At the turn of the XIX-XX centuries. in connection with the creation and entry into everyday life of new types of machines, installations and Vehicle operating under loads cyclically changing in time, it turned out that the existing calculation methods did not provide reliable results for the calculation of such structures. For the first time, such a phenomenon was encountered by railway transport when a series of disasters occurred associated with a break in the axles of wagons and steam locomotives.
Later it turned out that the cause of the destruction was the alternating stresses that arose during the movement train due to the rotation of the wagon axle along with the wheels. However, it was initially suggested that during long-term operation, the metal changes its crystal structure - tired. This assumption was not confirmed, however, the name "fatigue calculations" has been preserved in engineering practice.
Based on the results of further studies, it was found that fatigue failure is due to the accumulation of local damage in the material of the part and the development of cracks. It is these processes that occur during the operation of various machines, vehicles, machine tools and other installations subject to vibration and other types of time-varying loads that will be considered below.
Consider a cylindrical sample fixed in the spindle at one end, at the other, free, end of which a force is applied through the bearing F(Fig. 16.1).
Rice. 16.1.
The plot of the bending moment of the sample changes linearly, and its maximum value is equal to F.I. At the points of the cross section of the sample BUT and AT there are maximum absolute value voltage. The value of the normal stress at point L will be
In the case of rotation of the sample with an angular velocity from the point of the cross section, they change their position relative to the plane of action of the bending moment. During t characteristic point BUT rotates through an angle φ = ω/ and ends up in a new position BUT"(Fig. 16.2, a).
Rice. 16.2.
The stress in the new position of the same material point will be equal to
Similarly, we can consider other points and come to the conclusion that when the sample rotates due to a change in the position of the points, the normal stresses change according to the cosine law (Fig. 16.2, b).
To explain the process of fatigue failure, one will have to abandon the fundamental hypotheses about the material, namely the hypothesis of continuity and the hypothesis of homogeneity. Real materials are not ideal. As a rule, the material initially contains defects in the form of imperfections in the crystal lattice, pores, microcracks, foreign inclusions, which are the cause of the structural inhomogeneity of the material. Under conditions of cyclic loading, structural inhomogeneity leads to inhomogeneity of the stress field. In the weakest places of the part, microcracks are born, which, under the influence of time-varying stresses, begin to grow, merge, turning into main crack. Getting into the tension zone, the crack opens, and in the compression zone, on the contrary, it closes.
A small local area in which the first crack appears and from where its development begins is called focus of fatigue failure. Such an area, as a rule, is located near the surface of the parts, but its appearance in the depth of the material is not ruled out if there is any damage. The simultaneous existence of several such regions is not excluded, and therefore the destruction of the part can begin from several centers that compete with each other. As a result of the development of cracks, the cross section is weakened until fracture occurs. After failure, the fatigue crack propagation zone is relatively easy to recognize. In the section of the part destroyed from fatigue, there are two sharply different areas (Fig. 16.3).
Rice. 16.3.
1 - area of crack growth; 2 - region of brittle fracture
Region 1 characterized by a shiny smooth surface and corresponds to the beginning of the destruction process, which proceeds in the material at a relatively low speed. On the final stage process, when the section weakens sufficiently, a rapid avalanche-like destruction of the part occurs. This final stage in Fig. 16.3 corresponds area 2, which is characterized by a rough, rough surface due to the rapid final failure of the part.
It should be noted that theoretical study fatigue strength of metals is associated with significant difficulties due to the complexity and multifactorial nature of this phenomenon. For this reason essential tool becomes phenomenological approach. For the most part, the formulas for calculating parts for fatigue are obtained on the basis of experimental results.
In the vast majority of cases, calculations for the strength of parts operating under variable voltages, perform as verification. This is primarily due to the fact that the overall coefficient of reduction in the endurance limit or in the process of designing a part can only be chosen approximately, since the calculator (designer) at this stage of work has only very approximate ideas about the size and shape of the part. The design calculation of a part, which serves to determine its main dimensions, is usually performed approximately without taking into account the variability of stresses, but at reduced allowable stresses.
