it can be shown that the axial moments of the inertia of all sections shown in Fig. 14.5, the same. In general, the displacement of the parts of the section parallel to some axis does not affect the value of the axial moment of inertia relative to this axis.

Obviously, the sum of the axial moments of the inertia of the triangle relative to the axes shown in Fig. 13.5, a and 13.5, in, should be equal to the axial moment of the inertia of the rectangle relative to the axis shown in Fig. 11.5, a. This follows from the fact that the rectangle can be considered as two triangles, for one of which the axis passes through the base, and for the other - through the vertex parallel to its base (Fig. 15.5).

Indeed, according to formulas (17.5) and (19.5)

which coincides with the expression of the rectangle according to formula (12.5).

Circle

We define the axial moment of the inertia of the circle regarding any axis passing through its center of gravity. From fig. 16.5, and follows

Obviously, relative to any axis passing through the center of the circle, the axial moment of inertia will be equal and, therefore,

By formula (11.5), we find the polar moment of inertia of the circle relative to its center:

The axial torque formula of the circle can be obtained more simply, if you first output the formula for its polar moment inertia relative to the center (point O). To do this, select the elementary ring from the circle with a thickness of the radius and an area (Fig. 16.5, b).

The polar moment of inertia of the elementary ring relative to the center of the circle as all the elementary platforms of which consists of this ring are located at the same distance from the center of the circle. Hence,

This result coincides with the above.

Moments of inertia (polar and axial) section having a circular ring with an outer diameter D and inner (Fig. 17.5) can be defined as the difference between the corresponding moments of the inertia of the outer and inner circles.

Polar moment of inertia rings based on formula (21.5)

or if you designate

Similarly, for axial moments inertia rings

Moment of inertia and moment of resistance

When determining the cross section of building structures, it is very often necessary to know the moment of inertia and the moment of resistance for the cross section under consideration. What is the moment of resistance and how it is associated with the moment of inertia is set out separately. In addition, for compressible structures you also need to know the value of the radius of the inertia. Determine the moment of resistance and moment of inertia, and sometimes the radius of inertia for most transverse sections of a simple geometric shape can be known for long known formulas:

Table 1. Section forms, cross-sectional area, moments of inertia and moments of resistance for structures are sufficiently simple geometric forms.

Usually, these formulas are sufficient for most calculations, but there are no cases of the construction and the cross section of the design may not be such a simple geometric shape or the position of the axes relative to which it is necessary to determine the moment of inertia or the moment of resistance, may not be as possible, then you can use the following formulas:

Table 2. Section forms, cross sections, moments of inertia and resistance moments for constructions of more complex geometric shapes

As can be seen from Table 2, calculating the moment of inertia and the moment of resistance for non-equivalent corners is quite difficult, but there is no need for this. For non-equilibrium and equal return rolling corners, as well as for channels, ducts and profile pipes there are sorting. IN sortments The torque values \u200b\u200bof the inertia and the moment of resistance are given for each profile.

Table 3. Changes in moments inertia and resistance moments depending on the position of the axes.

Formulas from Table 3 may be needed to calculate the inclined elements of the roof.

It would be nice to explain on a visual example for especially gifted, like me, what is the moment of inertia and what it is eaten with. On specialized sites somehow everything is very confusing, and the dock has a clear talent to bring information, perhaps not the most complicated, but very competent and understandable

In principle, what is the moment of inertia and where he came from, explained quite in detail in the article "The foundations of the conversion, calculated formulas", here only I repeat: "W is the moment of resistance to the cross section of the beam, in other words, the area of \u200b\u200bthe compressible or stretched part of the beam section, Multipled to the shoulder of the actions of the resultant force. " The moment of resistance needs to be known to calculate the design for strength, i.e. on limit voltages. The moment of inertia needs to be known to determine the angles of rotation of the cross section and the deflection (displacement) of the center of gravity of the cross section, since maximum deformations occur in the highest and the lowest layer of the bent design, it is possible to determine the moment of inertia by multiplying the moment of resistance to the distance from the center of gravity Sections to the upper or lower layer, therefore, for rectangular sections I \u003d WH / 2. When determining the moment of inertia of the cross sections of complex geometric forms, the complex figure is broken down on the simplest, then the cross-sectional area of \u200b\u200bthese figures and the moments of the protozoa figures are determined, then the space of the simplest figures is multiplied by the square of the distance from the total severity center to the center of severity of the simplest figure. The moment of inertia of the simplest figure in the composition of complex section is equal to the moment of inertia of the figure + square distance multiplied to the square. Then the resulting moments of inertia are summed up and the moment of inertia of a complex section is obtained. But these are the most simplified wording (although, agreeing, it still looks quite wise).

