The law of a thread in tension and compression. Deformations longitudinal and transverse deformations

Voltage and deformation during tension and compression are related to the linear dependence, which is called law of Guka. , named English Physics R. Thick (1653-1703), which established this law.
Formulate the law of the throat like this: normal voltage is directly proportional to relative elongation or shortening .

Mathematically, this dependence is recorded as:

Σ \u003d E ε.

Here E. - the proportionality coefficient, which characterizes the rigidity of the material material, i.e. its ability to resist deformation; he's called module longitudinal elasticity , or Module elasticity of the first kind .
The modulus of elasticity, as well as the voltage, is expressed in pascal (PA) .

Values E. For various materials are established experimentally experimental, and their value can be found in the relevant reference books.
So, for steel E \u003d (1.96. ... 2,16) x 105 MPa, for copper E \u003d (1.00 ... 1.30) x 105 MPa, etc.

It should be noted that the bike law is valid only within certain limits of loading.
If in the formula of the law of the thief to substitute the previously obtained values \u200b\u200bof relative elongation and voltage: ε \u003d ΔL / L , Σ \u003d n / a You can get the following dependence:

ΔL \u003d n L / (E A).

Production of the modulus of elasticity on the cross section E. × BUT standing in the denominator, is called the rigidity of the cross section when stretching and compression; It characterizes both the physico-mechanical properties of the material of the bar and the geometric dimensions of the cross section of this bar.

The above formula can be read like this: the absolute elongation or shortening of the timber is directly proportional to the longitudinal strength and the length of the bar, and inversely proportional to the rigidity of the cross section of the bar.
Expression E A / L Call rigidity of timber during stretching and compression .

The above-mentioned formulas of the bike law are valid only for bars and their sites that have a permanent cross section made from one material and with constant strength. For a bar having several sections that differ in the material, the size of the section, the longitudinal force, the change in the length of the entire bar is determined as the allege amount of elongation or shortening of individual sections:

ΔL \u003d σ (ΔL i)

Deformation

Deformation (eng. dEFORMATION.) - This is a change in the shape and size of the body (or part of the body) under the action of external forces, with a change in temperature, humidity, phase transformations and other influences that cause a change in the position of body particles. With increasing voltage, the deformation may end with destruction. The ability of materials to resist deformation and destruction under the influence of various types of loads is characterized by the mechanical properties of these materials.

On the emergence of one or another species type The nature of the voltage applied to the body applied to the body has a great influence. Alone deformation processes Associated with the prevailing effect of the tangent component of the voltage, others - with the action of its normal component.

Types of deformation

By the nature of the load applied to the body types of deformation divided as follows:

Strain deformation;
Deformation of compression;
Deformation of the shift (or cut);
Deformation when crashes;
Wheel deformation.

TO simpled species : Deformation of stretching, deformation of compression, shift deformation. The following types of deformation are distinguished: the deformation of comprehensive compression, twist, bend, which are various combinations of the simplest types of deformation (shift, compression, stretching), since the force applied to the body subjected to deformation is usually not perpendicular to its surface, but directed at an angle What causes both normal and tangent tensions. Study of species of deformation There are such sciences like solid body physics, materials science, crystallography.

In solid bodies, in particular - metals, allocate two main types of deformations - Elastic and plastic deformation, whose physical essence is different.

Shift is called such a type of deformation, when only the reverse forces arise in transverse sections.. Such a stress state corresponds to the action on the rod of two equal, oppositely directed and infinitely closely arranged transverse forces (Fig. 2.13, a, B.) causing a slice on the plane located between the forces.

Fig. 2.13. Deformation and stress when shift

The slice is preceded by deformation - distortion of a direct angle between two mutually perpendicular lines. At the same time on the edges of the dedicated element (Fig. 2.13, in) Tangent stresses arise. The magnitude of the displacement of the faces is called absolute shift. The value of the absolute shift depends on the distance h. between the planes of the action F.. More complete shift deformation characterizes the angle to which the straight angles of the element change - relative shift:

. (2.27)

Using the previously considered method of sections, it is easy to make sure that only the release forces arise on the lateral faces of the dedicated element. Q \u003d F.which are referring to tangent stresses:

Considering that tangent stresses are distributed evenly in cross section BUTThe value is determined by the ratio:

. (2.29)

It is experimentally established that within the limits of elastic deformations, the value of tangent stresses is proportional to the relative shift (The law of the thread in the shift):

where G. - Module of elasticity during the shift (elastic modulus of the second kind).

