# Effects of torsion

- the right angle between the generatrix and directrix lines will be modified during torsion; this modification is smaller near the edges and larger near the midline of the plate face of the bar;
- if the length of the bar is greater in comparison with the dimensions of its basis, one small rotation suffices to rotate the sufficiently deformed sections with an important angle.

### Remark:

Based on the aforementioned observation, that the squares placed near the edges of the bar will remain undistorded and those from the middle of the outer plate faces will become rhombuses after twisting, we can conclude that the specific slips are maximum at the midpoints of the section sides and zero at the section corners.

All these aspects, which can be remarked in the case of the bars twisted under the limit of proportionality of their materials, lead to the following conclusions:

- the generatrixes of the bar, which were straight lines before the load application, will become parallel inclined and curved lines after twisting; in the case of round bars, the deformed generatrixes will have equal lengths, but in the case of the bars having non-circular cross-sections the length of the deformed generatrixes will be different from a helix to another;
- the transverse lines marked on the outer surface of the rectangular bar are distorded in the direction of the axis of bar, but the directrix circles marked on the outer surface of the round bar will remain undistorded after twisting;
- the angle between the directrix and generatrix lines, which was right before twisting, will modify its size during torsion, only, in the case of rectangular bar.

According to the laws and hypotheses of the Mechanics and Strength of Materials we can, also, make the following remarks relating to the effects of elastic torsion:

- the inclination of the generatrixes, as effect of the cross-sections rotation, leads to their helical orientation with the increase of the angle of twist, the resulted helical curves having a large pitch and a small curvature;
- the circular cross-sections, with respect to Bernoulli's hypotheses don't warp, but the non-circular cross-sections, according to the hypotheses concerning the displacement in torsion, will warp;
- in the case of the round bars, the non-existence of the differences in length between the deformed generatrixes leads to conclusion that the axial stresses don't occur during elastic torsion and the existence of the angular deformations shows that the twisting load determines the apparition of the shearing stresses on the cross-sections of bar; on title contrary, in the case of the non-circular bars, the differences between the length of the deformed generatrixes indicate the existence of the axial stresses and the change of the angles between the generatrix and directrix lines indicates the existence of the shearing stresses on the cross-sections, these stresses having a non-uniform distribution. [4] [6] [11]

The problem of the generatrixes inclination, in the case of bars twisted under the limit of proportionality of the materials, is, generally, treated by the technical literature from the theoretical point of view. From the well-known relation which expresses the angle of inclination y of the generatrix in the case of round bars:

where r is the radius of bar, f is the angle of rotation of the cross-section, 1 represents the length of bar and 8 is the angle of twist per unit length, we can assume that between the phenomena of the generatrixes inclination and cross-sections rotation there is a tight connection.

The problem relating to the warping of the non-circular cross-sections, was, also, examined from the theoretical point of view, the efforts being devoted in order to determine the warping functions for some cross-section shapes, such as: triangular, elliptical, rectangular or square sections. Thus, the problem of finding of the elliptical and triangular cross-sections warping in the elastic and elastic-plastic torsion is treated in [7] [13] [4]. The warping of the rectangular and square cross-sections for the elastic and elastic-plastic torsion is examined in [7] [4].

### Remarks:

In the case of prismatic bars having triangular and notched circular cross-sections, the problems of finding of the warping and center of twist is analysed in [13]. The main conclusions of the analysis is that the center of twist (which defines the points where the mean square of the warping is minimum and which is the, only, line that remains straigth in the twisted bar) is not coincident with the centroid if the section has, only, one axis of symmetry (notched circular section) and is coincident with the centroid if the section has more than one axis of symmmetry (triangular section). ii. By using the FEM analysis in [10] it is demonstrated that the rectangular section has two lines (parallel to x and y directions) and the square section has more than two lines (between the opposite corners of the section) which don't move in the direction of the axis of bar. In Fig. 4.4 it is shown the deformation of the square section at the midpoint of a center axis in the case of a bar twisted with an angle of twist per unit length equal to 90°.

### 4.1.2 Concluding Remarks

From this summary analysis we can conclude that the torsion will determine significant modifications of the initial configuration of bar.

In the elastic torsion the main effects of the twisting load on the initial configuration of bar are as follows:

- the inclination or helical orientation of the generatixes of bar; these generatrixes, which initially were straight lines, will become helical curves of large pitch and small curvature after twisting;
- the warping of the non-circular sections in the axial direction. These two effects are very important to be known in the designing phase of the parts, because in the servicing stage every change of their configuration can determine certain disturbances in the behaviour of the assembly or machine which contains them.

Relating to these effects, the technical literature offers an ample data basis which permits the accurate determination of the behaviour of different bars during elastic torsion.

In the plastic torsion, the effects of the twisting load on the configuration of bar are little known. Generally, the theoretical and experimental studies have been made in order to study the plastic torsion problem of the round bars, which are more frequently used in the torsion tests effected in order to establish the behaviour of the metallic materials; but, the results of these tests don't indicate the modification of the initial configuration of these bars during torsion.