The moment of inertia and the moment of resistance - the doctor

Chapter 5. Moments of inertia of flat sections

Any flat cross section is characterized by a number of geometric characteristics: area, coordinates of the center of gravity, the static moment, the moment of inertia, etc.

Static moments relative to the axes h. and y. equal:

Static moments are usually expressed in cubic centimeters or meters and can have both positive and negative values. Axis relative to which the static moment is zero, called central. The intersection point of the central axes is called center severity section. Formulas for determining the coordinates of the center of gravity x C. and y C complex section broken into the simplest components for which the area is known A I. and the situation of the center of gravity x CIand y Ci., have kind

The magnitude of the moment of inertia characterizes the resistance to the deformation (twist, bending) rod depending on the size and shape of the cross section. Distinguish the moments of inertia:

- axial defined by integrals of the form

Axial and polar moments of inertia are always positive and not

apply to zero. Polar moment inertia I P. equal to the amount of axial moments inertia I H. and I W. Regarding any pair of mutually perpendicular axes h. and w.:

The centrifugal moment of inertia can be positive, negative and equal to zero. The dimension of the moments of inertia is CM 4 or M 4. Formulas To determine the moments of inertia of simple sections relative to the central axes are given in reference books. When calculating the moments of the inertia of complex sections, the formulas of the transition from the central axes of simple sections to other axes parallel to the central one are often used.

where are the moments of inertia of simple sections relative to the central axes;

m, N. - distances between the axes (Fig. 18).

Fig. 18. By determining the moments of inertia with respect to the axes,

The main central axis of the section is important. The main central is the two mutually perpendicular axes passing through the center of severity, relative to which the centrifugal moment of the inertia is zero, and the axial moments of the inertia have extreme values. The main moments of inertia are designated I U. (MAX) and I V. (min) and are determined by the formula

The position of the main axes is determined by the angle α, which is from the formula

The angle α is deposited from the axis with a large inertia inertia; Positive value - counterclockwise.

If the cross section has a symmetry axis, then this axis is the main one. Another main axis perpendicular to the axis of symmetry. In practice, sections are often used, made up of several rolling profiles (2-way, channel, corner). The geometric characteristics of these profiles are given in the tables of the sorting. For unequal and equilibrium corners, the centrifugal moment of inertia relative to the central axes parallel to the shelves is determined by the formula

Pay attention to the designation of the main central axes in the sorting table for the corners. Sign I xy For a corner depends on its position in the section. Figure 19 shows the possible positions of the corner in cross section and signs for I xy.

Fig. 19. Possible positions of the corner in the section

Determine I U, I V and the position of the main central axes of section

A complex cross section consists of two rolling profiles. Extract from the sorting tables (adj. 5) is shown in Fig. 21.

As auxiliary assumptions of the axis passing by external

sides of the chawller (axis x B., y B.See fig. 20) .Copinates of the center of gravity section:

(Calculate yourself).

Fig. 20. Position of the main central axes of inertia

U. and V. Complex cross section

As an auxiliary, it would be possible to choose, for example, the central axes of the chawller. Then the scope of computations will slightly decrease.

Axial moments of inertia:

Please note that an unequal corner in the section is located

otherwise, what is shown in the table of the sorting. Calculate yourself.

№ 24 180 x 110 x 12

Fig. 21. The values \u200b\u200bof the geometric characteristics of rolling profiles:

but - Schawler number 24; b. - unequish corner 180 x 110 x 12

Centrifugal moments inertia:

- for a channel (there is a symmetry axis);

- for the corner

a minus sign - due to the position of the corner in the section;

- For the whole section:

Follow the appointment of signs from n. and m.. From the central axes of the channel, we turn to the general central axes of the section, so + m 2.

The main moments of the cross section:

The position of the main central axes of section:

; α \u003d 55 o 48 ';

Check the correctness of the calculation of values I U., I V. and α is produced by the formula

The angle α for this formula is counted from the axis u..

