eccentric compression. Construction of the section kernel. Bending with twist. Calculations for strength under complex stress state.

Eccentric compression- this is a type of deformation in which the longitudinal force in the cross section of the rod is not applied at the center of gravity. With eccentric compression, in addition to the longitudinal force (N), there are two bending moments (M x and M y).

It is considered that the rod has a high bending rigidity in order to neglect the deflection of the rod under eccentric compression.

Let's transform the formula of moments for eccentric compression by substituting the values ​​of bending moments:

Let us denote the coordinates of a certain point of the neutral (zero) line under eccentric compression xN, yN and substitute them into the formula for normal stresses under eccentric compression. Considering that the stresses at the points of the neutral line are equal to zero, after the reduction by P/F, we obtain the equation of the neutral line under eccentric compression:

(35)

The zero line for eccentric compression and the point of application of the load are always located on opposite sides of the center of gravity of the section.

Rice. 43. Eccentric compression

The segments cut off by the zero line from the coordinate axes, denoted ax and ay, can be easily found from the zero line equation for eccentric compression. If we first take xN = 0, yN = ay, and then take yN = 0, xN = ax, then we find the points of intersection of the zero line under eccentric compression with the main central axes:

Rice. 44. Neutral line with eccentric tension - compression

The neutral line under eccentric compression will divide the cross section into two parts. In one part, the stresses will be compressive, in the other - tensile. The strength calculation, as in the case of oblique bending, is carried out according to the normal stresses that occur at the dangerous point of the cross section (the furthest from the zero line).

(36)

Section core - a small area around the center of gravity of the cross section, characterized by the fact that any compressive longitudinal force applied inside the core causes compressive stresses at all points of the cross section.

Examples of the section kernel for rectangular and circular bar cross sections.

Rice. 45. Section Kernel Shape for Rectangle and Circle

Bending with twist. Shafts of machines and mechanisms are often subject to such loading (simultaneous action of torques and bending moments). To calculate the beam, it is necessary first of all to establish dangerous sections. To do this, plots of bending and torque moments are built.

Using the principle of independence of the action of forces, we determine the stresses that arise in the beam separately for torsion and for bending.

During torsion, shear stresses arise in the cross sections of the beam, reaching the greatest value at the points of the contour of the section. When bending, normal stresses arise in the cross sections of the beam, reaching the greatest value in the extreme fibers of the beam.

Consider a straight rod loaded at the end with forces directed parallel to the axis Oh. The resultant of these forces F applied at the point FROM. In the local right-handed coordinate system yOz, coinciding with the main central axes of the section, the coordinates of the point FROM equal but And b(Fig. 5.18).

Let us replace the applied load with a statically equivalent system of forces and moments. To do this, we transfer the resultant force F to the center of gravity of the section ABOUT and load the rod with two bending moments equal to the product of the force T^ on its arms with respect to the coordinate axes: M ff = Fa And Mz = Fb.

Note that according to the rule of the right-handed coordinate system for the point C, which lies in the first quadrant, the bending moments will formally receive the following

Rice. 5.18.Straight rod loaded at the end with forces directed parallel to the axisOh

blowing signs: M y \u003d Fa and M 7 = -Fb. In this case, in the elementary area lying in the first quarter, both moments cause tensile stress.

Using the principle of independence of the action of forces, we determine the stresses at the current point of the section with coordinates at And z from each power factor separately. The total voltage is obtained by summing all three voltage components:

Let us determine the position of the neutral axis. To do this, in accordance with formula (5.69), we equate to zero the value of the normal stress at the current point:

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

where i y And i z - main radii of inertia determined by formulas (3.14).

Thus, in the case of eccentric tension-compression, the neutral line does not pass through the center of gravity of the section (Fig. 5.19), as indicated by the presence in equation (5.70) of a free term that differs from zero.

The maximum stresses occur at the points of the section BUT And IN, farthest from the neutral line. Let's establish the relationship between the coordinates of the force application point and the position of the neutral line. To do this, we determine the points of intersection of the coordinate axes by this line:

Rice. 5.19.

The resulting formulas show that the coordinate of the force application point but and the coordinate of the point of intersection of the neutral line of the coordinate axis Oz(point r 0) have opposite signs. The same can be said about the quantities b And at 0 . Thus, the point of application of the resultant force and the neutral line are on opposite sides of the origin.

According to the obtained formulas, when the point of force application approaches the center of gravity of the section, the neutral line moves away from the central zone. In the limiting case (a = b = 0) we arrive at the case of central tension-compression.

