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Allowable (permissible) stress is the stress value that is considered extremely acceptable when calculating dimensions cross section element designed for a given load. We can talk about permissible tensile, compressive and shear stresses. The permissible stresses are either prescribed by the competent authority (say, the control bridge department railway), or are selected by a designer who knows well the properties of the material and the conditions of its use. The permissible voltage is limited to the maximum operating voltage designs.

When designing structures, the goal is to create a structure that, while being reliable, at the same time would be extremely light and economical. Reliability is ensured by the fact that each element is given such dimensions that the maximum operating stress in it will be to a certain extent less than the stress that causes the loss of strength of this element. Loss of strength does not necessarily mean destruction. Machine or building construction is considered to have failed when it cannot perform its function satisfactorily. A part made of a plastic material, as a rule, loses strength when the stress in it reaches the yield point, since due to too much deformation of the part, the machine or structure ceases to meet its intended purpose. If the part is made of brittle material, then it is almost not deformed, and its loss of strength coincides with its destruction.

The difference between the stress at which the material loses strength and the permissible stress is the “margin of safety” that must be provided for, taking into account the possibility of accidental overload, calculation inaccuracies associated with simplifying assumptions and uncertain conditions, the presence of undetected (or undetectable) material defects and subsequent reduction in strength due to metal corrosion, wood rotting, etc.

The safety factor of any structural element is equal to the ratio ultimate load, causing loss of strength of the element, to a load that creates permissible stress. In this case, the loss of strength means not only the destruction of the element, but also the appearance of residual deformations in it. Therefore, for a structural element made of plastic material, the ultimate stress is the yield strength. In most cases, operating stresses in structural elements are proportional to the loads, and therefore the safety factor is defined as the ratio of the ultimate strength to the permissible stress (safety factor for ultimate strength).

Permissible stresses. Condition of strength.

The tensile strength and yield strength determined experimentally are average statistical values, i.e. have deviations upward or downward, therefore, the maximum stresses in strength calculations are compared not with the yield strength and strength, but with slightly lower stresses, which are called permissible stresses.
Plastic materials work equally well in tension and compression. The dangerous stress for them is the yield point.
The permissible stress is indicated by [σ]:

where n is the safety factor; n>1. Brittle metals work worse in tension, but better in compression. That's why dangerous voltage for them the tensile strength σv. Allowable stresses for fragile materials are determined by the formulas: where n is the safety factor; n>1. Brittle metals work worse in tension, but better in compression. Therefore, the dangerous stress for them is the tensile strength σtemp. Allowable stresses for brittle materials are determined by the formulas:

where n is the safety factor; n>1.

Brittle metals work worse in tension, but better in compression. Therefore, the dangerous stress for them is the tensile strength σv.
Allowable stresses for brittle materials are determined by the formulas:

σtr - tensile strength;

σs - compressive strength;

nр, nс - safety factors for ultimate strength.

Strength condition for axial tension (compression) for plastic materials:

Strength conditions for axial tension (compression) for brittle materials:

Nmax is the maximum longitudinal force, determined from the diagram; A is the cross-sectional area of ​​the beam.

There are three types of strength calculation problems:
Type I tasks - verification calculation or stress check. It is produced when the dimensions of the structure are already known and assigned and only a strength test needs to be carried out. In this case, use equations (4.11) or (4.12).
Type II problems - design calculations. Produced when the structure is at the design stage and some characteristic dimensions must be assigned directly from the strength condition.

For plastic materials:

For fragile materials:

Where A is the cross-sectional area of ​​the beam. Of the two obtained area values, select the largest.
III type tasks - determination of permissible load [N]:

for plastic materials:

for brittle materials:

Of the two permissible load values, select the minimum.

Strength and stiffness calculations are carried out using two methods: permissible stresses, deformations And permissible load method.

Voltages, for which a sample from of this material collapses or in which significant plastic deformations develop are called extreme. These stresses depend on the properties of the material and the type of deformation.

Voltage, the value of which is regulated technical specifications, called permissible.

Allowable voltage– this is the highest stress at which the required strength, rigidity and durability of a structural element is ensured under the given operating conditions.

The permissible stress is a certain fraction of ultimate voltage:

where is normative safety factor, a number showing how many times the permissible voltage is less than the maximum.