The considered section has the greatest resistance of bending relative to the axis u. and the smallest - relative to the axis v..

We introduce the decartian rectangular coordinate system O XY. Consider in the plane of the coordinate arbitrary cross section (closed area) with an area A (Fig. 1).

Static moments

Point C with coordinates (X C, Y C)

called center severity section.

If the axes of coordinates pass through the severity center, the static moments of the section are zero:

Axial moments inertia Sections relative to the x and y axes are called the integrals of the form:

Polar moment inertia Sections regarding the start of coordinates is called the integral of the form:

Centrifugal torque inertia Sections are called the integral of the form:

The main axes of inertia section Two mutually perpendicular axes are called, relative to which I xy \u003d 0. If one of the mutually perpendicular axes is the axis of the symmetry of the section, then I xy \u003d 0 and, therefore, these axes are the main. The main axes passing through the center of severity are called the main central axes of the inertia section

2. Step by Steiner Guygens about parallel transfer of axes

Steiner-Guigens Theorem (Steiner Theorem).
The axial moment of the inertia of the section I relative to the arbitrary fixed axis x is equal to the sum of the axial torque of the inertia of this section I with the relative parallel axis x * passing through the center of the secting, and the production of the section A per square of the distance D between the two axes.

If the moments of the inertia i x and i y are known respect to the x and y axes, then relative to the axes ν and u, rotated at the angle α, the moments of the inertia axial and centrifugal are calculated by the formulas:

From the above formulas it can be seen that

Those. The sum of the axial moments of the inertia is not changed by rotation of mutually perpendicular axes, i.e.i u and v, with respect to which the centrifugal moment of the cross section is zero, and the axial moments of the inertia І u and i v have extreme Max or Min values, called the main axes of section . The main axes passing through the center of severity are called main central axes of section. For symmetric sections of the axis of their symmetry are always the main central axes. The position of the main axes of the cross section relative to other axes is determined using the ratio:

where α 0 is the angle to which the X and Y axes should be deployed so that they become the main ones (the positive angle is made to postpone against the clockwise progress, negative - along the clockwise arrow). Axial moments of inertia relative to the main axes are called the main moments of inertia:

the plus sign before the second term refers to the maximum moment of inertia, the minus sign is to the minimum.

In determining the moments of the inertia of the composite section, the latter is divided into simple figures, which are known for the position of the centers of gravity and moments of inertia with respect to its own central axes. According to formulas (2.5), the coordinates of the center of gravity of the entire section in the system of arbitrarily selected auxiliary axes are found. In parallel, these axes conduct central axes, relative to which the axial and centrifugal moments of inertia are determined by formulas (2.6). The moments of inertia relative to the main central axes are determined by formula (2.12), and the position of the main central axes by formulas (2.11).

Example 2.1. We define the moments of inertia with respect to the main central axes of the cross-section of the 2-way beam 130, reinforced by two steel sheets with a cross section of 200 x 20 mm (Fig. 2.12).

Axis symmetry Oh, OU They are the main central axes of the entire section. Drink out of the sorting (see Appendix) The values \u200b\u200bof the area and the moments of the inertia of the cross section of the bidaillion relative to the axes Oh, Oh:

The moments of the inertia of sections of sheets with respect to its own central axes will determine by formulas (2.14):

The area of \u200b\u200bthe total cross section is equal F \u003d. 46.5 + 2 20 2 \u003d 126.5 cm 2.

Moments of inertia sections relative to the main central axes Oh, OU Defined by formulas (2.6):

Example 2.2. We define the moments of the inertia with respect to the main central axes of the cross section of the rack of the rack of the rack of the two equilibrium corners 1_70х70х8, composed of cruciform (Fig. 2.13). The collaboration of the corners is provided by connecting spacers.

Coordinates of the center of severity section of the corner, the values \u200b\u200bof the area and moments of inertia relative to its own central axes Oh ^. and OU 0 are given in the assortment (see Appendix):

Distance from the center of gravity ABOUT just a cross section to the center of gravity corner equals but \u003d (2.02 + 0.4) l / 2 \u003d 3.42 cm.

The area of \u200b\u200bthe total cross section is equal F \u003d. 2 10.7 \u003d 21.4 cm 2.

