Study of an Annular Section
Abstract Generated abstract
This study investigates annular shearing in mechanical safety disks used in rubber-heating rolls, where failure load depends on the thickness and geometry of the sheared section. Experiments on grooved and grooveless steel disks recorded load and punch indentation while varying shear-section thickness, and results were compared with ordinary shear tests on rectangular specimens. The authors find that grooved disks show a complex dependence of nominal shear stress on thickness, attributed to changes in stress state, skewing, plastic deformation, and clamping effects, so standard shear strength values cannot reliably predict their limiting load. Grooveless disks behave more regularly, with limiting loads approximately proportional to shear area but typically exceeding values calculated from standard shear strength, while shear strength also decreases somewhat with increasing absolute specimen dimensions.
Full Text
INVESTIGATION OF AN ANNULAR SECTION
T. M. SEDOKOV, M. V. MOISEENKO, A. F. SELIN
(Presented by the scientific seminar of the Department of Strength of Materials)
In many machines mechanical safety devices are provided which are sheared off under overloads, thereby making it possible to avoid accidents and breakdowns of such machines. In rubber-heating rolls of the cable and rubber industries, mechanical safety devices in the form of a disk with annular grooves are used. The calculation of safety disks for the maximum load can be reduced to determining the thickness of the section, since the width of the section, equal to the perimeter of the cutting punch, remains constant for a given machine. But for such a calculation it is necessary to know the magnitude of the shear strength of the material from which the disks are made. As the first experiments showed, the shear strength for one and the same material is a variable quantity and depends on the thickness of the section. Therefore there arose the need for a detailed investigation of the process of annular shearing, which was carried out in the Strength of Materials Laboratory of the Tomsk Polytechnic Institute in 1959.
In the investigation indicated, the force and deformation were recorded during the process of annular shearing of disks of various thicknesses, which were made from two grades of steel: steel 6 and steel for wagon axles. The mechanical characteristics of these steels proved to be the following:
| Steel 6 | Axle steel | |
|---|---|---|
| Tensile strength, kg/mm² | 51 | 53 |
| Relative elongation at rupture, in % | 26 | 15 |
| Shear strength, kg/mm² | 32.1 | 34.5 |
The shear strength was determined on specimens of square cross section \(10 \times 10\) mm under single-sided shear. The shear strength of the material was taken to be the maximum tangential stress, equal to the ratio of the shearing force to the initial area of the transverse section of the specimen. The strength limit in annular shearing was likewise taken to be the maximum tangential stress in the plane of shear, which was determined by dividing the maximum load by the area of the shear plane.
area of the lateral surface of the cylinder punched out by the punch when the disk is pressed through.
The protective disk consists of a central part—a cylinder of diameter 82 mm and height 13 mm—and an outer projection 15.5 mm high, with diameters 88 and 115 mm.
Between the central part of the disk and the outer ring, grooves 3 mm wide are cut on both sides of the disk; the depth of the grooves was varied, thereby producing a change in the thickness of the sheared section.
The dependence of the maximum shear stresses for disks with annular grooves, made of axle steel and steel 6, on the thickness of the sheared section, shown in Fig. 1, has a rather complex character.
Fig. 1. Dependence of the maximum shear stresses on the thickness of the sheared section for disks with annular grooves.
As the thickness of the sheared section is increased from approximately 1.5 to 4 mm, the shear stresses increase from 22.8 to 32 kg/mm² (for axle steel); then they remain almost constant as the thickness is increased to 9 mm, and after 9 mm they rise sharply with a slight increase in thickness and reach almost 40 kg/mm².
Disks with a small thickness of the sheared section (deep grooves) were sheared at stresses considerably lower than the shear strength of the given steel, whereas disks of large thickness (shallow grooves) were sheared at stresses exceeding the strength limit.
Such a change in shear stresses in annular shearing of disks is explained by a change in the state of stress as the thickness of the sheared section increases.
As is known, two basic types of fracture are distinguished: fracture by tearing-off and fracture by shear [1].
Tearing-off, in principle, occurs without prior plastic deformation, but actual fracture by tearing-off is usually preceded by at least a very small plastic deformation; moreover, the resistance to tearing-off depends to a considerable degree on the magnitude of this plastic deformation preceding fracture [2].
In our experiments the width of the annular grooves was constant, while their depth varied with the change in the thickness of the layer being sheared. With a small shear thickness the cuts (grooves) proved to be deep and thereby created conditions that hindered the development of substantial plastic deformations preceding fracture.
Among these conditions is the considerable skewing of the weak link of the safety disk, which, in its results, may be compared with the tensile testing of notched specimens with skewing.
In tensile testing of notched specimens with small skewing, an increase in the nominal stress is observed in comparison with a smooth specimen; whereas at a skew angle of the order of \(8^\circ\), a considerable drop in the nominal stress is observed. In our experiments, at a small shear thickness, a considerable skewing of the weak element of the disk was observed, and as the shear thickness increased this skewing decreased sharply.
