The axes passing through the center of gravity of a plane figure are called central axes.
The moment of inertia about the central axis is called the central moment of inertia.
Theorem
The moment of inertia about any axis is equal to the sum of the moment of inertia about the central axis parallel to the given one and the product of the area of the figure and the square of the distance between the axes.
To prove this theorem, consider an arbitrary plane figure whose area is equal to A
, the center of gravity is located at the point WITH
, and the central moment of inertia about the axis x
will I x
.
Let's calculate the moment of inertia of the figure relative to a certain axis x 1
, parallel to the central axis and spaced from it at a distance A
(rice).
I x1 = Σ y 1 2 dA + Σ (y + a) 2 dA =
= Σ y 2 dA + 2a Σ y dA + a 2 Σ dA.
Analyzing the resulting formula, we note that the first term is the axial moment of inertia relative to the central axis, the second term is the static moment of the area of this figure relative to the central axis (hence, it is equal to zero), and the third term after integration can be represented as a product a 2 A , i.e., as a result we get the formula:
I x1 = I x + a 2 A- the theorem is proven.
Based on the theorem, we can conclude that of a series of parallel axes, the axial moment of inertia of a flat figure will be the smallest relative to the central axis .
Principal axes and principal moments of inertia
Let us imagine a flat figure whose moments of inertia relative to the coordinate axes I x And I y , and the polar moment of inertia relative to the origin is equal to I ρ . As was established earlier,
I x + I y = I ρ.
If the coordinate axes are rotated in their plane around the origin of coordinates, then the polar moment of inertia will remain unchanged, and the axial moments will change, while their sum will remain constant. Since the sum of variables is constant, one of them decreases and the other increases, and vice versa.
Consequently, at a certain position of the axes, one of the axial moments will reach the maximum value, and the other - the minimum.
The axes about which the moments of inertia have minimum and maximum values are called the main axes of inertia.
The moment of inertia about the main axis is called the main moment of inertia.
If the principal axis passes through the center of gravity of a figure, it is called the principal central axis, and the moment of inertia about such an axis is called the principal central moment of inertia.
We can conclude that if a figure is symmetrical about any axis, then this axis will always be one of the main central axes of inertia of this figure.
Centrifugal moment of inertia
The centrifugal moment of inertia of a flat figure is the sum of the products of elementary areas taken over the entire area and the distance to two mutually perpendicular axes:
I xy = Σ xy dA,
Where x
, y
- distances from the site dA
to axles x
And y
.
The centrifugal moment of inertia can be positive, negative or zero.
The centrifugal moment of inertia is included in the formulas for determining the position of the main axes of asymmetrical sections.
Standard profile tables contain a characteristic called radius of gyration of the section
, calculated by the formulas:
i x = √ (I x / A),i y = √ (I y / A) , (hereinafter the sign"√"- root sign)
Where I x , I y
- axial moments of inertia of the section relative to the central axes; A
- cross-sectional area.
This geometric characteristic is used in the study of eccentric tension or compression, as well as longitudinal bending.
Torsional deformation
Basic concepts about torsion. Torsion of a round beam.
Torsion is a type of deformation in which only a torque occurs in any cross section of the beam, i.e. a force factor that causes a circular movement of the section relative to an axis perpendicular to this section, or prevents such movement. In other words, torsional deformations occur if a pair or pairs of forces are applied to a straight beam in planes perpendicular to its axis.
The moments of these pairs of forces are called twisting or rotating. Torque is denoted by T
.
This definition conventionally divides the force factors of torsional deformation into external ones (torsional, torque T
) and internal (torques M cr
).
In machines and mechanisms, round or tubular shafts are most often subjected to torsion, so strength and rigidity calculations are most often made for such units and parts.
Consider the torsion of a circular cylindrical shaft.
Imagine a rubber cylindrical shaft in which one of the ends is rigidly fixed, and on the surface there is a grid of longitudinal lines and transverse circles. We will apply a couple of forces to the free end of the shaft, perpendicular to the axis of this shaft, i.e. we will twist it along the axis. If you carefully examine the grid lines on the surface of the shaft, you will notice that:
- the shaft axis, which is called the torsion axis, will remain straight;
- the diameters of the circles will remain the same, and the distance between adjacent circles will not change;
- longitudinal lines on the shaft will turn into helical lines.
