Consider a straight rod loaded on the end by the forms directed parallel to the axis Oh. Equality of these forces F. Applied to point FROM. In the local right-hand coordinate system yoz.coinciding with the main central axes of section, point coordinates FROM equal but and b. (Fig. 5.18).

Replace the applied load with a statically equivalent system of forces and moments. To do this, we will transfer the equality F.to the center of severition ABOUT And exhaust the rod with two bending moments equal to the work of force T ^ on its shoulders relative to the coordinate axes: M ff \u003d Fa and M z \u003d fb.

It should be noted that according to the rule of the right-hand coordinate system for a point with, lying in the first quarter, the bending moments will formally get

Fig. 5.18. Straight rod, loaded on the end by the forms directed parallel to the axisOh

misery: M y \u003d Fa and m 7 \u003d -FB. At the same time, in the elementary platform lying in the first quarter, both moments cause tensile tension.

Using the principle of independence of the action, we define the voltage at the current section of the cross section with coordinates w. and z. From each power factor separately. General voltage We obtain the summation of all three components of the stresses:

We define the position of the neutral axis. To do this, in accordance with formula (5.69), we equate to zero the value of normal voltage at the current point:

As a result of simple transformations, we obtain the neutral line equation

where i y. and i z - Main radii inertiadefined by formulas (3.14).

Thus, in the case of high-center stretching-compression, the neutral line does not pass through the severity center (Fig. 5.19), which indicates the presence in equation (5.70) differing from zero of a free member.

Maximum voltages occur at the sections BUT and IN, The most remote from the neutral line. We establish the ratio between the coordinates of the point of the application of the force and the position of the neutral line. To do this, we define the intersection points of this line coordinate axes:

Fig. 5.19.

The resulting formulas show that the coordinate point of the force application but and coordinate point of intersection of the neutral coordinate axis Oz. (Point g 0) have opposite signs. The same can be said about the values b. and in 0. Thus, the point of the application of the equal force and the neutral line are located on different directions relative to the origin.

According to the formulas obtained, when the point of the application is approached, the neutral line is distinguished from the central zone approaching the severity. In the limit case (A \u003d B \u003d 0) Come to the occasion of central stretching compression.

It is of interest to define the zone of the application of force, in which the stresses in the section will have the same sign. In particular, for materials, poorly resisting stretching, the compressive force is rational to make it in this zone, so that only compressive stresses acted in the cross section. Such a zone around the center of severity is called core of the section.

If the force is applied in the core of the section, the neutral line does not cross section. In the case of an application of force along the border of the cross-section core, the neutral line concerns the cross section contour. To determine the cross section core, you can use formula (5.71).

If a neutral line is represented as a tangent to the contour of the section and consider all possible positions tangent and corresponding to these provisions of the point of force, the point of application of the force of the core of the cross section.

Fig. 5.20.

but - ellipse; 6 - Rectangle

An extracentrous stretch or compression is called such a type of deformation, when a longitudinal (stretching or compressive) force and is simultaneously operating in the cross section of the bar. bending moment; In this section can act and transverse force.

Especially stretched or compressed bar, when calculating which one can not take into account additional bending moments, equal to the works of longitudinal external forces p to the deflection, is called tough, and the bar, when calculating them should be considered - flexible.

Hard are high-centerly compressed and stretched bars depicted in fig. 10.9, a, g, d, if the largest of their devices are small compared with the distances of the forces r from the axes of Bruusyev, and the bars depicted in fig. 10.9, B, B, in cases where the works are small compared to external moments

Consider the calculation of hard bars; The method of calculating flexible bars is presented below in § 5.13.

In fig. 11.9, and depicted a hard bar; In its upper cross section, the longitudinal force N and the bending moment M, which constitutes relative to the main axes and in the inertia of the section are equal to normal voltage in an arbitrary point from the cross section with coordinates of y and equal to the amount of stresses from the longitudinal force of N and bending moments, i.e. .

The longitudinal force n and moments can be considered as a result of the impact on the ram of the extracentrate attached force

That is why the case of the simultaneous action in the cross section of the longitudinal force and the bending moment is called an off-centrular stretching (with stretching longitudinal strength) or compression (with compression).

