The figure shows a simplified diagram of limiting amplitudes. Limit stress diagrams
During the operation of machines and engineering structures, stresses arise in their elements, changing over time in a wide variety of cycles. To calculate elements for strength, it is necessary to have data on the values of endurance limits during cycles with different asymmetry coefficients. Therefore, along with tests under symmetrical cycles, tests are also carried out under asymmetrical cycles.
It should be borne in mind that endurance tests under asymmetric cycles are performed on special machines, the designs of which are much more complex than the designs of machines for testing samples under a symmetric bending cycle.
The results of endurance tests for cycles with different asymmetry coefficients are usually presented in the form of diagrams (graphs) depicting the relationship between any two parameters of limit cycles.
These diagrams can be constructed, for example, in coordinates from, they are called limit amplitude diagrams, they show the relationship between average stresses and amplitudes of limit cycles-cycles for which the maximum stresses are equal to the endurance limits: Here and below are the maximum, minimum, average and amplitude limit stresses cycle we will denote
A diagram of the dependence between the parameters of a limit cycle can also be constructed in coordinates. Such a diagram is called a limit stress diagram.
When calculating steel structures in industrial and civil engineering, diagrams are used that give the relationship between the cycle asymmetry coefficient R and the endurance limit omax
Let us consider in detail the diagram of limiting amplitudes (it is sometimes called a diagram), which is subsequently used to obtain dependencies used in strength calculations under variable stresses.
To obtain one point of the diagram under consideration, it is necessary to test a series of identical samples (at least 10 pieces) and construct a Wöhler curve, which will determine the endurance limit for a cycle with a given asymmetry coefficient (this also applies to all other types of diagrams for limit cycles).
Let us assume that tests have been carried out under a symmetrical bending cycle; As a result, the value of the endurance limit is obtained. The coordinates of the point representing this limit cycle are equal to: [see. formulas (1.15) - (3.15)], that is, the point is on the ordinate axis (point A in Fig. 6.15). For an arbitrary asymmetric cycle, according to the endurance limit determined from experiments, it is not difficult to find from. According to formula (3.15),
but [see formula (5.15)], therefore,
In particular, for a zero cycle with an endurance limit equal to
This cycle corresponds to point C in the diagram shown in Fig. 6.15.
Having determined the experimental value for five or six different cycles, using formulas (7.15) and (8.15) the coordinates from and of individual points belonging to the limit curve are obtained. In addition, as a result of testing under constant load, the ultimate strength of the material is determined, which, for generality of reasoning, can be considered as the endurance limit for the cycle with . Point B corresponds to this cycle on the diagram. By connecting the points whose coordinates are found from experimental data with a smooth curve, a diagram of the limiting amplitudes is obtained (Fig. 6.15).
Reasoning about constructing a diagram carried out for cycles of normal stresses is applicable for cycles of tangential stresses (for torsion), but the notation changes instead of from, etc.).
The diagram presented in Fig. 6.15, was built for cycles with positive (tensile) average stresses from 0. Of course, it is in principle possible to construct a similar diagram in the region of negative (compressive) average stresses, but practically at present there is very little experimental data on fatigue strength at For low- and medium-carbon steels, it can be approximately assumed that in the region of negative average stresses the limit curve is parallel to the abscissa axis.
Let us now consider the question of using the constructed diagram. Let the working stress cycle correspond to point N with coordinates (i.e., when working at the considered point of the part, stresses arise, the cycle of change of which is specified by any two parameters, which allows you to find all the parameters of the cycle and, in particular, ).
Let us draw a ray from the origin through point N. The tangent of the angle of inclination of this ray to the abscissa axis is equal to the cycle characteristic:
Obviously, any other point lying in the same ray corresponds to a cycle similar to the given one (a cycle having the same values). So, any ray drawn through the origin is the locus of points corresponding to similar cycles. All cycles depicted by points of the beam lying no higher than the limit curve (i.e., points of the segment (G)) are safe with respect to fatigue failure. Moreover, the cycle depicted by the point KU is, for a given asymmetry coefficient, its maximum stress, defined as the sum of the abscissa and the ordinate of point K (otah) is equal to the endurance limit:
Similarly, for a given cycle, the maximum voltage is equal to the sum of the abscissa and ordinate of the point
Assuming that the operating stress cycle in the part being calculated and the limit cycle are similar, we determine the safety factor as the ratio of the endurance limit to the maximum stress of a given cycle:
As follows from the above, the safety factor in the presence of a diagram of limiting amplitudes constructed from experimental data can be determined by a graphical-analytical method. However, this method is suitable only on the condition that the part being calculated and the samples, as a result of which the diagram was obtained, are identical in shape, size and quality of processing (this is described in detail in § 4.15, 5.15).