There is a relationship between longitudinal elasticity and shift modules

where is the Poisson coefficient.

Approximate values \u200b\u200bof the modulus of elasticity during shift, MPa: steel - 0.8 · 10 5; cast iron - 0.45 · 10 5; Copper - 0.4 · 10 4; aluminum - 0.26 · 10 5; Rubber - 4.

2.4.1.1. Calculations for strength during shift

The net shift in real structures is extremely difficult to implement, since due to the deformation of the combined elements, an additional bending of the rod occurs, even with a relatively short distance between the action planes. However, in a number of structures, normal stresses in sections are small and they can be neglected. In this case, the condition of the strength reliability of the part has the form:

, (2.31)

where - permissible voltage on the slice, which are usually prescribed depending on the value of the allowed tension voltage:

- for plastic materials with static load \u003d (0.5 ... 0.6);

- for fragile - \u003d (0.7 ... 1.0).

2.4.1.2. Calculations for rigidity during shift

They reduce the limitation of elastic deformations. Solving together expression (2.27) - (2.30), determine the magnitude of the absolute shift:

, (2.32)

where is the stiffness during the shift.

Torsion

2.4.2.1. Building Torus Moment

2.4.2.2. Deformations when crashes

2.4.2.4. Geometrical characteristics of sections

2.4.2.5. Calculations for strength and rigidity

Cool is called such a type of deformation, when a single power factor appears in cross sections - torque.

Crane deformation occurs when the timber is loaded with pairs of forces, the plane of the actions of which are perpendicular to its longitudinal axis.

2.4.2.1. Building Torus Moment

To determine the voltages and deformations of the bar, build a plug of torque, showing the distribution of torque torque along the length of the bar. Applying the method of sections and examined in equilibrium, any part will be obvious that the moment of the internal forces of elasticity (torque) should balance the effect of external (rotating) moments on the part of the bar. Take the moment to be considered positive if the observer looks at the section under consideration from the outside of the normal normal and sees the torque T.directed against the movement of the clockwise. In the opposite direction, the moment is attributed to the minus sign.

For example, the equilibrium condition for the left side of the bar has a form (Fig. 2.14):

- in cross section A-A:

- in cross section BB:

The boundaries of the plots in constructing the plot are the planes of the action of torque.

Fig. 2.14. Estimated timber (shaft) circuit

2.4.2.2. Deformations when crashes

If on the side surface of the rod round cross section, apply a grid (Fig. 2.15, but) From equivalent circles and generators, and to the free ends to attach pairs of forces with moments T. In the planes perpendicular to the axis of the rod, then at low deformation (Fig. 2.15, b.) You can detect:

Fig. 2.15. Circular deformation scheme

· The forming cylinders turn into a large-step screw lines;

· Squares formed by a mesh turn into a rhombus, i.e. transverse sections occurs;

· Sections, round and flat to deformation, retain their shape and after deformation;

· The distance between cross-sections is practically not changed;

· There is a turn of one section relative to another to some angle.

Based on these observations, the brushing theory is based on the following assumptions:

· Cross cross sections, flat and normal to its axis to deformation, remain flat and normal to the axis and after deformation;

· Equal flow cross sections are rotated relative to each other at equal angles;

· The radii of transverse sections in the process of deformation is not curved;

· Only tangent stresses arise in transverse sections. Normal voltages are small. The length of the bar can be considered unchanged;

· The material of the bar during deformation obeys the law of the throat when shifting :.

In accordance with these hypothesis, the curtain of the rod round cross section is represented as a result of shifts caused by a mutual rotation of sections.

On the rod round cross section with radius r.embedded by one end and loaded torque T. at the other end (Fig. 2.16, but), denote by the side surface forming AD.which under the action of the moment will occupy AD 1.. On distance Z. From the seal to highlight the element length dZ.. The left end of this element as a result of a twist will turn to an angle, and the right - at an angle (). Forging Sun Element will take the position In 1 s 1By rejected from the initial position at the angle. Due to the smallness of this corner

The relation is the angle of twisting the unit of the length of the rod and is called relative spinning angle. Then

Fig. 2.16. Estimated voltage determination scheme
When cutting a rod round cross section

Taking into account (2.33), the law of the throat when crashed can be described by the expression:

. (2.34)

Due to the hypothesis that radii of round transverse sections are not twisted, tangent shift stresses in the vicinity of any point of the body located at the distance from the center (Fig. 2.16, b.) are equal to the work

those. Proportional to the distance to the axis.

