Hooke's law in tension and compression. Longitudinal and transverse deformations Normal tension and compression stresses

9. Absolute and relative strain in tension (compression). Poisson's ratio.

If, under the action of a force, a beam in length has changed its longitudinal value by , then this value is called absolute longitudinal deformation (absolute elongation or shortening). In this case, transverse absolute deformation is also observed.

The ratio is called the relative longitudinal strain, and the ratio is called the relative transverse strain.

The ratio is called Poisson's ratio, which characterizes the elastic properties of the material.

Poisson's ratio matters. (for steel it is equal to )

10. Formulate Hooke's law in tension (compression).

I form. In the cross sections of the beam with central tension (compression), normal stresses are equal to the ratio of the longitudinal force to the area cross section:

II form. The relative longitudinal strain is directly proportional to the normal stress, whence .

11. How are the stresses in the transverse and inclined sections of the beam determined?

- force equal to the product of stress and the area of the inclined section:

12. What formula can be used to determine the absolute elongation (shortening) of a beam?

The absolute elongation (shortening) of a beam (rod) is expressed by the formula:

, i.e.

Considering that the value represents the rigidity of the cross section of the beam with a length, we can conclude that the absolute longitudinal deformation is directly proportional to the longitudinal force and inversely proportional to the rigidity of the cross section. This law was first formulated by Hooke in 1660.

13. How are temperature strains and stresses determined?

With an increase in temperature, the mechanical strength characteristics of most materials decrease, and with a decrease in temperature, they increase. For example, steel grade St3 at and ;

for and , i.e. .

The elongation of the rod during heating is determined by the formula , where is the coefficient of linear expansion of the rod material, is the length of the rod.

The normal stress arising in the cross section. As the temperature decreases, the rod shortens and compressive stresses arise.

14. Give a description of the tension (compression) diagram.

The mechanical characteristics of materials are determined by testing samples and constructing appropriate graphs and diagrams. The most common is the static tensile (compression) test.

Limit of proportionality (up to this limit, Hooke's law is valid);

Material yield strength;

Ultimate strength of the material;

Destructive (conditional) stress;

Point 5 corresponds to the true breaking stress.

1-2 area of material flow;

2-3 material hardening zone;

and - the value of plastic and elastic deformation.

Modulus of elasticity in tension (compression), defined as: , i.e. .

15. What parameters characterize the degree of plasticity of a material?

The degree of plasticity of the material can be characterized by the following values:

Residual relative elongation - as the ratio of the residual deformation of the sample to its original length:

where is the length of the sample after rupture. The value for various steel grades ranges from 8 to 28%;

Residual relative narrowing - as the ratio of the cross-sectional area of the sample at the rupture site to the original area:

where is the cross-sectional area of the torn sample at the thinnest point of the neck. The value ranges from a few percent for brittle high carbon steel to 60% for mild steel.

16. Tasks to be solved in the calculation of tensile (compressive) strength.

Consider a straight beam of constant section with a length l, sealed at one end and loaded at the other end with a tensile force P (Fig. 2.9, a). Under the action of the force P, the beam lengthens by a certain amount? l, which is called the full, or absolute, elongation (absolute longitudinal deformation).

At any point of the beam under consideration, there is the same stress state, and, consequently, the linear deformations for all its points are the same. Therefore, the value can be defined as the ratio of the absolute elongation? l to the initial length of the beam l, i.e. . Linear deformation during tension or compression of the bars is usually called relative elongation, or relative longitudinal deformation, and is denoted

Hence,

Relative longitudinal deformation is measured in abstract units. Let us agree to consider the elongation deformation as positive (Fig. 2.9, a), and the compression deformation as negative (Fig. 2.9, b).

The greater the magnitude of the force that stretches the bar, the greater, ceteris paribus, the elongation of the bar; the larger the cross-sectional area of the beam, the lower the elongation of the beam. Bars from various materials lengthen differently. For cases where the stresses in the bar do not exceed the proportionality limit, the following dependence has been established by experience:

Here N is longitudinal force in the cross sections of the beam;

F - cross-sectional area of \u200b\u200bthe beam;

E - coefficient depending on physical properties material.

Taking into account that the normal stress in the cross section of the beam, we obtain

The absolute elongation of the beam is expressed by the formula

those. absolute longitudinal deformation is directly proportional to the longitudinal force.

For the first time the law of direct proportionality between forces and deformations was formulated by R. Hooke (in 1660).

More general is the following formulation of Hooke's law: the relative longitudinal strain is directly proportional to the normal stress. In this formulation, Hooke's law is used not only in the study of tension and compression of the bars, but also in other sections of the course.

