The given length of the rod is equal to. Scientific electronic library. Euler's formula for a rod with pinched and free ends

Longitudinal bend

When calculating the strength, it was assumed, what structural balance Under the influence external forces is sustainable. However, failure of the structure may occur due to the fact that equilibrium structures for one reason or another. turn out to be unsustainable. In many cases, in addition to strength testing, it is also necessary to stability check structural elements.

The state of equilibrium is considered sustainable, if for any possible deviation of the system from the equilibrium position forces arise that strive to return it to its original position.

Consider the known types of equilibrium.

unstable equilibrium condition will be in the case when, at least one of the possible deviations of the system from the equilibrium position, forces arise, seeking to remove it from its original position.

The state of equilibrium will indifferent, if, with different deviations of the system from the equilibrium position, forces arise that tend to return it to its initial position, but at least for one of the possible deviations, the system continues to remain in equilibrium in the absence of forces that seek to return it to its initial position or remove it from this position.

At loss of stability, the nature of the work of the structure changes, since this type of deformation passes into another, more dangerous one, capable of leading it to destruction under a load much lower than that followed from the calculation of strength. It is very significant that loss of stability is accompanied by an increase in large deformations, so this phenomenon is catastrophic in nature.

In the transition from a stable equilibrium state to an unstable one, the structure passes through a state of indifferent equilibrium. If a structure in this state is informed of some slight deviation from the initial position, then upon termination of the cause that caused this deviation, the structure will no longer return to its original position, but will be able to maintain the new position given to it due to the deviation.

The state of indifferent equilibrium, representing, as it were, the boundary between two basic states - stable and unstable, is called critical condition. The load at which the structure maintains a state of indifferent equilibrium is called critical load.

Experiments show that it is usually enough to slightly increase the load compared to its critical value for the structure to lose its bearing capacity due to large deformations and fail. In construction engineering, the loss of stability of even one structural element causes a redistribution of forces throughout the structure and often leads to an accident.

The bending of the rod associated with the loss of stability is called buckling.

Critical Power. Critical stress

The smallest value of the compressive force, at which the initial shape of the balance of the rod - rectilinear becomes unstable - curved, is called critical.

In the study of the stability of the equilibrium forms of elastic systems, the first steps were taken Euler.

AT elastic stage deformation of the rod under stress, not exceeding the limit of proportionality, the critical force is calculated from Euler formula:

where Imin – minimum moment of inertia of the bar section(due to the fact that the bending of the rod occurs in the plane with the least rigidity), however, exceptions can only be in cases where the conditions for fixing the ends of the rod are different in different planes, ℓ - geometric length rod, μ - or (depending on the methods of fixing the ends of the rod), Values μ shown under the respective rod fixing diagram

Critical voltage is calculated as follows

, where flexibility rod,

a radius of inertia of the section.

We introduce the concept ultimate flexibility.

Value λ before depends only on the type of material:

If steel 3 E\u003d 2 10 11 Pa, and σ pc \u003d 200 MPa, then ultimate flexibility

For wood (pine, spruce) ultimate flexibilityλ prev=70, for cast iron λ prev=80

Thus, for rods of great flexibility λ≥λ before critical force is determined by Euler's formula.

In the elastic-plastic stage of the rod deformation, when the value of flexibility is in the range λ 0 ≤λ≤λ pr,(rods of medium flexibility) the calculation is carried out according to empirical formulas, for example, you can use the formula of Yasinsky F.S. The values ​​of the parameters entered into it are determined empirically for each material.

σ to \u003d a-bλ, or F cr= A(a— bλ)

where a and b- constants determined experimentally ().So, for steel3 a=310MPa, b\u003d 1.14 MPa.

With values ​​of rod flexibility 0≤λ≤λ0(rods of low flexibility) no loss of stability is observed.

Thus, the limits of applicability Euler formulas — applied only in the zone of elastic deformations.

Stability condition. Types of problems in the calculation of stability.

Stability condition compressed rod is the inequality:

Here allowable stability stress [σ mouth] is not a constant, as it was under strength conditions, and depending on the following factors:

1) on the length of the rod, on the dimensions and even on the shape of the cross sections,

2) on the method of fixing the ends of the rod,

3) from the material of the rod.