From this we can conclude that when a round cylindrical beam (shaft) is torsioned, the hypothesis of flat sections is valid, and we can also assume that the radii of the circles remain straight during deformation (since their diameters have not changed). And since there are no longitudinal forces in the shaft sections, the distance between them is maintained.
Consequently, the torsional deformation of a round shaft consists in the rotation of the cross sections relative to each other around the torsion axis, and their rotation angles are directly proportional to the distances from the fixed section - the farther any section is from the fixed end of the shaft, the greater the angle relative to the shaft axis it twists .
For each section of the shaft, the angle of rotation is equal to the angle of twist of the part of the shaft enclosed between this section and the seal (fixed end).
Corner ( rice. 1) rotation of the free end of the shaft (end section) is called the full angle of twist of the cylindrical beam (shaft).
Relative twist angle φ 0
called the torsion angle ratio φ 1
to the distance l 1
from a given section to the embedment (fixed section).
If the cylindrical beam (shaft) is long l
has a constant cross-section and is loaded with a torsional moment at the free end (i.e., consists of a homogeneous geometric section), then the following statement is true:
φ 0 = φ 1 / l 1 = φ / l = const
- the value is constant.
If we consider a thin layer on the surface of the above rubber cylindrical bar ( rice. 1), limited by a grid cell cdef , then we note that this cell warps during deformation, and its side, remote from the fixed section, shifts towards the twist of the beam, occupying the position cde 1 f 1 .
It should be noted that a similar picture is observed during shear deformation, only in this case the surface is deformed due to translational movement of sections relative to each other, and not due to rotational movement, as in torsional deformation. Based on this, we can conclude that during torsion in cross sections, only tangential internal forces (stresses) arise, forming a torque.
So, the torque is the resulting moment relative to the axis of the beam of internal tangential forces acting in the cross section.
Let z With, y s– the central axes of the sections, – the moments of inertia of the section relative to these axes. Let us determine the moments of inertia of the section relative to the new axes z 1, at 1, parallel to the central axes and displaced relative to them by distances a And d. Let dA– an elementary area in the vicinity of a point M with coordinates y And z in the central coordinate system. From Fig. 4.3 it is clear that the coordinates of point C in the new coordinate system will be equal to, .
Let us determine the moment of inertia of the section relative to the y-axis 1 :
|
| z c |
| y c |
| z 1 |
| y 1 |
| d |
| a |
| C |
Thus,
Likewise
Changing the moments of inertia of the section when rotating the axes
Let's find the relationship between the moments of inertia about the axes y, z and moments of inertia about the axes y 1, z 1, rotated at an angle a. Let Jy> Jz and positive angle a measured from the axis y counterclock-wise. Let the coordinates of the point M before the turn - y, z, after turning – y 1, z 1(Fig. 4.4).
From the figure it follows:
Now let's determine the moments of inertia about the axes y 1 And z 1:
|
| M |
| z |
| z 1 |
| y 1 |
| y |
| a |
| y |
| y 1 |
| z 1 |
| z |
Likewise:
Adding equations (4.13) and (4.14) term by term, we obtain:
those. the sum of moments of inertia about any mutually perpendicular axes remains constant and does not change when the coordinate system is rotated.
Principal axes of inertia and principal moments of inertia
With changing the angle of rotation of the axes a Each of the quantities changes, but their sum remains unchanged. Therefore, there is such a meaning
a = a 0, at which the moments of inertia reach extreme values, i.e. one of them reaches its maximum value, and the other reaches its minimum. To find the value a 0 take the first derivative of (or) and equate it to zero:
Let us show that relative to the obtained axes the centrifugal moment of inertia is equal to zero. To do this, we equate the right side of equation (4.15) to zero: , from where, i.e. got the same formula for a 0 .
The axes about which the centrifugal moment of inertia is zero and the axial moments of inertia take extreme values are called principal axes. If these axes are also central, then they are called main central axes. Axial moments of inertia about the principal axes are called principal moments of inertia.
Let us denote the main axes by y 0 And z 0. Then
If a section has an axis of symmetry, then this axis is always one of the main central axes of inertia of the section.
Let Ix, Iy, Ixy also be known. Let's draw a new axis x 1, y 1 parallel to the xy axes.
And let's determine the moment of inertia of the same section relative to the new axes.