It is of interest to determine the zone of force application, in which the stresses in the section will have the same sign. In particular, for materials that poorly resist stretching, it is rational to apply a compressive force precisely in this zone, so that only compressive stresses act in the section. Such a zone around the center of gravity of the section is called section core.

If the force is applied at the core of the section, then the neutral line does not intersect the section. If a force is applied along the boundary of the section core, the neutral line touches the section contour. Formula (5.71) can be used to determine the core of the section.

If we represent the neutral line as a tangent to the contour of the section and consider all possible positions of the tangent and the points of application of the force corresponding to these positions, then the points of application of the force will outline the core of the section.

Rice. 5.20.

but - ellipse; 6 - rectangle

Eccentric tension this type of loading of a beam is called, in which external forces act along the longitudinal axis of the beam, but do not coincide with it (Fig. 8.4). Stresses are determined using the principle of independence of the action of forces. Eccentric tension is a combination of axial tension and oblique (in particular cases - flat) bending. The formula for normal stresses can be obtained as the algebraic sum of normal stresses arising from each type of loading:

where ; ;

y F , z F– coordinates of the force application point F.

To determine the dangerous points of the section, it is necessary to find the position of the neutral line (n.l.) as the locus of points at which the stresses are equal to zero.

.

Equation n.l. can be written as the equation of a straight line in segments:

,

where And are segments cut off by n.l. on the coordinate axes,

, are the main radii of inertia of the section.

The neutral line divides the cross section into zones with tensile and compressive stresses. The diagram of normal stresses is presented in fig. 8.4.

If the section is symmetrical about the main axes, then the strength condition is written for plastic materials, in which [ s c] = [s p] = [s], as

. (8.5)

For brittle materials with [ s c]¹[ s p], the strength condition should be recorded separately for the dangerous point of the section in the tension zone:

and for the dangerous point of the section in the compressed zone:

,

where z1, y 1 And z2, y2- the coordinates of the points of the section most distant from the neutral line in the stretched 1 and compressed 2 zones of the section (Fig. 8.4).

Zero line properties

1. The zero line divides the entire section into two zones - tension and compression.

2. The zero line is straight, since the x and y coordinates are in the first degree.

3. The zero line does not pass through the origin (Fig. 8.4).

4. If the point of force application lies on the main central inertia of the section, then the zero line corresponding to it is perpendicular to this axis and passes on the other side of the origin (Fig. 8.5).

5. If the point of application of the force moves along the ray emerging from the origin, then the zero line corresponding to it moves behind it (Fig. 8.6):

n.l

Rice. 8.5 Fig. 8.6

a) when the point of force application moves along the beam emanating from the origin from zero to infinity (y F ®∞, z F ®∞), but at ®0; but z ®0. The limit state of this case: the zero line will pass through the origin (bend);

b) when the point of application of the force (t. K) moves along the beam emanating from the origin from infinity to zero (y F ® 0 and z F ® 0), but y ®∞; but z ®∞. The limiting state of this case: the zero line is removed to infinity, and the body will experience a simple stretching (compression).

6. If the point of force application (point K) moves along a straight line intersecting the coordinate axes, then in this case the zero line will rotate around a certain center located in the opposite quadrant from point K.

8.2.3. Section kernel

Some materials (concrete, masonry) can absorb very small tensile stresses, while others (such as soil) cannot resist stretching at all. Such materials are used for the manufacture of structural elements in which tensile stresses do not occur, and are not used for the manufacture of instruction elements that experience bending, torsion, central and eccentric tension.

Only centrally compressed elements can be made from these materials, in which tensile stresses do not occur, as well as eccentrically compressed elements, if tensile stresses do not form in them. This occurs when the point of application of the compressive force is located inside or on the border of some central region of the cross section, called the core of the section.

Section kernel a beam is called its some central area, which has the property that the force applied at any of its points causes stresses of the same sign at all points of the cross section of the beam, i.e. the zero line does not pass through the section of the beam.

If the point of application of the compressive force is located outside the core of the section, then compressive and tensile stresses arise in the cross section. In this case, the zero line crosses the cross section of the beam.

If the force is applied at the boundary of the core of the section, then the zero line touches the contour of the section (at a point or along a line); at the point of contact, the normal stresses are equal to zero.

When calculating eccentrically compressed rods made from a material that poorly perceives tensile stresses, it is important to know the shape and dimensions of the section core. This allows, without calculating stresses, to establish whether tensile stresses arise in the cross section of the beam (Fig. 8.7).

It follows from the definition that the kernel of a section is some area that is inside the section itself.

For brittle materials, a compressive load should be applied in the core of the section in order to exclude tensile zones in the section (Fig. 8.7).