The value of the relative spinning angle by formula (2.35) can be found from the condition that the elementary circumferential force () on the elementary area of \u200b\u200bsize dAlocated at a distance from the axis of the bar creates relative to the axis of the elementary moment (Fig. 2.16, b.):

The amount of elementary moments acting throughout the cross section BUTis equal to the torque M Z.. Considering that:

The integral is a purely geometric characteristic and is called polar moment inertia sections.

Under the action of stretching forces along the axis of the bar, its length increases, and the transverse dimensions decrease. With the action of compressive efforts there is a reverse phenomenon. In fig. 6 shows a timber stretched by two forces R. As a result of stretching, the timber lengthened by the value of Δ l., which is called absolute elongation and get absolute transverse narrowing ΔA. .

The ratio of the magnitude of the absolute elongation and shortening to the initial length or width of the bar is called relative deformation. In this case, relative deformation is called longitudinal deformation, but - relative transverse deformation. The ratio of relative transverse deformation to relative longitudinal deformation is called the coefficient of Poisson: (3.1)

Poisson coefficient for each material as an elastic constant is determined by the experimental way and is within: ; For steel.

Within the limits of elastic deformations, it was established that the normal voltage is directly proportional to the relative longitudinal deformation. This dependence is called dungal Law:

, (3.2)

where E. - proportionality coefficient called module normal elasticity.

Consider a direct bar for a constant cross section close to one end and loaded at the other end of the tensile force p (Fig. 8.2, a). Under the action of strength p, the bar is lengthened on some value called full, or absolute, elongation (absolute longitudinal deformation).

At any points of the case under consideration, there is the same intense state and, therefore, linear deformations (see § 5.1) for all its points are the same. Therefore, the value can be defined as the ratio of absolute elongation to the initial length of the timber i, i.e.. Linear deformation during tension or compression of BRUSEV is usually called relative elongation, or relative longitudinal deformation, and denote.

Hence,

Relative longitudinal deformation is measured in distracted units. Deformation of the elongation will agree to be considered positive (Fig. 8.2, a), and the deformation of compression is negative (Fig. 8.2, b).

The greater the value of the force stretching the timber, the greater, with other things being equal, the elongation of the bar; The larger the cross section of the bar, the extension of the bar less. The bars from various materials are lengthened differently. For cases when voltages in a bar do not exceed the proportionality limit (see § 6.1, p. 4), the following dependence is established:

Here n is the longitudinal force in the cross sections of the bar; - cross-sectional area of \u200b\u200btimber; E is a coefficient depending on the physical properties of the material.

Considering that the normal stress in the cross section of the bar is obtained

The absolute lengthening of the bar is expressed by the formula

i.e. absolute longitudinal deformation is directly proportional to the longitudinal force.

For the first time, a law on direct proportionality between forces and deformations formulated (in 1660). Formulascence (10.2) - (13.2) are mathematical expressions of the law of the throat during the tension and compression of the bar.

More general is the following formulation of the law of the throat [see Formulascence (11.2) and (12.2)]: relative longitudinal deformation is directly proportional to normal voltage. In such a wording, the law of the throat is used not only in the study of the stretching and compression of the BRUSEV, but also in other sections of the course.

The E which is included in formula (10.2) - (13.2), is called the modulus of the first kind of elasticity (abbreviated module of elasticity) this value is a physical constant material characterizing its rigidity. The greater the value of E, the less, with other things being equal, the longitudinal deformation.

The product is called the rigidity of the cross section of the bar during stretching and compression.

Appendix I shows the values \u200b\u200bof the moduli of elasticity E for various materials.

Formula (13.2) can be used to calculate the absolute longitudinal deformation of the area of \u200b\u200bthe bar length only under the condition that the cross section of the beam within this section is constantly and the longitudinal force N in all cross-sections is the same.

In addition to the longitudinal deformation, a transverse deformation is also observed under the action on the barrier of the compressive or tensile force. When compressing a bar, the transverse dimensions increase it, and when stretched, it decreases. If the transverse size of the bar before the application to it squeezed the strength p designate B, and after the application of these forces (Fig. 9.2), then the value will denote the absolute transverse deformation of the bar.

The attitude is relative transverse deformation.

Experience shows that at stresses that do not exceed the limit of elasticity (see § 6.1, p. 3), relative transverse deformation is directly proportional to the relative longitudinal deformation, but has a reverse sign:

The coefficient of proportionality in formula (14.2) depends on the material of the bar. It is called a transverse deformation coefficient, or a Poisson coefficient, and is the ratio of relative transverse deformation to the longitudinal, taken in absolute value, i.e.

