Inside any material there are internal interatomic forces, the presence of which determines the body’s ability to perceive external forces acting on it, resist destruction, change in shape and size. The application of an external load to a body causes a change in internal forces. Additional internal forces are studied in the strength of materials. In strength of materials they are simply called internal forces.

Internal forces are interaction forces between individual structural elements or between individual parts of an element that arise under the influence of external forces.

To numerically determine the magnitude of internal forces, the method of sections is used.

Section method comes down to four steps:

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Any cut off part of the body (preferably the most complex one) is discarded, and its action on the remaining part is replaced by internal forces so that the remaining part under study is in balance (Fig. 8);

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The resulting forces (N, Qy, Qz) (Fig. 9) and moments (Mk, My, Mz) are called internal force factors in the section

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The following names are accepted for internal force factors:

-longitudinal or axial force;

And -shear forces;

-torque;

And
-bending moments.

Internal force factors are found by composing six static equilibrium equations for the considered part of the dissected body.

Voltage

If we select an infinitesimal area in the section
and assume that the internal forces applied to its various points are identical in magnitude and direction, then their resultant
will pass through the element's center of gravity
(Fig. 10).

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Projections
on the axis ,And there will be an elementary longitudinal force
, and elementary shear forces
And
.

Let's divide these elementary forces by the area
, we obtain values ​​called stresses at the point of the drawn section.

;
;
,

Where - normal voltage; - tangential stress.

Stress is an internal force per unit area at a given point of the section under consideration.

Stress is measured in stress units - pascals (Pa) and its multiples - (kPa, MPa)

Sometimes, in addition to normal and tangential stresses, total stress is also considered

The concept " voltage» plays a very important role in strength calculations. Therefore, a significant part of the strength of materials course is devoted to studying methods for calculating stresses And .

Tension and compression

Central tension (compression) This type of deformation is called in which only longitudinal force (tensile and compressive) occurs in the cross section of the beam and all other internal force factors are equal to zero.

Longitudinal forces are determined using the section method.

Example

Let there be a stepped rod loaded with forces
,
And
along the axis of the rod shown in Fig. 11, a. Determine the magnitude of longitudinal forces.

Solution. The rod can be divided into sections according to the places where loads are applied and where the cross section changes.

The first section is limited by the points of application of forces And . Let's direct the axis (beginning of the first section). Mentally cut the first section with a cross section at a distance from the beginning of the first section. Moreover, the coordinate can be taken in the interval
, Where - length of the first section.

;
, kN

A positive sign of the longitudinal force indicates that the first section is stretched.

The value of the longitudinal force does not depend on the coordinate , therefore, throughout the entire section the value of the longitudinal force is constant and equal .

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The second section is limited by the points of application of forces And . Let's direct the axis along the axis of the section upwards with the origin at the point of application of the force (beginning of the second section).

Mentally cut the second section with a cross section at a distance from the beginning of the second section. Moreover, the coordinate can be taken in the interval
, Where - length of the second section.

Let us consider the equilibrium of the lower part of the rod, replacing the action of the upper part on the lower part of the rod with a longitudinal force
, having previously directed it in the direction of stretching the part in question.

From the static equilibrium condition:

;

The minus sign indicates that the second section is compressed.

Similarly for the third section:

;

For greater clarity, it is more convenient to present the results obtained in the form of a graph ( diagramsN), showing the change in longitudinal force along the axis of the rod. To do this, we draw a zero (base) line parallel to the axis of the rod, perpendicular to which we will plot the values ​​of the axial forces on a scale (Fig. 1.11, e). We put positive values ​​in one direction and negative values ​​in the other. The diagram is shaded perpendicular to the zero line, and a sign of the deferred value is placed inside the diagram. The values ​​of the deferred quantities are indicated next to them. Next to the diagram, the name of the diagram (“N”) is indicated in quotation marks and the units of measurement (kN) are indicated, separated by commas.

Section Method Steps

The cutting method consists of four successive steps: cut, discard, replace, balance.

Let us cut the rod, which is in equilibrium under the action of a certain system of forces (Fig. 1.3, a), into two parts by a plane perpendicular to its z-axis.

Let's discard one of the parts of the rod and consider the remaining part.

Since we, as it were, cut an infinite number of springs connecting infinitely close particles of the body, now divided into two parts, at each point of the cross section of the rod it is necessary to apply elastic forces, which, during the deformation of the body, arose between these particles. In other words, let's replace the action of the discarded part with internal forces (Fig. 1.3, b).