To construct the core of the section, it is necessary to sequentially combine the zero line with the contour of the cross section so that the zero line does not intersect the section, and at the same time calculate the corresponding point

application of compressive force K with

Rice. 8.7 dinami y F And z F according to the formulas:

; .

The resulting points of force application with coordinates y F , z F must be connected by straight lines. The area bounded by the resulting polyline will be the core of the section.

The sequence of constructing the section kernel

1. Determine the position of the center of gravity of the cross section and the main central axes of inertia y and z, as well as the values ​​of the squared radii of inertia i y , i z .

2. Show all possible positions of the n.l. relating to the contour of the section.

3. For each position of n.l. define segments a y And a z, cut off by it from the main central axes of inertia y and z.

4. For each position of n.l. set the coordinates of the center of pressure y F, And z F .

5. The obtained centers of pressure are connected by line segments, inside which the core of the section will be located.

Torsion with bend

The type of loading in which the bar is subjected to the action of twisting and bending moments at the same time is called bending with torsion.

When calculating, we use the principle of independence of the action of forces. Let's determine the stresses separately during bending and torsion (Fig. 8.8) .

When bending in the cross section, normal stresses arise, reaching a maximum value in the outermost fibers

.

During torsion in the cross section, shear stresses arise, reaching the highest value at the points of the section near the shaft surface

.

s

t

C

B

x

y

z

Rice. 8.9

s

s

t

t

Rice. 8.10

C

x

z

y

M

T

Rice. 8.8

Normal and shear stresses simultaneously reach their maximum value at the points FROM And IN shaft section (Fig. 8.9). Consider the stress state at the point FROM(Fig. 8.10). It can be seen that the elementary parallelepiped selected around the point FROM, is in a plane stress state.

Therefore, to test the strength, we apply one of the strength hypotheses.

Strength condition according to the third strength hypothesis (the hypothesis of the largest shear stresses)

.

Given that , , we obtain the condition of the shaft strength

. (8.6)

If the shaft bends in two planes, then the strength condition will be

.

Using the fourth (energy) strength hypothesis

,

after substitution s And t we get

. (8.7)

Questions for self-examination

1. What kind of bend is called oblique?

2. What combination of types of bend is an oblique bend?

3. What formulas are used to determine the normal stresses in the cross sections of a beam during oblique bending?

4. How is the position of the neutral axis in oblique bending?

5. How are dangerous points determined in a section with oblique bending?

6. How are the displacements of the beam axis points determined during oblique bending?

7. What kind of complex resistance is called eccentric tension (or compression)?

8. What formulas are used to determine the normal stresses in the cross sections of the rod during eccentric tension and compression? What is the form of the diagram of these stresses?

9. How is the position of the neutral axis determined in eccentric tension and compression? Write down the corresponding formulas.

10. What stresses occur in the cross section of the beam when bending with torsion?

11. How are the dangerous sections of a round beam in bending with torsion?

12. Which points of a circular cross section are dangerous when bending with torsion?

13. What stress state occurs at these points?

Many elements of building structures (columns, racks, supports) are under the influence of compressive forces applied not in the center of gravity of the section. On fig. 12.9 shows the column on which the floor beam rests. As you can see, the force acts with respect to the axis of the column with an eccentricity e, and thus, in an arbitrary section ah columns along with longitudinal force N = -R there is a bending moment, the magnitude of which is equal to Re. Eccentric tension (compression) of a rod is a type of deformation in which the resultant of external forces act along a straight line parallel to the axis of the rod. In what follows, we will mainly consider problems of eccentric compression. In case of eccentric tension, in all the calculation formulas given, the sign in front of the force should be changed R to the opposite.

Let a rod of arbitrary cross section (Fig. 12.10) be loaded at the end with an eccentrically applied compressive force R, directed parallel to the axis Oh. Accept positive

directions of the main axes of inertia of the section OU And Oz so that the point of application of force R was in the first quarter of the coordinate axes. We denote the coordinates of the point of application of force R across y r And z P -

Internal forces in an arbitrary section of the rod are equal to

The minus signs of the bending moments are due to the fact that in the first quarter of the coordinate axes these moments cause compression. The values ​​of internal forces in this example do not change along the length of the rod, and thus, the distribution of stresses in sections sufficiently remote from the place of application of the load will be the same.