Moments of inertia relative to the main central axes, which are the axis of symmetry Oh, Oh, Defined by formulas (2.6):

Example 2.3. We define the position of the center of gravity and moments of inertia relative to the main central axes of the transverse section of the beam made up of two channels x] and Oh y (. Then, by formulas (2.5) we will get:

These values \u200b\u200band coordinates of the centers of severity of the channel and a corner in the coordinate system Ohu Showing in Fig. 2.16 and accordingly are equal:

Determine by formulas (2.6) moments of inertia section relative to central axes Oh and OU

According to formulas (2.12) and (2.11) we will find the magnitudes of the main moments of inertia and the angles of inclination of the main axes 1 and 2 to the axis Oh:

The axial (or equatorial) moment of inertia of the cross section relative to some axis is called the amount of elementary sites on the squares of their distances from this axis, i.e.

The polar moment of the inertia of the section relative to some point (pole) is called the amount of blocks of elementary sites on the squares of their distances from this point, i.e.

The centrifugal moment of inertia of the section relative to some two mutually perpendicular axes is called the amount of blocks of elementary sites on their distances from these axes, i.e.

Moments of inertia are expressed in, etc.

Axial and polar moments of inertia are always positive, since in their expressions under the integral signs, the values \u200b\u200bof the sites are included (always positive) and the squares of the distances of these sites from this axis or pole.

In fig. 9.5, and depicted section F and shows the axis y and z. Axial moments of inertia of this section relative to the axes of the:

The sum of these moments of inertia

and, therefore,

Thus, the sum of the axial moments of the cross section of the cross section relative to two mutually perpendicular axes is equal to the polar moment of the inertia of this section relative to the intersection point of the specified axes.

Centrifugal moments of inertia can be positive, negative or equal to zero. For example, the centrifugal moment of the inertia of the section shown in Fig. 9.5, and, relative to the axes of y and positive, as for the main part of this section, located in the first quadrant, and therefore positive.

If you change the positive direction of the axis of the y or on the opposite (Fig. 9.5, b) or turn both of these axes 90 ° (Fig. 9.5, c), then the centrifugal moment of the inertia will become negative (the absolute value does not change it), as the main part The cross sections will then be located in the quadrant, for the points of which the coordinates are positive, and the z coordinates are negative. If you change the positive directions of both axes to the inverse, then this will not change a sign nor the centrifugal moment of inertia.

Consider a figure, symmetric with respect to one or more axes (Fig. 10.5). We carry out the axes so that at least one of them (in this case, the axis y) coincided with the axis of the symmetry of the figure. Each site located to the right of the axis corresponds to in this case the same platform located symmetrically the first, but to the left of the axis. The centrifugal moment of the inertia of each pair of such symmetrically located sites is:

Hence,

Thus, the centrifugal moment of inertia of the section relative to the axes, of which one or both coincide with its axes of symmetry is zero.

The axial moment of the inertia of the complex cross section relative to some axis is equal to the sum of the axial moments of the inertia of the components of its parts relative to the same axis.

Similarly, the centrifugal moment of the inertia of the complex cross section relative to any two mutually perpendicular axes is equal to the sum of the centrifugal moments of the inertia of the components of its parts relative to the same axes. Also, the polar moment of inertia of the complex section relative to a certain point is equal to the sum of the polar moments of the inertia of the components of its parts relative to the same point.

It should be borne in mind that it is impossible to summarize the moments of inertia calculated relative to different axes and points.

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Geometrical characteristics of flat sections

Area:, DF - Elementary Playground.

Static moment of the element of the areadf. Regarding the 0x axis
- The product of the element area at the distance "Y" from the 0x axis: ds x \u003d ydf

Having aroused (confirming) such works across the entire area of \u200b\u200bthe figure, we get static moments relative to the axes y and x:
;
[cm 3, m 3, etc.].

Coordinates of the center of gravity:
. Static moments about central axes (axes passing through the severity center) are zero. When calculating the static moments of a complex figure, it is divided into simple parts, with known areas F I and the coordinates of the centers of gravity X I, Y i. The fact of the area of \u200b\u200bthe whole figure \u003d the sum of the static moments of each part of it:
.

Coordinates of the center of gravity of a complex figure:

M.
omter inertia section

Axial (equatorial) moment of inertia section - The sum of the works of the elementary DF sites on the squares of their distances to the axis.

;
[cm 4, m 4, etc.].

