Allowable stresses and mechanical properties of materials. Limit and allowable stress What is the allowable stress

Allow to define limit stresses(), in which the sample material is directly destroyed or large plastic deformations occur in it.

Ultimate stress in strength calculations

As limit stress in strength calculations it is assumed:

yield point for a plastic material (it is believed that the destruction of a plastic material begins when noticeable plastic deformations appear in it)

,

tensile strength for a brittle material, the value of which is different at:

To ensure a real part, it is necessary to choose its dimensions and material in such a way that the largest that arises at a certain point during operation is less than the limit:

However, even if the highest design stress in the part is close to the ultimate stress, its strength cannot yet be guaranteed.

Acting on a part cannot be established accurately enough,

design stresses in a part can sometimes be calculated only approximately,

deviations from the calculated characteristics are possible.

The part must be designed with some calculated safety factor:

.

It is clear that the larger n, the stronger the part. However, very large safety factor leads to waste of material, and this makes the part heavy and uneconomical.

The required safety factor is set depending on the purpose of the structure.

Strength condition: the strength of the part is considered to be ensured if. Using the expression , rewrite strength condition as:

From here you can get another form of recording strength conditions:

The relation on the right-hand side of the last inequality is called permissible voltage:

If the limiting and, therefore, allowable stresses in tension and compression are different, they are denoted by and. Using the concept allowable voltage, can strength condition be formulated as follows: the strength of a part is ensured if the arising in it greatest stress does not exceed allowable voltage.

Limiting voltage consider the stress at which a dangerous state occurs in the material (destruction or dangerous deformation).

For plastic materials, the limiting stress is considered yield point, since the arising plastic deformations do not disappear after removing the load:

For fragile materials where plastic deformations are absent, and fracture occurs in a brittle type (necks are not formed), the ultimate stress is taken tensile strength:

For plastic-brittle materials, the limiting stress is the stress corresponding to the maximum deformation of 0.2% (one hundred, 2):

Allowable voltage- the maximum stress at which the material should work normally.

Allowable stresses are obtained according to the limiting ones, taking into account the safety factor:

where [σ] is the permissible stress; s- safety factor; [s] - permissible safety factor.

Note. In square brackets, it is customary to denote the permissible value of the quantity.

Allowable safety factor depends on the quality of the material, the working conditions of the part, the purpose of the part, the accuracy of processing and calculation, etc.

It can range from 1.25 for simple parts to 12.5 for complex parts operating under varying loads, shock and vibration.

Features of the behavior of materials during compression tests:

1. Plastic materials perform almost equally in tension and compression. The tensile and compressive mechanical properties are the same.

2. Brittle materials usually have higher compressive strength than tensile strength: σ bp< σ вс.

If the allowable tensile and compressive stresses are different, they are denoted by [σ ​​p] (tension), [σ c] (compression).

Tensile and Compressive Strength Calculations

Strength calculations are carried out according to strength conditions - inequalities, the fulfillment of which guarantees the strength of the part under these conditions.

To ensure strength, the calculated stress should not exceed the permissible stress:

Rated voltage a depends on load and size cross-section allowed only from part material and working conditions.

There are three types of strength calculation.

1. Design calculation - the design scheme and loads are set; material or dimensions of the part are selected:

Determination of cross-sectional dimensions:

Selection of material

according to the value of σ before, you can select the grade of the material.

2. Verification calculation - the loads, material, dimensions of the part are known; necessary check if the strength is ensured.

The inequality is checked

3. Determination of load capacity(maximum load):

Examples of problem solving

A straight beam is stretched with a force of 150 kN (Fig. 22.6), the material is steel σ t = 570 MPa, σ b = 720 MPa, safety factor [s] = 1.5. Determine the dimensions of the cross-section of the timber.

Solution

1. Condition of strength:

2. The required cross-sectional area is determined by the ratio

3. The allowable stress for the material is calculated from the specified mechanical properties. The presence of the yield point means that the material is plastic.

4. Determine the size of the required cross-sectional area of ​​the bar and select the dimensions for two cases.

Section - a circle, define the diameter.

The resulting value is rounded up d = 25 mm, A = 4.91 cm 2.

Section - equal angle corner No. 5 in accordance with GOST 8509-86.

The closest cross-sectional area of ​​the corner is A = 4.29 cm 2 (d = 5 mm). 4.91> 4.29 (Appendix 1).

