Stability of compressed rods critical stress Euler formula. Euler's Formula for Critical Force

Lecture 7

STABILITY OF COMPRESSED RODS

The concept of the stability of a compressed rod. Euler formula. The dependence of the critical force on the method of fixing the rod. Limits of applicability of the Euler formula. Yasinsky formula. Sustainability calculation.

The concept of the stability of a compressed rod

Let us consider a rod with a straight axis loaded with a longitudinal compressive force F. Depending on the magnitude of the force and the parameters of the rod (material, length, shape and dimensions of the cross section), its rectilinear equilibrium shape may be stable or unstable.

To determine the type of equilibrium of the rod, let us act on it with a small transverse load Q. As a result, the rod will move to a new equilibrium position with a curved axis. If, after the termination of the transverse load, the rod returns to its original (rectilinear) position, then the rectilinear form of equilibrium is stable (Figure 7.1a). In the case when, after the termination of the action of the transverse force Q, the rod does not return to its original position, the rectilinear form of equilibrium is unstable (Fig. 7.1b).

Thus, stability is the ability of the rod, after some deviation from its original position as a result of the action of some disturbing load, to spontaneously return to its original position when this load is terminated. The smallest longitudinal compressive force at which the rectilinear equilibrium shape of the rod becomes unstable is called the critical force.

The considered scheme of operation of the central compressed rod is theoretical. In practice, the compressive force may act with some eccentricity, and the rod may have some (albeit small) initial curvature. Therefore, from the very beginning of the longitudinal loading of the rod, its bending is observed. Research shows that as long as the compressive force is less than the critical force, the bar deflections will be small. When the force approaches the critical value, the deflections begin to increase indefinitely. This criterion (an unlimited increase in deflections with a limited increase in compressive force) is taken as the criterion for buckling.

The loss of stability of elastic equilibrium occurs not only during the compression of the rod, but also during its torsion, bending, and more complex types of deformation.

Euler formula

Consider a rod with a straight axis, fixed by means of two hinged supports (Fig. 7.2). Let us assume that the longitudinal compressive force acting on the rod has reached a critical value, and the rod is bent in the plane of least rigidity. The plane of least rigidity is located perpendicular to that main central axis of the section, relative to which the axial moment of inertia of the section has a minimum value.

(7.1)

where M is the bending moment; I min is the minimum moment of inertia of the section.

From fig. 7.2 find the bending moment

(7.2)

On fig. 7.2 the bending moment due to the action of the critical force is positive, and the deflection is negative. In order to agree on the accepted signs, a minus sign is put in dependence (7.2).

Substituting (7.2) into (7.1), to determine the deflection function, we obtain the differential equation

(7.3)

(7.4)

From the course of higher mathematics it is known that the solution of equation (7.3) has the form

where A, B are integration constants.

To determine the constants of integration in (7.5), we use the boundary conditions

For a bent rod, the coefficients A and B cannot be equal to zero at the same time (otherwise the rod will not be bent). So

Equating (7.6) and (7.4), we find

(7.7)

Of practical importance is the smallest non-zero value of the critical force. Therefore, substituting n=1 into (7.7), we finally have

(7.8)

Dependence (7.8) is called the Euler formula.

Critical force dependence

from the method of fixing the rod

Formula (7.8) was obtained for the case of a rod being fixed by means of two hinged supports located at its edges. For other methods of fixing the rod, the generalized Euler formula is used to determine the critical force

(7.9)

where μ is the length reduction factor, taking into account the method of fixing the rod.

The most common ways of fixing the rod and the corresponding length reduction coefficients are shown in fig. 7.3.

Limits of applicability of the Euler formula. Yasinsky's formula

P When deriving the Euler formula, the condition was used that Hooke's law is satisfied at the moment of loss of stability. The stress in the rod at the moment of buckling is equal to

where
- rod flexibility; A is the cross-sectional area of ​​the rod.

At the moment of loss of stability, Hooke's law will be satisfied under the condition

where σpc is the limit of proportionality of the rod material;
- the first ultimate flexibility of the rod. For steel St3 λ pr1 = 100.

Thus, the Euler formula is valid when condition (7.10) is satisfied.

If the flexibility of the rod is in the interval
then the rod will lose stability in the area of ​​elastic-plastic deformations and the Euler formula cannot be used. In this case, the critical force is determined by the experimental formula of Yasinsky

where a, b are experimental coefficients. For steel St3 a = 310 MPa, b = 1.14 MPa.

