Permissible (permissible) stress is the value of stress, which is considered to be the maximum acceptable when calculating the dimensions of the cross-section of the element, calculated for a given load. We can talk about the allowable tensile, compressive and shear stresses. Allowable voltages are either prescribed by a competent authority (say, the department of control bridges railway), or are selected by a designer who is well aware of the properties of the material and the conditions for its use. The allowable stress limits the maximum operating stress 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 at which 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 failure. A machine or building structure is considered to have failed when it cannot satisfactorily perform its function. A part made of plastic material, as a rule, loses strength when the stress in it reaches the yield strength, because in this case, due to too much deformation of the part, the machine or structure ceases to be suitable for its intended purpose. If the part is made of a brittle material, then it almost does not deform, and its loss of strength coincides with its destruction.
The difference between the stress at which the material loses strength and the allowable stress is the "margin of safety" that must be taken into account, 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 decrease in strength due to metal corrosion, wood decay, etc.
The safety factor of any structural element is equal to the ratio ultimate load, causing the loss of strength of the element, to the load that creates the allowable stress. In this case, the loss of strength is understood not only as 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, the working stresses in the structural elements are proportional to the loads, and therefore the safety factor is defined as the ratio of the ultimate strength to the allowable stress (the safety factor for the ultimate strength).
Permissible (permissible) voltage- this is the value of stress, which is considered to be the maximum acceptable when calculating the dimensions of the cross-section of the element, calculated for a given load. We can talk about the allowable tensile, compressive and shear stresses. Allowable stresses are either prescribed by a competent authority (say, the department of bridges of the railway control), or are selected by a designer who knows the properties of the material well and the conditions for its use. The allowable stress limits the maximum operating stress 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 at which 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 failure. A machine or building structure is considered to have failed when it cannot satisfactorily perform its function. A part made of plastic material, as a rule, loses strength when the stress in it reaches the yield strength, because in this case, due to too much deformation of the part, the machine or structure ceases to be suitable for its intended purpose. If the part is made of a brittle material, then it almost does not deform, 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 allowable stress is the “margin of safety” that must be taken into account, 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 decrease in strength due to metal corrosion, wood decay, etc.
stock factor. The safety factor of any structural element is equal to the ratio of the ultimate load that causes the element's strength loss to the load that creates the allowable stress. In this case, the loss of strength is understood not only as 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, the working stresses in the structural elements are proportional to the loads, and therefore the safety factor is defined as the ratio of the ultimate strength to the allowable stress (the safety factor for the ultimate strength). So, if the tensile strength of structural steel is 540 MPa, and the allowable stress is 180 MPa, then the safety factor is 3.
To determine the allowable stresses in mechanical engineering, the following basic methods are used.
1. A differentiated margin of safety is found as a 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 working conditions of the parts.
2. Tabular - allowable stresses are taken according to the standards systematized in the form of tables
(Tables 1 - 7). This method is less accurate, but the simplest and most convenient for practical use in design and verification strength calculations.
In the work of design bureaus and in the calculation of machine parts, both differentiated and tabular methods, as well as their combination. In table. 4 - 6 shows the allowable stresses for non-standard cast parts for which special calculation methods have not been developed and the allowable stresses corresponding to them. Typical parts (for example, gears and worm wheels, pulleys) should be calculated according to the methods given in the relevant section of the handbook or special literature.
The given allowable stresses are intended for approximate calculations only for the main loads. For more accurate calculations, taking into account additional loads (for example, dynamic), the table values \u200b\u200bshould be increased by 20 - 30%.
Permissible 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 calculated part, it is necessary to multiply the rated stresses σ nom and τ nom by the concentration factor k σ or k τ:
1. Permissible stresses*
for carbon steels ordinary quality hot rolled
2. Mechanical properties and allowable stresses
carbon quality structural steels
3. Mechanical properties and allowable stresses
alloy structural steels
4. Mechanical properties and allowable stresses
for castings made of carbon and alloy steels
5. Mechanical properties and allowable stresses
for gray iron castings
6. Mechanical properties and allowable stresses
for ductile iron castings
For ductile (non-hardened) steels at static stresses (I type of load), the concentration factor 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 factor, in the presence of stress concentration, is also taken into account under loads I of the form (k > 1). For ductile steels under the action of variable loads and in the presence of stress concentration, these stresses must be taken into account.