For plastic materials the permissible stress is chosen so that in case of any calculation inaccuracies or unforeseen operating conditions, residual deformations do not occur in the material, i.e. (yield strength):

Where - safety factor in relation to .

For brittle materials, permissible stresses are assigned based on the condition that the material does not collapse, i.e. (tensile strength):

Where - safety factor in relation to .

In mechanical engineering (under static loading), safety factors are taken: for plastic materials =1,4 – 1,8 ; for fragile ones - =2,5 – 3,0 .

Strength calculation based on permissible stresses is based on the fact that the maximum design stress in the dangerous section of the rod structure does not exceed the permissible value (less than - no more than 10%, more - no more than 5%):

Stiffness rating the rod structure is carried out on the basis of checking the conditions of tensile rigidity:

The amount of permissible absolute deformation [∆l] assigned separately for each design.

Permissible load method is that the internal forces arising in the most dangerous section of the structure during operation should not exceed the permissible load values:

, (2.23)

where is the breaking load obtained as a result of calculations or experiments taking into account manufacturing and operating experience;

– safety factor.

In the future we will use the method of permissible stresses and deformations.

2.6. Checking and design calculations

for strength and rigidity

The strength condition (2.21) makes it possible to carry out three types of calculations:

– check– according to the known dimensions and material of the rod element (the cross-sectional area is specified A And [σ] ) check whether it is able to withstand the given load ( N):

; (2.24)

– design– according to known loads ( N– given) and the material of the element, i.e., according to the known [σ], pick up required dimensions cross section providing it safe work:

– determination of permissible external load– according to known sizes ( A– given) and the material of the structural element, i.e., according to the known [σ], find the permissible value of the external load:

Stiffness rating rod structure is carried out on the basis of checking the stiffness condition (2.22) and formula (2.10) under tension:

. (2.27)

The amount of permissible absolute deformation [∆ l] is assigned separately for each structure.

Similar to calculations for the strength condition, the stiffness condition also involves three types of calculations:

– hardness check of a given structural element, i.e. checking that condition (2.22) is met;

– calculation of the designed rod, i.e. selection of its cross section:

– performance setting of a given rod, i.e. definition permissible load:

. (2.29)

Strength analysis any design contains the following main steps:

1. Determination of all external forces and support reaction forces.

2. Construction of graphs (diagrams) of force factors acting in cross sections along the length of the rod.

3. Constructing graphs (diagrams) of stresses along the axis of the structure, finding the maximum stress. Checking the strength conditions in places of maximum stress values.

4. Constructing a graph (diagram) of the deformation of the rod structure, finding the maximum deformation. Checking stiffness conditions in sections.

Example 2.1. For the steel rod shown in rice. 9a, determine the longitudinal force in all cross sections N and voltage σ . Also determine vertical displacements δ for all cross sections of the rod. Display the results graphically by constructing diagrams N, σ And δ . Known: F 1 = 10 kN; F 2 = 40 kN; A 1 = 1 cm 2; A 2 = 2 cm 2; l 1 = 2 m; l 2 = 1 m.

Solution. For determining N, using the ROZU method, mentally cut the rod into sections I−I And II−II. From the condition of equilibrium of the part of the rod below the section I−I (Fig. 9.b) we get (stretching). From the condition of equilibrium of the rod below the section II−II (Fig. 9c) we get

from where (compression). Having chosen the scale, we build a diagram of longitudinal forces ( rice. 9g). In this case, we consider the tensile force to be positive and the compressive force to be negative.

The stresses are equal: in the sections of the lower part of the rod ( rice. 9b)

(stretch);

in sections of the upper part of the rod

(compression).

On the selected scale we construct a stress diagram ( rice. 9d).

To plot a diagram δ determine the displacements of characteristic sections B−B And S−S(section movement A−A equals zero).

Section B−B will move up because top part shrinks:

The displacement of the section caused by tension is considered positive, and that caused by compression - negative.

Moving a section S−S is the algebraic sum of displacements B−B (δ V) and lengthening part of the rod with a length l 1:

On a certain scale, we plot the values ​​of and , connect the resulting points with straight lines, since under the action of concentrated external forces the displacements linearly depend on the abscissa of the sections of the rod, and we obtain a graph (diagram) of displacements ( rice. 9e). From the diagram it is clear that some section D–D doesn't move. Sections located above the section D–D, move upward (the rod is compressed); the sections located below move downwards (the rod is stretched).