X 1 = x-a; y 1 =y-b
I x 1 = ∫ y 1 dA = ∫ (y-b) 2 dA = ∫ (y 2 - 2by + b 3)dA = ∫ y 2 dA – 2b ∫ ydA + b 2 ∫dA=
Ix – 2b Sx + b 2 A.
If the x axis passes through the center of gravity of the section, then the static moment Sx =0.
I x 1 = Ix + b 2 A
Similar to the new y 1 axis, we will have the formula I y 1 = Iy + a 2 A
Centrifugal moment of inertia about new axes
Ix 1 y 1 = Ixy – b Sx –a Sy + abA.
If the xy axes pass through the center of gravity of the section, then Ix 1 y 1 = Ixy + abA
If the section is symmetrical, at least one of the central axes coincides with the axis of symmetry, then Ixy =0, which means Ix 1 y 1 = abA
Changing moments of inertia when turning axes.
Let the axial moments of inertia about the xy axes be known.
We obtain a new xy coordinate system by rotating the old system by an angle (a > 0), if the rotation is counterclockwise.
Let's establish the relationship between the old and new coordinates of the site
y 1 =ab = ac – bc = ab- de
from triangle acd:
ac/ad =cos α ac= ad*cos α
from triangle oed:
de/od =sin α dc = od*sin α
Let's substitute these values into the expression for y
y 1 = ad cos α - od sin α = y cos α - x sin α.
Likewise
x 1 = x cos α + y sin α.
Let's calculate the axial moment of inertia relative to the new axis x 1
Ix 1 = ∫y 1 2 dA = ∫ (y cos α - x sin α) 2 dA= ∫ (y 2 cos 2 α - 2xy sin α cos α + x 2 sin 2 α)dA= =cos 2 α ∫ y 2 dA – sin2 α ∫xy dA + sin 2 α ∫x 2 dA = Ix cos 2 α - Ixy sin2 α + Iy sin 2 α .
Similarly, Iy 1 = Ix sin 2 α - Ixy sin2 α + Iy cos 2 α.
Let's add the left and right sides of the resulting expressions:
Ix 1 + Iy 1 = Ix (sin 2 α + cos 2 α) + Iy (sin 2 α + cos 2 α) + Ixy (sin2 α - cos2 α).
Ix 1 + Iy 1 = Ix + Iy
The sum of the axial moments of inertia during rotation does not change.
Let us determine the centrifugal moment of inertia relative to the new axes. Let's imagine the values x 1 ,y 1 .
Ix 1 y 1 = ∫x 1 y 1 dA = (Ix – Iy)/2*sin 2 α + Ixy cos 2 α .
Main moments and main axes of inertia.
Main moments of inertia they are called extreme values.
The axes about which the extreme values were obtained are called the main axes of inertia. They are always mutually perpendicular.
The centrifugal moment of inertia relative to the main axes is always equal to 0. Since it is known that there is an axis of symmetry in the section, the centrifugal moment is equal to 0, which means the axis of symmetry is the main axis. If we take the first derivative of the expression I x 1, then equate it to “0”, we obtain the value of the angle = corresponding to the position of the main axes of inertia.
tan2 α 0 = - 
If α 0 >0, then for a certain position of the main axes the old axis must be rotated counterclockwise. One of the main axes is max, and the other is min. In this case, the max axis always corresponds to a smaller angle with that random axis relative to which it has a larger axial moment of inertia. Extreme values of the axial moment of inertia are determined by the formula:
Chapter 2. Basic concepts of strength of materials. Objectives and methods.
When designing various structures, it is necessary to solve various issues of strength, rigidity, and stability.
Strength– the ability of a given body to withstand various loads without destruction.
Rigidity– the ability of a structure to absorb loads without large deformations (displacements). Preliminary permissible deformation values are regulated by building codes and regulations (SNIP).
Sustainability
Consider the compression of a flexible rod
If the load is gradually increased, the rod will first shorten. When the force F reaches a certain critical value, the rod will buckle. - absolute shortening.
In this case, the rod does not collapse, but sharply changes its shape. This phenomenon is called loss of stability and leads to destruction.
Sopromat– these are the fundamentals of the sciences of strength, rigidity, and stability of engineering structures. Strength materials use methods of theoretical mechanics, physics, and mathematics. Unlike theoretical mechanics, strength resistance takes into account changes in the size and shape of bodies under the influence of load and temperature.
