At what number of cycles is the limiting amplitude determined? Limit stress diagrams

1.3.4. METHOD OF “STEP AMPLITUDES” (MSTA). The essence of this method is that in the process of strength training, when performing each individual strength exercise, the movement of the weight would not occur through the full working amplitude, but stepwise, in

To determine the endurance limit under the action of stresses with asymmetric cycles, diagrams of various types are constructed. The most common ones are:

diagram of ultimate stresses, in coordinates dmax - dm (Smith diagram);

diagram of limiting amplitudes, in coordinates yes - dt (Hay diagram).

Let's look at these limit stress diagrams. In the Smith chart, the maximum cycle stress corresponding to the endurance limit is plotted vertically, the average stress along the horizontal axis (Fig. 12.6).

First, point C is plotted on the dmax axis, the ordinate of which represents the endurance limit for a symmetric cycle d-1 (with a symmetric cycle, the average stress is zero). Then the endurance limit is experimentally determined for some asymmetric load, for example, for zero load, in which the maximum stress is always twice the average. Let us plot point P on the diagram, the ordinate of which represents the endurance limit for the zero cycle d0. For many materials, the values ​​of d-1 and d0 are determined and given in reference books.

Similarly, the endurance limit for asymmetric cycles with other parameters is determined experimentally.

The results are plotted on the diagram in the form of points A, B, etc., the ordinates of which are the endurance limits for the corresponding stress cycles. Point D, which also lies on the bisector OD, characterizes the ultimate stress (ultimate strength) for a constant load for which dmax = dt.

Since for plastic materials the dangerous stress is also the yield strength o*., the horizontal line KL is plotted on the diagram, the ordinate of which is equal to dt. (For plastic materials, the tensile diagram of which does not have a yield plateau, the role of dt is played by the conditional yield strength d0.2.) Consequently, the diagram of the ultimate stress will finally have a value of CAPKL.

Usually this diagram is simplified by replacing it with two straight lines CM and ML, with the straight line CM drawn through point C (corresponding to a symmetric cycle) and point P (corresponding to a zero cycle).

The indicated method of schematizing the limit stress diagram was proposed by S. V. Serensen and R. S. Kinasoshvili.

In this case, within the limits of direct SM, the maximum cycle stress (endurance limit) will be expressed by the equation

The coefficient characterizes the sensitivity of the material to cycle asymmetry.

When calculating endurance, a diagram of limiting amplitudes is often used, which is plotted in coordinates - (Hay diagram). To do this, the amplitude voltage is plotted along the vertical axis, and the average voltage is plotted along the horizontal axis (Fig. 12.7).

Point A of the diagram corresponds to the endurance limit for a symmetrical cycle, since with such a cycle dt = 0.

Point B corresponds to the ultimate strength at constant stress, since in this case yes = 0.

Point C corresponds to the endurance limit for a pulsating cycle, since with such a cycle yes = dt.

Other points in the diagram correspond to endurance limits for cycles with different ratios of dm and dm.

The sum of the coordinates of any point of the limit curve of the ASV gives the value of the endurance limit at a given average cycle stress

For plastic materials, the ultimate stress should not exceed the yield strength

Therefore, on the limit stress diagram we plot a straight line DE, constructed according to the equation

The final ultimate stress diagram is AKD.

In practice, they usually use an approximate da--dt diagram, constructed from three points A, C and D and consisting of two straight sections AL and LD (Sørensen-Kinaso-Shvili method). Point L is obtained as a result of the intersection of two lines: line DE and line AC. Calculations using the Smith and Hay charts with the same approximation methods lead to the same results.

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.

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, carburization, 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.