Limit and permissible stresses. Allowable stresses Allowable stress is determined by the formula

To determine permissible stresses in mechanical engineering, the following basic methods are used.
1. A differentiated safety factor is found as the product of a number of partial coefficients that take into account the reliability of the material, the degree of responsibility of the part, the accuracy of the calculation formulas and the acting forces and other factors that determine the operating conditions of the parts.
2. Tabular - permissible voltages are taken according to standards systematized in the form of tables
(Tables 1 – 7). This method is less accurate, but is the simplest and most convenient for practical use in design and testing strength calculations.

In the work of design bureaus and in the calculations of machine parts, both differentiated and tabular methods, as well as their combination. In table 4 – 6 show the permissible stresses for non-standard cast parts for which special calculation methods and the corresponding permissible stresses have not been developed. Typical parts (for example, gears and worm wheels, pulleys) should be calculated using the methods given in the corresponding section of the reference book or specialized literature.

The permissible stresses given are intended for approximate calculations only for basic loads. For more accurate calculations taking into account additional loads (for example, dynamic), the table values ​​should be increased by 20 - 30%.

Allowable stresses are given without taking into account the stress concentration and dimensions of the part, calculated for smooth polished steel samples with a diameter of 6-12 mm and for untreated round cast iron castings with a diameter of 30 mm. When determining the highest stresses in the part being calculated, it is necessary to multiply the nominal stresses σ nom and τ nom by the concentration factor k σ or k τ:

1. Permissible stresses*
for carbon steels of ordinary quality in hot-rolled condition

2. Mechanical properties and permissible stresses
carbon quality structural steels

3. Mechanical properties and permissible stresses
alloyed structural steels

4. Mechanical properties and permissible stresses
for castings made of carbon and alloy steels

5. Mechanical properties and permissible stresses
for gray cast iron castings

6. Mechanical properties and permissible stresses
for ductile iron castings

For ductile (unhardened) steels for static stresses (I type of load), the concentration coefficient is not taken into account. For homogeneous steels (σ in > 1300 MPa, as well as in the case of their operation at low temperatures), the concentration coefficient, in the presence of stress concentration, is introduced into the calculation under loads I type (k > 1). For ductile steels under variable loads and in the presence of stress concentrations, these stresses must be taken into account.

For cast iron in most cases, the stress concentration coefficient is approximately equal to unity for all types of loads (I – III). When calculating strength to take into account the dimensions of the part, the given tabulated permissible stresses for cast parts should be multiplied by a scale factor equal to 1.4 ... 5.

Approximate empirical dependences of endurance limits for cases of loading with a symmetrical cycle:

for carbon steels:
– when bending, σ -1 =(0.40÷0.46)σ in;
σ -1р =(0.65÷0.75)σ -1;
– during torsion, τ -1 =(0.55÷0.65)σ -1;

for alloy steels:
– when bending, σ -1 =(0.45÷0.55)σ in;
- when stretched or compressed, σ -1р =(0.70÷0.90)σ -1;
– during torsion, τ -1 =(0.50÷0.65)σ -1;

for steel casting:
– when bending, σ -1 =(0.35÷0.45)σ in;
- when stretched or compressed, σ -1р =(0.65÷0.75)σ -1;
– during torsion, τ -1 =(0.55÷0.65)σ -1.

Mechanical properties and permissible stresses of anti-friction cast iron:
– ultimate bending strength 250 – 300 MPa,
– permissible bending stresses: 95 MPa for I; 70 MPa – II: 45 MPa – III, where I. II, III are designations of types of load, see table. 1.

Approximate permissible stresses for non-ferrous metals in tension and compression. MPa:
– 30…110 – for copper;
– 60…130 – brass;
– 50…110 – bronze;
– 25…70 – aluminum;
– 70…140 – duralumin.

The online calculator determines the estimated permissible stresses σ depending on the design temperature for various grades of materials of the following types: carbon steel, chromium steel, austenitic class steel, austenitic-ferritic class steel, aluminum and its alloys, copper and its alloys, titanium and its alloys according to 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

1. At design temperatures below 20°C, the permissible stresses are taken to be the same as at 20°C, subject to the permissible use of the material at a given temperature.
2. For steel grade 20 at R e/20
3. For steel grade 10G2 at R р0.2/20
4. For steel grades 09G2S, 16GS, strength classes 265 and 296 according to 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 service life of no more than 10 5 hours. For a design service life of up to 2 * 10 5 hours, the permissible stress located below the horizontal line is multiplied by the coefficient: 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

1. At design temperatures below 20 °C, the permissible stresses are taken to be the same as at 20 °C, subject to the permissible use of the material at a given temperature.
2. For intermediate design wall temperatures, the permissible stress is determined by linear interpolation with rounding the results down to 0.5 MPa.
3. The permissible stresses located below the horizontal line are valid for a service life of 10 5 hours. For a design service life of up to 2 * 10 5 hours, the permissible stress located below the horizontal line is multiplied by a factor of 0.85.

for heat-resistant, heat-resistant and corrosion-resistant austenitic steels

1. For intermediate design wall temperatures, the permissible stress is determined by interpolating the two closest values ​​​​indicated in the table, with the results rounded down to the nearest 0.5 MPa.
2. For forgings made of steel grades 12Х18Н10Т, 10Х17Н13M2T, 10Х17Н13М3Т, the permissible stresses at temperatures up to 550 °C are multiplied by 0.83.
3. For long rolled steel grades 12Х18Н10Т, 10Х17Н13M2T, 10Х17Н13М3Т, permissible stresses at temperatures up to 550 °C are multiplied by the ratio (R* p0.2/20) / 240.
   (R* p0.2/20 - the yield strength of the rolled steel material is determined according to GOST 5949).
4. For forgings and long products made of steel grade 08X18H10T, 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 03Х21Н21М4ГБ (ZI-35), the permissible stresses are multiplied by 0.88.
8. For forgings made of steel grade 03Х21Н21М4ГБ (ZI-35), the permissible stresses are multiplied by the ratio (R* p0.2/20) / 250.
   (R* p0.2/20 is the yield strength of the forging material, determined according to GOST 25054).
9. The permissible stresses located below the horizontal line are valid for a service life of no more than 10 5 hours.