Deformations of the body (structural elements) under consideration arise from the application of an external force. In this case, the distances between the particles of the body change, which in turn leads to a change in the forces of mutual attraction between them. Hence, as a consequence, internal efforts arise. In this case, internal forces are determined by the universal method of sections (or cutting method).

It is known that there are external forces and internal forces. External forces (loads) are a quantitative measure of the interaction of two different bodies. These also include reactions in connections. Internal forces are a quantitative measure of the interaction of two parts of one body located on opposite sides of the section and caused by the action of external forces. Internal forces arise directly in the deformable body.

Figure 1 shows the design diagram of a beam with an arbitrary combination of external load forming an equilibrium system of forces:

From top to bottom: elastic body, left cut-off part, right cut-off part
Fig.1. Section method.

In this case, the bond reactions are determined from the known equilibrium equations of the statics of a solid body:

where x 0, y 0, z 0 is the base coordinate system of the axes.

Mentally cutting a beam into two parts with an arbitrary section A (Fig. 1 a) leads to equilibrium conditions for each of the two cut parts (Fig. 1 b, c). Here ( S') And ( S"} - internal forces arising respectively in the left and right cut off parts due to the action of external forces.

When composing mentally cut-off parts, the condition of body equilibrium is ensured by the relation:

Since the initial system of external forces (1) is equivalent to zero, we obtain:

{S ’ } = – {S ” } (3)

This condition corresponds to the fourth axiom of statics about the equality of action and reaction forces.

Using the general methodology of the theorem Poinsot on bringing an arbitrary system of forces to a given center and choosing the center of mass as the reduction pole, sections A " , point WITH " , system of internal forces for the left side ( S') we reduce to the main vector and the main moment of internal efforts. The same is done for the right cut off part, where the position of the center of mass of the section A"; is determined, respectively, by the point WITH" (Fig. 1 b,c).

Here, in accordance with the fourth axiom of statics, the following relations still hold:

Thus, the main vector and the main moment of the system of internal forces arising in the left, conditionally cut off part of the beam are equal in magnitude and opposite in direction to the main vector and the main moment of the system of internal forces arising in the right conditionally cut off part.

The graph (diagram) of the distribution of the numerical values ​​of the main vector and the main moment along the longitudinal axis of the beam determines, first of all, specific issues of strength, rigidity and reliability of structures.

Let us determine the mechanism for the formation of components of internal forces that characterize simple types of resistance: tension-compression, shear, torsion and bending.

At the centers of mass of the sections under study WITH" or WITH" let's ask accordingly to the left (c", x", y", z") or right (c", x", y", z”) systems of coordinate axes (Fig. 1 b, c), which, unlike the base coordinate system x, y, z We will call them “followers”. The term is due to their functional purpose. Namely: tracking changes in the position of section A (Fig. 1 a) when it is conditionally displaced along the longitudinal axis of the beam, for example when: 0 x’ 1 a, a x’ 2 b etc., where A And b- linear dimensions of the boundaries of the studied sections of the timber.

Let us set the positive directions of the projections of the main vector or and the main moment or on the coordinate axes of the tracking system (Fig. 1 b, c):

In this case, the positive directions of projections of the main vector and the main moment of internal forces on the axis of the servo coordinate system correspond to the rules of statics in theoretical mechanics: for force - along the positive direction of the axis, for moment - counterclockwise rotation when observed from the end of the axis. They are classified as follows:

Nx- normal strength, a sign of central tension or compression;

M x - internal torque, occurs during torsion;

Q z , Q y- transverse or shearing forces – a sign of shear deformations,

M y, M z- internal bending moments, corresponding to bending.

The connection of the left and right mentally cut off parts of the beam leads to the well-known (3) principle of equality in magnitude and opposite direction of all components of the same name of internal forces, and the condition for the equilibrium of the beam is defined as:

Taking into account the equivalence to zero of the original system of forces (1), the following holds:

As a natural consequence of relations 3,4,5, the resulting condition is necessary for the same components of internal forces to form subsystems of forces equivalent to zero in pairs:

The total number of internal forces (six) in statically definable problems coincides with the number of equilibrium equations for a spatial system of forces and is associated with the number of possible mutual movements of one conditionally cut-off part of the body in relation to another. z ( P i) = Mz + Mz(P i) + … + Mz(Pk) = 0 > Mz

Here, for simplicity of coordinate system notation c" x" y" z" And c"x"y"t" replaced by a single oxyz.

In order to judge the strength of the body under study, which is in equilibrium under the influence of external forces, it is first necessary to be able to determine the internal forces caused by them.

External forces deform the body; internal efforts, resisting this deformation, strive to maintain the original shape and volume of the body.

