THE EFFECT OF ELASTIC RECOIL OF CLOSED-TYPE LIFTING ROPES AFTER THEIR MANUFACTURE AND DRAWING

Abstract. A study was made of stresses and factors of internal forces in the elements of ropes of a closed structure during their manufacture. It is established that when twisting a rope of closed construction, its shaped wires suffer from bending deformation, twisting and stretching. In this case, the shaped wires are subjected to a complex load with the concomitant rotation of the axes of the stress tensor. The stresses in the elastic cross-sectional area of the rope wire are considered and its limit is determined for shaped cross-sections and for asymmetric shaped cross-sections. Formulas for approximate determination of tangential and normal stresses in the elements of a closed rope of noncircular profile are obtained: (wedge-shaped, zeta-shaped and x-shaped).

Experience in the manufacture and operation of ropes of closed design shows that immediately after twisting the closed rope, which is an elastic-plastic system, at the first load acquires significant elongations, and its stress-strain state changes significantly. As a result, a number of serious structural defects (bundles, "waves", wire breaks, etc.) often appear during the first cycles of rope operation, which is the reason for the failure of a new rope [1,2].
The influence of the presence of gap between the wires in the outer layers and its value on the compatibility of operation of the layers in the radial direction and the preservation of the structural integrity of the rope during operation is revealed [3]. Analysis of the stress-strain state of closed rope elements under axial tension and torsion [4]. The closed rope consists of an outer layer of Z-profile wires, a subsurface layer of alternating round and H-profile wires [5,6,7]. Evaluation of extended structural elements using non-contact mobile systems using body waves and directional waves (piezoelectric, electromagnetic-acoustic transducers) [8]. The axial forces and torques in the cross sections of the layers are found to be redistributed when a rope turns under an external torque, which leads to a decrease in the safety factor of the rope, a violation of the compatibility of the axial and radial displacements of the layers, and a violation of the structural integrity of the rope in the form of breaks in the outer layer wires [9,10]. The influence of wire cracks on the amplitude distribution of the generated field is specified for two steel rope kinds assuming surface and inner defects [11]. The behavior of short and very short fatigue cracks emanating from so-called "smooth" specimens with stress concentration is described [12].
To assess the reliability of the rope and prevent the occurrence of these defects, as well as to assess its strength and durability, it is necessary to know the stress-strain state of its constituent elements of wires, both during its manufacture and after the first axial load. It is established that when twisting a rope of closed construction shaped wires suffer together with bending and torsional deformation, and then during extraction during operation -tensile deformation. In this case, the shaped wires are subjected to a complex load with the concomitant rotation of the axes of the stress tensor. It should be noted that the bending of shaped wires of some profiles is oblique [3,4]. For example, when twisting z-shaped and 8-shaped wires, the plane of the bending moment, which contains the normal of the helix, does not coincide with the main axis of the wire cross section y0 or z0 (Fig. 1). These wires experience oblique bending, and wires of round, x-shaped and wedge-shaped profiles undergo flat bending. Thus, the material of the wires is subjected to a complex load, the analysis of which the existing theories of plasticity do not solve. There are known developments of some methods for solving this problem, but these methods can be obtained only local solutions that meet the rigid arameters [1, 2, 5].
Thus, the study of technological stresses and internal power factors in the shaped wires of closed ropes in general form is an urgent scientific and applied task.
Given the complexity of the task of analysis of technological stresses and internal force factors in the shaped wires of closed ropes, the study is proposed to approximate its solution under the general assumption that the wire material is idealelastic-plastic, and the entire cross section of wires covered by elasticplastic deformation.
Consider the stress in the elastic cross-sectional area of the rope wire and determine its limit. Normal bending stress is determined by: for shaped cross-sections symmetrical with respect to the bending plane xy (wedge-shaped, 8-shaped) wire ropes (see Fig. 1, a and c) by expression where be x,y σnormal bending stress; -curvature of the centerline of the wire; Rwire twisting radius; Е -Jung's module.
-(z-shaped) wire ropes (see Fig. 1, b) by expression for asymmetric shaped crosssections where z0 and y0the main central axes of the wire. In Figure 1, b φ0 is the polar angle, which is calculated from the main central axis z0.
Analyzing the plots of tangential stresses during torsion of non-round rods, for which exact solutions are obtained, it can be noted that they have slight nonlinearity ( Fig. 1, a). This nonlinearity is smaller the greater the sloping delineation of the wire profile, and for elliptical and round cross-sections of wires the nature of the change in tangential stresses across the section is linear.
This fact gives the right to make a few additional assumptions. First, for smoothly delineated profiles of sections of shaped wires of a closed rope, the displacements γxy and γxz with a sufficiently probable approximation can be calculated from the linear functions of the z and y coordinates (Fig. 2).
where a and bsegments on the main axes of the cross section of the wire. Secondly, the torque in the cross section of the wire with a smoothly delineated profile is determined by the expression where τxy and τxzprojections of full stress on the corresponding planes; Gshear strength modulus. Substitute formula (3) into expression (5) and, given equality (4) The twisting angle of the shaped wires is determined by the formula where Itorsthe moment of inertia at pure torsion with a sufficient degree of accuracy can be determined by the formula of Saint-Venan [6].
where F and Ipplane and polar moment of inertia of the wire cross section, respectively. From formula (8) we obtain: Thirdly, the moment of elastic recoil of each layer of wires of the closed rope, taking into account expression (1)  where nthe number of wires in each layer; αthe twist angle of the wires; Mzthe moment of bending relative to the z-axis of the cross section of each wire; Mxtorque in the cross section of each wire. Based on the accepted assumptions and mathematical studies of stresses arising in the process of making ropes, simplified formulas for determining the bending and torques in the cross sections of closed rope wires, taking into account the coefficients A1, B1, N1, N2, N3. Formulas for their determination in the polar coordinate system starting at the center of gravity of the wire cross section were obtained in [3].
With an approximate solution of the problem obtained: for shaped cross-sections of wires symmetrical about the plane of bending xy (x-shaped):  (12) The obtained results can be compared with the coefficients A0, B0 and C0, which characterize the degree of plastic deformation under uniaxial loading (stretching) and have the same intensity as under complex loading.
where φ and ρі (φ)current polar angle and radius-vector of points of the contour line of section; Δ(φ)increase in the current polar angle; A0, C0characterize the degree of plastic deformation under uniaxial loading (stretching).

Conclusions.
1. Further study of the elastic recoil after pulling the closed rope using expressions (3) and (4) shows that the stress-strain state of the components of the closed ropes changes, while the redistribution of stresses in the cross sections of wires, and the moment of elastic recoil of the rope as a whole decreases. This significantly improves its operating conditions and increases service life.
2. The main criterion that determines the change in the moment of elastic recoil is the symmetry or asymmetry of the shaped cross section of the wire. In order to evaluate the effect of stretching on the magnitude of the moment of elastic recoil of the rope, a comparative analysis of the obtained experimental results was carried out, as a result of which a significant difference was observed in the distribution of forces over the layers of an unstretched and pre-stretched rope with a subsequent nominal load.
3. Calculations of technological internal force factors in the cross sections of wires and moments of elastic recoil on the layers of the rope make it possible not only to assess the degree of technological imbalance of the rope, but also rationally choose the direction of twisting in the layers. As a result, reduce the moment of elastic recoil and ensure the reliability and durability of the rope structure during its operation.