The second ultimate flexibility of the rod is determined by the formula

where σ t is the yield strength of the rod material. For steel St3 λ pr2 = 60.

When the condition λ ≤ λ pr2 is met, the critical stress (according to Yasinsky) will exceed the yield strength of the rod material. Therefore, in this case, to determine the critical force, the relation is used

(7.12)

AT as an example in Fig. 7.4 shows the dependence of the critical stress on the flexibility of the rod for steel St3.

Sustainability calculation

Stability analysis is performed using the stability condition

(7.13)

Permissible stress when calculating stability;

- stability factor.

The allowable stress in the stability analysis is found from the allowable stress in the compression analysis

(7.14)

where φ is the coefficient of buckling (or reduction of the main allowable stress). This coefficient varies within 0 ≤ φ ≤ 1.

Considering that for plastic materials

formulas (7.13) and (7.14) imply

(7.15)

The values ​​of the coefficient of buckling depending on the material and flexibility of the rod are given in the reference literature.

The most interesting is the design calculation from the stability condition. With this type of calculation, the following are known: the calculation scheme (coefficient μ), external compressive force F, material (permissible stress [σ]) and length l of the rod, the shape of its cross section. It is necessary to determine the dimensions of the cross section.

The difficulty lies in the fact that it is not known by which formula to determine the critical stress, because. without cross-sectional dimensions, it is impossible to determine the flexibility of the bar. Therefore, the calculation is performed by the method of successive approximations:

1) We accept the initial value = 0.5. Determine the cross-sectional area

2) By area we find the dimensions of the cross section.

3) Using the obtained cross-sectional dimensions, we calculate the flexibility of the rod, and by flexibility - the final value of the buckling coefficient .

4) If the values ​​do not match and perform the second approximation. The initial value of φ in the second approximation is taken equal to
. Etc.

We repeat the calculations until the initial and final values ​​of the coefficient φ differ by no more than 5%. As an answer, we accept the values ​​of the dimensions obtained in the last approximation.

To find the critical stresses, it is necessary to calculate the critical force, i.e., the smallest axial compressive force that can keep a slightly curved compressed rod in balance.

This problem was first solved by Academician of the St. Petersburg Academy of Sciences L. Euler in 1744.

Note that the very formulation of the problem is different than in all previously considered sections of the course. If earlier we determined the deformation of the rod under given external loads, then here we pose the inverse problem: given the curvature of the axis of the compressed rod, it is necessary to determine at what value of the axial compressive force R such distortion is possible.

Consider a straight rod of constant cross section, hinged at the ends; one of the supports allows the possibility of longitudinal movement of the corresponding end of the rod (Fig. 3). We neglect the self-weight of the rod.

Fig.3. Calculation scheme in the "Euler problem"

We load the rod with centrally applied longitudinal compressive forces and give it a very slight curvature in the plane of least rigidity; the rod is held in a bent state, which is possible because .

The bending deformation of the rod is assumed to be very small, therefore, to solve the problem, we can use the approximate differential equation for the bent axis of the rod. Selecting the origin of coordinates at a point BUT and the direction of the coordinate axes, as shown in Fig. 3, we have:

(1)

Take a section at a distance X from the origin; the ordinate of the curved axis in this section will be at, and the bending moment is

According to the original scheme, the bending moment turns out to be negative, while the ordinates for the chosen direction of the axis at turn out to be positive. (If the rod were curved with a bulge downwards, then the moment would be positive, and at- negative and .)

The differential equation just given takes the form:

dividing both sides of the equation by EJ and denoting the fraction through we bring it to the form:

The general integral of this equation has the form:

This solution contains three unknowns: constants of integration a and b and value , since the magnitude of the critical force is unknown to us.

The boundary conditions at the ends of the rod give two equations:

at point A at x = 0 deflection at = 0,

AT X= 1 at = 0.

It follows from the first condition (since cos kx =1)

So the bent axis is a sinusoid with the equation

(2)

Applying the second condition, we substitute into this equation

at= 0 and X = l

we get:

It follows from this that either a or kl are equal to zero.

If a a is equal to zero, then from equation (2) it follows that the deflection in any section of the rod is equal to zero, i.e., the rod remained straight. This contradicts the initial premises of our conclusion. Therefore sin kl= 0, and the value can have the following infinite series of values:

where is any integer.

Hence, and since then

In other words, the load that can keep a slightly curved rod in balance can theoretically have a number of values. But since it is sought, and it is interesting from a practical point of view, the smallest value of the axial compressive force at which buckling becomes possible, it should be taken.