The coordinates of the points and the application of force p are called the eccentricity of this force relative to the main axes of inertia and y, respectively:

Point A POP of POPs call the pressure center or pole.

Substitute in formula (10.9) of expressions [on the basis of formulas (11.9) and Fig. 1.9, b]:

The places plus in front of all members of this formula are supplied because the positive longitudinal force as well as the bending moments (with positive eccentricity) is caused at the cross-sectional points with positive coordinates of y and z stretching (positive) voltages.

In formula (12.9), the magnitude of the tensile force P is substituted with a plus sign, and compressing - with a minus sign; Coordinates of y and z in this formula are substituted with their signs. The sign of normal stresses arising at any point of section from the bending moment caused by eccentric (echocently) applied force p can also be installed by presenting a cross section in the form of a plate fixed on the shaft, the axis of which coincides with the axis; The plate relies on the rigid base through the springs system (Fig. 12.9).

The moment from the force p shown, for example, in Fig. 12.9, it causes a turn of the plates around the z axis, as a result of which the springs located under the shaded part of the plate are compressed; Consequently, in this part of the cross section of the bar from the moment compressive stresses occur. Similarly, in order to establish a voltage sign from the moment it is necessary to submit a plate fixed on the shaft, the axis of which coincides with the axis of the.

Formula (12.9) is used to determine normal stresses at any point of cross-section during embetrous stretching and compression.

Formula (12.9) can be represented as follows:

where - the radii of the inertia of the cross section of the bar with respect to the main central axes of the inertia of the GNU, respectively.

It should be borne in mind that in formulations (10.9) - (14.9), the axes of the axis (10.9) are the main central axes of the inertia of the cross section of the bar.

Formulascence (12.9) - (14.9) It is convenient to use when the resultant internal force in the cross section of the bar is known (i.e. strength p) and the coordinates of its application (pole). The formula (10.9) is conveniently used when the internal efforts acting in cross section are known.

EPUR Options for normal stresses arising in the cross section of a bar during an off-centrular compression (i.e., with a negative force of P), depicted in axonometry in Fig. 13.9.

They are limited on one side by the transverse sections with 1-2-3-4, and on the other - a plane of 1-2-3-4. Ordinates of Epur in the center of severity (at y \u003d z \u003d 0) are equal

All ordinates of the Epura shown in Fig. 13.9, and, negative, since the plane limiting them, does not cross the plane 1-2-3-4 within the cross section of the bar. The ordents of the same plot shown in Fig. 13.9, B, one side from the straight line, and the other is positive.

Direct PP is a line intersection line 1-2-3-4 with a cross-sectional plane of a bar. At all points located on direct PP, the voltage is zero, and, therefore, this straight line is a neutral axis (zero line).

We define the position of the neutral axis (Fig. 14.9). To do this, we equate zero by the right side of the expression (14.9):

Since, then

The expression (15.9) is the equation of straight (as coordinates of the coordinates and are part of it in the first degree) and is the neutral axis equation. To determine the position of the neutral axis, we find the ordinate of the point in its intersection with the axis of y (Fig. 14.9); The abscissa of this point and therefore on the basis of the expression (15.9)

The abscissa point with the intersection of the neutral axis with the axis is equal to (Fig. 14.9), and the ordinate of this point substituting the values \u200b\u200bin the expression (15.9), we find

Thus, the values \u200b\u200bof segments cut off by a neutral axis (zero line) on the coordinate axes are determined by expressions:

From these expressions it follows:

1) the position of the zero line does not depend on the magnitude and sign of the force p;

2) the zero line and pole lie on different directions from the origin of the coordinates;

4) If the pole is located on one of the main central axes of inertia, then the zero line is perpendicular to this axis; For example, when the pole is located on the axis, then i.e. the neutral axis is parallel to the axis.

With non-centcented tension and compression, normal voltages at each point of the cross section of the bar, as well as in bending, are directly proportional to the distance from this point to the neutral axis. The largest stresses occur at the cross-section points, most remote from the neutral axis.

Epur of normal stresses, the values \u200b\u200bof which are postponed from the line, perpendicular to the neutral axis, is shown in Fig. 14.9.