The value of E, included in the formulas, is called the modulus of longitudinal elasticity (abbreviated as the modulus of elasticity). This value is the physical constant of the material, which characterizes its rigidity. The larger the value of E, the smaller, ceteris paribus, the longitudinal deformation.

The product EF is called the cross-sectional stiffness of the beam in tension and compression.

If the transverse dimension of the beam before the application of compressive forces P to it, denote b, and after the application of these forces b +? b (Fig. 9.2), then the value? b will indicate the absolute transverse deformation of the beam. The ratio is the relative transverse strain.

Experience shows that at stresses not exceeding the elastic limit, the relative transverse strain is directly proportional to the relative longitudinal strain e, but has the opposite sign:

The coefficient of proportionality in formula (2.16) depends on the material of the beam. It is called the transverse strain ratio, or Poisson's ratio, and is the ratio of transverse strain to longitudinal strain, taken in absolute value, i.e.

Poisson's ratio, along with the modulus of elasticity E, characterizes the elastic properties of the material.

The value of Poisson's ratio is determined experimentally. For various materials, it has values from zero (for cork) to a value close to 0.50 (for rubber and paraffin). For steel, Poisson's ratio is 0.25-0.30; for a number of other metals (cast iron, zinc, bronze, copper), it has values from 0.23 to 0.36.

Table 2.1 Values of the modulus of elasticity.

Table 2.2 Values of transverse strain coefficient (Poisson's ratio)

Let, as a result of deformation, the initial length of the rod l will become equal. l 1. Changing the length

is called the absolute elongation of the bar.

The ratio of the absolute elongation of the rod to its original length is called relative elongation (- epsilon) or longitudinal deformation. Longitudinal deformation is a dimensionless quantity. Dimensionless deformation formula:

In tension, the longitudinal deformation is considered positive, and in compression, negative.

The transverse dimensions of the rod as a result of deformation also change, while they decrease during tension, and increase during compression. If the material is isotropic, then its transverse deformations are equal to each other:

It has been experimentally established that during tension (compression) within the limits of elastic deformations, the ratio of transverse to longitudinal deformation is a constant value for a given material. The modulus of the ratio of transverse to longitudinal strain, called Poisson's ratio or transverse strain ratio, is calculated by the formula:

For different materials, Poisson's ratio varies within . For example, for cork, for rubber, for steel, for gold.

Longitudinal and transverse deformations. Poisson's ratio. Hooke's law

Under the action of tensile forces along the axis of the beam, its length increases, and the transverse dimensions decrease. Under the action of compressive forces, the opposite occurs. On fig. 6 shows a beam stretched by two forces P. As a result of tension, the beam lengthened by Δ l, which is called absolute elongation, and get absolute transverse constriction Δа .

The ratio of the magnitude of absolute elongation and shortening to the original length or width of the beam is called relative deformation. In this case, the relative deformation is called longitudinal deformation, A - relative transverse deformation. The ratio of relative transverse strain to relative longitudinal strain is called Poisson's ratio: (3.1)

Poisson's ratio for each material as an elastic constant is determined empirically and is within: ; for steel.

Within the limits of elastic deformations, it is established that the normal stress is directly proportional to the relative longitudinal deformation. This dependency is called Hooke's law:

, (3.2)

Where E is the coefficient of proportionality, called modulus of normal elasticity.

If we substitute the expression into the formula of Hooke's law and , then we get the formula for determining the elongation or shortening in tension and compression:

, (3.3)

where is the product EF is called tensile and compressive stiffness.

Longitudinal and transverse deformations. Hooke's law

Have an idea about longitudinal and transverse deformations and their relationship.

Know Hooke's law, dependencies and formulas for calculating stresses and displacements.

To be able to carry out calculations on the strength and stiffness of statically determinate bars in tension and compression.

Tensile and Compressive Deformations

Consider the deformation of the beam under the action of the longitudinal force F(Fig. 4.13).

The initial dimensions of the beam: - initial length, - initial width. The beam is extended by the amount Δl; Δ1- absolute elongation. When stretched, the transverse dimensions decrease, Δ A- absolute narrowing; ∆1 > 0; Δ A 0.

In the resistance of materials, it is customary to calculate deformations in relative units: fig.4.13

- relative extension;

Relative contraction.

There is a relationship between longitudinal and transverse strains ε'=με, where μ is the coefficient of transverse strain, or Poisson's ratio, is a characteristic of the plasticity of the material.

Encyclopedia of Mechanical Engineering XXL

Equipment, materials science, mechanics and.