The first root =0 requires that it be equal to zero, which does not correspond to the initial data of the problem; so this root must be discarded and the value taken as the smallest root. Then we get the expression for the critical force:

Thus, the more inflection points the sinusoidally curved axis of the rod has, the greater the critical force should be. More complete studies show that the forms of equilibrium defined by formulas (1) are unstable; they pass into stable forms only in the presence of intermediate supports at the points AT and With(Fig. 1).

Fig.1

Thus, the task is solved; for our rod, the smallest critical force is determined by the formula

and the curved axis represents a sinusoid

The value of the constant of integration a remained undefined; its physical meaning will be found out if we put in the sinusoid equation; then (i.e., in the middle of the length of the rod) will receive the value:

Means, a- this is the deflection of the rod in the section in the middle of its length. Since at the critical value of the force R the equilibrium of a curved rod is possible with various deviations from its rectilinear shape, if only these deviations were small, then it is natural that the deflection f remained undefined.

At the same time, it must be so small that we have the right to use the approximate differential equation of the curved axis, i.e., so that it is still small compared to unity.

Having obtained the value of the critical force, we can immediately find the value of the critical stress by dividing the force by the cross-sectional area of ​​​​the rod F; since the value of the critical force was determined from the consideration of the deformations of the rod, on which local weakening of the cross-sectional area has an extremely weak effect, then the formula for includes the moment of inertia, therefore, it is customary when calculating the critical stresses, as well as when compiling the stability condition, to enter into the calculation the full, and not the weakened, cross-sectional area of ​​the rod. Then it will be equal

Thus, if the area of ​​a compressed rod with such flexibility was selected only according to the strength condition, then the rod would collapse from the loss of stability of a rectilinear shape.

For the first time, the problem of the stability of compressed rods was posed. Euler derived a calculation formula for the critical force and showed that its value essentially depends on the method of fixing the rod. The idea of ​​the Euler method is to establish the conditions under which, in addition to the rectilinear one, an adjacent (i.e., arbitrarily close to the original) curvilinear equilibrium form of the rod under a constant load is also possible.

Let us assume that a straight rod hinged at the ends, compressed by a force P= Pk, was brought out of rectilinear equilibrium by some horizontal force and remained bent after the removal of the horizontal force (Fig. 13.4). If the deflections of the rod are small, then the approximate differential equation of its axis will have the same form as in the case of transverse bending of the beam:

Combining the origin of coordinates with the center of the lower section, we direct the axis at towards the deflections of the rod, and the axis X- along the axis of the rod.

In the theory of buckling, it is customary to consider the compressive force to be positive. Therefore, determining the bending moment in the current section of the considered rod, we obtain

But, as follows from Fig. 13.4, with the selected direction of the axes at // <0, поэтому знаки левой и правой частей уравнения (17.2) будут одинаковыми, если в правой части сохранить знак минус. Если изменить направление оси at to the opposite, then the signs will change simultaneously at and at// and the minus sign on the right side of equation (13.2) will remain.

Therefore, the equation of the elastic line of the rod has the form

.

Assuming α 2 =Rk/EI, we obtain a linear homogeneous differential equation

,

whose general integral

Here A and B- constants of integration, determined from the conditions of fixing the rod, the so-called boundary or boundary conditions.

The horizontal displacement of the lower end of the rod, as seen from Fig. 13.4, is equal to zero, i.e. when X=0 deflection at=0. This condition will be met if B=0. Therefore, the bent axis of the rod is a sinusoid

.

The horizontal displacement of the top end of the bar is also zero, so

.

Constant A, which is the maximum deflection of the rod, cannot be equal to zero, since when A=0, only a rectilinear form of equilibrium is possible, and we are looking for a condition under which a curvilinear form of equilibrium is also possible. Therefore it must be sinα l=0. It follows that curvilinear equilibrium forms of the rod can exist if α l takes values π ,2π ,.nπ . Value α l cannot be equal to zero, since this solution corresponds to the case

Equating α l= nπ and substituting

we get

.

Expression (13.5) is called the Euler formula. It can be used to calculate the critical force Rk when the rod buckles in one of its two main planes, since only under this condition is equation (13.2) valid, and hence formula (13.5).

The buckling of the rod occurs in the direction of the least rigidity, if there are no special devices that prevent the rod from bending in this direction. Therefore, in the Euler formula it is necessary to substitute Imin- the smallest of the main central moments of inertia of the cross section of the rod.