The polar moment of inertia of the section relative to some point (pole) is the amount of the works of elementary sites on the squares of their distances from this point.
; [cm 4, m 4, etc.]. J y + j x \u003d j p.

Centrifugal moment of inertia sections - The sum of the works of elementary sites at their distances from two mutually perpendicular axes.
.

The centrifugal moment of the inertia of the section relative to the axes, of which one or both coincide with the axes of symmetry is zero.

Axial and polar moments of inertia are always positive, inertia centrifugal moments can be positive, negative or equal to zero.

The moment of the inertia of the complex figure is equal to the sum of the moments of the inertia of the compound parts.

Moments of inertia sections of simple form

P
Lougonal cross section circle

TO


oltso.

T.
reegon

r
Avnobed

Rectangular

t.
reegon

C. machine Circle

J y \u003d j x \u003d 0,055r 4

J xy \u003d 0,0165r 4

in fig. (-)

Semicircle

M.

ovens of inertia standard profiles are from the colors' tables:

D.
voltavr Channel Corner

M.

J. x1 \u003d j x + a 2 f;

J y1 \u003d j y + b 2 f;

the moment of inertia relative to any axis is equal to the moment of inertia with respect to the central axis parallel to this, plus the product of the figure of the figure on the square of the distance between the axes. J y1x1 \u003d j yx + ABF; ("A" and "B" are substituted into the formula, taking into account their sign).

Dependence between moments of inertia when turning axes:

J. x1 \u003d j x cos 2  + j y sin 2  - j xy sin2; J y1 \u003d j y cos 2  + j x sin 2  + j xy sin2;

J x1y1 \u003d (j x - j y) sin2 + j xy cos2;

The angle \u003e 0 if the transition from the old coordinate system to the new occurs against the hour. J y1 + j x1 \u003d j y + j x

Extreme (maximum and minimum) Inertia moments are called the main moments of inertia. Axis relative to which the axial moments of inertia have extreme values, are called the main axes of inertia. The main axes of the inertia are mutually perpendicular. Centrifugal moments of inertia relative to the main axes \u003d 0, i.e. The main axes of inertia are axis relative to which the centrifugal moment of inertia \u003d 0. If one of the axes coincides or both coincide with the axis of symmetry, then they are the main. The angle determining the position of the main axes:
if  0\u003e 0  axes turn against an hour. The maximum axis is always a smaller angle with that of the axes relative to which the moment of inertia is greater value. The main axes passing through the center of gravity are called the main central axes of inertia. Moments inertia relative to these axes:

J max + j min \u003d j x + j y. The centrifugal moment of inertia relative to the main central axes of inertia is 0. If the main moments of inertia are known, then the transition formulas to the rotated axes:

J x1 \u003d j max cos 2  + j min sin 2 ; J y1 \u003d j max cos 2  + j min sin 2 ; J x1y1 \u003d (j max - j min) sin2;

The ultimate goal of calculating the geometric characteristics of the section is the determination of the main central moments of the inertia and the position of the main central axes of inertia. R adiius inertia -
; J x \u003d fi x 2, j y \u003d fi y 2.

If j x and j y Main moments of inertia, then i x and i y - main radii inertia. Ellipse, built on the main radius of inertia as on the semi-axes, is called ellipse inertia. With the help of an inertia ellipse, you can graphically find the radius of inertia I x1 for any axis x 1. To do this, we need to tangent to the ellipse, parallel axis X 1, and measure the distance from this axis to tangential. Knowing the radius of inertia, you can find the moment of inertia section relative to the axis X 1:
. For sections with more than two axes of symmetry (for example: a circle, square, ring, etc.), the axial moments of the inertia relative to all central axes are equal to each other, J xy \u003d 0, the inertia ellipse appeals to the circle of inertia.

Moments of resistance.

Axial moment of resistance - The ratio of the moment of inertia relative to the axis to the distance from it to the most remote section of the cross section.
[cm 3, m 3]

The moments of resistance regarding the main central axes are especially important:

rectangle:
; Circle: W x \u003d W y \u003d
,

tubular cross section (ring): W x \u003d W y \u003d
, where  \u003d d n / d b.

The polar moment of resistance is the ratio of the polar moment of inertia by the distance from the pole to the most remote point of section:
.

For a circle w p \u003d
.