Longitudinal deformation in tension (compression)

It has been experimentally established that the ratio of transverse strain ej. to longitudinal deformation e under tension (compression) up to the limit of proportionality for a given material is a constant value. Denoting the absolute value of this ratio (X), we get

Experiments have established that the relative transverse strain eo in tension (compression) is a certain part of the longitudinal strain e, i.e.

The ratio of transverse to longitudinal strain in tension (compression), taken as an absolute value.

In the previous chapters of the strength of materials have been considered simple views beam deformations - tension (compression), shear, torsion, direct bending, characterized by the fact that in the cross sections of the beam there is only one internal force factor in tension (compression) - longitudinal force, in shear - transverse force, in torsion - torque, with pure straight bending - the bending moment in the plane passing through one of the main central axes of the beam cross section. With direct transverse bend there are two internal force factors - a bending moment and a transverse force, but this type of beam deformation is referred to as simple, since the combined effect of these force factors is not taken into account when calculating the strength.

When stretched (compressed), the transverse dimensions also change. The ratio of the relative transverse strain e to the relative longitudinal strain e is a physical constant of the material and is called Poisson's ratio V = e/e.

When stretching (compressing) the beam, its longitudinal and transverse dimensions receive changes characterized by deformations of the longitudinal prod (bg) and transverse (e, e). which are related by the relation

As experience shows, when the beam is stretched (compressed), its volume changes somewhat with an increase in the length of the beam by the value Ar, each side of its section decreases by We will call the relative longitudinal deformation the value

Longitudinal and transverse elastic deformations that occur during tension or compression are related to each other by the dependence

So, consider a beam of isotropic material. The hypothesis of flat sections establishes such a geometry of deformations in tension and compression that all longitudinal fibers of the beam have the same deformation x, regardless of their position in the cross section F, i.e.

An experimental study of volumetric deformations was carried out during tension and compression of glass-reinforced plastic samples with simultaneous registration on the K-12-21 oscilloscope of changes in the longitudinal and transverse deformations of the material and the force under loading (on the testing machine TsD-10). The test until reaching the maximum load was carried out at almost constant loading speeds, which was ensured by a special regulator that the machine is equipped with.

As experiments show, the ratio of transverse strain b to longitudinal strain e in tension or compression for a given material within the application of Hooke's law is a constant value. This ratio, taken in absolute value, is called the transverse strain ratio or Poisson's ratio.

Here /p(compressive) - longitudinal deformation in tension (compression) /u - transverse deformation in bending I - length of the deformable beam P - area of its cross section / - moment of inertia of the cross-sectional area of the sample relative to the neutral axis - polar moment of inertia P - applied force -torsion moment - coefficient, uchi-

The deformation of the rod during tension or compression consists in changing its length and cross section. Relative longitudinal and transverse deformations are determined, respectively, by the formulas

The ratio of the height of the side plates (tank walls) to the width in batteries of significant dimensions is usually more than two, which makes it possible to calculate the tank walls using the formulas for cylindrical bending of the plates. The tank lid is not rigidly fastened to the walls and cannot prevent their buckling. Neglecting the influence of the bottom, it is possible to reduce the calculation of the tank under the action of horizontal forces on it to the calculation of a closed statically indeterminate frame-strip separated from the tank by two horizontal sections. The modulus of normal elasticity of glass-reinforced plastic is relatively small; therefore, structures made of this material are sensitive to buckling. The strength limits of fiberglass in tension, compression and bending are different. A comparison of the calculated stresses with the limiting stresses should be made for the deformation that is predominant.

Let us introduce the notation used in the algorithm, the values with indices 1,1-1 refer to the current and previous iterations at the time stage m - Am, m and 2 - respectively, the rate of longitudinal (axial) deformation in tension (i > > 0) and compression (2 deformations are related by the relation

Dependencies (4.21) and (4.31) were checked for large numbers materials and various conditions loading. The tests were carried out in tension-compression at a frequency of about one cycle per minute and one cycle per 10 minutes over a wide range of temperatures. Both longitudinal and transverse strain gauges were used to measure strains. At the same time, solid (cylindrical and corset) and tubular samples were tested from boiler steel 22k (at temperatures of 20-450 C and asymmetries - 1, -0.9 -0.7 and -0.3, in addition, the samples were welded and with notch), heat-resistant steel TS (at temperatures of 20-550 ° C and asymmetries -1 -0.9 -0.7 and -0.3), heat-resistant nickel alloy EI-437B (at 700 ° C), steel 16GNMA, ChSN , Х18Н10Т, steel 45, aluminum alloy AD-33 (with asymmetries -1 0 -b0.5), etc. All materials were tested as delivered.