Questions for self-control

1. How are the values ​​of axial force in the cross sections of a rod calculated?

2. What is a diagram of longitudinal forces and how is it constructed?

3. How are normal stresses distributed in the cross sections of a centrally stretched (compressed) rod and what are they equal to?

4. How is the diagram of normal stresses under tension (compression) constructed?

5. What is called absolute and relative longitudinal deformation? Their dimensions?

6. What is the cross-sectional stiffness under tension (compression)?

8. How is Hooke's law formulated?

9. Absolute and relative transverse deformations of the rod. Poisson's ratio.

10. What is the permissible stress? How is it selected for ductile and brittle materials?

11. What is called the safety factor and what main factors does its value depend on?

12. Name the mechanical characteristics of strength and ductility construction materials.

Table 2.4

Fig.2.22

Fig.2.18

Fig.2.17

Rice. 2.15

For tensile tests, tensile testing machines are used, which make it possible to record a diagram in “load – absolute elongation” coordinates during testing. The nature of the stress-strain diagram depends on the properties of the material being tested and on the rate of deformation. Typical view Such a diagram for low-carbon steel with a static load is shown in Fig. 2.16.

Let us consider the characteristic sections and points of this diagram, as well as the corresponding stages of sample deformation:

OA – Hooke’s law is valid;

AB – residual (plastic) deformations have appeared;

BC – plastic deformations increase;

SD – yield plateau (increase in strain occurs at constant load);

DC – area of ​​strengthening (the material again acquires the ability to increase resistance to further deformation and accepts a force that increases to a certain limit);

Point K – the test was stopped and the sample was unloaded;

KN – unloading line;

NKL – line of repeated loading of the sample (KL – hardening section);

LM is the area where the load drops, at this moment a so-called neck appears on the sample - a local narrowing;

Point M – sample rupture;

After rupture, the sample has the appearance approximately shown in Fig. 2.17. The fragments can be folded and the length after the test ℓ 1, as well as the diameter of the neck d 1, can be measured.

As a result of processing the tensile diagram and measuring the sample, we obtain a number of mechanical characteristics that can be divided into two groups - strength characteristics and plasticity characteristics.

Strength characteristics

Proportionality limit:

The maximum voltage up to which Hooke's law is valid.

Yield Strength:

The lowest stress at which deformation of the sample occurs under constant tensile force.

Tensile strength (temporary strength):

The highest voltage observed during the test.

Voltage at break:

The stress at break determined in this way is very arbitrary and cannot be used as a characteristic of the mechanical properties of steel. The convention is that it is obtained by dividing the force at the moment of rupture by the initial cross-sectional area of ​​the sample, and not by its actual area at rupture, which is significantly less than the initial one due to the formation of a neck.

Plasticity characteristics

Let us recall that plasticity is the ability of a material to deform without fracture. Plasticity characteristics are deformation, therefore they are determined from measurement data of the sample after fracture:

∆ℓ ос = ℓ 1 - ℓ 0 – residual elongation,

– neck area.

Relative elongation after break:

. (2.25)

This characteristic depends not only on the material, but also on the ratio of the sample dimensions. That is why standard samples have a fixed ratio ℓ 0 = 5d 0 or ℓ 0 = 10d 0 and the value of δ is always given with an index - δ 5 or δ 10, and δ 5 > δ 10.

Relative narrowing after rupture:

. (2.26)

Specific work of deformation:

where A is the work spent on destruction of the sample; is found as the area bounded by the stretch diagram and the x-axis (area of ​​the figure OABCDKLMR). Specific work of deformation characterizes the ability of a material to resist the impact of a load.

Of all the mechanical characteristics obtained during testing, the main characteristics of strength are the yield strength σ t and the tensile strength σ pch, and the main characteristics of plasticity are the relative elongation δ and the relative contraction ψ after rupture.