For parts made of plastic materials, not only fatigue failure is dangerous, but also the occurrence of noticeable residual deformations, i.e., the onset of yield. Therefore, from the area limited by line AB (Fig. 7.15), all points of which correspond to cycles that are safe with respect to fatigue failure, it is necessary to select a zone corresponding to cycles with maximum stresses less than the yield strength. To do this, from point L, the abscissa of which is equal to the yield strength, draw a straight line inclined to the abscissa axis at an angle of 45°. This straight line reading on the ordinate axis is the segment OM, equal (on the scale of the diagram) to the yield strength. Therefore, the equation of straight line LM (equation in segments) will have the form
that is, for any cycle represented by the points on the LM line, the maximum stress is equal to the yield strength. The points lying above the LM line correspond to cycles with maximum stresses greater than the yield strength. Thus, cycles that are safe both with respect to fatigue failure and the occurrence of yield are depicted by points in the region
To construct a diagram of limiting amplitudes, it is necessary to have endurance limits at different values of the parameter “ ” (asymmetry coefficient). The introduction significantly complicates the experiment, because now it is necessary to have several dozen samples, each ten of which are tested at . By setting a constant value, we find, through successive tests of samples, the highest amplitude value at which the material is still able to withstand an unlimited number of cycles. As a result of testing one dozen samples, we obtain one point on the diagram of limiting amplitudes. Having tested the next group of samples, we get another point, etc. (Fig. 11.7).
The meaning of the limit amplitude diagram is obvious. Let the cycle be characterized by voltages and , which we will consider as the coordinates of the operating point. By plotting the operating point on the diagram, we can judge the strength of the sample. If the operating point is located below the limit curve, then the sample will withstand an infinitely large number of cycles (not less than the base one). If R.T. is above the curve, then the sample will fail at a certain number of cycles less than the base one.
Constructing a diagram of limiting amplitudes is very labor-intensive, so it is often schematized with straight line segments. The dot reflects the corresponding testing of samples in a symmetrical cycle. The point corresponds to static testing of samples. For brittle materials it is determined by tensile strength. For plastic materials, the limitation can be both in terms of yield strength and tensile strength.
To construct the left side of the diagram, you need at least one more point, for example, for a pulsation cycle, or you need to know the slope of the straight line. Let's introduce the concept of angular coefficient = . Experiments have proven that the value of the angular coefficient for carbon steels lies in the range of 0.1÷0.2 and for alloyed steels 0.2÷0.3.
Thus, the equation of the left line has the form 
. The right side of the diagram is approximated by a straight line passing through the point and making an angle of 45 with the axes and
Consequently, when schematizing, the diagram of limiting amplitudes is replaced by two straight lines and .
The constructed diagram does not yet allow us to calculate the strength of parts, because Fatigue strength depends on many other factors.
Factors affecting fatigue strength
Stress concentration
Concentration is the phenomenon of an abrupt increase in stress near sudden changes in the shape of a part, holes, recesses (Fig. 11.8)
The concentration measure is the theoretical stress concentration coefficient equal to:
When stretching, bending, torsion,
The so-called rated voltage, determined by the formulas for the resistance of materials, is the highest local stress. Data on the theoretical stress concentration coefficient are given in mechanical engineering reference books. Stress concentration has different effects on the strength of a part depending on the properties of the material and loading conditions. Therefore, instead of the theoretical stress concentration factor, the effective stress concentration factor and is introduced.
For a symmetrical cycle, the effective stress concentration coefficient is determined by the relation
where are the endurance limits of a smooth sample,
Fatigue limits calculated from nominal stresses for samples having a stress concentration, but the same cross-sectional dimensions as a smooth sample. determined from tables.
In cases where there is no experimental data, by direct definition they resort to approximate estimates. For example, according to the formula
The coefficient of sensitivity of a material to stress concentration. It depends mainly on the material. For structural steels.
Scale effect
If several samples of different diameters are made from the same material, then after a fatigue test it can be found that the endurance limit decreases with increasing diameter. The decrease in endurance limit with increasing part size is called the scale effect.
The measure of this reduction is the scale factor
Fatigue limit of a sample with a diameter similar to the part
Sample endurance limit d= 7.5mm.
In Fig. 11.9 gives an approximate dependence of the scale factor on the shaft diameter for the case of bending and torsion.
Curve 1 was obtained for carbon steel, 2 for alloy steel.