The coefficient of proportionality E, linking both normal stress and longitudinal deformation, is called the modulus of elasticity in tension-compression of the material. This coefficient has other names, the modulus of elasticity of the 1st kind, Young's modulus. Elastic modulus E is one of the most important physical constants characterizing the ability of a material to resist elastic deformation. The larger this value, the less the beam is stretched or compressed when the same force P is applied.

If we assume that in Fig. 2-20, and the shaft O is the leading one, and the shafts O1 and O2 are driven, then when the disconnector is turned off, the thrust LL1 and L1L2 will work in compression, and when turned on, in tension. While the distances between the axes of the shafts O, 0 and O2 are small (up to 2000 mm), the difference between the deformation of the thrust in tension and compression (longitudinal bending) does not affect the operation of the synchronous transmission. In a disconnector for 150 kV, the distance between the poles is 2800 mm, for 330 kV - 3500 mm, for 750 kV - 10,000 mm. With such large distances between the centers of the shafts and significant loads that they must transmit, they say /> d. This length is chosen for reasons of greater stability, since a long sample, in addition to compression, may experience buckling deformation, which will be discussed in the second part of the course. Samples from building materials are made in the form of a cube with dimensions of 100 X YuO X YuO or 150 X X 150 X 150 mm. During the compression test, the cylindrical sample assumes an initially barrel-shaped shape. If it is made of a plastic material, then further loading leads to flattening of the sample; if the material is brittle, then the sample suddenly cracks.

At any point of the beam under consideration, there is the same stress state and, therefore, the linear deformations (see 1.5) are the same for all its currents. Therefore, the value can be defined as the ratio of the absolute elongation A/ to the original length of the beam /, i.e. e, = A///. Linear deformation during tension or compression of the beams is usually called relative elongation (or relative longitudinal deformation) and is denoted e.

See pages where the term is mentioned Longitudinal deformation in tension (compression) : Technical Handbook of the Railwayman Volume 2 (1951) - [ c.11 ]

Longitudinal and transverse deformations in tension - compression. Hooke's Law

When tensile loads are applied to the rod, its initial length / increases (Fig. 2.8). Let us denote the length increment by A/. The ratio of the increase in the length of the rod to its original length is called elongation or longitudinal deformation and is denoted by g:

Relative elongation is a dimensionless value, in some cases it is customary to express it as a percentage:

When stretched, the dimensions of the rod change not only in the longitudinal direction, but also in the transverse direction - the rod narrows.

Rice. 2.8. Tensile deformation of the rod

Change ratio A A cross-sectional size to its original size is called relative transverse narrowing or transverse deformation.

It has been experimentally established that there is a relationship between the longitudinal and transverse deformations

where p is called Poisson's ratio and are constant for a given material.

Poisson's ratio is, as can be seen from the above formula, the ratio of transverse to longitudinal deformation:

For various materials, Poisson's ratio values range from 0 to 0.5.

On average, for metals and alloys, Poisson's ratio is approximately 0.3 (Table 2.1).

The value of Poisson's ratio

When compressed, the picture is reversed, i.e. in the transverse direction, the initial dimensions decrease, and in the transverse direction, they increase.

Numerous experiments show that, up to certain loading limits for most materials, the stresses arising from the tension or compression of the rod are in a certain dependence on the longitudinal deformation. This dependency is called Hooke's law, which can be formulated as follows.

Within known loading limits, there is a directly proportional relationship between the longitudinal deformation and the corresponding normal stress

Proportionality factor E called modulus of longitudinal elasticity. It has the same dimension as the voltage, i.e. measured in Pa, MPa.

The modulus of longitudinal elasticity is a physical constant of a given material, which characterizes the ability of a material to resist elastic deformations. For a given material, the modulus of elasticity varies within narrow limits. Yes, for steel different brands E=(1.9. 2.15) 10 5 MPa.

For the most commonly used materials, the modulus of elasticity has the following values in MPa (Table 2.2).