For cast iron in most cases, the stress concentration factor is approximately taken equal to unity for all types of loads (I - III). When calculating the strength to take into account the dimensions of the part, the given tabular allowable stresses for cast parts should be multiplied by a scale factor equal to 1.4 ... 5.
Approximate empirical dependencies of fatigue limits for loading cases with a symmetrical cycle:
for carbon steels:
- when bending σ -1 \u003d (0.40 ÷ 0.46) σ in;
σ -1р = (0.65÷0.75)σ -1;
- when twisting τ -1 =(0.55÷0.65)σ -1;
for alloy steels:
- when bending σ -1 \u003d (0.45 ÷ 0.55) σ in;
- in tension or compression, σ -1р = (0.70÷0.90)σ -1;
- when twisting τ -1 =(0.50÷0.65)σ -1;
for steel casting:
- when bending σ -1 \u003d (0.35 ÷ 0.45) σ in;
- in tension or compression, σ -1р = (0.65÷0.75)σ -1;
- when twisting τ -1 =(0.55÷0.65)σ -1.
Mechanical properties and allowable stresses of anti-friction cast iron:
- ultimate strength in bending 250 - 300 MPa,
– allowable bending stresses: 95 MPa for I; 70 MPa - II: 45 MPa - III, where I. II, III - designations of types of load, see table. one.
Approximate allowable 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.
Ultimate voltage consider the stress at which a dangerous state occurs in the material (destruction or dangerous deformation).
For plastic materials, the ultimate stress is considered yield strength, because the resulting plastic deformations do not disappear after the load is removed:
For fragile materials where there are no plastic deformations, and the fracture occurs according to the brittle type (necks are not formed), the ultimate stress is taken tensile strength:
For plastic-brittle materials, the limiting stress is considered to be the stress corresponding to the maximum deformation of 0.2% (one hundred.2):
Allowable voltage- the maximum voltage at which the material should work normally.
Permissible stresses are obtained according to the limiting ones, taking into account the margin of safety:
where [σ] - allowable stress; s- safety factor; [s] - allowable safety factor.
Note. In square brackets, it is customary to designate the permissible value of a quantity.
Permissible 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 variable loads under shock and vibration conditions.
Features of the behavior of materials during compression tests:
1. Plastic materials work almost equally in 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 allowable stress in tension and compression is different, they are designated [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 given conditions.
To ensure strength, the design stress should not exceed the allowable stress:
Rated voltage a depends on load and size cross section, allowed only from the material of the part and working conditions.
There are three types of strength calculations.
1. Design calculation - design scheme and loads are set; material or dimensions of the part are selected:
Determining the dimensions of the cross section:
Material selection
according to the value of σ, it is possible to choose the grade of the material.
2. Check calculation - loads, material, dimensions of the part are known; necessary check if the durability is guaranteed.
Inequality is checked
3. Determination of load capacity(maximum load):
Examples of problem solving
A straight bar is stretched with a force of 150 kN (Fig. 22.6), the material is steel σ t \u003d 570 MPa, σ w \u003d 720 MPa, safety factor [s] \u003d 1.5. Determine the dimensions of the cross section of the beam.
Solution
1. Strength condition:
2. The required cross-sectional area is determined by the ratio
3. The allowable stress for the material is calculated from the given mechanical characteristics. The presence of a yield strength means that the material is ductile.
4. Determine the value of the required cross-sectional area of \u200b\u200bthe beam and select the dimensions for two cases.
The section is a circle, we determine the diameter.
The resulting value is rounded up d= 25 mm, A \u003d 4.91 cm 2.
Section - equal-shelf corner No. 5 according to GOST 8509-86.
The closest cross-sectional area of the corner is A \u003d 4.29 cm 2 (d \u003d 5 mm). 4.91 > 4.29 (Appendix 1).
Control questions and tasks
1. What phenomenon is called fluidity?
2. What is a "neck", at what point of the tension diagram is it formed?
3. Why are the mechanical characteristics obtained during testing conditional?
4. List strength characteristics.
5. List the characteristics of plasticity.
6. What is the difference between an automatically drawn stretch chart and a shown 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 limit and allowable stresses?
9. Write down the condition of tensile and compressive strength. Are the strength conditions different in tensile and compressive calculations?