Diagram characterizing the ratio of maximum amplitudes and average cycle stresses for a given durability;
See also:
- Diagram
- Scheffler diagram
- fatigue chart
- thermokinetic diagram
- recrystallization diagram
- solubility diagram
- plasticity diagram
- slag fusibility diagram
- kinetic fatigue diagram
- deformation diagram
- diagram of isothermal transformation (C-diagram)
- cyclic deformation diagram
- phase diagram of iron - carbon
- state diagram
- pressing diagram
- Keller-Goodwin limit plasticity diagram
- Kolmogorov-Bogatov plasticity diagram
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To determine the endurance limit under the action of stresses with asymmetric cycles, diagrams of various types are constructed. The most common ones are:
1) diagram of cycle limit stresses in coordinates max - m
2) diagram of the limiting amplitudes of the cycle in coordinates a - m.
Let's consider a diagram of the second type.
To construct a diagram of the limiting amplitudes of a cycle, the amplitude of the stress cycle a is plotted along the vertical axis, and the average stress of the cycle m is plotted along the horizontal axis (Fig. 8.3).
Dot A diagram corresponds to the endurance limit for a symmetric cycle, since with such a cycle m = 0.
Dot IN corresponds to the ultimate strength at constant stress, since in this case a = 0.
Point C corresponds to the endurance limit for a pulsating cycle, since with such a cycle a = m .
Other points in the diagram correspond to endurance limits for cycles with different ratios of a and m.
The sum of the coordinates of any point on the limit curve of the DIA gives the endurance limit at a given average cycle stress
.
For plastic materials, the ultimate stress should not exceed the yield strength i.e. Therefore, we plot the straight line DE on the limit stress diagram , built according to the equation
The final limit stress diagram looks like AKD .
Workloads must be within the diagram. The endurance limit is less than the strength limit, for example, for steel σ -1 = 0.43 σ in.
In practice, they usually use an approximate diagram a - m, constructed from three points A, L and D, consisting of two straight sections AL and LD. Point L is obtained as a result of the intersection of two straight lines DE and AC . The approximate diagram increases the fatigue strength margin and cuts off the area with a scatter of experimental points.
Factors affecting endurance limit
Experiments show that the endurance limit is significantly influenced by the following factors: stress concentration, cross-sectional dimensions of parts, surface condition, nature of technological processing, etc.
Effect of stress concentration.
TO 
concentration (local increase) of stress occurs due to cuts, sudden changes in size, holes, etc. In Fig. Figure 8.4 shows voltage diagrams without a concentrator and with a concentrator. The influence of the concentrator on strength takes into account the theoretical stress concentration coefficient.
Where 
- voltage without concentrator.
Kt values are given in reference books.
Stress concentrators significantly reduce the fatigue limit compared to the fatigue limit for smooth cylindrical samples. At the same time, concentrators have different effects on the fatigue limit depending on the material and the loading cycle. Therefore, the concept of an effective concentration coefficient is introduced. The effective stress concentration factor is determined experimentally. To do this, take two series of identical samples (10 samples in each), but the first without a stress concentrator, and the second with a concentrator, and determine the endurance limits in a symmetrical cycle for samples without a stress concentrator σ -1 and for samples with a stress raiser σ -1 ".
Attitude
determines the effective stress concentration coefficient.
The values of K - are given in reference books
Sometimes the following expression is used to determine the effective stress concentration factor
where g is the coefficient of sensitivity of the material to stress concentration: for structural steels - g = 0.6 0.8; for cast iron - g = 0.
Influence of surface condition.
Experiments show that rough surface treatment of a part reduces endurance limit . The influence of surface quality is associated with changes in micro-geometry (roughness) and the state of the metal in the surface layer, which, in turn, depends on the method of machining.
To assess the influence of surface quality on the endurance limit, the coefficient p is introduced, called the surface quality coefficient and equal to the ratio of the endurance limit of a sample with a given surface roughness σ -1 n to the endurance limit of a sample with a standard surface σ -1
N 
and fig. 8.5 shows a graph of values p
depending on the tensile strength σ in
steel and type of surface treatment. In this case, the curves correspond to the following types of surface treatment: 1 - polishing, 2
-
grinding, 3
-
fine turning, 4 - rough turning, 5 - presence of scale.
Various methods of surface hardening (hardening, carburizing, nitriding, surface hardening with high-frequency currents, etc.) greatly increase the endurance limit values. This is taken into account by introducing the coefficient of influence of surface hardening . By surface hardening of parts, the fatigue resistance of machine parts can be increased by 2-3 times.
Influence of part dimensions (scale factor).
