When determining internal forces new factors are considered to be applied at the center of gravity of the section. In reality, internal forces, resulting from the interaction of body particles, are continuously distributed over the cross section. The intensity of these forces at different points of the section can be different. As the load on a structural element increases, the internal forces increase and, accordingly, their intensity increases at all points of the section. If at some point the intensity of internal forces reaches a value determined for a given material, a crack appears at this point, the development of which will lead to the destruction of the element, or unacceptable plastic deformations will occur. Consequently, the strength of structural elements should be judged not by the value of internal force factors, but by their intensity. The measure of the intensity of internal forces is called voltage.

In the vicinity of an arbitrary point belonging to the section of a certain loaded body, we select an elementary area within which the internal force acts (Fig. 1.6, A).

The average value of the intensity of internal forces on the site, called the average stress, is determined by the formula

Reducing the area, in the limit we obtain the true stress at a given point of the section

The vector quantity is called total voltage at point. In the International System of Units (SI), the unit of voltage is taken to be pascal(Pa) is the stress at which an internal force of 1 N acts on an area of 1 m2.

Since this unit is very small, a multiple unit of stress is used in calculations - megapascal (1 MPa = 10 6 Pa).

Let us decompose the total voltage vector into two components (Fig. 1.6, b).

The projection of the total stress vector onto the normal to a given area is denoted by and is called normal voltage.

Rice. 1.6

The component lying in the section in a given area is denoted by and is called shear stress.

The normal stress directed away from the section is considered positive, and that directed towards the section is considered negative.

Normal stresses arise when, under the influence of external forces, particles located on both sides of the section tend to move away from one another or move closer together. Shear stresses arise when particles tend to move relative to each other in the section plane.

Shear stress can be decomposed along the coordinate axes into two components and (Fig. 1.6, V). The first index at shows which axis is perpendicular to the section, the second - parallel to which axis the stress acts. If the direction of the shear stress is not important in the calculations, it is indicated without subscripts.

There is a relationship between the total voltage and its components

An infinite number of sections can be drawn through a point of a body, and for each of them the stresses have their own meaning. Consequently, when determining stresses, it is necessary to indicate the position of not only a point of the body, but also the section drawn through this point.

The set of stresses for many areas passing through this point, forms stress state at this point.

Stresses in cross sections are related to internal force factors by certain dependencies.

Let us take an infinitesimal cross-section of area . In the general case, infinitesimal (elementary) internal forces act along this area (Fig. 1.7)

Fig.1.7

The corresponding elementary moments about the coordinate axes , , have the form.

Classification of forces

Forces are divided into external and internal. External forces characterize the interaction between bodies, internal forces characterize the interaction between particles of one body.

External forces acting on structural elements are divided into active, called load, and reactive(connection reactions). Load is divided into surface and volumetric. Surface load refers to contact forces that arise when two structural elements are mated or interact; to volumetric (mass) forces - forces acting on each infinitesimal element of volume. Examples of volumetric forces are inertial forces, gravity forces, and magnetic interaction forces.

Based on the nature of the action on the structure, the load is distinguished:

static– changes slowly and smoothly from zero to the final value so that the accelerations of the points of the system that arise in this case are very small, therefore inertial forces can be neglected compared to the load;
dynamic– applied to the body in a short period of time or instantly with the formation of significant accelerations;
re-variable– changing according to an arbitrary periodic law.

Internal force factors (section method)

Let a free body under the influence of a system of forces be in equilibrium (Fig. 2.1). It is required to determine the internal forces in the section. Let us mentally cut the body into two parts along a given section and consider the conditions of equilibrium of one (any) part of the body. Both parts after the cut, generally speaking, will not be in balance, since the internal connections are broken. Let us replace the action of the left part of the body with the right and the right with the left by some system of forces in the section, i.e. internal forces (Fig. 2.2). The nature of the distribution of internal forces in the cross section is unknown, but they must ensure the balance of each part of the body. To create the equilibrium condition for the cut-off part, we bring the internal forces in the form of the main vector and the main moment to the center of gravity of the section and project them onto the coordinate axes (Fig. 2.3). We get three projections of the main vector and three projections of the main moment, which are called internal force factors:– longitudinal force; – shear forces; – torque; – bending moments.