Often, when solving practical problems, it is necessary to determine the moments of inertia of a section relative to axes oriented in different ways in its plane. In this case, it is convenient to use the already known values of the moments of inertia of the entire section (or its individual constituent parts) relative to other axes, given in the technical literature, special reference books and tables, as well as calculated using available formulas. Therefore, it is very important to establish the relationships between the moments of inertia of the same section relative to different axes.
In the most general case, the transition from any old to any new coordinate system can be considered as two successive transformations of the old coordinate system:
1) by parallel transfer of coordinate axes to a new position and
2) by rotating them relative to the new origin. Let's consider the first of these transformations, i.e. parallel translation of coordinate axes.
Let us assume that the moments of inertia of a given section relative to the old axes (Fig. 18.5) are known.
Let's take a new coordinate system whose axes are parallel to the previous ones. Let us denote a and b the coordinates of the point (i.e., the new origin) in the old coordinate system
Let's consider an elementary platform. Its coordinates in the old coordinate system are equal to y and . In the new system they are equal
Let us substitute these coordinate values into the expression for the axial moment of inertia relative to the axis
In the resulting expression, the moment of inertia, the static moment of the section relative to the axis, is equal to the area F of the section.
Hence,
If the z axis passes through the center of gravity of the section, then the static moment and
From formula (25.5) it is clear that the moment of inertia about any axis that does not pass through the center of gravity is greater than the moment of inertia about the axis passing through the center of gravity, by an amount that is always positive. Consequently, of all the moments of inertia relative to parallel axes, the axial moment of inertia has the smallest value relative to the axis passing through the center of gravity of the section.
Moment of inertia about the axis [by analogy with formula (24.5)]
In the particular case when the y-axis passes through the center of gravity of the section
Formulas (25.5) and (27.5) are widely used in calculating axial moments of inertia of complex (composite) sections.
Let us now substitute the values into the expression for the centrifugal moment of inertia relative to the axes
If the axes are central, then the moment axes will look like: 
15.Dependency between moments of inertia when turning the axes:
J x 1 =J x cos 2 a + J y sin 2 a - J xy sin2a; J y 1 =J y cos 2 a + J x sin 2 a + J xy sin2a;
J x 1 y1 = (J x - J y)sin2a + J xy cos2a ;
Angle a>0, if the transition from the old coordinate system to the new one occurs counterclockwise. J y 1 + J x 1 = J y + J x
Extreme (maximum and minimum) values of moments of inertia are called main moments of inertia. The axes about which the axial moments of inertia have extreme values are called main axes of inertia. The main axes of inertia are mutually perpendicular. Centrifugal moments of inertia about the main axes = 0, i.e. main axes of inertia - axes about which the centrifugal moment of inertia = 0. If one of the axes coincides or both coincide with the axis of symmetry, then they are the main ones. Angle defining the position of the main axes: 
, if a 0 >0 Þ the axes rotate counterclockwise. The maximum axis always makes a smaller angle with that of the axes relative to which the moment of inertia has a greater value. The main axes passing through the center of gravity are called main central axes of inertia. Moments of inertia about these axes: 
J max + J min = J x + J y . The centrifugal moment of inertia relative to the main central axes of inertia is equal to 0. If the main moments of inertia are known, then the formulas for transition to rotated axes are:
J x 1 =J max cos 2 a + J min sin 2 a; J y 1 =J max cos 2 a + J min sin 2 a; J x 1 y1 = (J max - J min)sin2a;
The ultimate goal of calculating the geometric characteristics of the section is to determine the main central moments of inertia and the position of the main central axes of inertia. 
Radius of inertia - 
; J x =F×i x 2 , J y =F×i y 2 .
If J x and J y are the main moments of inertia, then i x and i y - principal radii of inertia. An ellipse built on the main radii of inertia as on the semi-axes is called ellipse of inertia. Using the ellipse of inertia, you can graphically find the radius of inertia i x 1 for any axis x 1. To do this, you need to draw a tangent to the ellipse, parallel to the x1 axis, and measure the distance from this axis to the tangent. Knowing the radius of inertia, you can find the moment of inertia of the section relative to the x axis 1: . For sections with more than two axes of symmetry (for example: circle, square, ring, etc.), the axial moments of inertia about all central axes are equal, J xy =0, the ellipse of inertia turns into a circle of inertia.