Unloading and reloading

When describing the tensile diagram, it was indicated that at point K the test was stopped and the sample was unloaded. The unloading process was described by straight line KN (Fig. 2.16), parallel to the straight section OA of the diagram. This means that the elongation of the sample ∆ℓ′ P, obtained before the start of unloading, does not completely disappear. The disappeared part of the elongation in the diagram is depicted by the segment NQ, the remaining part by the segment ON. Consequently, the total elongation of a sample beyond the elastic limit consists of two parts - elastic and residual (plastic):

∆ℓ′ P = ∆ℓ′ up + ∆ℓ′ os.

This will happen until the sample ruptures. After rupture, the elastic component of the total elongation (segment ∆ℓ up) disappears. The residual elongation is depicted by the segment ∆ℓ os. If you stop loading and unload the sample within the OB section, then the unloading process will be depicted by a line coinciding with the load line - the deformation is purely elastic.

When a sample of length ℓ 0 + ∆ℓ′ oc is re-loaded, the loading line practically coincides with the unloading line NK. The limit of proportionality increased and became equal to the voltage from which the unloading was carried out. Next, straight line NK turned into curve KL without a yield plateau. The part of the diagram located to the left of the NK line turned out to be cut off, i.e. the origin of coordinates moved to point N. Thus, as a result of stretching beyond the yield point, the sample changed its mechanical properties:

1). the limit of proportionality has increased;

2). the turnover platform has disappeared;

3). the relative elongation after rupture decreased.

This change in properties is called hardened.

When hardened, elastic properties increase and ductility decreases. In some cases (for example, when machining) the phenomenon of hardening is undesirable and is eliminated by heat treatment. In other cases, it is created artificially to improve the elasticity of parts or structures (shot processing of springs or stretching of cables of lifting machines).

Stress diagrams

To obtain a diagram characterizing the mechanical properties of the material, the primary tensile diagram in coordinates Р – ∆ℓ is reconstructed in coordinates σ – ε. Since the ordinates σ = Р/F and abscissas σ = ∆ℓ/ℓ are obtained by dividing by constants, the diagram has the same appearance as the original one (Fig. 2.18,a).

From the σ – ε diagram it is clear that

those. modulus of normal elasticity equal to tangent the angle of inclination of the straight section of the diagram to the abscissa axis.

From the stress diagram it is convenient to determine the so-called conditional yield strength. The fact is that most structural materials do not have a yield point - a straight line smoothly turns into a curve. In this case, the stress at which the relative permanent elongation is equal to 0.2% is taken as the value of the yield strength (conditional). In Fig. Figure 2.18b shows how the value of the conditional yield strength σ 0.2 is determined. The yield strength σ t, determined in the presence of a yield plateau, is often called physical.

The descending section of the diagram is conditional, since the actual cross-sectional area of ​​the sample after necking is significantly less than the initial area from which the coordinates of the diagram are determined. The true stress can be obtained if the magnitude of the force at each moment of time P t is divided by the actual cross-sectional area at the same moment of time F t:

In Fig. 2.18a, these voltages correspond to the dashed line. Up to the ultimate strength, S and σ practically coincide. At the moment of rupture, the true stress significantly exceeds the tensile strength σ pc and, even more so, the stress at the moment of rupture σ r. Let us express the area of ​​the neck F 1 through ψ and find S r.

Þ Þ .

For ductile steel ψ = 50 – 65%. If we take ψ = 50% = 0.5, then we get S р = 2σ р, i.e. the true stress is greatest at the moment of rupture, which is quite logical.

2.6.2. Compression test various materials

A compression test provides less information about the properties of a material than a tensile test. However, it is absolutely necessary to characterize the mechanical properties of the material. It is carried out on samples in the form of cylinders, the height of which is not more than 1.5 times the diameter, or on samples in the form of cubes.

Let's look at the compression diagrams of steel and cast iron. For clarity, we depict them in the same figure with the tensile diagrams of these materials (Fig. 2.19). In the first quarter there are tension diagrams, and in the third – compression diagrams.

At the beginning of loading, the steel compression diagram is an inclined straight line with the same slope as during tension. Then the diagram moves into the yield area (the yield area is not as clearly expressed as during tension). Further, the curve bends slightly and does not break off, because the steel sample is not destroyed, but only flattened. The modulus of elasticity of steel E under compression and tension is the same. The yield strength σ t + = σ t - is also the same. It is impossible to obtain compressive strength, just as it is impossible to obtain plasticity characteristics.

