Calculation of permissible stress for steel. Safety factor, permissible voltage
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.
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?
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Answer the test questions.
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. 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 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.
Type III 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, in which a sample of a given material is destroyed 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 by technical conditions, is 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 the maximum stress:
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], select the required cross-sectional dimensions to ensure its safe operation:
– 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. determining the 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 upward as the top is compressed:
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 of structural materials.
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.