Experiments show that the larger the absolute dimensions cross-section of the part, the lower the endurance limit , since with increasing size increases the likelihood of defects entering the hazardous area . Fatigue limit ratio of a part with diameter d σ -1 d to the endurance limit of a laboratory sample with a diameter d 0 = 7 – 10 σ -1 mm is called the scale factor
experimental data to determine m not enough yet.
It has been experimentally established that the endurance limit with an asymmetrical cycle is greater than with a symmetrical one, and depends on the degree of asymmetry of the cycle:
When graphically depicting the dependence of the endurance limit on the asymmetry coefficient, it is necessary for each R Determine your endurance limit. This is difficult to do, since in the range from a symmetrical cycle to simple stretching there is an infinite number of very diverse cycles. An experimental determination for each type of cycle is almost impossible due to the large number of samples and the long time they are tested.
Due to specified reasons for the limited number of experiments for three to four values R construct a limit cycle diagram.
| Rice. 445 |
A limit cycle is one in which the maximum stress is equal to the endurance limit, i.e. . On the ordinate axis of the diagram we plot the value of the amplitude, and on the abscissa axis we plot the average voltage of the limit cycle. Each pair of voltages and , defining the limit cycle, is depicted by a certain point on the diagram (Fig. 445). Experience has shown that these points are generally located on the curve AB, which on the ordinate axis cuts off a segment equal to the endurance limit of the symmetrical cycle (with this cycle = 0), and on the abscissa axis – a segment equal to the strength limit. In this case, voltages that are constant over time apply:
Thus, the limit cycle diagram characterizes the relationship between the values of average stresses and the values of the limit amplitudes of the cycle.
Any point M, located inside this diagram corresponds to a certain cycle determined by the quantities (CM) And (ME).
To define a cycle from a point M carry out segments MN And M.D. until it intersects with the x-axis at an angle of 45° to it. Then (Fig. 445):
Cycles whose asymmetry coefficients are the same (similar cycles) will be characterized by points located on a straight line 01, the angle of inclination of which is determined by the formula
| Rice. 446 |
Dot 1 corresponds to a limit cycle from all specified similar cycles. Using the diagram, you can determine the limiting voltages for any cycle, for example, for a pulsating (zero) cycle, for which , a (Fig. 446). To do this, draw a straight line from the origin of coordinates (Fig. 445) at an angle α 1 = 45°() until it intersects the curve at the point 2. Coordinates of this point: ordinate H2 is equal to the maximum amplitude voltage, and the abscissa K2– the maximum average voltage of this cycle. The maximum maximum voltage of the pulsating cycle is equal to the sum of the coordinates of the point 2:
In a similar way, the question of the limiting stresses of any cycle can be resolved.
If a machine part experiencing alternating stresses is made of a plastic material, then not only fatigue failure will be dangerous, but also the occurrence of plastic deformations. The maximum cycle stresses in this case are determined by the equality
where - betrayed to fluidity.
Points satisfying this condition are located on a straight line DC, inclined at an angle of 45° to the abscissa axis (Fig. 447, A), since the sum of the coordinates of any point on this line is equal to .
If straight 01 (Fig. 447, a), corresponding to this type of cycle, with increasing loads on the machine part, intersects the curve AC, then fatigue failure of the part will occur. If the line 01 crosses the line CD, then the part will fail as a result of plastic deformation.
Often in practice, schematized diagrams of limiting amplitudes are used. curve ACD(Fig. 447, a) for plastic materials approximately replace the straight line A.D. This straight line cuts off the segments and on the coordinate axes. The equation is
| Rice. 447 |
For brittle materials diagram limit straight A B with equation
The most widely used are diagrams of limiting amplitudes, constructed based on the results of three series of tests of samples: with a symmetrical cycle ( point A), at zero cycle (point C) and static break (point D)(Fig. 447, b). Connecting the dots A And WITH straight and drawing out D straight line at an angle of 45°, we obtain an approximate diagram of the limiting amplitudes. Knowing the coordinates of the point A And WITH, you can create an equation of the straight line AB. Let's take an arbitrary point on the line TO with coordinates and . From the similarity of triangles ASA 1 And KSK 1 we get
from where we find the equation of the line AB in form
End of work -
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It must be borne in mind that not any variable stress causes fatigue failure. It can occur if the alternating stresses at one point or another on the part exceed
Factors influencing the endurance limit
Many factors influence the endurance limit. Let us consider the influence of the most important of them, which are usually taken into account when assessing fatigue strength. Stress concentration. Mouth
Calculation of strength under variable stresses
In strength calculations under variable stresses, the strength of a part is usually assessed by the value of the actual safety factor n, comparing it with the permissible safety factor n)