Detection of internal forces and their calculation constitute the first and main problem of the strength of materials, which is solved using the method of sections, the essence of this method is as follows:

• - first operation. We cut (mentally) the rod along the cross section in which the magnitude of the internal forces should be determined.
• - second operation. We discard any part of the rod, for example, part 1. Usually the part to which a greater number of forces is applied is discarded.
• - third operation. We replace the forces acting on the remaining part with the main vector and the main moment, aligning the center of reduction O with the center of gravity (c.t.) of the section (in Fig. 1, b M not shown).
• - fourth operation. We balance the remaining part, since before the dissection it was in equilibrium. To do this, at point O we apply a force R and a moment M, equal and oppositely directed to the main vector and the main moment. The forces and and are those internal forces that were transmitted from the thrown side to the remaining part of the rod.
• - The method of sections is only the first step towards studying internal forces, since with its help it is not possible to find out the law of distribution of internal forces in a section.

By composing equilibrium equations for a cut-off part of a body, it is possible to obtain projections onto the coordinate axes of both the main vector and the main moment.

When calculating beams, the origin of coordinates is placed at the center of gravity of the cross section under consideration. The “Z” axis in a straight beam is aligned with its longitudinal axis, in a curved beam it is directed tangentially to its axis at the point where the origin of coordinates is located.

The “X” and “Y” axes are aligned with the directions of the main central axes of inertia of the section under consideration. Projections onto the coordinate axes of the main vector and the main moment of internal forces in the beam are denoted respectively: , N, M x , M y , and are called internal force factors (internal efforts).

Represent shear forces in the direction of the "X" or "Y" axis (N)

N - normal (longitudinal) force (n.).

M x , M y - bending moments relative to the “X” or “Y” axes, respectively (nm)

M z - torque (nm).

Having examined the cut-off part of the beam (for example, the right one) (Fig. 1, b) and compiled the equilibrium equation based on the method of sections, we can say the following: normal force N is an internal force, numerically equal to the sum of the projection onto the longitudinal axis of the beam of all external forces located on one side of the section under consideration.

• -the transverse force in the direction of the “X” axis is numerically equal to the sum of the projections onto the “X” axis of all external forces located on one side of the section under consideration.
• - the transverse force in the direction of the “Y” axis is numerically equal to the sum of the projections onto the “Y” axis of all external forces located on one side of the section under consideration

M x - bending moment relative to the “X” axis is numerically equal to the sum of the moments of all external forces located on one side of this section.

M Y - bending moment relative to the “Y” axis is numerically equal to the sum of the moments of all external forces located on one side of this section.

M z - bending moment relative to the “Z” axis is numerically equal to the sum of the moments of all external forces located on one side of this section.

So, in the general case of loading a beam, the internal forces in its cross sections are reduced to the indicated six internal force factors.

Types of loads, types of supports and beams.

Any rod that bends is called a beam.

Active forces are assumed to be known and are reduced to concentrated forces F(H), force pairs m (nm) and loads distributed along the length of the beam q (n/m). The magnitude and direction of reactions R 1, R 2 are determined from the equilibrium condition of the beam and the type of its supporting fastenings.

Beams can have the following three types of supports:

• 1. Hard pinching or embedding. The end of the beam is deprived of three degrees of freedom. It cannot move in either vertical or horizontal directions and has no ability to rotate. Consequently, three reactions occur in this support: two forces R 1 and R 2, preventing linear displacements of the end of the beam, and one reactive moment M R, preventing rotation.
• 2. Hinged-fixed support.

Such a support deprives the beam of two degrees of freedom: vertical and horizontal displacement, but does not prevent the beam from rotating around the hinge. Consequently, in this support two components of the support reaction R 1 and R 2 arise.

3. A hinged-movable support is the least rigid support; it deprives the end of the beam of only one degree of freedom - vertical linear movement. In the articulated movable support, one reaction occurs.

It should be noted that this support prevents the end of the beam from moving both down and up. It should be noted that in practice the rolling plane of the movable support is always made parallel to the axis of the beam. Then the reaction of the movable support should have a direction perpendicular to the axis of the beam.

By using different types of supports, we get different types of beams. Since the beam in the plane has three degrees of freedom, then in order to be fixed, the beam must be deprived of all three degrees of freedom.

The first type of beam is a cantilever. The console has a seal at one end that takes away all three degrees of freedom, and its other end is free. In the embedment, the following occur: reactive moment, vertical reaction and, in the presence of a horizontal or inclined load, horizontal reaction. The console is used in technology in the form of brackets, masts, etc.