Poisson coefficient along with the elastic modulus E characterizes the elastic properties of the material.

The value of the Poisson coefficient is determined experimentally. For various materials, it has zero values \u200b\u200b(for cork) to a value close to 0.50 (for rubber and paraffin). For steel, the coefficient of Poisson is equal to 0.25-0.30; For a number of other metals (cast iron, zinc, bronze, copper), it has a value from 0.23 to 0.36. The estimated values \u200b\u200bof the Poisson coefficient for various materials are shown in Appendix I.

Changes in size, volume and possibly body shape, with external influence on it, called deformation in physics. The body is deformed when stretching, compression, or (s), when it changes its temperature.

The deformation appears when different parts of the body make different movements. For example, if the rubber cord pull in the ends, then it will be shown different parts relative to each other, and the cord will be deformed (stretching, lengthens). During deformation, the distances between atoms or molecules of bodies change, so the forces of elasticity occur.

Let a straight timber, length and having a constant section, are fixed in one end. Over the end, it stretches it, applying force (Fig. 1). In this case, the body is lengthened by the value, which is called absolute elongation (or absolute longitudinal deformation).

At any point of the body under consideration there is the same intense state. Linear deformation () when tensile and compression such objects are called relative elongation (relative longitudinal deformation):

Relative longitudinal deformation

Relative longitudinal deformation - the magnitude is dimensionless. As a rule, relative elongation is much less than one ().

The deformation of the elongation is usually considered positive, and the deformation of the compression is negative.

If the voltage in the timber does not exceed a certain limit, dependence is experimentally established:

where is the longitudinal force in the cross sections of the bar; S is the cross-sectional area of \u200b\u200bthe bar; E is the modulus of elasticity (Jung module) - the physical value, the characteristic of the stiffness of the material. Taking on attention to the fact that normal stress in cross section ():

The absolute lengthening of the bar can be expressed as:

The expression (5) is a mathematical record of the Law R. Thick, which reflects the direct relationship between force and deformation at low loads.

In the following wording, the thrust law is used not only when considering the stretching (compression) of the timber: the relative longitudinal deformation is directly proportional to the normal voltage.

Relative deformation during shift

When shift, relative deformation is characterized by formula:

where is the relative shift; - the absolute shift of the layers parallel to each other; h - distance between layers; - Angle of shift.

The law of a shift knuckle is written as:

where G is the shift module, F is the force causing a shift parallel to the shifting layers of the body.

Examples of solving problems

Example 1.

The task

What is the relative elongation of the steel rod, if its upper end is fixed fixed (Fig.2)? Cross cross section of the rod. The load is attached to the lower end of the rod. Consider that the rod's own mass is much less than the mass of the cargo.

Decision

The force that makes the rod stretch, equal to the strength of the severity of the load, which is located at the lower end of the rod. This force acts along the rod axis. Relative rod lengthening as:

where. Before calculating, the Jung module for steel should be found in reference books. PA.

Answer

Example 2.

The task

The bottom base of the metal parallelepiped with the base in the form of a square with a side A and high height is fixed. On the top base parallel to the base is the force F (Fig. 3). What is the relative shift deformation ()? Shift module (G) Consider known.

Consider a direct bar of constant cross section (Fig. 1.5), close to one end and loaded at the other end of the tensile force R. Under the action of power R The timber is lengthened on some value , which is called full (or absolute) elongation (absolute longitudinal deformation).

Fig. 1.5. Deformation of Bruus

At any points of the case under consideration, there is the same intense state and, therefore, linear deformations for all its points are the same. Therefore, the value of E can be defined as the ratio of absolute elongation to the initial length of the bar, i.e.

The bars from various materials are lengthened differently. For cases where voltages in the bar do not exceed the limit of proportionality, the experiment is established as follows:

where N- longitudinal force in cross sections of timber; F- Cross cross section area; E- The coefficient depends on the physical properties of the material.

Considering that the normal voltage in the cross section of the bar σ \u003d N / f, Receive ε \u003d σ / e. From σ \u003d ε.

The absolute lengthening of the bar is expressed by the formula

A more general is the following wording of the thread of the throat: relative longitudinal deformation is directly proportional to normal voltage. In such a wording, the law of the throat is used not only in the study of the stretching and compression of the BRUSEV, but also in other sections of the course.

Value E. called the first kind of elastic modulus. This is a physical constant material that characterizes its rigidity. The greater the value E, the less, with other things being equal, the longitudinal deformation. The modulus of elasticity is expressed in the same units as the voltage, i.e. in Pascal (PA) (Steel E \u003d 2 *10 5 MPa, copper E \u003d. 1 * 10 5 MPa).