The effects of the plastic torsion on the configuration of the non-circular bars (such as: the helical orientation of the longitudinal profiles and fibres of material), are used in some technical applications of torsion such as: the generating of the twist-drills helical profile, the improvement of certain strength characteristics of the bars, the improvement of the mechanical properties of the materials, the achievement of the ornamental profiles, a.s.o. Relating to these effects the technical literature offers few studies and results which don't permit to determine the parameters that characterize the changes of the bar configuration and to explain the mechanism of their apparition.

In this chapter are analysed the main aspects concerning the changes which take place in the configuration of the non-circular bars twisted in the cold range of temperature over the yield limit of their materials.

straight lines contained in normal planes, will suffer after twisting small inclination in comparison with the normal plane and a distortion in axial direction;

- the edges of the longitudinal sections cutted along the axis of the twisted bars are, alternativelly, composed of convex and concave curved lines (Fig. 4.9); the correspondent points of the concave curved lines, which initially were disposed along straight lines parallel to the axis of the bar, will be orientated along a direction inclined with an angle equal to the angle of inclination of the resulted helix after twisting.

The comparative analysis, between the mode of deformation of these bars, points out the following different aspects:

- in the case of the bars having longitudinal grooves, the outer surface, which initially was composed of concave and convex circular surfaces, will be formed by concave and convex helical surfaces after twisting;
- in the case of the bars having square and rectangular cross-section shapes, their outer surfaces, which were composed of rectangular plate faces before twisting, will be formed by concave helical surfaces after twisting;
- the transverse lines marked on the outer surfaces of the bars, which were contained in planes normal to the neutral axis before twisting, in the case of bars having longitudinal grooves will result after twisting more much inclined and distorded than in the case of square and rectangular bars.

From this analyses we can, also, conclude that the torsion over the yield limit will determine the following changes in the axial configuration of the bars having non-circular cross-section shapes:

- in all the analysed cases, the longitudinal geometric forms will be helical orientated after twisting, the shape of the resulted helical surfaces depending oh the initial configuration of bars;
- the cross-sections of bars, which initially were contained in planes normal to the neutral axis, will warp in the axial direction during torsion;
- the transverse lines and the cross-sections of the bar will result inclined in comparison with their initial positions, the angle of inclination depending on the configuration of bar and on the shape of the resulted helical surface.

This complex mode of deformation of the bars during torsion, which has as results the helical orientation of the longitudinal geometric forms, the inclination and warping of the cross-sections in comparison with the neutral axis of bar, is caused by certain specific phenomena that occur in the bars twisted over the yield limit of their materials.

Thus, the helical orientation of the longitudinal generatrixes and geometric forms is the result of the twisting load applied to the ends of bar. According to the hypotheses concerning the deformations in torsion, the application of the twisting

On the basis of the afore mentioned aspects relating to the changes in the configuration of bar in axial direction, we can conclude that during plastic torsion the helical orientation of the longitudinal geometric forms has as result the elongation of the generatrixes of bars and hence, of the material fibres. The size of the elongation is a function of the angle of twist, the shape and dimensions of the cross-section and the size of the contractions in length which take place during torsion. The variation of the helix length takes place in the sense of its increase to the increase of the angle of twist and depends on the position and distance of the generatrix, which generates the helix, in comparison with the neutral axis of bar; thus, the helical curve of maximum length will be generated by the generatrix placed to a maximum distance from the neutral axis. These aspects of variation of the helix length and generatrix elongation are very important to be known because they can clarify some unknown problems such as: the warping of the cross-sections, the crack initiation and the fracture of bars during torsion. One of the conclusions which results from this analysis is that the cracks and fracture can start from the points contained by the most elongated generatrix and material fibre. Another conclusion, which was, also, formulated in the previous paragraph, is that in the case of non-circular bars the length of the resulted helixes differs from a helix to another; these diffeences in length explain the effect of torsion on the cross-sections of bar in direction of the neutral axis, i.e. the warping of the cross-section.

The variation of the angle of twist will determine the variation of the resulted helix pitch in the sense of its decreasing to the increase of the number of rotations. The variation of the helix pitch is, also, influenced by the dimensions of the bar, the initial cross-section shape and the changes in length which take place during torsion.Over certain limits of the angle of twist and depending on the above presented influences, the material of bar has the tendency to work-hardening. The work-hardening phenomena, which is, also, influenced by the structure and chemical composition of the material, can determine after the cessation of the twisting load application, the elastic recovery of the bar, and, hence, the difference between the calculated and the real parameters of the resulted helical profile.

The aspects relating to the variation of the helix pitch are very important to be known for the following applications of torsion as working procedure: the generating of the helical profiles of the twist drills and the achievement of the ornamental profiles.

According to the last remark formulated in the paragraph 4.2.2, that, the cross-section are elastic-plastic deformed and cannot touch the fully plastic state, because