The value of the greatest deflection of the rod A in the given solution remains undefined, it is taken arbitrary, but it is assumed to be small.

The value of the critical force, determined by formula (13.5), depends on the coefficient n. Let us find out the geometric meaning of this coefficient.

Above, we established that the bent axis of the rod is a sinusoid, the equation of which, after substitution α =π n/l into expression (13.4) takes the form

.

Sinusoids for n=1, n=2 are shown in fig. 13.5. It is easy to see that the value n represents the number of half-waves of the sinusoid along which the rod will bend. Obviously, the rod will always bend according to the smallest number of half-waves allowed by its supporting devices, since according to (13.5) the smallest n corresponds to the smallest critical force. Only this first critical force has a real physical meaning.

For example, a rod with hinged ends will bend as soon as the smallest value of the critical force is reached, corresponding to n=1, since the support devices of this rod allow it to bend along one half-wave of a sinusoid. Critical forces corresponding n=2, n\u003d 3, and more, can only be achieved if there are intermediate supports (Fig. 13.6). For a rod with hinged end supports without intermediate fastenings, the first critical force has a real meaning

.

Formula (13.5), as follows from its derivation, is valid not only for a rod with hinged ends, but also for any rod that bends during buckling along an integer number of half-waves. Let us apply this formula, for example, when determining the critical force for a rod, the support devices of which allow only longitudinal displacements of its ends (stand with embedded ends). As can be seen from figure 13.7, the number of half-waves of the curved axis in this case n=2 and, consequently, the critical force for the rod with given support devices

.

Let us assume that a rack with one pinched and the other free end (Fig. 13.8) is compressed by a force R.

If strength P= Pk, then in addition to the rectilinear one, there can also exist a curvilinear form of the balance of the rack (dotted line in Fig. 13.8).

The differential equation of the bent axis of the rack in the one shown in fig. 13.8 the system of coordinate axes has the same form.

The general solution to this equation is:

Subordinating this solution to the obvious boundary conditions: y=0 at x=0 and y/ =0 at x= l, we get B=0, Aα cosα l= 0.

We assumed that the post is curved, so the value A cannot be equal to zero. Hence, cosα l= 0. The smallest non-zero root of this equation α l= π /2 defines the first critical force

,

which corresponds to the bending of the rod along the sinusoid

.

Values α l=3π /2, α l=5π /2, etc., as shown above, correspond to large values Pk and more complex forms of the curved axis of the rack, which can practically exist only in the presence of intermediate supports.

As a second example, consider a rack with one pinched and a second hinged end (Fig. 13.9). Due to the curvature of the axis of the rod at P= Pk from the side of the hinged support, a horizontal reactive force arises R. Therefore, the bending moment in the current section of the rod

.α :

The smallest root of this equation determines the first critical force. This equation is solved by the selection method. It is easy to believe that the smallest non-zero root of this equation α l= 4.493=1.43 π .

Taking α l= 1.43 π , we obtain the following expression for the critical force:

Here μ =1/n- the reciprocal of the number of half-waves n sinusoid along which the rod will bend. Constant μ is called the length reduction factor, and the product μ l- reduced length of the rod. The reduced length is the half-wave length of the sinusoid along which this rod is bent.

The case of hinged fastening of the ends of the rod is called the main case. It follows from the above that the critical force for any case of fixing the rod can be calculated by the formula for the main case when the actual length of the rod is replaced in it by its reduced length μ l.

Reduction coefficients μ for some racks are given in fig. 17.10.

The concept of stability and critical power. Design and verification calculations.

In structures and structures, parts that are relatively long and thin rods, in which one or two cross-sectional dimensions are small compared to the length of the rod, are of great use. The behavior of such rods under the action of an axial compressive load turns out to be fundamentally different than when short rods are compressed: when the compressive force F reaches a certain critical value equal to Fcr, the rectilinear shape of the equilibrium of a long rod turns out to be unstable, and when Fcr is exceeded, the rod begins to intensively bend (bulge). In this case, a new (momentary) equilibrium state of the elastic long becomes some new already curvilinear form. This phenomenon is called stability loss.

Rice. 37. Loss of stability

Stability - the ability of a body to maintain a position or shape of balance under external influences.

Critical force (Fcr) - load, the excess of which causes loss of stability of the original shape (position) of the body. Stability condition:

Fmax ≤ Fcr, (25)

Stability of a compressed rod. Euler problem.