Test questions and tasks

1. What phenomenon is called fluidity?

2. What is a “neck”, at what point of the tension diagram does it form?

3. Why are the mechanical characteristics obtained during the tests conditional?

4. List the strength characteristics.

5. List the characteristics of plasticity.

6. What is the difference between an auto-drawn stretch chart and a referenced stretch chart?

7. Which of the mechanical characteristics is chosen as the ultimate stress for ductile and brittle materials?

8. What is the difference between ultimate and allowable stress?

9. Record the tensile and compressive strength condition. Are the tensile strength and compression strength conditions different?

Answer the questions of the test item.

The online calculator determines the calculated allowable stresses σ depending on the design temperature for various grades of materials of the following types: carbon steel, chromium steel, austenitic steel, austenite-ferritic steel, aluminum and its alloys, copper and its alloys, titanium and its alloys in accordance with GOST-52857.1-2007.

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I. Calculation method:

Allowable stresses were determined according to GOST-52857.1-2007.

for carbon and low alloy steels

St3, 09G2S, 16GS, 20, 20K, 10, 10G2, 09G2, 17GS, 17G1S, 10G2S1:

1. At design temperatures below 20 ° C, the permissible stresses are assumed to be the same as at 20 ° C, subject to the permissible use of the material at this temperature.
2. For steel grade 20 at R e / 20
3. For steel grade 10G2 at R p0.2 / 20
4. For steel grades 09G2S, 16GS, strength classes 265 and 296 in accordance with GOST 19281, the permissible stresses, regardless of the sheet thickness, are determined for thicknesses over 32 mm.
5. The permissible stresses located below the horizontal line are valid for a resource of not more than 10 5 hours. For an estimated service life of up to 2 * 10 5 hours, the permissible stress located below the horizontal line is multiplied by a factor: for carbon steel by 0.8; for manganese steel by 0.85 at a temperature< 450 °С и на 0,8 при температуре от 450 °С до 500 °С включительно.

for heat-resistant chromium steels

12XM, 12MX, 15XM, 15X5M, 15X5M-U:

1. At design temperatures below 20 ° C, the permissible stresses are assumed to be the same as at 20 ° C, provided that the material is used at a given temperature.
2. For intermediate design wall temperatures, the permissible stress is determined by linear interpolation with results rounded to 0.5 MPa downward.
3. The permissible stresses located below the horizontal line are valid for a resource of 10 5 hours. For an estimated service life of up to 2 * 10 5 hours, the permissible voltage located below the horizontal line is multiplied by a factor of 0.85.

for heat-resistant, heat-resistant and corrosion-resistant steels of the austenitic class

03X21H21M4GB, 03X18H11, 03X17H14M3, 08X18H10T, 08X18H12T, 08X17H13M2T, 08X17H15M3T, 12X18H10T, 12X18H12T, 10X17H13M2T, 10X17H13M3T, 10X14Г14H4:

1. For intermediate design wall temperatures, the permissible stress is determined by interpolation of the two nearest values ​​indicated in the table, with the results rounded to 0.5 MPa towards a lower value.
2. For forgings made of steel grades 12X18H10T, 10X17H13M2T, 10X17H13M3T, the permissible stresses at temperatures up to 550 ° C are multiplied by 0.83.
3. For long products made of steel grades 12X18H10T, 10X17H13M2T, 10X17H13M3T, the permissible stresses at temperatures up to 550 ° C are multiplied by the ratio (R * p0.2 / 20) / 240.
   (R * p0.2 / 20 - the yield point of the long product material is determined in accordance with GOST 5949).
4. For forgings and rolled sections made of 08X18H10T steel, the permissible stresses at temperatures up to 550 ° C are multiplied by 0.95.
5. For forgings made of steel grade 03X17H14M3, the permissible stresses are multiplied by 0.9.
6. For forgings made of steel grade 03X18H11, the permissible stresses are multiplied by 0.9; for long products made of steel grade 03X18H11, the permissible stresses are multiplied by 0.8.
7. For pipes made of steel grade 03X21N21M4GB (ZI-35), the permissible stresses are multiplied by 0.88.
8. For forgings made of steel grade 03X21N21M4GB (ZI-35), the permissible stresses are multiplied by the ratio (R * p0.2 / 20) / 250.
   (R * p0.2 / 20 is the yield point of the forgings material, determined in accordance with GOST 25054).
9. The permissible voltages below the horizontal line are valid for a resource of no more than 10 5 hours.