For a design service life of up to 2*10 5 hours, the permissible voltage located below the horizontal line is multiplied by a factor of 0.9 at 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

1. At design temperatures below 20 °C, the permissible stresses are taken to be the same as at 20 °C, subject to the permissible use of the material at a given temperature.
2. For intermediate design wall temperatures, the permissible stress is determined by interpolating the two closest values ​​indicated in this table, rounding down to the nearest 0.5 MPa.

for aluminum and its alloys

1. Allowable stresses are given for aluminum and its alloys in the annealed state.
2. The permissible stresses are given for the thickness 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

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 calculated wall temperatures, the permissible stresses are determined by linear interpolation with rounding the results to 0.1 MPa towards the lower value.

for titanium and its alloys

1. At design temperatures below 20 °C, the permissible stresses are taken to be the same as at 20 °C, subject to the permissibility of using the material at a given temperature.
2. For forgings and rods, the permissible stresses are multiplied by 0.8.

II. Definitions and notations:

R e/20 - minimum value of the yield strength at a temperature of 20 °C, MPa; R р0.2/20 - the minimum value of the conditional yield strength at a permanent elongation of 0.2% at a temperature of 20 °C, MPa. permissible
tension - the highest stresses that can be allowed in a structure, subject to its safe, reliable and durable operation. The value of the permissible stress is established by dividing the tensile strength, yield strength, etc. by a value greater than one, called the safety factor. calculated
temperature - the temperature of the wall of equipment or pipeline, equal to the maximum arithmetic mean value of the 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 releases 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 permissible stresses, as well as when calculating strength taking into account temperature effects.
• ,Clause 5.2. The design temperature is determined on the basis of thermal 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, a temperature of 20 °C is taken as the design temperature when determining permissible stresses.
• ,section 5.3. If it is impossible to carry out thermal calculations or measurements and if during operation the wall temperature rises to the temperature of the medium in contact with the wall, then the highest temperature of the medium, but not lower than 20 °C, should be taken as the design temperature.
• When heating with an open flame, exhaust gases or electric heaters, the design temperature is taken equal to the ambient temperature increased by 20 °C for closed heating and by 50 °C for direct heating, unless more accurate data are available.
• ,section 5.4. If a vessel or apparatus is operated under several different loading modes or different elements of the apparatus operate under different conditions, for each mode its own design temperature can be determined (GOST-52857.1-2007, clause 5).

III. Note:

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

Ultimate voltage They consider the stress at which a dangerous condition occurs in a material (fracture or dangerous deformation).

For plastic materials the ultimate stress is considered yield strength, because the resulting plastic deformations do not disappear after removing the load:

For fragile materials where there are no plastic deformations, and fracture occurs of the brittle type (no necking is formed), the ultimate stress is taken tensile strength:

For ductile-brittle materials, the ultimate stress is considered to be the stress corresponding to a maximum deformation of 0.2% (one hundred.2):

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

The permissible stresses are obtained according to the limit values, taking into account the safety factor:

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

Note. It is customary to indicate the permissible value of a quantity in square brackets.

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

It can range from 1.25 for simple parts to 12.5 for complex parts operating under variable loads under conditions of shock and vibration.

Features of the behavior of materials during compression tests:

1. Plastic materials work almost equally under tension and compression. The mechanical characteristics in tension and compression are the same.

2. Brittle materials usually have greater compressive strength than tensile strength: σ vr< σ вс.

If the permissible stress in tension and compression is different, they are designated [σ р ] (tension), [σ с ] (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 given conditions.

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

Design voltage A depends on load and size cross-section, permitted only from the material of the part and working conditions.

There are three types of strength calculations.

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

Determination of cross-section dimensions:

Material selection

Based on the value of σ, it is possible to select the grade of material.

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

Inequality is checked

3. Determination of load capacity(maximum load):

Examples of problem solving

The 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 cross-sectional dimensions of the beam.

Solution

1. Strength condition:

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

3. The permissible stress for the material is calculated from the specified mechanical characteristics. The presence of a yield point means that the material is plastic.

4. We determine the required cross-sectional area of ​​the beam and select dimensions for two cases.

The cross section is a circle, we determine the diameter.

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

Section - equal angle angle No. 5 according to 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 assignments

1. What phenomenon is called fluidity?

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

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

4. List the strength characteristics.

5. List the characteristics of plasticity.

6. What is the difference between an automatically drawn stretch diagram and a given stretch diagram?

7. Which mechanical characteristic is chosen as the limiting stress for ductile and brittle materials?

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

9. Write down the condition for tensile and compressive strength. Are the strength conditions different for tensile and compressive calculations?

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

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. A machine or building structure 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 of the maximum load causing the loss of strength of the element to the load creating the 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).

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

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. A machine or building structure 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.

Margin of safety. 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) defects in the material and subsequent reduction in strength due to metal corrosion, wood rotting, etc.

Safety factor. The safety factor of any structural element is equal to the ratio of the maximum load causing the loss of strength of the element to the load creating the 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). So, if the tensile strength of structural steel is 540 MPa, and the permissible stress is 180 MPa, then the safety factor is 3.