After completing the working drawing of the part, its refined verification calculation is carried out, taking into account the variability of stresses, as well as design and technological factors affecting the fatigue strength of the part. At the same time, the calculated safety factors for one or more supposedly dangerous sections of the part are determined. These safety factors are compared with those that are assigned or recommended for parts similar to those designed under given operating conditions. With such a verification calculation, the strength condition has the form
The value of the required safety factor depends on a number of circumstances, the main of which are: the purpose of the part (the degree of its responsibility), working conditions; the accuracy of determining the loads acting on it, the reliability of information about mechanical properties ah of its material, the values of stress concentration factors, etc. Usually
If the calculated safety factor is lower than required (i.e., the strength of the part is insufficient) or significantly higher than required (i.e., the part is uneconomical), it is necessary to make certain changes in the dimensions and design of the part, and in some cases even change her material.
Let us consider the determination of the safety factors for a uniaxial stress state and for pure shear. The first of these types of stress state, as is known, occurs during tension (compression), direct or oblique bending and joint bending and tension (or compression) of the beam. Recall that shear stresses during bending (straight and oblique) and combination of bending with axial loading at the dangerous point of the beam, as a rule, are small and are neglected when calculating the strength, i.e., it is believed that a uniaxial stress state occurs at the dangerous point.
Pure shear occurs at the points of a torsion bar with a circular cross section.
In most cases, the safety factor is determined on the assumption that the duty cycle of stresses that occur in the calculated part during its operation is similar to the limit cycle, i.e., the asymmetry coefficients R and the characteristics of the working and limit cycles are the same.
The most simple safety factor can be determined in the case of a symmetrical stress change cycle, since the endurance limits of the material during such cycles are usually known, and the fatigue limits of the calculated parts can be calculated from the values of the fatigue limit reduction coefficients taken from reference books. The safety factor is the ratio of the endurance limit, defined for the part, to the nominal value of the maximum voltage that occurs at the dangerous point of the part. The nominal value is the stress value determined by the basic formulas of the resistance of materials, i.e. without taking into account the factors affecting the value of the endurance limit (stress concentration, etc.).
Thus, to determine the safety factor for symmetrical cycles, we obtain the following dependencies:
when bending
in tension-compression
in torsion
When determining the safety factor in the case of an asymmetric cycle, difficulties arise due to the lack of experimental data necessary to construct a segment of the limit stress line (see Fig. 7.15). Note that there is practically no need to build the entire diagram limiting amplitudes, since for cycles with endurance limits greater than the yield strength, the safety factor must be determined by yield (for plastic materials), i.e., the calculation should be carried out as in the case of a static load action.
In the presence of an experimentally obtained section AD of the limit curve, the safety factor could be determined by a graph-analytical method. As a rule, these experimental data are absent and the AD curve is approximately replaced by a straight line constructed from any two points, the coordinates of which are determined experimentally. As a result, a so-called schematized diagram of limiting amplitudes is obtained, which is used in practical strength calculations.
Let us consider the main ways of schematizing the safe zone of the diagram of limiting amplitudes.
In modern calculation practice, the Serensen-Kinasoshvili diagram is most often used, in the construction of which the section AD is replaced by a straight line drawn through points A and C, corresponding to the limiting symmetric and zero cycles (Fig. 9.15, a). The advantage of this method is its relatively high accuracy (the approximating straight line AC is close to the curve; its disadvantage is that, in addition to the endurance limit value for a symmetrical cycle, it is necessary to have experimental data on the endurance limit value) also with a zero cycle.