Having drawn up the equilibrium conditions for the cut-off part, we obtain

(2.1)

Equations (2.1) are called the relationship between the external load on the cut-off part and internal force factors (static equivalents of internal

Rice. 2.1

Rice. 2.2

strength). If the external loads are known, then with their help it is possible to determine the internal force factors.

The following main types of deformations are distinguished:

Rice. 2.3

Rice. 2.4

Concept of voltage

According to hypothesis 1 (see paragraph 2.1.1), it can be assumed that internal forces are continuously distributed over the cross-sectional area of the beam. Let it be on a small but final platform A(Fig. 2.5) internal elementary force acts R. Having spread out R into components along the axes we obtain its components Relation of the form

determines the average voltage on a given site at a given point.

The total, or true, voltage at a point is the ratio

which determines the intensity of internal forces at a given point of the section under consideration. Since an infinite number of sections can be drawn through a point on a body, at a given point there are an infinite number of stresses associated with the action areas. The set of all stresses acting on different areas at a given point is called stressed state of the point. The unit of stress is N/m2 or Pa. By analogy with expression (2.3), we can write:

Expression (2.4) determines normal stress σ x (Fig. 2.6), the vector of which is directed in the same way as the normal force vector Ν x. Expressions (2.5) and (2.6) determine shear stress; their vectors have the same directions as, respectively, and. The first index for τ indicates which axis the normal to the area of action of the stress under consideration is parallel to, the second index shows which axis the given stress is parallel to.

Relationship between total voltage TO and its components is expressed by the formula

Let us consider the relationship between stresses and internal force factors in the cross section of the beam.

Rice. 2.5

Rice. 2.6

The components of the main vector and the main moment of internal forces will have the following form.

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1.6 Voltages. Relationship between stress and internal force factors. Saint-Venant's principle

Voltages

E if you select a small area
in cross section and designate the internal force acting on it
(Fig. 1.6), the total stress vector at a point on the body will be determined by the formula

, (1.1)

The total voltage vector is set by its projections on the axis
, , . To do this, let us denote the projections of the vector on the axis
,
,
(Fig. 1.7) and find the corresponding projections of the total stress:

Normal voltage

, (1.2)

Rice. 1.7 - shear stress along the axis

, (1.3)

Shear stress along the axis

. (1.4)

If the law of stress distribution over the cross section is known, then using formulas (1.2) – (1.4) and figures (1.8), (1.5) one can obtain feedback between stresses and internal force factors

, (1.5)
The stresses caused by a local load at points of the body sufficiently distant from the place where this load is applied to it depend little on the specific nature of the load distribution, but are determined only by its main vector and moment.

A load is called local if the dimensions of the area to which it is applied are small compared to the dimensions of the body.

The relationship between the moments of inertia during parallel translation of the axes and when turning the axes.

When moving axes in parallel:

If S x and S y are equal to zero, then: ;

When turning the axes:

and for the centrifugal moment of inertia:

Principal axes, principal moments of inertia. Determination of the direction of the main axes. Determination of the value of the main moments of inertia.

The axes relative to which centrifugal moment the inertia of the section becomes zero, called the principal axes. The moments of inertia about the principal axes of inertia are called the principal moments of inertia of the section. To determine the position of the main central axes of an asymmetrical figure, we rotate an arbitrary initial system of central axes z, y to a certain angle at which the centrifugal moment of inertia becomes equal to zero.

Where .

Determination of the values of the main moments of inertia:

Moreover, the upper signs should be taken at .

Types of stress. Stress tensor. Law of pairing of tangential stresses.

The stressed state of a body at a point is the totality of normal and tangential stresses acting on all areas containing the point.