The second type of beam is a two-support beam. The beam is supported at two points by using one movable and one fixed hinged support, which together take away all three degrees of freedom from the beam. In a movable support, only a vertical reaction occurs, in a fixed one - vertical and horizontal (in the presence of horizontal components of the loads).

The distance between supports is called span. If one of the supports is displaced by a certain distance, then the beam is called single-cantilever. Beams of the listed types have the minimum required number of supports; therefore, they are statically determinable, i.e. their support reactions can be found from the equilibrium equation.

The installation of additional supports makes the beam statically indeterminate: the calculation of such beams is possible only taking into account their deformations.

The method of sections is that the body is mentally cut by a plane into 2 parts, any of which is discarded and in its place the forces acting before the cut are applied to the remaining section, the remaining part is considered as an independent body that is in equilibrium under the influence of external and internal forces applied to the section . According to Newton’s 3rd law, the internal forces acting in the section of the remaining and discarded parts of the body are equal in magnitude, but opposite; therefore, when we consider the equilibrium of any of the 2 parts of the dissected body, we get the same value of the internal forces.

Bending is a type of loading of a beam in which a moment is applied to it, lying in a plane passing through the longitudinal axis. Bending moments occur in the cross sections of the beam. When bending, deformation occurs in which the axis of a straight beam bends or the curvature of a curved beam changes.

A beam that bends is called beam. A structure consisting of several bendable rods, most often connected to each other at an angle of 90°, is called a frame.

A bend is called flat or straight if the plane of action of the load passes through the main central axis of inertia of the section.

With plane transverse bending, two types of internal forces arise in the beam: transverse force Q and bending moment M. In the frame with plane transverse bending, three forces arise: longitudinal N, transverse force Q and bending moment M.

If the bending moment is the only internal force factor, then such bending is called clean(Fig. 6.2). When there is a shear force, bending is called transverse. Strictly speaking, simple types of resistance include only pure bending; transverse bending is conventionally classified as a simple type of resistance, since in most cases (for sufficiently long beams) the effect of transverse force can be neglected when calculating strength.

Oblique bending is a bending in which the loads act in one plane that does not coincide with the main planes of inertia.

Complex bending is a bending in which loads act in different (arbitrary) planes.

Constructing diagrams of shear force and bending moment

In order to calculate a beam for bending, it is necessary to know the magnitude of the maximum bending moment M and the position of the section in which it occurs. In the same way, you need to know the greatest shear force Q. For this purpose, diagrams of bending moments and shear forces are constructed. From the diagrams it is easy to judge where the maximum value of the moment or shear force will be.

Before determining internal forces (transverse forces and bending moments) and constructing diagrams, as a rule, it is necessary to find the support reactions that arise in the fastening of the rod. If support reactions and internal forces can be found from static equations, then the structure is called statically determinate. Most often we encounter three types of support fastenings for rods: rigid pinching (embedding), hinged-fixed support and hinged-movable support. In Fig. Figure 6.5 shows these fastenings. For fixed (Fig. 6.5, b) and movable (Fig. 6.5, c) supports, two equivalent designations for these fastenings are given. Let us recall that when a load is applied in one plane, three support reactions arise in the embedment (vertical, horizontal reactions and concentrated reactive moment) (Fig. 6.5, a); in the articulated fixed support - two reactive forces (Fig. 6.3, b); in a hinged-movable support there is one reaction - a force perpendicular to the plane of support (Fig. 6.5, c).

If an external force rotates the cut part of the beam clockwise, then the force is positive; if an external force rotates the cut part of the beam counterclockwise, then the force is negative.

If, under the influence of an external force, the curved axis of the beam takes the form of a concave bowl, such that rain falling from above will fill it with water, then the bending moment is positive. If, under the influence of an external force, the curved axis of the beam takes the form of a convex bowl, such that rain falling from above will not fill it with water, then the bending moment is negative.

It is quite obvious and confirmed by experience that when bending a beam, it is deformed in such a way that the fibers located in the convex part are stretched, and in the concave part they are compressed. Between them lies a layer of fibers, which only bends without changing its original length (Fig. 6.8). This layer is called neutral or zero, and its trace on the cross-sectional plane is called the neutral (zero) line or axis.

When constructing diagrams of Q and M, we will agree on plotting Q to put positive values ​​on top of the zero line. On the M diagram, it is customary for builders to put positive ordinates from below. This rule for constructing a diagram of M is called constructing a diagram from the side of stretched fibers, i.e., positive values ​​of M are deposited towards the convexity of the curved beam.