When using this diagram, the safety factor is determined by endurance (fatigue failure), if the beam of cycles similar to the given one intersects the straight line and by yield, if the specified beam intersects the line
A slightly lower, but in many cases sufficient for practical calculations, accuracy is given by a method based on the approximation of the section AD of the limit curve by a straight line segment (Fig. 9.15, b) drawn through points A (corresponding to a symmetrical cycle) and B (corresponding to limiting constant stresses) .
The advantage of the method under consideration is the smaller amount of required experimental data compared to the previous one (data on the value of the endurance limit at zero cycle are not needed). Which of the safety factors, by fatigue failure or by yield, is less, is determined in the same way as in the previous case.
In the third type of schematized diagrams (Fig. 9.15, c), an approximating straight line is drawn through point A and some point P, the abscissa of which is determined as a result of processing the available experimentally obtained limit stress diagrams. For steel, with sufficient accuracy, it can be assumed that the segment OP - s is equal. The accuracy of such diagrams is almost the same as the accuracy of diagrams constructed using the Sørensen-Kinasoshvili method.
The schematized diagram is especially simple, in which the safe zone is limited by the straight line AL (Fig. 9.15, d). It is easy to see that the calculation according to such a diagram is very uneconomical, since in the schematized diagram the limit stress line is located much lower than the actual limit stress line.
In addition, such a calculation does not have a definite physical meaning, since it is not known which safety factor, fatigue or yield, will be determined. Despite these serious shortcomings, the diagram in Fig. 9.15 and sometimes used in foreign practice; in domestic practice in recent years, such a diagram is not used.
Let us derive an analytical expression for determining the safety factor for fatigue failure based on the considered schematized diagrams of limiting amplitudes. At the first stage of the derivation, we will not take into account the influence of factors that reduce the endurance limit, i.e., first we will obtain a formula suitable for normal laboratory samples.
Let us assume that the point N, representing the working cycle of stresses, is in the area (Fig. 10.15) and, therefore, when the stresses increase to the value determined by the point, fatigue failure will occur (as already indicated, it is assumed that the working and limiting cycles are similar). The fatigue safety factor for the cycle represented by point N is defined as the ratio
Draw through the point N a line parallel to the line and a horizontal line NE.
It follows from the similarity of triangles that
As follows from Fig. 10.15,
Let us substitute the obtained values of OA and into equality (a):
Similarly, in the case of variable shear stresses
The values depend on the type of schematized limit stress diagram adopted for the calculation and on the material of the part.
So, if we accept the Sorensen-Kinasoshvili diagram (see Fig. 9.15, a), then
likewise,
According to the schematized diagram shown in Fig. 9.15, b,
(20.15)
likewise,
(21.15)
The values and when calculating by the method of Serensen - Kinasoshvili can be taken according to the given data (Table 1.15).
Table 1.15
Coefficient values for steel
When determining the safety factor for a particular part, it is necessary to take into account the influence of the fatigue limit reduction factor. Experiments show that stress concentration, scale effect and surface condition affect only the values of limiting amplitudes and practically do not affect the values of limiting average stresses. Therefore, in design practice, it is customary to refer the coefficient of fatigue limit reduction only to the amplitude stress of the cycle. Then the final formulas for determining the safety factors for fatigue failure will have the form: in bending
(22.15)
in torsion
(23.15)
In tension-compression, formula (22.15) should be used, but instead of substituting in it the endurance limit for a symmetrical tension-compression cycle.
Formulas (22.15), (23.15) are valid for all the indicated methods of schematization of limit stress diagrams; only the values of the coefficients change
Formula (22.15) was obtained for cycles with positive average stresses for cycles with negative (compressive) average stresses should be assumed, i.e., proceed from the assumption that in the compression zone the limit stress line is parallel to the abscissa axis.
Most machine parts under operating conditions experience variable stresses that change cyclically over time. Breakage analysis shows that the materials of machine parts operating for a long time under the action of variable loads can fail at stresses lower than the tensile strength and yield strength.
The destruction of a material caused by repeated action of variable loads is called fatigue failure or material fatigue.