For an estimated service life of up to 2 * 10 5 h, the permissible voltage located below the horizontal line is multiplied by a factor of 0.9 at a temperature< 600 °С и на коэффициент 0,8 при температуре от 600 °С до 700 °С включительно.

for heat-resistant, heat-resistant and corrosion-resistant steels of austenitic and austenitic-ferritic class

08X18G8N2T (KO-3), 07X13AG20 (ChS-46), 02X8N22S6 (EP-794), 15X18N12S4TYu (EI-654), 06XN28MDT, 03XN28MDT, 08X22N6T, 08X21N6M2T:

1. At design temperatures below 20 ° C, the permissible stresses are assumed to be the same as at 20 ° C, subject to the permissible use of the material at this temperature.
2. For intermediate design wall temperatures, the permissible stress is determined by interpolation of the two nearest values ​​indicated in this table, rounded to 0.5 MPa towards a lower value.

for aluminum and its alloys

A85M, A8M, ADM, AD0M, AD1M, AMtsSM, AMr2M, AMr3M, AMr5M, AMr6M:

1. Allowable stresses are given for aluminum and its alloys in the annealed state.
2. Allowable stresses are given for thicknesses of sheets and plates of aluminum grades A85M, A8M no more than 30 mm, other grades - no more than 60 mm.

for copper and its alloys

M2, M3, M3r, L63, LS59-1, LO62-1, LZhMts 59-1-1:

1. Allowable stresses are given for copper and its alloys in the annealed state.
2. Allowable stresses are given for sheet thicknesses from 3 to 10 mm.
3. For intermediate values ​​of the design wall temperatures, the permissible stresses are determined by linear interpolation with the results rounded to 0.1 MPa downward.

for titanium and its alloys

VT1-0, OT4-0, AT3, VT1-00:

1. At design temperatures below 20 ° C, the permissible stresses are assumed to be the same as at 20 ° C, provided that the material can be used at a given temperature.
2. For forgings and rods, the allowable stresses are multiplied by 0.8.

II. Definitions and notation:

R e / 20 - the minimum value of the yield point at a temperature of 20 ° C, MPa; R p0.2 / 20 - the minimum value of the conventional yield stress at a residual elongation of 0.2% at a temperature of 20 ° C, MPa. permissible
stress - the highest stresses that can be tolerated in a structure, provided it is safe, reliable and durable. The value of the allowable stress is set by dividing the tensile strength, yield strength, etc., by a value greater than one, called the safety factor. calculated
temperature - the temperature of the equipment wall or pipeline, equal to the maximum arithmetic mean value of temperatures on its outer and inner surfaces in one section under normal operating conditions (for parts of nuclear reactor vessels, the design temperature is determined taking into account internal heat release as the average integral value of the temperature distribution over the thickness of the vessel wall (PNAE G-7-002-86, clause 2.2; PNAE G-7-008-89, appendix 1).

Design temperature

• , clause 5.1. The design temperature is used to determine the physical and mechanical characteristics of the material and the permissible stresses, as well as when calculating the strength, taking into account the temperature effects.
• , item 5.2. The design temperature is determined on the basis of heat engineering calculations or test results, or operating experience of similar vessels.
• The highest wall temperature is taken as the design temperature of the wall of the vessel or apparatus. At temperatures below 20 ° C, the temperature of 20 ° C is taken as the design temperature when determining the permissible stresses.
• , p.5.3. If it is impossible to carry out thermal calculations or measurements and if during operation the temperature of the wall rises to the temperature of the medium in contact with the wall, then the highest temperature of the medium should be taken as the design temperature, but not lower than 20 ° C.
• When heating with an open flame, exhaust gases or electric heaters, the design temperature is taken equal to the temperature of the environment, increased by 20 ° C with closed heating and by 50 ° C with direct heating, if there are no more accurate data.
• , item 5.4. If a vessel or apparatus is operated under several different loading conditions or different elements of the apparatus operate under different conditions, for each mode, you can determine its own design temperature (GOST-52857.1-2007, p. 5).

III. Note:

The source data block is highlighted in yellow, block of intermediate calculations is highlighted in blue, the decision block is highlighted in green.

To assess the strength of structural elements, the concepts of operating (design) stresses, ultimate stresses, permissible stresses and safety margins are introduced. They are calculated according to the dependencies presented in clauses 4.2, 4.3.

Working (design) voltages  and  characterize the stress state of structural elements under the action of an operational load.