Substituting (12.11) into (12.1), we obtain the formula for normal stresses under eccentric compression:

This formula can be converted to the form

where i , i- main radii of inertia of the section. Wherein

Putting in (12.12) o = 0, we obtain the equation zero line:

Here at 0 and z 0 - coordinates of points of the zero line (Fig. 12.11). Equation (12.14) is the equation of a straight line that does not pass through the center of gravity of the section. To draw a zero line, we find the points of its intersection with the coordinate axes. Assuming in (12.14) successively y 0 = 0 and z0= 0, respectively, we find

where a z And and y- segments cut off by the zero line on the coordinate axes (Fig. 12.11).

Let us establish the features of the position of the zero line under eccentric compression.

• 1. It follows from formulas (12.15) that and at And a z have opposite signs respectively y r And z P - Thus, the zero line passes through those quarters of the coordinate axes that do not contain the point of application of the force (Fig. 12.12).
• 2. As the point of application of force approaches R in a straight line to the center of gravity of the section, the coordinates of this point y r And z P decrease. From (12.15) it follows that in this case the absolute values ​​of the lengths of the segments and at And a z increase, that is, the zero line moves away from the center of gravity, remaining parallel to itself (Fig. 12.13). In the limit at Z P = y P = 0 (force is applied at the center of gravity) the zero line is removed to infinity. In this case, the stresses in the cross section will be constant and equal to o = -P/F.
• 3. If the point of application of force R located on one of the main axes, the zero line is parallel to the other axis. Indeed, putting in (12.15), for example, y r= 0, we get that and at= that is, the zero line does not cross the axis OU(Fig. 12.14).
• 4. If the point of application of the force moves along a straight line that does not pass through the center of gravity, then the zero line rotates around a certain point. Let's prove this property. Points of application of forces R x And R 2, located on the coordinate axes correspond to the zero lines 1 - 1 and 2-2, parallel to the axes (Fig. 12.15), which intersect at the point D. Since this point belongs to two zero lines, the stresses at this point from simultaneously applied forces R x And R 2 will be equal to zero. Since any force R 3 , the point of application of which is located on a straight line R ( R 2 , can

decompose into two parallel components applied at the points Pj and R 2, then it follows that the stresses at the point D from the force R 3 are also equal to zero. Thus, the zero line 3-3, corresponding to the strength R 3 , passes through a point D.

In other words, a set of points R, located on a straight line R ( R 2 , corresponds to a pencil of lines passing through a point D. The converse statement is also true: when the zero line rotates around a certain point, the point of application of the force moves along a straight line that does not pass through the center of gravity.

If the zero line crosses the section, then it divides it into zones of compression and tension. As in oblique bending, it follows from the flat section hypothesis that the stresses reach their greatest values ​​at the points farthest from the zero line. The nature of the stress diagram in this case is shown in Fig. 12.16, but.

If the zero line is located outside the section, then at all points of the section the stresses will be of the same sign (Fig. 12.16, b).

Example 12.3. Let us construct a diagram of normal stresses in an arbitrary section of an eccentrically compressed rectangular column with dimensions b X h(Fig. 12.17). The squares of the radii of inertia of the section according to (12.22) are

The segments cut off by the zero line on the coordinate axes are determined by the formulas (12.15):

Substituting sequentially into (12.12) the coordinates of the points C and the most distant from the zero line IN(Fig. 12.18)

find

Plot o is shown in fig. 12.18. The highest compressive stresses in absolute value are four times higher than the stresses that would be in the case of a central application of force. In addition, significant tensile stresses appeared in the cross section. Note that from (12.12) it follows that at the center of gravity (y = z\u003d 0) the stresses are equal o \u003d -P/F.

Example 12.4. Cutout strip loaded with tensile force R(Fig. 12.19, but). Compare the stresses in the section lv, far enough from the end and the place of the cut, with stresses in the section CD at the cutout.

in section AB(Fig. 12.19, b) strength R causes central tension and the stresses are a = P/F = P/bh.

in section CD(Fig. 12.19, in) line of force R does not pass through the center of gravity of the section, and therefore eccentric tension occurs. By changing the sign in formula (12.12) to the opposite and taking y r= 0, we obtain for this section

Taking

Zero line in section CD parallel to axis OU and crosses the axis Oz on distance a =-i 2 y /z P- b/ 12. At the points of the section furthest from the zero line C(z - -b/ 4) and D(z - b/ 4) stresses according to (12.16) are equal

Diagrams of normal stresses for sections LV And CD shown in fig. 12.19, b, c.

Thus, despite the fact that the cross section CD has an area two times smaller than the cross section AB, due to the eccentric application of the force, the tensile stresses in the weakened section increase not by a factor of two, but by a factor of eight. In addition, significant compressive stresses appear in this section.