Fatigue failure is caused by the appearance of microcracks in the material, the heterogeneity of the structure of materials, the presence of traces of machining and surface damage, and the result of stress concentration.
Endurance called the ability of materials to resist destruction under the action of alternating stresses.
The periodic laws of change in variable voltages may be different, but all of them can be represented as a sum of sinusoids or cosine waves (Fig. 5.7).
Rice. 5.7. Variable voltage cycles: a- asymmetric; b- pulsating; in - symmetric
The number of voltage cycles per second is called loading frequency. Stress cycles can be of constant sign (Fig. 5.7, a, b) or alternating (Fig. 5.7, in).
The cycle of alternating voltages is characterized by: maximum voltage a max, minimum voltage a min, average voltage a t =(a max + a min)/2, cycle amplitude s fl = (a max - a min)/2, cycle asymmetry coefficient r G= a min / a max.
With a symmetrical loading cycle a max = - ci min ; a t = 0; g s = -1.
With a pulsating voltage cycle a min \u003d 0 and \u003d 0.
The maximum value of periodically changing stress at which the material can resist destruction indefinitely is called endurance limit or fatigue limit.
To determine the endurance limit, samples are tested on special machines. The most common bending tests are under a symmetrical loading cycle. Tensile-compressive and torsional endurance tests are less frequently performed because they require more complex equipment than in the case of bending.
For endurance testing, at least 10 identical samples are selected. Tests are carried out as follows. The first sample is installed on the machine and loaded with a symmetrical cycle with a stress amplitude of (0.5-0.6)st (o in - tensile strength of the material). At the moment of destruction of the sample, the number of cycles is fixed by the counter of the machine N. The second sample is tested at a lower voltage, and the destruction occurs at more cycles. Then the following samples are tested, gradually reducing the voltage; they break down with more cycles. Based on the data obtained, an endurance curve is built (Fig. 5.8). There is a section on the endurance curve tending to a horizontal asymptote. This means that at a certain voltage a, the sample can withstand an infinitely large number of cycles without being destroyed. The ordinate of this asymptote gives the endurance limit. So, for steel, the number of cycles N= 10 7, for non-ferrous metals - N= 10 8 .
Based on a large number of tests, approximate relationships have been established between the bending endurance limit and the endurance limits for other types of deformation.
where st_ |p - endurance limit for a symmetrical cycle of tension-compression; t_j - torsional endurance limit under symmetrical cycle conditions.
Bending stress
where W = / / u tah - moment of resistance of the rod in bending. Torsional stress
where T - torque; Wp- polar torsional moment of resistance.
At present, endurance limits for many materials are defined and are given in reference books.
Experimental studies have shown that in zones of sharp changes in the shape of structural elements (near holes, grooves, grooves, etc.), as well as in contact zones, stress concentration- high voltage. The reason causing the stress concentration (hole, undercut, etc.) is called stress concentrator.
Let the steel strip stretch by force R(Fig. 5.9). A longitudinal force acts in the cross section /' of the strip N= R. Rated voltage, i.e. calculated under the assumption that there is no stress concentration, is equal to a = R/F.
Rice. 5.9.
The stress concentration decreases very quickly with distance from the hub, approaching the nominal voltage.
Qualitatively, the stress concentration for various materials is determined by the effective stress concentration factor
where about _ 1k, t_ and - endurance limits determined by nominal stresses for samples having stress concentration and the same cross-sectional dimensions as a smooth sample.
The numerical values of the effective stress concentration factors are determined on the basis of fatigue tests of the specimens. For typical and most common forms of stress concentrators and basic structural materials, graphs and tables are obtained, which are given in reference books.
It has been empirically established that the endurance limit depends on the absolute dimensions of the cross section of the sample: with an increase in the cross section, the endurance limit decreases. This pattern has been named scale factor and is explained by the fact that with an increase in the volume of the material, the probability of the presence of structural inhomogeneities in it (slag and gas inclusions, etc.) increases, causing the appearance of foci of stress concentration.