Limiting stresses  lim and  lim characterize the mechanical properties of the material and are dangerous for a structural element in terms of its strength.

Allowable voltages [  ] and [  ] are safe and ensure the strength of the structural element in the given operating conditions.

Margin of safety n establishes the ratio of limiting and permissible stresses, taking into account the negative impact on the strength of various unaccounted for factors.

For the safe operation of parts of mechanisms, it is necessary that the maximum stresses arising in the loaded sections do not exceed the value allowed for a given material:

;
,

where
and
- the highest stresses (normal and tangential ) in the dangerous section;
and - permissible values ​​of these stresses.

With complex resistance, equivalent voltages are determined
in a dangerous section. The strength condition has the form

.

Allowable stresses are determined depending on the limiting stresses  lim and  lim obtained during material tests: under static loads - ultimate strength
and τ V for brittle materials, yield strength
and τ T for plastic materials; at cyclic loads - endurance limit and τ r :

;
.

Safety factor appointed on the basis of experience in the design and operation of similar structures.

For machine parts and mechanisms operating under cyclic loads and having a limited service life, the calculation of the permissible stresses is carried out according to the dependencies:

;
,

where
- coefficient of service life, taking into account the specified service life.

Calculate the coefficient of durability according to the dependence

,

where
- the base number of test cycles for a given material and type of deformation;
- the number of loading cycles of the part, corresponding to a given service life; m - an indicator of the degree of the endurance curve.

When designing structural elements, two methods of strength calculations are used:

design calculation for permissible stresses to determine the basic dimensions of the structure;

verification calculation to assess the performance of an existing structure.

5.5. Calculation examples

5.5.1. Calculation of stepped bars for static strength

R

Let us consider the stress state of stepped bars for simple types of deformations. In fig. 5.3 shows three schemes (cx. 1, 2, 3) of loading by forces F of round bars of variable cross-section, cantilevered in a rigid support, and three stress diagrams (ep. 1, 2, 3) acting in the cross-sections of loaded bars. Force F = 800 N is applied at a distance h = 10 mm from the axis of the rod. The smaller diameter of the rods is d = 5 mm, the larger one is D = 10 mm. Rod material - Art. 3 with permissible voltages
= 160 MPa and = 100 MPa.

For each of the presented schemes, we determine:

1. Type of deformation:

cx. 1 - stretching; cx. 2 - torsion; cx. 3 - pure bend.

2. Internal force factor:

cx. 1 - normal strength

N = 2F = 2800 = 1600 H;

cx. 2 - torque М Х = T = 2Fh = 280010 = 16000 N mm;

cx. 3 - bending moment M = 2Fh = 280010 = 16000 N mm.

3. Type of stresses and their magnitude in sections A and B:

cx. 1 - normal
:

MPa;

MPa;

cx. 2 - tangents
:

MPa;

MPa;

cx. 3 - normal
:

MPa;

MPa.

4. Which of the stress diagrams corresponds to each loading scheme:

cx. 1 - ep. 3; cx. 2 - ep. 2; cx. 3 - ep. one.

5. Fulfillment of the strength condition:

cx. 1 - the condition is met:
MPa
MPa;

cx. 2 - the condition is not met:
MPa
MPa;

cx. 3 - the condition is not met:
MPa
MPa.

6. The minimum permissible diameter that ensures the fulfillment of the strength condition:

cx. 2:
mm;

cx. 3:
mm.

7. Maximum allowable forceFfrom the strength condition:

cx. 2:
H;

cx. 3:
N.

The main task of calculating a structure is to ensure its strength under operating conditions.

The strength of a structure made of brittle metal is considered to be ensured if, in all cross-sections of all its elements, the actual stresses are less than the ultimate strength of the material. The magnitudes of loads, stresses in the structure and the ultimate strength of the material cannot be established absolutely accurately (due to the approximation of the calculation method, methods for determining the ultimate strength, etc.).

Therefore, it is necessary that the highest stresses obtained as a result of structural analysis (design stresses) do not exceed a certain value less than the ultimate strength, called the permissible stress. The value of the allowable stress is established by dividing the ultimate strength by a value greater than one, called the safety factor.

In accordance with the above, the condition for the strength of a structure made of a brittle material is expressed as

where are the highest calculated tensile and compressive stresses in the structure; and [are the permissible tensile and compressive stresses, respectively.