When determining the critical force causing the buckling of a compressed rod, it is assumed that the rod is perfectly straight and the force F is applied strictly centrally. The problem of the critical load of a compressed rod, taking into account the possibility of the existence of two forms of equilibrium at the same value of the force, was solved by L. Euler in 1744.

Rice. 38. Compressed rod

Consider a rod pivotally supported at the ends, compressed by a longitudinal force F. Suppose that for some reason the rod received a small curvature of the axis, as a result of which a bending moment M appeared in it:

where y is the deflection of the rod in an arbitrary section with the x coordinate.

To determine the critical force, you can use the approximate differential equation of an elastic line:

(26)

Having carried out the transformations, it can be seen that the critical force will take on a minimum value at n = 1 (one half-wave of the sinusoid fits along the length of the rod) and J = Jmin (the rod is bent about the axis with the smallest moment of inertia)

(27)

This expression is Euler's formula.

Dependence of the critical force on the conditions for fixing the rod.

Euler's formula was obtained for the so-called basic case - assuming the hinged support of the rod at the ends. In practice, there are other cases of fastening the rod. In this case, one can obtain a formula for determining the critical force for each of these cases by solving, as in the previous paragraph, the differential equation of the bent axis of the beam with the appropriate boundary conditions. But you can use a simpler technique, if you remember that, in the event of loss of stability, one half-wave of a sinusoid should fit along the length of the rod.

Let us consider some characteristic cases of fastening the rod at the ends and obtain a general formula for various types of fastening.

Rice. 39. Various cases of fastening the rod

Euler's general formula:

(28)

where μ·l = l pr - reduced length of the rod; l is the actual length of the rod; μ is the coefficient of the reduced length, showing how many times it is necessary to change the length of the rod so that the critical force for this rod becomes equal to the critical force for the hinged beam. (Another interpretation of the reduced length coefficient: μ shows on which part of the length of the rod for a given type of fastening one half-wave of the sinusoid fits in the event of buckling.)

Thus, the final stability condition takes the form

(29)

Let us consider two types of calculation for the stability of compressed rods - verification and design.

Check calculation

The stability check procedure looks like this:

Based on the known dimensions and shape of the cross section and the conditions for fixing the rod, we calculate the flexibility;

According to the reference table, we find the reduction factor for the allowable stress, then we determine the allowable stress for stability;

Compare the maximum stress with the allowable stability stress.

Design calculation

In the design calculation (to select a section for a given load), there are two unknown quantities in the calculation formula - the desired cross-sectional area A and the unknown coefficient φ (since φ depends on the flexibility of the rod, and hence on the unknown area A). Therefore, when selecting a section, it is usually necessary to use the method of successive approximations:

Usually, in the first attempt, φ 1 \u003d 0.5 ... 0.6 is taken and the cross-sectional area is determined in the first approximation

According to the found area A1, the section is selected and the flexibility of the rod is calculated in the first approximation λ1. Knowing λ, find a new value φ′1;

The choice of material and the rational shape of the section.

Material selection. Since only Young's modulus is included in the Euler formula of all mechanical characteristics, it is not advisable to use high-strength materials to increase the stability of highly flexible rods, since Young's modulus is approximately the same for all steel grades.

For rods of low flexibility, the use of high-grade steels is justified, since with an increase in the yield strength of such steels, critical stresses increase, and hence the stability margin.

Irkutsk State Transport University

Lab #16

by discipline "Strength of materials"

EXPERIMENTAL DETERMINATION OF CRITICAL FORCES

FOR LONGITUDINAL BENDING

Department of PM

Lab #16

Experimental determination of critical forces in buckling

Objective: study of the phenomenon of buckling of a compressed steel rod in an elastic

stages. Experimental determination of the values ​​of critical loads of compressed

rods with various methods of fastening and comparing them with theoretical

values.

General provisions

Compressed rods are not enough to test for strength according to the well-known condition:

,

where [σ] is the allowable stress for the rod material, P - compressive force F - cross-sectional area.

In practice, engineers deal with flexible rods subjected to compression, thin compressed plates, thin-walled structures, the failure of which is caused not by loss of bearing capacity, but by loss of stability.

Loss of stability is understood as the loss of the original form of equilibrium.

The resistance of materials considers the stability of structural elements working in compression.

Consider a long thin rod (Fig. 1) loaded with an axial compressive force P .

P< P kr P > P kr

Rice. one. Rod loaded with axial compressive force P .