The influence of the absolute dimensions of the part is taken into account by introducing the coefficient into the calculation formulas G, equal to the ratio of endurance limit o_ld given sample of given diameter d to the endurance limit a_j of a geometrically similar laboratory sample (usually d=l mm):
So, for steel accept e a\u003d e t \u003d e (usually r \u003d 0.565-1.0).
The endurance limit is affected by the cleanliness and condition of the surface of the part: with a decrease in surface cleanliness, the fatigue limit decreases, since stress concentration is observed near its scratches and scratches on the surface of the part.
Surface quality factor is the ratio of the endurance limit st_, a sample with a given surface condition to the endurance limit st_, a sample with a polished surface:
Usually (3 \u003d 0.25 -1.0, but with surface hardening of parts using special methods (hardening with currents high frequency, cementation, etc.) may be greater than one.
The values of the coefficients are determined according to tables from reference books on strength calculations.
Strength calculations at alternating voltages, in most cases, they are performed as test ones. The result of the calculation is the actual safety factors n, which are compared with the required (permissible) for a given design safety factors [P], moreover, the condition l > [n J] must be satisfied. Usually for steel parts [l] = 1.4 - 3 or more, depending on the type and purpose of the part.
With a symmetrical cycle of stress changes, the safety factor is:
0 for stretch (compress)
0 for twist
0 for bend
where a their - the nominal values of the maximum normal and shear stresses; K SU, K T- effective stress concentration factors.
When parts are operated under conditions of an asymmetric cycle, the safety factors n a along normal and tangent n x stresses are determined by the Serensen-Kinasoshvili formulas
where |/ st, |/ t - coefficients of reduction of an asymmetric cycle to an equally dangerous symmetric one; t, x t- medium stresses; st th, x a- cycle amplitudes.
In the case of a combination of basic deformations (bending and torsion, torsion and tension or compression), the overall safety factor is determined as follows:
The obtained safety factors should be compared with their allowable values, which are taken from the strength standards or reference data. If the condition is met n>n then the structural element is recognized as reliable.
The calculation of metal structures should be carried out according to the method of limit states or permissible ones. stresses. In complex cases, it is recommended to solve the issues of calculating structures and their elements through specially designed theoretical and experimental studies. The progressive method of calculation by limit states is based on a statistical study of the actual loading of structures under operating conditions, as well as the variability of the mechanical properties of the materials used. In the absence of a sufficiently detailed statistical study of the actual loading of the structures of certain types of cranes, their calculations are carried out according to the method of permissible stresses, based on the safety factors established by practice.
Under a plane stress state, in the general case, the plasticity condition according to the modern energy theory of strength corresponds to the reduced stress
where σ x and σy- stresses along arbitrary mutually perpendicular coordinate axes X and at. At σy= 0
σ pr = σ Т, (170)
what if σ = 0, then the limit shear stresses
τ = = 0.578 σ Т ≈ 0,6σ Т. (171)
In addition to strength calculations for certain types of cranes, there are limitations on deflection values, which have the form
f/l≤ [f/l], (172)
where f/l and [ f/l] - calculated and permissible values of the relative static deflection f in relation to span (departure) l.Significant deflections may occur. safe for the structure itself, but unacceptable from an operational point of view.
The calculation according to the method of limit states is carried out according to the loads given in Table. 3.
Table notes:
1. Combinations of loads provide for the following operation of mechanisms: . Ia and IIa - the crane is stationary; smooth (Ia) or sharp (IIa) lifting of the load from the ground or its braking when lowering; Ib and IIb - crane in motion; smooth (Ib) and abrupt (IIb) starting or braking of one of the mechanisms. Depending on the type of crane, load combinations Ic and IIc etc. are also possible.
2. In the table. 3 shows the loads that are constantly acting and regularly arising during the operation of structures, forming the so-called main combinations of loads.