This, in our opinion, explains the low value of the nominal destructive stresses in tests of disks with a small thickness of the layer being sheared. We determined tangential stresses, while fracture proceeds by separation; consequently, the nominal tangential stresses cannot be the cause determining the fracture process of such disks.
With a gradual increase in the thickness of the layer being sheared, the groove depth decreased and conditions were created under which fracture was preceded by considerable plastic deformation, which, as is known, leads to fracture by shear.
Thus, the growth of the maximum tangential stresses along the shear plane with increasing shear thickness can be explained, first, by an increase in resistance to separation with the growth of plastic deformation and, second, by the transition to fracture by shear.
In testing disks with a large shear thickness, the outer ring bent strongly and clamped the central pressed-through part of the disk. The presence of radial compressive stresses along the shear plane also explains why the maximum tangential stresses for such disks turn out to be higher than the shear strength for the given steel.
As indicated above, in testing the disks the shear deformation was measured; in this case, the depth of indentation of the punch into the specimen was taken as the shear deformation. Measurement of the depth of indentation of the punch into the specimen up to the moment of final single rapid shear was carried out with an indicator.
The dependence of the absolute deformation of punch indentation on the shear thickness has the same character as the dependence of the maximum tangential stresses on the shear thickness.
A certain discrepancy in the indicated graphs is observed in the range of shear thickness from 4 to 9 mm.
In this range the shear strength remains almost constant (Fig. 1), while the shear deformation increases, although considerably more slowly than for shear thicknesses from 1.5 to 4 mm and above 9 mm.
The relative deformation of punch indentation decreases with increasing shear thickness. The increase in the relative deformation of punch indentation with decreasing shear thickness is explained by the fact that, in testing disks of small thickness, there was considerable bending deformation of the layer being sheared (Fig. 2). The indicator measured the displacement of the punch, and for disks of small thickness this displacement occurred mainly because of the skewing of the layer being sheared in the disk and, to a lesser extent, because of shear deformation.
Fig. 2. Safety disk with annular grooves.
Fig. 3. Dependence of the maximum shear stresses on the thickness of the sheared layer for disks without grooves under annular shearing.
Vertical axis: Ultimate strength under annular shear, \(\tau_{\text{shear}}\), kg/mm\(^2\).
Horizontal axis: Thickness of the sheared layer, \(\delta\), mm.
Inset:
\[ \tau_{\text{av}}=\frac{P_{\max}}{\pi D h}. \]
In the course of investigating the operation of disks with annular grooves, a design for a grooveless disk was proposed.
The dependence of the maximum shear stresses in annular shear of grooveless disks made of axle steel on the thickness of the shear section is presented in Fig. 3, curve 1. The strength limit for grooveless disks is also a function of the shear-section thickness, but varies within smaller limits. When the shear-section thickness is 2 mm, the stress is about 36 kg/mm²; then it rises rapidly to 42 kg/mm² at a shear-section thickness of 4 mm, and subsequently slowly falls to 38 kg/mm².
All grooveless disks were sheared at stresses exceeding the shear strength limit for the given steel.
The decrease in the annular-shear stresses for grooveless disks with increasing shear-section thickness is associated with the influence of the absolute dimensions of the specimen.
To verify this proposition, a series of ordinary rectangular-section specimens for shear was made. The width of all specimens was constant, while the height varied from 2 to 11 mm. The tests showed that the change in stresses in this case has the same character (Fig. 3, curve 2) as for grooveless disks. The decrease in the strength limit when the specimen thickness changes from 4 to 10 mm in simple shear is about 10%. For grooveless disks, when the shear-section thickness changes within the same limits, a stress drop of the same order, about 10%, is also observed.
Consequently, the noted decrease in stresses with increasing thickness for grooveless disks is associated with the influence of the absolute dimensions of the specimen.
In the experiments, a certain increase in the strength limit is noted at a shear-section thickness of about 4 mm, both for grooveless disks and in simple shear of rectangular-section specimens.
On the basis of the investigations carried out, the following conclusions may be drawn:
-
Calculation for the maximum load of safety elements having a complex shape (grooves, cuts, recesses) cannot be carried out according to the shear strength limit determined on ordinary specimens in mechanical tests, since the actual maximum stress in shear of such elements may be considerably less or substantially greater than the shear strength limit.
-
Safety elements in machines and structures should have a very simple shape (without grooves, cuts, or recesses); the limiting load for such elements will be approximately proportional to the shear area, but the actual limiting load will be 20–25% greater than that calculated from the shear strength limit for the given material.
-
The shear strength limit depends on the absolute dimensions of the shear area and decreases somewhat as this area increases.
LITERATURE
- N. N. Davidenkov, Dynamic Testing of Metals. 2nd ed., 1936.
- B. Fridman, Mechanical Properties of Metals. Oborongiz, 1952.