Each ordinate of this plot determines the magnitude of normal stresses arising at the cross-section points located on a direct DD passing through this ordinate in parallel neutral axis. To build this plot, it is sufficient to determine the position of the neutral axis and calculate the normal voltages in one of the cross-section points (not located on this axis), for example, in the center of severition. Using such a plot, the values \u200b\u200bof normal voltages are most simply defined at any points of the cross section.

Calculation of the strength of the rod, compressed or stretched by the extracently applied longitudinal external forces (i.e., in the absence of transverse forces), it is most simply produced, since in this case the internal efforts are the same in all transverse sections of each rod section. This eliminates the need to determine the hazardous cross section, since with a rod with constant transverse dimensions within each section, all sections of one section are equivalent. When the rod with variable transverse dimensions is dangerous within each site, the smaller size section is.

In the presence of cross-sections, the rod of the transverse forces, bending moments are continuously changed along the length of the rod, and therefore the determination of a hazardous section becomes more complex. Typically, in such cases, the strength is carried out, determining the normal stresses in a number of sections (which may be expected to be dangerous) and comparing them with allowable stresses.

To determine the position of hazardous points in a section, a parallel neutral axis should carry out the line concerning the cross section circuit. In this way, the sections are found on both sides of the neutral axis and the most distant from it, which can be dangerous.

The extracentral stretching (compression) is caused by the force parallel to the axis of the bar, but not coinciding with it (Fig. 9.4).

The projection of the point of the application of force on the cross section is called a pole or power point, and a straight, passing through the pole and the center of the section - the power line.

Especially stretching (compression) can be reduced to axial stretching (compression) and oblique bending, if you transfer the force of the force of the severity. So, the force p, marked in Fig. 9.4 One dash r will cause axial stretching of the bar, and a pair of forces marked with two dashes is oblique bending.

Based on the principle of independence of the voltage forces at the cross-sectional points during incantented tension (compression) are determined by the formula

In this formula, the axial force of the bending moments as well as the coordinates of the point of section, in which the voltage is determined, it is necessary to substitute with their signs. For bending moments, we will take the same rule of signs, as in oblique bending, and we will consider the axial strength positive when it causes a stretching.

If the coordinates of the pole designate through, then the moment of formula (9.5) takes the view

It can be seen from this equation that the ends of the voltage vectors at the sections are located on the plane. The line intersection line with a cross-sectional plane is a neutral line, the equation of which we find, equating the right-hand side of the equality (9.6) zero. After cutting on the p we get

Thus, the neutral line with non-centrular tension (compression) does not pass through the center of severity and is not perpendicular to the plane of the bending moment. The neutral line cuts off on the axes of the segments coordinates

Imagine the moments of inertia as the work of the cross-section area on the square of the corresponding radius of inertia

Then the expressions (9.8) can be written as follows:

From formulas (9.8) it can be seen that the pole and the neutral line are always located along different directions from the center of secting, the position of the neutral line is determined by the coordinates of the pole.

When the pole approaches the power line to the severity center, the neutral line will be removed from the center, while remaining parallel to its original direction. In the limit with a neutral line will be removed in infinity. In this case, there will be a central stretching (compression) of the bar.

On the power line you can always find this position of the pole, at which the neutral line will touch the contour of the section, without crossing it anywhere. If you have all possible neutral lines so that they concern the section circuit, without crossing it, and find them the corresponding poles, it turns out that the poles will be located on a closed line quite defined for each cross section. The area bounded by this line is called the core of the section. In the circular section, for example, the kernel is a circle of diameter of 4 times smaller diameter of the section, and in rectangular and altar sections, the kernel has a parallelogram form (Fig. 9.5).

From the very construction of the cross section, it follows that as long as the pole is inside the nucleus, the neutral line will not cross the circuit of the section and the voltage throughout the cross section will be one sign. If, the pole is located outside the nucleus, the neutral line will cross the section circuit, and then in the section there will be voltages of different signs. This circumstance must be taken into account when calculating the subsistence compression of racks from fragile materials. Since fragile materials are poorly perceived by tensile loads, it is preferably the external forces to apply to the rack so that only compression voltages acted throughout the cross section. For this purpose, the point of the application of the equal external forces compressing the rack must be inside the cross section core.