To take into account the lower probability of coincidence of design loads with more complex combinations, combination coefficients are introduced n s < 1, на которые умножаются коэффициенты перегрузок всех нагрузок, за исключением постоянной. Коэффициент сочетаний основных и дополнительных нерегулярно возникающих нагрузок, к которым относятся технологические, транспортные и монтажные нагрузки, а также нагрузки от температурных воздействий, принимается равным 0,9; коэффициент сочетаний основных, дополнительных и особых нагрузок (нагрузки от удара о буфера и сейсмические) – 0,8.
3. For some structural elements, the total effect of both the combination of loads Ia with its own number of cycles, and the combination of loads Ib with its own number of cycles should be taken into account.
4. Angle of deviation of the load from the vertical a. can also be seen as the result of an oblique lift.
5. Working wind pressure R b II and non-operating - hurricane R b III - per design is determined according to GOST 1451-77. With a combination of loads Ia and Ib, the wind pressure on the structure is usually not taken into account due to the low frequency of design wind speeds per year. For tall cranes with a period of free oscillation low frequency more than 0.25 s and installed in wind regions IV-VIII according to GOST 1451-77, the wind pressure on the structure is taken into account with a combination of loads Ia and Ib.
6. Technological loads can refer both to the case of loads II and to the case of loads III.
Table 3
Loads in calculations by the method of limit states
The limit states are the states in which the structure ceases to satisfy the operational requirements imposed on it. The limit state calculation method aims to prevent the occurrence of limit states during operation during the entire service life of the structure.
Metal structures of TT (hoisting and transport machines) must meet the requirements of two groups of limit states: 1) loss of bearing capacity of crane elements in terms of strength or loss of stability from a single action of the largest loads in working or non-working condition. The working state is the state in which the crane performs its functions (Table 3, load case II). The state is considered inoperative when the crane without load is subject only to loads from its own weight and wind or is in the process of installation, dismantling and transportation (Table 3, load case III); loss of bearing capacity of crane elements due to fatigue failure under repeated action of loads of various sizes over the estimated service life (Table 3, case of loads I, and sometimes II); 2) unsuitability for normal operation due to unacceptable elastic deformations or vibrations that affect the operation of the crane and its elements, as well as service personnel. For the second limit state for the development of excessive deformations (deflections, angles of rotation), the limit condition (172) is set for individual types of cranes.
Calculations for the first limit state are of the greatest importance, since in rational design, structures must satisfy the requirements of the second limit state.
For the first limit state in terms of bearing capacity (strength or stability of elements), the limit condition has the form
N ≤ F,(173)
where N- design (maximum) load in the element under consideration, expressed in force factors (force, moment, stress); F- design bearing capacity (smallest) of the element according to force factors.
When calculating the first limit state for the strength and stability of elements to determine the load N in formula (171) the so-called normative loads R H i(for hoisting-and-transport machine designs, these are the maximum loads of the working state, entered into the calculation as based on specifications, and based on design and operating experience) are multiplied by the overload factor of the corresponding standard load n i , after which the work P Hi p i represents the greatest possible load during the operation of the structure, called the design load. Thus, the design force in the element N in accordance with the design combinations of loads given in table. 3 can be represented as
, (174)
where a i is the force in the element at Р Н i= 1, and the calculated moment
, (175)
where M H i- the moment from the standard load.
To determine the overload coefficients, a statistical study of the variability of loads based on experimental data is necessary. Let for a given load Pi its distribution curve is known (Fig. 63). Since the distribution curve always has an asymptotic part, when assigning the calculated load, it should be borne in mind that loads that are greater than the calculated ones (the area of \u200b\u200bthese loads is shaded in Fig. 63) can cause damage to the element. The adoption of large values for the design load and overload factor reduces the likelihood of damage and reduces losses from breakdowns and accidents, but leads to an increase in the weight and cost of structures. The question of the rational value of the overload factor should be decided taking into account economic considerations and safety requirements. Let the calculated force distribution curves be known for the element under consideration N and bearing capacity F. Then (Fig. 64) the shaded area, within whose boundaries the limit condition (173) is violated, will characterize the failure probability.