Composition EF. It is called the rigidity of the cross section of a bar during stretching and compression.

In addition to the longitudinal deformation under the action on the bar of the compressive or tensile force, transverse deformation is also observed. When compressing a bar, the transverse dimensions increase it, and when stretched, it decreases. If the transverse size of the bar before the application to it compressive forces Rdenote IN, And after the application of these forces In - Δv, then the amount ΔB Will denote the absolute transverse deformation of the bar.

The attitude is relative transverse deformation.

Experience shows that at stresses that do not exceed the limit of elasticity, the relative transverse deformation is directly proportional to the relative longitudinal deformation, but has a reverse sign:

The proportionality coefficient is depends on the material of the bar. It is called a transverse deformation coefficient (or the coefficient of Poisson ) and represents the ratio of relative transverse deformation to the longitudinal, taken in absolute value, i.e. Poisson coefficient Along with the elastic module E.characterizes the elastic properties of the material.

The Poisson coefficient is determined experimentally. For different materials, it has zero values \u200b\u200b(for cork) to a value close to 0.50 (for rubber and paraffin). For steel, the coefficient of Poisson is 0.25 ... 0.30; For a number of other metals (cast iron, zinc, bronze, copper) he

It values \u200b\u200bfrom 0.23 to 0.36.

Fig. 1.6. Variable cross section

The definition of the cross section of the rod is performed on the basis of the strength condition

where [σ] is the allowable voltage.

Determine the longitudinal movement Δ A. Points but axis of timber stretched by force R( Fig. 1.6).

It is equal to the absolute deformation of the part of the bar ad, prisonered between the sealing and cross section spent through the point d, those. Longitudinal deformation of the bar is determined by the formula

This formula is applicable only when within the entire portion of the longitudinal force of N and rigidity EF. Brous cross sections are constant. In the case under consideration on the site aBlongitudinal force N. equal to zero (own bruse weights do not take into account), and on the site bD. It is equal R, In addition, the cross-sectional area of \u200b\u200bthe bar on the plot aC Differs from the cross section on the site cD. Therefore, the longitudinal deformation of the site aD should be determined as the amount of longitudinal deformations of three sites aB, LS and cD, For each of which values N. and EF.permanent along its entire length:

Longitudinal forces on the sections of the bar

Hence,

Similarly, you can determine the movement Δ of any points of the bar axis, and by their values \u200b\u200bto build a lot longitudinal movements (Eppuraδ), i.e. The graph depicting the change in these movements along the length of the axis of the bar.

4.2.3. Strength conditions. Calculation of rigidity.

When checking the stresses of cross-sections area F.and the longitudinal forces are known and the calculation is to calculate the estimated (actual) stresses σ in the characteristic sections of the elements. The highest voltage obtained is compared with allowable:

When selecting sections Determine the required space [F] cross sections of the element (according to the well-known longitudinal forces N. and allowable voltage [σ]). Accepted areas of sections F.must meet the condition of strength expressed in the following form:

When determining the lifting capacity According to famous values F. And the allowable voltage [σ] calculate the permissible values \u200b\u200bof [n] of the longitudinal forces:

According to the obtained values \u200b\u200bof [n], the permissible values \u200b\u200bof external loads are determined [ P.].

For this case, the condition of strength is

The values \u200b\u200bof the regulatory reserve factors are set by the rules. They depend on the class class (capital, temporary, etc.), the outlined period of its operation, load (static, cyclic, etc.), the possible inhomogeneity of the manufacture of materials (for example, concrete), on the type of deformation (stretching, compression , bending, etc.) and other factors. In some cases, it is necessary to reduce the reserve coefficient in order to reduce the weight of the structure, and sometimes increase the reserve coefficient - if necessary, take into account the wear of the driving parts of the machines, corrosion and rewarding the material.

The magnitudes of the regulatory reserve coefficients for various materials, structures and loads have in most cases the meanings: - 2.5 ... 5 and - 1.5 ... 2.5.

Under the testing of the stiffness of the design element, which is in a state of pure stretching - compression, it is understood as the search for the answer to the question: are the values \u200b\u200bof the tough characteristics of the element (material elastic module E. and cross-sectional area F) In order to maximize from all the values \u200b\u200bof the movements of the points of the element caused by the external forces, U Max did not exceed some given limit value [U]. It is believed that when violating the inequality U Max< [u] конструкция переходит в предельное состояние.