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Answer the test questions.
The main task of design calculation is to ensure its strength under operating conditions.
The strength of a structure made of brittle metal is considered to be ensured if the actual stresses in all cross sections of all its elements are less than the tensile strength of the material. The magnitude of the loads, stresses in the structure and the tensile strength of the material cannot be established exactly (due to the approximation of the calculation methodology, methods for determining the tensile strength, etc.).
Therefore, it is necessary that the highest stresses obtained as a result of the design calculation (design stresses) do not exceed a certain value that is less than the ultimate strength, called the allowable stress. The value of the allowable stress is set by dividing the tensile strength by a value greater than one, called the safety factor.
In accordance with the above, the strength condition for a structure made of brittle material is expressed as
where - the highest design tensile and compressive stresses in the structure; and [-permissible stresses in tension and compression, respectively.
Permissible stresses depend on the tensile and compressive strength of the material stvs and are determined by the expressions
where is the normative (required) safety factor in relation to the ultimate strength.
The absolute values of stresses are substituted into formulas (39.2) and (40.2)
For structures from plastic materials(whose tensile and compressive strengths are the same) the following strength condition is used:
where a is the largest absolute value compressive or tensile design stress in the structure.
The allowable stress for plastic materials is determined by the formula
where is the normative (required) safety factor in relation to the yield strength.
The use of the yield strength in determining the allowable stresses for ductile materials (rather than the tensile strength, as for brittle materials) is due to the fact that after the yield strength is reached, the deformations can increase very sharply even with a slight increase in the load and the structures may no longer satisfy their operating conditions.
The strength analysis performed using the strength conditions (39.2) or (41.2) is called the allowable stress analysis. The load at which the greatest stresses in the structure are equal to the allowable stresses is called allowable.
The deformations of a number of structures made of plastic materials after reaching the yield strength 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 that experience bending or torsion deformations.
The calculation of these structures is carried out either according to the allowable stresses, i.e., using the strength condition (41.2), or according to the so-called limit state. In the latter case, the allowable load is called the maximum allowable load, and its value is determined by dividing the maximum load by the standard bearing capacity safety factor. The two simplest examples of limit state analysis of a structure are given below in § 9.2 and calculation example 12.2.
It should be strived to ensure that the allowable stresses are fully used, i.e. 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 allowable ones. A slight excess of the calculated allowable stresses is possible and, consequently, a slight decrease in the actual safety factor (compared to the standard one).
The calculation of a centrally tensioned or compressed structural element for strength must ensure that the strength condition is met for all cross sections of the element. At the same time, it is of great importance correct definition the so-called dangerous sections of the element, in which the greatest tensile and greatest compressive stresses occur. In cases where the allowable tensile or compressive stresses are the same, it is enough to find one dangerous section in which there are normal stresses of the highest 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 bar of constant 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) voltage test (test calculation);
b) selection of sections (design calculation);
c) determination of carrying capacity (determination of permissible load). Let us consider these types of problems on the example of a stretched rod 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 maximum voltage obtained in this case is then compared with the allowable:
When selecting sections, the required cross-sectional areas of the element are determined (according to known longitudinal forces N and allowable stress ). The accepted cross-sectional areas F must satisfy the strength condition expressed in the following form:
When determining the carrying capacity according to known values F and the allowable stress, the allowable values of the longitudinal forces are calculated: Based on the values obtained, the allowable values of external loads [P] are then determined.
For this case, the strength condition has the form
The values of the normative safety factors are established by the norms. They depend on the class of construction (capital, temporary, etc.), the intended period of its operation, the load (static, cyclic, etc.), possible heterogeneity 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 increase the safety factor - if necessary, take into account the wear of the rubbing parts of machines, corrosion and decay of the material.
The values of standard safety factors for various materials, structures and loads in most cases have the following values: - from 2.5 to 5 and - from 1.5 to 2.5.
Safety factors and, consequently, allowable stresses for building structures are regulated by the relevant standards for their design. In mechanical engineering, the required safety factor is usually chosen, focusing on the experience of designing and operating machines. similar designs. In addition, a number of advanced machine-building plants have in-plant allowable stress standards, which are often used by other related enterprises.
Approximate values of allowable stresses in tension and compression for a number of materials are given in annex II.