Given in table. 3 overload factors n> 1, since they take into account the possibility of exceeding their actual loads normative values. In the event that it is dangerous not to exceed, but to reduce the actual load compared to the standard one (for example, the load on the beam consoles, unloading the span, with the design section in the span), the overload factor for such a load should be taken equal to reciprocal, i.e. n"= 1/n< 1.
For the first limit state for the loss of bearing capacity due to fatigue, the limit condition has the form
σ pr ≤ m K R,(176)
where σ pr is the reduced voltage, and m K– see formula (178).
Calculations for the second limit state according to condition (172) are made at overload factors equal to one, i.e., according to standard loads (the weight of the load is assumed to be equal to the nominal).
Function F in formula (173) can be represented as
F= Fm K R , (177)
where F- the geometric factor of the element (area, moment of resistance, etc.).
Under design resistance R should be understood in the calculations:
for fatigue resistance - the endurance limit of the element (taking into account the number of cycles of load changes and the concentration and cycle asymmetry factors), multiplied by the corresponding uniformity coefficient for fatigue tests, characterizing the spread of test results, k 0= 0.9, and divided by k m is the reliability coefficient for the material in strength calculations, characterizing both the possibility of changing the mechanical qualities of the material in the direction of their reduction, and the possibility of reducing the cross-sectional areas of rolled products due to the minus tolerances established by the standards; in appropriate cases, the reduction of the initial endurance limit by the loads of the second design case should be taken into account;
strength at constant stress R= R P /k m - quotient from dividing the normative resistance (normative yield strength) by the corresponding safety factor for the material; for carbon steel k m = 1.05, and for low-alloyed - k m = 1.1; thus, in relation to the work of the material, the limit state is not the complete loss of its ability to perceive the load, but the onset of large plastic deformations that prevent the further use of the structure;
stability - the product of the design resistance to strength by the coefficient of reduction in the bearing capacity of compressible (φ, φ int) or bending (φ b) elements.
Working conditions coefficients m K depend on the circumstances of the operation of the element, which are not taken into account by the calculation and quality of the material, i.e. are not included in the force N, nor in design resistance R.There are three such main circumstances, and therefore we can accept
m K = m 1 m 2 m 3 , (178)
where m 1 - coefficient taking into account the responsibility of the calculated element, i.e., the possible consequences of destruction; the following cases should be distinguished: destruction does not cause the crane to stop working, causes the crane to stop without damage or with damage to other elements, and finally causes the destruction of the crane; coefficient m 1 can be in the range of 1–0.75, in special cases (brittle fracture) m 1 = 0,6; m 2 - coefficient taking into account possible damage to structural elements during operation, transportation and installation, depends on the types of cranes; can be taken t 2 = 1.0÷0.8; t 3 - coefficient taking into account the imperfections of the calculation associated with inaccurate determination of external forces or design schemes. It should be set for individual types of structures and their elements. Can be taken for flat statically determinate systems t 3 = 0.9, .and for statically indeterminate -1, for spatial -1.1. For bending elements compared to those experiencing tension-compression t 3 = 1.05. Thus, the calculation for the first limit state for strength at constant stresses is carried out according to the formula
σ II<. m K R,(179)
and for fatigue resistance, if the transition to the limit state is carried out by increasing the level of variable tension, - according to the formula (176), where the design resistance R determined by one of the following formulas:
R= k 0 σ -1K/k m;(180)
R N= k 0 σ -1K N/k m; (181)
R*= k 0 σ -1K/k m;(182)
R* N= k 0 σ -1K N/k m; (183)
where k 0 , k m - uniformity coefficients for fatigue tests and reliability for the material; σ –1K , σ –1KN , σ * –1K , σ * –1KN– endurance limits unlimited, limited, reduced unlimited, reduced limited, respectively.