For simplicity, let us consider a beam with a rectangular cross section (Fig. 6.9). Following the section method, we mentally make a cut and discard some part of the beam and leave the other. On the remaining part we will show the forces acting on it and in the cross section - internal force factors that are the result of bringing the forces acting on the rejected part to the center of the cross section. Considering that external forces and distributed loads lie in the same plane and act perpendicular to the axis of the beam, we obtain a transverse force and a bending moment in the section. These internal force factors are unknown in advance, so they are shown in a positive direction in accordance with the accepted rules of signs.

Internal forces. Section method

External forces acting on a real object are most often known. Usually it is necessary to determine internal forces (the result of interaction between individual parts of a given body) that are unknown in magnitude and direction, but knowledge of which is necessary for strength and deformation calculations. The determination of internal forces is carried out using the so-called section method, the essence of which is as follows:

Mentally cut the body along the section that interests us.

Discard one of the parts (no matter which one).

The action of the discarded part of the body is replaced by the remaining one with a system of forces, which in this case become external. According to the principle of action and reaction, elastic forces are always mutual and represent a system of forces continuously distributed over the cross section. Their value and orientation at each point of the section are arbitrary and depend on the orientation of the section relative to the body, the magnitude and direction of external forces, and the geometric dimensions of the body. Internal forces can be reduced to the main vectorR and the main moment M. The center of gravity of the section is usually taken as the reference point. Having chosen the X, Y, Z coordinate system (Z is the longitudinal axis normal to the cross section, X and Y are in the plane of this section) and the origin of the system at the center of gravity, we denote the projections of the main vector R onto the coordinate axes by N, Q x, Q y, and the projections of the main moment M are M x, M y, M k. These three forces and three moments are called internal force factors in the section:

N – longitudinal force,

Q x , Q y – transverse forces,

M k – torque,

M x , M y – bending moments.

4. Since internal forces are in equilibrium with external forces, they can be determined from the static equilibrium equations:

P z =0, P y =0, P x =0,

M x =0, M y =0, M z =0.

Any internal force factor in a section is equal to the algebraic sum of the corresponding external force factors acting on one side of the section.

The internal force factor in a section is numerically equal to the integral sum of the corresponding elementary internal forces or moments over the entire cross-sectional area:

The classification of the main types of loading is associated with the internal force factor arising in the section. Thus, if only longitudinal force N occurs in cross sections, and other internal force factors become zero, then tension or compression occurs in this section, depending on the direction of force N. Loading, when only transverse force Q occurs in the cross section, called a shift.

If only a torque Mk occurs in the cross section, then the rod works in torsion. In the case when only a bending moment M x (or M y) arises from external forces applied to the rod, then this type of loading is called pure bending in the yz (or xz) plane. If in a cross section, along with a bending moment (for example, M x), a transverse force Q y occurs, then this type of loading is called flat transverse bending (in the yz plane). The type of loading, when only bending moments M x and M y occur in the cross section of the rod, is called oblique bending (plane or spatial). When a normal force N and bending moments M x and M y are applied in the cross section, a loading occurs called complex bending with tension-compression or eccentric tension (compression). When a bending moment and a torque act in a section, bending with torsion occurs.

The general case of loading is the case when all six internal force factors arise in the cross section.

Special types of loading include crushing, when the deformation is local in nature, not spreading to the entire body, and longitudinal bending (a special case of the general phenomenon of loss of stability).

The concept of stress

IN the magnitude of internal force factors does not reflect the intensity
tense state of the body, proximity to a dangerous state (destruction). To assess the intensity of internal forces, a criterion (numerical measure) called stress is introduced. If in cross section F of a certain body, we select an elementary area F, Fig. 1.1, within which the internal force R is identified, then the ratio can be taken as the average stress on the area F:

The true stress at a point can be determined by reducing the area:

IN vector quantity R represents the total stress at a point. The voltage dimension is taken in Pa (Pascal) or MPa (Megapascal). The total stress is usually not used in calculations, but its component normal to the section  is determined - normal stress, and tangential  ,   - tangential stresses (Fig. 1.2). Total stresses per unit area can be expressed in terms of normal and shear stresses:

There is the following relationship between the acting stresses and internal force factors:

;

Normal and shear stresses are a function of internal force factors and geometric characteristics of the section. These voltages, calculated using the appropriate formulas, can be called actual or operating.

The highest value of actual stresses is limited by the limiting stress at which the material fails or unacceptable plastic deformations occur. The first of these boundaries exists for any brittle material and is called the tensile strength ( in,  in), the second occurs only in plastic materials and is called the yield strength ( t,  t). Under the action of cyclically changing stresses, destruction occurs when the so-called endurance limit ( R,  R) is reached, which is significantly less than the corresponding strength limits.