It should be noted that the above calculation does not take into account additional local stresses that occur near point C due to the presence of a recess. These stresses depend on the radius of the undercut (they increase with decreasing radius) and can significantly exceed the value found a c = 8P/bh. In this case, the nature of the stress diagram near point C will differ significantly from the linear one. The definition of local stresses (stress concentration) is discussed in Chapter 18.

Many building materials (concrete, brickwork, etc.) do not resist stretching well. Their tensile strength is many times less than that of compression. Therefore, the appearance of tensile stresses in structural elements made of such materials is undesirable. For this condition to be satisfied, it is necessary that the zero line is outside the section. Otherwise, the zero line will cross the section and tensile stresses will appear in it. If the zero line is tangent to the section contour, then the corresponding position of the force application point is the limit. In accordance with property 2 of the zero line, if the point of application of force approaches the center of gravity of the section, the zero line will move away from it. The locus of limit points corresponding to different tangents to the section contour is the boundary section kernels. The core of the section is a convex area around the center of gravity, which has the following property: if the point of application of the force is inside or on the border of this area, then at all points of the section the stresses have the same sign. The core of the section is a convex figure, since the zero lines must touch the envelope of the section contour and not cross it.

Through the dot BUT(Fig. 12.20) you can draw an infinite number of tangents (zero lines); while only tangent AC is tangent to the envelope, and a certain point of the section core contour must correspond to it. At the same time, for example, it is impossible to draw a tangent to the segment AB section contour because it intersects the section.

Let's build a section kernel for a rectangle (Fig. 12.21). For tangent 1 - 1 a 7 - b/ 2; but= . From (12.15) we find for point 1 corresponding to this tangent, z P \u003d -i 2 y / a 7 \u003d -b / 6; y r - 0. For tangent 2-2 and y - k / 2; a 7 \u003d ° °, and the coordinates of point 2 will be equal atR- -h/6; z P - 0. According to property 4 of the zero line, the points of force application corresponding to different tangents to the lower right corner point of the section are located on the straight line 1-2. The position of points 3 and 4 is determined from the symmetry conditions. Thus, the section kernel for a rectangle is a rhombus with diagonals b/3 and FROM.

To build a section kernel for a circle, it is enough to draw one tangent (Fig. 12.22). Wherein a = R; but= °o.

"U U ^^

Considering that for a circle i y - J y /F - R / 4, from (12.15) we get

Thus, the section kernel for a circle is a circle with a radius R/4.

On fig. 12.23, a, 6 section cores for an I-beam and a channel are shown. The presence of four corner points of the core of the section in each of these examples is due to the fact that the envelope of the contour for both the I-beam and the channel is a rectangle.

eccentric compression. Building section kernels. Bending with twist. Calculations for strength under complex stress state.

Off-center compression is a type of deformation in which the longitudinal force in the cross section of the rod is not applied at the center of gravity. At eccentric compression, in addition to the longitudinal force (N), there are two bending moments ( and ).

It is considered that the rod has a high bending rigidity in order to neglect the deflection of the rod under eccentric compression.

Let's transform the formula of moments for eccentric compression , substituting the values ​​of bending moments: .

Let us denote the coordinates of a certain point of the zero line under eccentric compression , and substitute them into the formula for normal stresses under eccentric compression. Considering that the stresses at the points of the zero line are equal to zero, after the reduction by , we obtain the equation of the zero line for eccentric compression: .

The zero line for eccentric compression and the point of application of the load are always located on opposite sides of the center of gravity of the section.

The segments cut off by the zero line from the coordinate axes, denoted by and , can be easily found from the zero line equation for eccentric compression. If we first accept and then accept , then we find the points of intersection of the zero line under eccentric compression with the principal central axes:

Zero line under eccentric compression will divide the cross section into two parts. In one part, the stresses will be compressive, in the other - tensile. The strength calculation, as in the case of oblique bending, is carried out according to the normal stresses that occur at the dangerous point of the cross section (the furthest from the zero line).

Section core - a small area around the center of gravity of the cross section, characterized by the fact that any compressive longitudinal force applied inside the core causes compressive stresses at all points of the cross section.

Examples of the section kernel for rectangular and circular bar cross sections.

Bending with twist. Shafts of machines and mechanisms are often subject to such loading (simultaneous action of torques and bending moments). To calculate the beam, it is necessary first of all to establish dangerous sections. To do this, plots of bending and torque moments are built.

Using the principle of independence of the action of forces, we determine the stresses that arise in the beam separately for torsion and for bending.

During torsion in the cross sections of the beam, shear stresses arise, reaching the highest value at the points of the section contour When bending in the cross sections of the beam, normal stresses arise, reaching the highest value in the extreme fibers of the beam .