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When analyzing the strength of cast iron pressure bearing components, they are generally simplified as members. The servo electromechanical controller will alarm and stop; It is considered that when this kind of member is damaged by axial pressure, the failure section and the cross section of the member roughly become an inclination angle of about 45. The reason is that at this time, the maximum working normal stress on the cross section is, while the normal stress and shear stress on any inclined section are

= Cos2 a = sin cos (1)

respectively. According to formula (1), the shear stress on the inclined section of = 45 is the maximum, and its value is 45 =/2. It is this maximum shear stress that causes the member to be destroyed (sheared) along its action surface. If [] is set as the allowable shear stress of the material, the strength condition established according to the maximum shear stress theory is

2 [] (2)

the above strength conditions are widely used in the design and calculation of cast iron pressure bearing components. However, a large number of engineering examples show that this strength condition is rough and too safe. It is necessary to correct it

1 compression test of cast iron

compression tests were carried out on 360 small cylindrical specimens of grey cast iron with a diameter length ratio of 1: (1.5 ~ 2) on the WE-30 universal testing machine. In the test, the diameter length ratio is small, so as to eliminate the influence of bending deformation as much as possible; Enter 5% into the upper and lower ends of the test piece Overload protection: after the sensor is overloaded, it will stop automatically and grind it smoothly to eliminate the influence of tilt deformation; Lubricants are used between the upper and lower ends of the test piece and the anvil block of the testing machine to reduce the influence of friction; A loose fit steel sleeve with a height of 10mm is added to the middle section of some specimens to reduce the influence of bulging deformation

after the test, the microscopic observation of the broken fracture of the specimen confirmed that the fracture particles were relatively smooth, with slip marks, showing the typical characteristics of shearing. It can be seen that the maximum shear stress theory is applicable to such problems. The distribution of the included angle between the failure section and the cross section of the specimen is shown in the table below

table distribution

it can be seen from table 1 that the number of specimens with an included angle of about 45 between the failure section and the cross section accounts for only 7.2%. However, the number of specimens of about 55 accounts for a large proportion, which is 88.9%. Excluding the residual effects of various non ideal random factors and the above-mentioned influencing factors, from the statistical point of view of experimental mechanics, the following conclusions can be drawn: when the cast iron specimen is damaged by axial pressure, the included angle between the failure section and the cross section of the specimen is generally about 55. This is very different from the previous conclusion that it should be about 45. It is necessary to explore the reason of (3) small crack stage in theory

2 stress analysis on any inclined section of pressure bearing member

at present, the theoretical analysis methods used in engineering generally only consider the tangential component of the total stress P on the failure section, but do not consider the role of the normal component. Figure 1 is the stress analysis diagram on any inclined section of the test piece. Assuming that the axial pressure is p, the normal stress on the cross section is, and the internal friction angle of the material is, then in addition to the normal stress and shear stress as shown in formula (1), there is friction shear stress f on any inclined section, and its expression is

(3)

Fig. 1. The direction of stress analysis diagram

and F on any inclined section is shown in Fig. 1. The expression of the total shear stress on the inclined section is

= - f (4)

from equations (1) and (3), we can get the law that changes with and f changes with, as shown in Figure 2 (set = 100 stress units)

if the influence of internal friction in the material is not considered (Fig. 2), the shear stress on any inclined section of the bar is a curve, and its maximum value is the extreme point 45 of the curve, which occurs on the inclined section of = 45. If the influence of internal friction is taken into account, then: when>, f decreases with the increase of, and increases with the increase of, = - F, its max must be reduced to a considerable extent compared with 45; When =, substitute the first formula in formula (1) into formula (3) to obtain f = sin cos =, so the f curve is the same as the curve, and the total shear stress on the inclined section = - F = 0; When ＜, the result of ＜ f will appear in the theoretical calculation, which is inconsistent with the fact. In fact, the frictional shear stress f is the static sliding friction constraint reaction caused by the active force. At this time, its value is no longer determined by Coulomb's law, but should be determined as F =, according to the static equilibrium condition. Therefore, the f curve is the same as the curve, and the total shear stress on the inclined section = - F = 0

Fig. 2, variation rule of F

1:f (= 10) 2:f (= 20) 3:f (= 30)

in conclusion, it can be seen that the influence of internal friction must be considered when analyzing the strength of cast iron pressure bearing components

3 position of failure section and maximum value of total shear stress

replace formula (1) and formula (3) into formula (4), and get

(5)

take the derivative of formula (5) and make it equal to zero, that is, the included angle between failure section and member cross section is

= 45 +/2 (6)

in formula (6), which is a range variable with the change of member size and shape, the change of normal stress on inclined section and the change of test conditions. It is noted that the shear strength of the material is composed of bonding force and internal friction force. When the internal friction force is considered alone, the bonding force between the two parts of the material along the shear plane can be roughly regarded as zero in the limit case. Based on this, several groups of cast iron cast iron friction pairs with different surface conditions were measured by tilt method. The minimum value of the static sliding friction angle measured without lubrication is 10. Obviously, from the perspective of engineering application, this can be considered as the lower limit of the internal friction angle of cast iron. In addition, from the molecular theory of friction, the formula for calculating the friction coefficient f is f =/PC, where is the shear strength of the micro adhesive node and PC is the yield pressure of the softer material in the friction pair. For isotropic materials, when adhesive friction occurs, from the relationship between Y and Y in the yield condition,

thus = 30. This is the upper limit of the internal friction angle of cast iron

substitute the lower and upper limits of the value into equation (6) respectively, and get

5060 (7)

this proves theoretically that when the cast iron rod is damaged by the same direction pressure, the included angle between the failure section and the cross section of the rod should generally be about 55. The theoretical analysis conclusion is consistent with the above test results. If formula (6) is substituted into formula (5), the maximum value of the total shear stress on the failure section is

but the performance stability is poor

(8)

4 the correction of the strength condition of cast iron pressure bearing members

the theoretical analysis method used in the current project does not consider the influence of the internal friction force of the material, then the maximum shear stress on the dangerous section is max = 45 =/2, and the strength condition established according to the maximum shear stress theory is shown in formula (2)

considering the influence of internal friction, the maximum shear stress on the dangerous section is shown in formula (8). The strength condition established according to the maximum shear stress theory is

(9)

if K is used to represent the internal friction influence coefficient of the material and

(10)

, the strength condition formula (9) can also be expressed as

(11)

formula (11) is a supplement to the current strength condition formula (2) of cast iron pressure bearing members. The two forms are basically the same, and the difference is only whether the internal friction influence coefficient K is included. Equation (11) has the advantages of being more accurate and more in line with engineering practice than equation (2). As mentioned above, the value range of internal friction angle of cast iron is 1030, so from equation (10), K is a positive number that is always greater than 1. It can be seen that the compressive capacity of cast iron is significantly improved after taking into account the influence of internal friction. If the upper limit of internal friction angle = 30 is substituted into equation (10), after calculation, its compressive capacity can be increased to 1.73 times of the original. In general engineering practical problems, if the internal friction angle of the material cannot be determined, it is suggested to take the middle smaller value as = 15 ~ 20, at this time, k = 1.30 ~ 1.43, and substitute into formula (8) and formula (11). The calculation results are in good agreement with the actual situation. For parts with particularly high safety requirements, it is recommended to take the minimum value of = 10. At this time, k = 1.19, and its compressive capacity can still be increased by nearly 20%

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