The tension and compression diagrams of cast iron are similar in shape: they bend from the very beginning and break off when the maximum load is reached. However, cast iron works better in compression than in tension (σ inch - = 5 σ inch +). Tensile strength σ pch is the only mechanical characteristic of cast iron obtained during compression testing.

The friction that occurs during testing between the machine plates and the ends of the sample has a significant impact on the test results and the nature of destruction. The cylindrical steel sample takes on a barrel shape (Fig. 2.20a), cracks appear in the cast iron cube at an angle of 45 0 to the direction of the load. If we exclude the influence of friction by lubricating the ends of the sample with paraffin, cracks will appear in the direction of the load and greatest strength will be less (Fig. 2.20, b and c). Most brittle materials (concrete, stone) fail under compression in the same way as cast iron and have a similar compression diagram.

It is of interest to test wood - anisotropic, i.e. having different strength depending on the direction of the force in relation to the direction of the fibers of the material. More and more widely used fiberglass plastics are also anisotropic. When compressed along the fibers, wood is much stronger than when compressed across the fibers (curves 1 and 2 in Fig. 2.21). Curve 1 is similar to the compression curves of brittle materials. Destruction occurs due to the displacement of one part of the cube relative to the other (Fig. 2.20, d). When compressed across the fibers, the wood does not collapse, but is pressed (Fig. 2.20e).

When testing a steel sample for tension, we discovered a change in the mechanical properties as a result of stretching until noticeable residual deformations appeared - cold hardening. Let's see how the sample behaves after hardening during a compression test. In Fig. 2.19 the diagram is shown with a dotted line. Compression follows the NC 2 L 2 curve, which is located above the compression diagram of the sample that was not subjected to work hardening OC 1 L 1 , and almost parallel to the latter. After hardening by tension, the limits of proportionality and compressive yield decrease. This phenomenon is called the Bauschinger effect, named after the scientist who first described it.

2.6.3. Hardness determination

A very common mechanical and technological test is the determination of hardness. This is due to the speed and simplicity of such tests and the value of the information obtained: hardness characterizes the state of the surface of a part before and after technological processing (hardening, nitriding, etc.), from which one can indirectly judge the magnitude of the tensile strength.

Hardness of the material called the ability to resist the mechanical penetration of another, more solid. The quantities characterizing hardness are called hardness numbers. Determined by different methods, they differ in size and dimension and are always accompanied by an indication of the method for their determination.

The most common method is the Brinell method. The test consists of pressing a hardened steel ball of diameter D into the sample (Fig. 2.22a). The ball is held for some time under load P, as a result of which an imprint (hole) of diameter d remains on the surface. The ratio of the load in kN to the surface area of ​​the print in cm 2 is called the Brinell hardness number

. (2.30)

To determine the Brinell hardness number, special testing instruments are used; the diameter of the indentation is measured with a portable microscope. Usually HB is not calculated using formula (2.30), but is found from tables.

Using the hardness number HB, it is possible to obtain an approximate value of the tensile strength of some metals without destroying the sample, because there is a linear relationship between σ inch and HB: σ inch = k ∙ HB (for low-carbon steel k = 0.36, for high-strength steel k = 0.33, for cast iron k = 0.15, for aluminum alloys k = 0.38 , for titanium alloys k = 0.3).

A very convenient and widespread method for determining hardness according to Rockwell. In this method, a diamond cone with an apex angle of 120 degrees and a radius of curvature of 0.2 mm, or a steel ball with a diameter of 1.5875 mm (1/16 inch) is used as an indenter pressed into the sample. The test takes place according to the scheme shown in Fig. 2.22, b. First, the cone is pressed in with a preliminary load P0 = 100 N, which is not removed until the end of the test. With this load, the cone is immersed to a depth h0. Then the full load P = P 0 + P 1 is applied to the cone (two options: A – P 1 = 500 N and C – P 1 = 1400 N), and the indentation depth increases. After removing the main load P 1, the depth h 1 remains. The indentation depth obtained due to the main load P 1, equal to h = h 1 – h 0, characterizes the Rockwell hardness. The hardness number is determined by the formula

, (2.31)

where 0.002 is the scale division value of the hardness tester indicator.

There are other methods for determining hardness (Vickers, Shore, microhardness), which are not discussed here.