Allowable stresses depend on the tensile and compressive strength of the material sts and are determined by the expressions

where is the standard (required) safety factor in relation to the ultimate strength.

Absolute values ​​of stresses are substituted into formulas (39.2) and (40.2)

For structures made of plastic materials (for which the tensile and compressive strengths are the same), the following strength condition is used:

where a is the largest in absolute value compressive or tensile design stress in the structure.

The permissible stress for plastic materials is determined by the formula

where is the standard (required) safety factor in relation to the yield point.

The use of the yield point (and not the ultimate strength, as for brittle materials) in determining the permissible stresses for plastic materials is due to the fact that after reaching the yield point, deformations can increase very sharply even with a slight increase in the load and structures may cease to meet the conditions of their operation.

Strength analysis performed using strength conditions (39.2) or (41.2) is called allowable stress analysis. The load at which the highest stresses in the structure are equal to the permissible stresses is called permissible.

Deformations of a number of structures made of plastic materials after reaching the yield point do not increase sharply even with a significant increase in the load, if it does not exceed the value of the so-called ultimate load. Such, for example, are statically indeterminate structures (see § 9.2), as well as structures with elements undergoing bending or torsion deformations.

The design of these structures is carried out either according to the permissible stresses, that is, using the strength condition (41.2), or according to the so-called limit state. In the latter case, the permissible load is called the maximum permissible load, and its value is determined by dividing the maximum load by the standard safety factor of the bearing capacity. The two simplest examples of ultimate limit state analysis of a structure are given below in § 9.2 and calculation example 12.2.

One should strive to ensure that the permissible stresses are fully used, that is, the condition is satisfied if this fails for a number of reasons (for example, due to the need to standardize the dimensions of structural elements), then the calculated stresses should differ as little as possible from the permissible ones. It is possible that the calculated permissible stresses are slightly exceeded and, consequently, a slight decrease in the actual safety factor (in comparison with the standard).

The strength analysis of a centrally stretched or compressed structural member must ensure that the strength condition is met for all cross-sections of the member. In this case, the correct definition of the so-called dangerous sections of the element, in which the greatest tensile and greatest compressive stresses arise, is of great importance. In cases where the permissible tensile or compressive stresses are the same, it is sufficient to find one dangerous section in which there are the highest normal stresses in absolute value.

With a constant value of the longitudinal force along the length of the beam, the cross-section, the area of ​​which has the smallest value, is dangerous. With a beam of constant cross-section, the cross-section in which the greatest longitudinal force occurs is dangerous.

When calculating structures for strength, there are three types of problems that differ in the form of using strength conditions:

a) verification of stresses (verification calculation);

b) selection of sections (design calculation);

c) determination of carrying capacity (determination of permissible load). Let us consider these types of problems using the example of a stretched bar made of plastic material.

When checking the stresses, the cross-sectional areas F and the longitudinal forces N are known and the calculation consists in calculating the calculated (actual) stresses a in the characteristic sections of the elements.

The highest stress obtained in this case is then compared with the allowable one:

When selecting sections, the required cross-sectional areas of the element are determined (according to the known longitudinal forces N and the allowable stress). The accepted cross-sectional areas F must satisfy the strength condition expressed in the following form:

When determining the carrying capacity from the known values ​​of F and the permissible stress, the permissible values ​​of the longitudinal forces are calculated: From the obtained values, the permissible values ​​of external loads [P] are then determined.

For this case, the strength condition has the form

The values ​​of the standard safety factors are established by the standards. They depend on the class of the structure (capital, temporary, etc.), the intended life of its operation, load (static, cyclic, etc.), possible inhomogeneity in the manufacture of materials (for example, concrete), on the type of deformation (tension, compression , bending, etc.) and other factors. In some cases, it is necessary to reduce the safety factor in order to reduce the weight of the structure, and sometimes to increase the safety factor - if necessary, take into account the wear of rubbing parts of machines, corrosion and material decay.

The values ​​of the standard safety factors for various materials, structures and loads have in most cases the values: - from 2.5 to 5 and - from 1.5 to 2.5.

The safety factors and, consequently, the permissible stresses for building structures are regulated by the corresponding design standards. In mechanical engineering, the required safety factor is usually chosen, focusing on the experience in the design and operation of machines of similar structures. In addition, a number of advanced machine-building plants have in-house standards for permissible voltages, which are often used by other related enterprises.

Guideline values ​​for allowable tensile and compressive stresses for a range of materials are given in annex II.