The value of the modulus of elasticity for the most commonly used materials

Moral and patriotic education can become an element of the educational process. Measures have been developed to ensure the patriotic and moral education of children and youth. The relevant draft law 1 was submitted to the State Duma by Federation Council member Sergei […]
How to apply for a dependency? Questions about the need to register a dependency do not arise often, since most of the dependents are such by virtue of the law, and the problem of establishing the fact of dependency disappears by itself. However, in some cases, the need to issue […]
Urgent registration and obtaining a passport No one is immune from the situation when there is a sudden need to quickly issue a passport in Moscow or any other Russian city. What to do? Where to apply? And how much would such a service cost? Necessary […]
Taxes in Sweden and business prospects Before you go to Sweden as a business migrant, it is useful to learn more about the country's tax system. Taxation in Sweden is a complex and, as our compatriots would say, tricky system. She […]
Tax on winnings: size in 2017 In previous years, you can clearly see the trend followed by public authorities. Increasingly strict measures are being taken to control the income of the gaming business, as well as the population receiving the winnings. So, in 2014 […]
Clarification of claims After the court accepts the claim and even during the trial, the plaintiff has the right to declare the clarification of the claims. As clarifications, you can indicate new circumstances or supplement old ones, increase or decrease the amount of the claim, […]
How to uninstall programs from a computer? It would seem that it is difficult to remove programs from a computer? But I know a lot of novice users have problems with this. Here, for example, is an excerpt from one letter that I received: “... I have a question for you: […]
WHAT IS IMPORTANT TO KNOW ABOUT THE NEW DRAFT ON PENSIONS From 01.01.2002, labor pensions are assigned and paid in accordance with federal law"On labor pensions in Russian Federation"No. 173-FZ dated December 17, 2001. When establishing the size of the labor pension in accordance with the […]

Consider a straight rod of constant cross section, rigidly fixed from above. Let the rod have a length and be loaded with a tensile force F . From the action of this force, the length of the rod increases by a certain amount Δ (Fig. 9.7, a).

When the rod is compressed by the same force F the length of the rod will be reduced by the same amount Δ (Fig. 9.7, b).

Value Δ , equal to the difference between the lengths of the rod after deformation and before deformation, is called the absolute linear deformation (elongation or shortening) of the rod during its tension or compression.

Absolute linear strain ratio Δ to the initial length of the rod is called relative linear deformation and is denoted by the letter ε or ε x ( where index x indicates the direction of deformation). When the rod is stretched or compressed, the value ε simply referred to as the relative longitudinal strain of the bar. It is determined by the formula:

Multiple studies of the process of deformation of a stretched or compressed rod in the elastic stage have confirmed the existence of a direct proportional relationship between normal stress and relative longitudinal deformation. This dependence is called Hooke's law and has the form:

Value E is called the modulus of longitudinal elasticity or the modulus of the first kind. It is a physical constant (constant) for each type of rod material and characterizes its rigidity. The larger the value E , the smaller will be the longitudinal deformation of the rod. Value E measured in the same units as voltage, that is, in Pa , MPa , etc. The values of the modulus of elasticity are contained in the tables of reference and educational literature. For example, the value of the modulus of longitudinal elasticity of steel is taken equal to E = 2∙10 5 MPa , and wood

E = 0.8∙10 5 MPa.

When calculating rods for tension or compression, it often becomes necessary to determine the value of absolute longitudinal deformation if the value of the longitudinal force, the cross-sectional area and the material of the rod are known. From formula (9.8) we find: . Let's replace in this expression ε its value from formula (9.9). As a result, we get = . If we use the normal stress formula , we get the final formula for determining the absolute longitudinal strain:

The product of the modulus of elasticity and the cross-sectional area of the rod is called its rigidity in tension or compression.

Analyzing formula (9.10), we will make a significant conclusion: the absolute longitudinal deformation of the rod in tension (compression) is directly proportional to the product of the longitudinal force and the length of the rod and inversely proportional to its rigidity.

Note that formula (9.10) can be used in the case when the cross section of the rod and the longitudinal force have constant values along its entire length. In the general case, when the rod has stepwise variable stiffness and is loaded along the length by several forces, it is necessary to divide it into sections and determine the absolute deformations of each of them using the formula (9.10).

The algebraic sum of the absolute deformations of each section will be equal to the absolute deformation of the entire rod, that is:

Longitudinal deformation of the rod from the action of a uniformly distributed load along its axis (for example, from the action of its own weight), is determined by the following formula, which is given without proof:

In the case of tension or compression of the rod, in addition to longitudinal deformations, transverse deformations also occur, both absolute and relative. Denote by b the size of the cross section of the rod before deformation. When the rod is stretched by force F this size will be reduced by Δb , which is the absolute transverse strain of the bar. This value has a negative sign. In compression, on the contrary, the absolute transverse deformation will have a positive sign (Fig. 9.8).

Poisson's ratio for each material as an elastic constant is determined empirically and is within: ; for steel.

Within the limits of elastic deformations, it is established that the normal stress is directly proportional to the relative longitudinal deformation. This dependency is called Hooke's law:

, (3.2)

Where E is the coefficient of proportionality, called modulus of normal elasticity.