Linear - if one main voltage is non-zero, and the other 2 are equal to 0.

Flat - if 2 main stresses are non-zero and one is equal to zero.

Volumetric – if all 3 main stresses are different from zero.

– stress tensor.

Law of pairing of tangential stresses:

Plane stress state. Stresses along inclined platforms. Determination of stresses using Mohr's circles. Direct and inverse problem.

A stress state in which one of the three principal stresses is zero is called flat.

Stresses on inclined platforms:

Determination of stresses using Mohr circles: ;

The coordinates of the circle points correspond to normal and shear stresses at various sites. We lay off the ray from the axis from center C at an angle of 2 ( , then counterclockwise), find point D, the coordinates of which are: , . You can solve both direct and inverse problems graphically.

Direct task: , ,

Let us determine the stresses and acting on any inclined platform using the known principal stresses and .

Inverse problem: ,

Using the known normal tangential stresses acting in two mutually perpendicular areas, find the main (max and min, 1 and 2) stresses and the position of the main areas. Tangential stresses along the main areas are equal to 0). The angle determining the position of the main platforms: . If one of the main stresses turns out to be negative, then they should be designated , , if both are negative, then , .

Oblique bend. Determination of stresses, strength conditions.

Bending with torsion of rods of round cross section. Determination of design stress and strength testing.

σ=√(Mx^2+My^2)/Wno; τ=Mcr/Wρ; According to the fourth energy theory: σmax^IV=√(σ^2+3*τ^2)

Internal power factors. Section method. The concept of stress. Relationship between internal force factors and stresses.

To find the internal forces, we will use the ROZU method of sections. P - cut an arbitrary plane into A and B. O - discard one of these parts, for example B. Consider the remaining part. Z – we replace. We replace internal forces with the main vector and the main moment. We expand the main vector and main point in a plane on an axis. Internal power factors:

Qx, Qy – cause shear – shearing shear forces; N – normal longitudinal tire, tension, compression of timber; Mz – torque; Mx, My – bending moment. The graph of changes in the internal factor when moving along the axis of the rod is called a diagram. U - balancing.

Let us select point B in the section under consideration, and in the vicinity of this point we select an elementary area with area . Let be the resultant of all internal forces acting on the site. The ratio is called average stress on the site, which characterizes the average intensity of the distribution of internal forces on this site. The limit of this ratio is called the total stress at point B. This stress can be decomposed into components: normal and tangential to the section plane. The normal component is called normal voltage; the component lying in the section plane is called shear stress. The tangent component is decomposed into 2 perpendicular components along the x and y axes - and . The magnitude of the total voltage. The relationship between stress and internal force factors can be described by the following relationships: 2. Tension and compression. Voltage. Deformation. Conditions of strength and rigidity. Tension (compression) is understood as a type of loading in which only longitudinal forces arise in the cross sections of the rod, and other force factors are equal to zero. Deformation is a change in the shape and size of a body under the influence of stress. Stress is a force acting per unit cross-sectional area of a part. Strength condition: , rigidity condition: . 3. Mechanical characteristics of materials. Tensile and compression testing of materials. Under mechanical characteristics refers to the values of stresses and deformations corresponding to certain points on the conditional stress diagram. The proportionality limit is the maximum stress up to which the deformations are directly proportional to the stresses. The elastic limit is the stress up to which the material does not receive residual deformations. The yield limit is the stress at which the deformations increase without noticeable increase in load. The tensile strength is the maximum stress that a material can withstand when stretched. The elastic limit is considered to be the stress at which residual deformations reach a predetermined value. 4. Geometric characteristics flat sections. Determination of the center of gravity of complex sections. Geometric characteristics – numerical quantities that determine the dimensions, shape, location of the cross section of a deformable structural element that is homogeneous in elastic properties.

The center of gravity of a complex section is determined from the condition

Strength of materials. Concept of stresses, strains and displacements Normal bending stress