For small values ​​of force F the rod is compressed while remaining straight. Moreover, if the rod is deflected from this position by a small transverse load, then it will bend, but when it is removed, the rod returns to a rectilinear state. This means that for a given force P the rectilinear form of equilibrium of the rod is stable.

If we continue to increase the compressive force P , then at a certain value, the rectilinear form of equilibrium becomes unstable and a new form of equilibrium of the rod arises - curvilinear (Fig. 1, b) . Due to the bending of the rod, a bending moment will appear in its sections, which will cause additional stresses, and the rod may suddenly collapse.

The curvature of a long rod compressed by a longitudinal force is called buckling .

The greatest value of the compressive force at which the rectilinear form of equilibrium of the rod is stable is called critical - P kr.

When the critical load is reached, there is a sharp qualitative change in the original form of equilibrium, which leads to failure of the structure. Therefore, the critical force is considered as a breaking load.

Euler and Yasinsky formulas

The problem of determining the critical force of a compressed rod was first solved by a member of the St. Petersburg Academy of Sciences L. Euler in 1744. The Euler formula has the form

(1)

where E – modulus of elasticity of the rod material; J min- the smallest moment of inertia of the cross section of the rod (since the bending of the rod during buckling occurs in the plane of least rigidity, i.e., the cross sections of the rod rotate around the axis, relative to which the moment of inertia is minimal, i.e. either around the axis x , or around the axis y );

(μ· l ) is the reduced length of the rod, this is the product of the length of the rod l by the coefficient μ, which depends on the methods of fixing the ends of the rod.

Coefficient μ called length reduction factor ; its value for the most common cases of fixing the ends of the rod are shown in fig. 2:

a- both ends of the rod are hinged and can approach each other;

b- one end is rigidly clamped, the other is free;

in- one end is hinged, the other has a "cross-floating seal";

G - one end is rigidly clamped, the other has a "cross-floating seal";

d- one end is fixed rigidly, on the other is a hinged-movable support;

e- both ends are rigidly clamped, but can approach each other.

It can be seen from these examples that the coefficient μ is the reciprocal of the number of half-waves of the elastic line of the rod during buckling.

Rice. 2. Coefficient μ for the most frequently

occurring cases of fixing the ends of the rod.

Normal stress in the cross section of a compressed rod, corresponding to the critical value of the compressive force, is also called critical.

We define it based on the Euler formula:

(2)

The geometric characteristic of the section i min, determined by the formula

called radius of gyration of the section (with respect to the c-axis J min). For rectangular section

Taking into account (3), formula (2) will take the form:

(4)

The ratio of the reduced length of the rod to the minimum radius of gyration of its cross section, at the suggestion of the professor of the St. Petersburg Institute of Railway Engineers F.S. Yasinsky (1856-1899) is called rod flexibility and denoted by the letter λ :

This dimensionless value simultaneously reflects the following parameters: the length of the rod, the method of its fastening and the characteristic of the cross section.

Finally, substituting (5) into formula (4), we obtain

When deriving Euler's formula, it was assumed that the material of the rod is elastic and follows Hooke's law. Therefore, the Euler formula can only be applied at stresses less than the limit of proportionality σ hc, i.e. when

This condition determines the applicability limit of the Euler formula:

The quantity on the right side of this inequality is called ultimate flexibility :

its value depends on the physical and mechanical properties of the rod material.

For mild steel St. 3, for which σ hc= 200 MPa, E = 2· 10 5 MPa:

Similarly, you can calculate the value of ultimate flexibility for other materials: for cast iron λ before= 80, for pine λ before = 110.

Thus, the Euler formula is applicable for rods whose flexibility is greater than or equal to the ultimate flexibility, i.e.

λ ≥ λ before

This should be understood as follows: if the flexibility of the rod is greater than the ultimate flexibility, then the critical force must be determined by the Euler formula.

At λ < λ before Euler's formula for rods is not applicable. In these cases, when the flexibility of the rods is less than the limiting one, the empirical Yasinsky's formula :

σ kr = a – b λ , (7)

where a and b - experimentally determined coefficients that are constant for a given material; they have the dimension of stress.

For some value of flexibility λ about stress σ kr, calculated by formula (7), becomes equal to the ultimate compressive stress, i.e., the yield strength σ t for ductile materials or compressive strength σ sun- for brittle materials. Rods of low flexibility ( λ < λ about) do not count on stability, but on strength under simple compression.

Thus, depending on the flexibility, the calculation of the compressed rods for stability is carried out differently.