Like any allowed value, [σ mouth] is determined by the ratio of the voltage dangerous for the compressed rod to the safety factor. For a compressed rod, the so-called critical stress σ kr, at which the rod loses the stability of the original form of equilibrium.

So

The value of the safety factor in stability problems is taken slightly larger than the value , that is, if k=1÷2, then kmouth=2÷5.

The allowable stability stress can be related to the allowable strength stress:

In this case ,

where σt- a stress that is dangerous from the point of view of strength (for ductile materials this is the yield strength, and for brittle materials it is the compressive strength σ sun ).

Coefficient φ<1 and that's why it's called reduction factor of the main allowable stress, i.e. [σ] strength, or else

With that said stability condition for a compressed rod takes the form:

The numerical values ​​of the coefficient φ are chosen from tables depending on material and degree of flexibility rod, where:

μ – reduced length factor(depending on how the ends of the rod are fixed), ℓ - geometric length rod,

i – radius of gyration cross section relative to that of the main central axes of the section, around which the rotation of the cross sections will occur after the load reaches a critical value.

Coefficient φ changes in the range 0≤φ≤1, depends, as already mentioned, both on the physical and mechanical properties of the material and on the flexibility λ. Dependencies between φ and λ for various materials are usually presented in tabular form with a step ∆λ=10.

When calculating φ values ​​for bars with slenderness values ​​that are not multiples of 10, the linear interpolation rule.

The values ​​of the coefficient φ depending on the flexibility λ for materials

Based on the stability condition, we solve three types of tasks:

1. Stability check.
2. Section selection.
3. Determining the load capacity(or safe load, or rod load capacity: [F]=φ[σ] BUT .

The most difficult is the solution of the problem of selection of the section, since the required value of the cross-sectional area is included in both the left and right parts of the stability condition:

Only on the right side of this inequality, the cross-sectional area is in an implicit form: it is included in the formula for the radius of gyration, which in turn is included in the flexibility formula, on which the value of the buckling coefficient depends φ . Therefore, here we have to use the trial and error method, clothed in the form method of successive approximations:

1 try: ask φ1 from the middle zone of the table, find, determine the dimensions of the section, calculate, then flexibility, according to the table we determine and compare with the value φ1. If , then.

Lecture 7

STABILITY OF COMPRESSED RODS

The concept of the stability of a compressed rod. Euler formula. Addiction critical force from 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 (Fig. 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 calculation of stability is based on the allowable stress in the calculation of compression

(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.

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 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.

ROD LENGTH REDUCED conditional length of a compressed rod with given conditions for fixing its ends, the length of which, by the value of the critical force, is equivalent to the length of a rod with hinged ends

(Bulgarian; Bulgarian) - given length on prt

(Czech; Čeština) - vzpěrná delka prutu

(German language; Deutsch) - reduzierte Stablänge; ideelle Stablange

(Hungarian; Magyar) - rud kihajlas! hosza

(Mongolian) - tuyvangiin khorvuulsen urt

(Polish language; Polska) - długość sprowadzona pręta

(Romanian; Român) - lungime conventională a barei

(Serbo-Croatian; Srpski jezik; Hrvatski jezik) - redukovana dužina stapa

(Spanish; Español) - luz efectiva de una barra

(English language; English) - reduced length of bar

(French language; Français) - longueur reduite d "une barre

Construction dictionary.

See what the "ROD LENGTH REDUCED" is in other dictionaries:

reduced rod length- The conditional length of a compressed rod with given conditions for fixing its ends, the length of which, by the value of the critical force, is equivalent to the length of a rod with hinged ends [Terminological dictionary for construction in 12 languages ​​(VNIIIS ... ...

reduced bar length- The nominal length of a single-span rod, the critical force of which, when its ends are hinged, is the same as for a given rod. [Collection of recommended terms. Issue 82. Structural mechanics. USSR Academy of Sciences. Scientific Committee ... ... Technical Translator's Handbook

Deformation schemes and coefficients for various fastening conditions and the method of applying the load Rod flexibility The ratio of the effective length of the rod ... Wikipedia

- (silomer). This name is called spring scales in physics courses, and in mechanics instruments for measuring mechanical work (cm). The oldest image of a spring balance, according to Karsten, was printed in 1726, without description, in the book: Leupold, ... ... Encyclopedic Dictionary F.A. Brockhaus and I.A. Efron

MEASURES- MEASURES defined by physical. quantities against which other quantities are compared in order to measure them. The main measures of the most common metric system: the meter length at 0 ° of a platinum rod stored in the International Bureau of Measures and ... ... Big Medical Encyclopedia

Let us determine the critical force for a centrally compressed rod hinged at the ends (Fig. 13.4). For small forces R the axis of the rod remains straight and central compression stresses arise in its sections o = P/F. At the critical value of the force P = P, a curved form of equilibrium of the rod becomes possible.