The calculation of the strength during non-centrular stretching and compression is made in the same way as in the oblique bending - by voltage at a dangerous point of the cross section. The cross section is dangerous, the most remote from its neutral line. However, in cases where the compression voltage is acting at this point, and the rack material is fragile, the dangerous point in which the largest tensile voltage is acting.

The stress plot is built on the axis perpendicular to the neutral cross section line, and is limited to the straight line (see Fig. 9.4).

The condition of strength will be recorded so.

Essentrated compression. Buildingcore sections. Bending with mold. Calculations for strength with complex stress state.

Essentren with compression is The type of deformation, in which the longitudinal force in the cross section of the rod is not applied in the center of gravity. For entheccently compression, in addition to the longitudinal force (N), there are two bends (s).

It is believed that the rod has a large rigidity to bend to neglect the barb of the rod with an outcidently compression.

We transform the formula of the moments during a non-centcenary compression, substituting the values \u200b\u200bof bending moments :.

Denote the coordinates of a certain point of the zero line during an outcidentren compression, and substitute them in the formula of normal stresses during an off-centrular compression. Considering that the voltages at the zero line points are zero, after cutting on, we obtain the zero line equation during an off-centrular compression: .

The zero line with an off-centrular compression and the point of the load application is always located on different sides of the center of severity.

Segments that are cut off by the zero line from the axes of coordinate, designated and, easily found from the zero line equation during an off-centrular compression. If you first take and then take , I will find the intersection points of the zero line during an off-centrular compression with the main central axes:

The zero line during an off-centrular compression will split the cross section into two parts. In one part of the voltage will be compressing, in the other - tensile. Calculation of strength, as in the case of oblique bending, is carried out according to normal stresses arising from a dangerous point of cross-section (the most remote from the zero line).

The core of the section is a small area around the center of gravity of the cross section, characteristic of the fact that any compressive longitudinal force applied inside the nucleus causes compressive voltages in all points of the cross section.

Examples of the cross-section core for rectangular and round cross sections of the rod.

Bending with mold. Such loading (simultaneous action of torque and bending moments) are often susceptible to the shafts of machines and mechanisms. To calculate the bar, you must first install hazardous sections. For this, the plots of bending and torque are being built.

Using the principle of independence of the forces, we define the voltages arising in a timber separately for twist, and for bending.

When drying in transverse sections of the bar arise by tangent stresses, reaching the greatest value at the dots of the cross section circuit With bending in cross sections of a bar arise normal stresses reaching the greatest value in the extreme fibers of the bar .

Non-centreneous stretch or compression This type of the deformation of the rod is called, in which the longitudinal force and bending moments arise in its cross section (and perhaps transverse forces).

The longitudinal force and bending moments can be considered as the result of the impact on the stem of the extracently applied force (Fig. 25). That is why this type of complex resistance is called an outcidentren stretching or compression.

The bending moments are associated with the coordinates of the point of the application of force by relations therefore from (1), formulas (1) ch. 3 and the principle of independence of the action for normal stresses at an arbitrary point of any cross section with coordinates x, we get

Neutral axis with outcidentren stretching or compression. The equation of the neutral axis of the cross section, at the points of which the voltages are zero, has a view in this case

It is easy to see that the neutral axis does not pass through the severity center. The remaining properties are the same as during braid bending. In addition, we point out another property of a neutral axis with an off-centrular stretching or compression: the neutral axis does not cross the sections in which the force is applied

Core section. The position of the neutral axis, as can be seen from the equation (4), depends on the coordinates of the point of the application of the force if the point of the application of force is located quite close to the severity center, in the area called the core of the section, then the neutral axis passes outside the cross section, i.e. . All points of section are experiencing normal voltages of one sign. In fig. 26 shows the kernel for rectangular and circular sections.

Strength conditions for incantented tension or compression have the form of restrictions on maximum normal voltages.

Example. Calculate the maximum normal stresses in the cross section of an echocently compressed rod of rectangular cross section at (Fig. 27). The point to the application of force has coordinates (Fig. 27, b).

Decision. We calculate the geometric characteristics of the section:

The neutral axis equation (4) takes the view from its location (Fig. 27, b) it can be seen that in and c - the most stressful points