Calculation according to the method of permissible stresses is carried out according to the loads given in Table 4. It is necessary to take into account all the notes to the table. 3, except note 2.
The values of safety factors are given in table. 5 and depend on the circumstances of the operation of the structure, not taken into account by the calculation, such as: responsibility, bearing in mind the consequences of destruction; calculation imperfections; deviations in size and quality of the material.
Calculation by the method of allowable stresses is carried out in cases where there are no numerical values for the overload coefficients of the design loads to perform the calculation by the method of limit states. Strength calculation is made according to the formulas:
σ II ≤ [ σ ] = σ T / n II , (184)
σ III ≤ [ σ ] = σ T / n III , (185)
where n II and n III - see table. 5. In this case, the allowable bending stresses are assumed to be 10 MPa (about 5%) more than for tension (for St3 180 MPa), given that during bending, fluidity first appears only in the outermost fibers and then gradually spreads over the entire section of the element , increasing its load-bearing capacity, i.e., during bending, there is a redistribution of stresses over the cross section due to plastic deformations.
When calculating for fatigue resistance, if the transition to the limit state is carried out by increasing the level of variable stress, one of the following conditions must be met:
σ pr ≤ [ σ –1K ]; (186)
σ pr ≤ [ σ –1K N]; (187)
σ pr ≤ [ σ * –1K ]; (188)
σ pr ≤ [ σ * –1KN ]; (189)
where σ pr - reduced voltage; [ σ –1K ], [σ –1K N], [σ * –1K ], [σ * –1KN] - allowable stresses, which are determined using the expression [ σ ] = σ –1K /n 1 or similarly to formulas (181) - (183) instead of σ –1K are used σ –1KN , σ * –1K and σ * –1KN. Margin of safety n I is the same as in the calculation of static strength.
Figure 65 - Scheme for calculating the fatigue life margin
If the transition to the limit state is carried out by increasing the number of cycles of repetition of alternating stresses, then when calculating for limited durability, the margin for fatigue life (Fig. 65) n d = Np/N. Because σ t etc Np = σ t –1K N b = σ t –1K N N,
n q = ( σ –1K N / σ etc) t = p t 1 (190)
and at n l = 1.4 and To= 4 n d ≈ 2.75, and at To= 2 n e ≈ 7.55.
In a complex stress state, the hypothesis of the highest tangential octahedral stresses is most consistent with the experimental data, according to which
(191)
and 
.
Then the margin of safety for symmetrical cycles
| |
i.e. P= n σ n τ /, (192)
where σ-IK and τ-l To- limiting stresses (endurance limits), and σ a and τ a are the amplitude values of the current symmetrical cycle. If the cycles are asymmetric, they should be reduced to symmetric by a formula like (168).
The progressiveness of the method of calculation by limit states lies in the fact that in calculations by this method, the actual work of structures is better taken into account; overload factors are different for each of the loads and are determined based on a statistical study of load variability. In addition, the mechanical properties of materials are better taken into account using the material safety factor. While in the calculation by the method of allowable stresses, the reliability of the structure is ensured by a single safety factor, in the calculation by the method of limit states, instead of a single safety factor, a system of three factors is used: reliability by material, overload and operating conditions, established on the basis of statistical accounting of the operating conditions of the structure.
Thus, the calculation for allowable stresses is a special case of calculation for the first limit state, when the overload factors for all loads are the same. However, it should be emphasized that the method of calculation by limit states does not use the concept of safety margin. It is also not used by the probabilistic calculation method currently being developed for crane construction. Having performed the calculation according to the method of limit states, it is possible to determine the value of the resulting safety factor according to the method of permissible stresses. Substituting into formula (173) the values N[cm. formula (174)] and F[cm. formula (177)] and passing to stresses, we obtain the value of the safety factor
n =Σ σ i n i k M / (m K Σ i). (193)