There is a longitudinal bend. The bending moment in an arbitrary section x of the rod is equal to

It is important to note that the bending moment is determined for the deformed state of the bar.

If we assume that the bending stresses arising in the cross sections of the rod from the action of the critical force do not exceed the proportionality limit of the material o pc and the deflections of the rod are small, then we can use the approximate differential equation for the bent axis of the rod (see § 9.2)

By introducing the notation

instead of (13.2) we get the following equation:

The general solution of this equation has the form

This solution contains three unknowns: the integration constants Cj, С2 and the parameter to, since the magnitude of the critical force is also unknown. To determine these three quantities, there are only two boundary conditions: u(0) = 0, v(l) = 0. It follows from the first boundary condition that C 2 = 0, and from the second we obtain

It follows from this equality that either C (= 0 or sin kl = 0. In the case C, = 0, deflections in all sections of the rod are equal to zero, which contradicts the initial assumption of the problem. In the second case kl = pc, where P - arbitrary integer. With this in mind, by formulas (13.3) and (13.5) we obtain

The considered problem is an eigenvalue problem. Found numbers to = pc/1 called own numbers, and their corresponding functions are own functions.

As can be seen from (13.7), depending on the number P the compressive force P (i), at which the rod is in a bent state, can theoretically take on a number of values. In this case, according to (13.8), the rod is bent along P half-waves of a sinusoid (Fig. 13.5).

The smallest value of the force will be at P = 1:

This force is called first critical force. Wherein kl = to and the curved axis of the rod is one half-wave of a sinusoid (Fig. 13.5, a):

where C( 1)=/ - deflection in the middle of the rod length, which follows from (13.8) with P= 1 of them = 1/2.

Formula (13.9) was obtained by Leonhard Euler and is called the Euler formula for the critical force.

All forms of equilibrium (Fig. 13.5), except for the first (P= 1), are unstable and therefore are of no practical interest. Forms of equilibrium corresponding P - 2, 3, ..., will be stable if at the inflection points of the elastic line (points C and C "in Fig. 13.5, b, c) introduce additional hinged supports.

The resulting solution has two features. First, the solution (13.10) is not unique, since the arbitrary constant Cj (1) =/ remains undefined, despite the use of all boundary conditions. As a result, the deflections were determined to within a constant factor. Secondly, this solution does not make it possible to describe the state of the rod at P > P cr. From (13.6) it follows that for P = P cr the rod can have a curved equilibrium shape provided that kl = k. If R > R cr, then kl F p, and then it should be Cj (1) = 0. This means that v= 0, that is, the bar after bending at P = P cr reverts to a straight line R > R. It is obvious that this contradicts the physical concepts of rod bending.

These features are due to the fact that the expression (13.1) for the bending moment and the differential equation (13.2) were obtained for the deformed state of the rod, while when setting the boundary condition at the end X= / axial movement and in this end (Fig. 13.6) due to bending was not taken into account. Indeed, if we neglect the shortening of the rod due to central compression, then it is easy to imagine that the deflections of the rod will have quite definite values ​​if we set the value and in.

From this reasoning, it becomes obvious that in order to determine the dependence of deflections on the magnitude of the compressive force R necessary instead of the boundary condition v(l)= 0 use refined boundary condition v(l - and v) = 0. It was found that if the force exceeds the critical value by only 1 + 2%, the deflections become large enough and it is necessary to use exact nonlinear differential buckling equation

This equation differs from the approximate equation (13.4) by the first term, which is an exact expression for the curvature of the bent axis of the rod (see § 9.2).

The solution of equation (13.11) is quite complicated and is expressed in terms of a complete elliptic integral of the first kind.