STUDY OF STRESSES IN THE CONTACT ZONE DURING RADIAL COMPRESSION OF A DISK

Yu. S. Bogomolov, L. M. Sedokov

(Presented by the scientific seminar of the Department of Strength of Materials)

Contact stresses, which inevitably arise when a load is transmitted through a finite area, are of great practical interest. A number of works by such scientists as H. Hertz [1], A. N. Dinnik [2], N. M. Belyaev [3], and others have been devoted to their study.

In our work, the study of stresses in the contact zone of a disk compressed along its radius was carried out experimentally by means of static electrical strain measurement. The developed method for calibrating low-base wire strain gauges [4] made it possible to obtain sufficiently reliable results.

Fig. 1. Study of stresses in a compressed disk by means of strain gauges

On a disk made of organic glass, 40 cm in diameter and 4.5 cm thick, two groups of strain gauges were placed in the contact zone (Fig. 1). The comparatively large dimensions of the disk made it possible, by means of

with strain gauges having a base of 5 mm, to determine the stresses at a distance of 5 mm from the point (area) of action of the compressive force, which amounts to 0.0125 of the disk diameter.

The first group of 9 strain gauges is arranged along the vertical diameter in the immediate vicinity of the point of application of the force. To determine the principal stresses, it is necessary to attach the gauges at the point under investigation in such a way that their axes are oriented in the direction of these stresses. Accordingly, 6 gauges are placed along the line of action of the compressive force (the vertical axis of symmetry) on one side, and three along the horizontal on the other side of the disk.

The first vertical strain gauge, nearest to the point of application of the force, is located at a distance of 5 mm; the remaining ones are attached at intervals of 2 mm. In order to maintain such a spacing, which is necessary for determining stresses at as many points as possible, it was necessary to attach part of the grid of the second gauge onto the grid of the first, to shift the third and fourth gauges to one side relative to the first, and the fifth and sixth to the other. In this arrangement the fourth gauge is attached onto the third, and the fifth onto the sixth.

The purpose of the first group of strain gauges is the quantitative determination of stresses at points of the vertical diameter located in the contact zone, the boundaries of which were determined (approximately) in [4].

The second group of 12 strain gauges is arranged along the trajectories of the principal stresses in the vicinity of the point of application of the force, in the region between the vertical diameter and the circumference of the disk. The trajectories of the principal stresses were determined by the method proposed by M. Frocht [5].

On one side of the disk, three gauges are attached along the trajectories of tensile stresses, and three along the trajectories of compressive stresses. Gauges are likewise arranged on the other side of the disk. The purpose of the second group of strain gauges is to determine the width of the zone in which the actual stresses deviate from the theoretical ones at the point of application of a concentrated load.

In addition to the strain gauges listed above, to check the correctness of the disk installation, 4 control gauges attached along the vertical diameter were used.

The coordinates of the points at which the strain gauges are located are given in the table.

The disk was subjected to radial compression between parallel press plates. The readings of the strain gauges were measured by means of a highly stable 40-point strain-measuring installation VST-3 of the Ural Polytechnic Institute; the electrical circuit of this installation is based on the “zero method” for measuring a small increment of resistance on a direct-current bridge, with the increment read from the scale of a measuring slide-wire.

Figures 2 and 3 show the stress diagrams of $\sigma_x$ and $\sigma_y$ for the vertical diameter of the compressed disk. The diagrams are plotted for loads of 1000 and 1800 kg.

Comparison of the measured stresses with the theoretical ones shows that the former, within an interval of 0.1 of the diameter from the point of application of the force, differ substantially from the latter.

The tensile stresses $\sigma_x$ in this zone give a peak (maximum), unlike the theoretical stresses, whose diagram should have the form of a straight line parallel to the coordinate axis. The magnitude of the maximum exceeds the theoretical stresses by a factor of 12–13.

Table

I group of strain gauges

Control gauges / II group of strain gauges

At a point located at a distance of 5 mm from the point of application of the force, the tensile stresses are 5–6 times greater than the theoretical values.

With increasing distance from the point of application of the force toward the center, the tensile stresses increase, reaching a maximum at a distance of 1.3 cm, or 0.0325 of the diameter, from the circumference of the disk. After the maximum is reached,

Fig. 2. Stress envelope \(\sigma_x\) along the vertical diameter of the disk.

Fig. 3. Stress envelope \(\sigma_y\) along the vertical diameter of the disk.

the magnitude of the tensile stresses decreases, and at a distance of 0.1 of the disk diameter from the point of application of the compressive force, the envelope of the stresses obtained experimentally coincides with the envelope of the theoretical stresses.

The compressive stresses \(\sigma_y\) in the contact zone are smaller than the theoretical ones by approximately a factor of 3. At the point corresponding to the maximum tensile stresses, the compressive stresses have a local minimum. The magnitude of these minimum stresses is 4 times smaller than the theoretical ones.

After the local minimum is reached, a certain increase in the compressive stresses is observed, and at a distance equal to 0.1 of the diameter, the epure of the actual stresses does not differ from the theoretical epure.

To compute the compressive stresses at the midpoint of the displacement of the strain gauges, one may use the formula [5]

\[ \sigma_{A-B}=\sigma_1+\frac{\sigma_1-\sigma_2}{a}\cdot\frac{l-a}{2}. \]

Here \(\sigma_{A-B}\) is the average stress on the segment between the points
\(A(x=0,\ y=19.75\ \text{cm})\) and \(B(x=0,\ y=19.55\ \text{cm})\);

\(\sigma_1\) is the stress determined from the readings of gauge 1;
\(\sigma_2\) is the stress determined from the readings of gauge 2;
\(l\) is the gauge base \((l=5\ \text{mm})\);
\(a\) is the displacement of one gauge relative to the other \((a=2\ \text{mm})\).

In this case the compressive stresses can be determined at the point with coordinates \(x=0,\ y=19.65\ \text{cm}\), i.e., at a distance of 0.00875 of the disk diameter from the circumference.

For loads of 1000 and 1800 kg they are, respectively, equal to 112 and \(182.5\ \text{kg}/\text{cm}^2\) and fit well on the experimental curves (Fig. 3).

Changing the compressive load from 200 to 2000 kg has practically no effect on the position of the peak of the principal stresses in the contact zone.

The readings of the strain gauges of the second group indicate the same tendency: an increase in tensile stresses and a decrease in compressive stresses for the zone under consideration, as compared with the theoretical values. Thus, the points investigated by means of the gauges of the second group belong to the contact zone. Evidently, the dimensions of the contact zone are bounded by a circle of radius about 4 cm, i.e., one tenth of the disk diameter.

Conclusions

1. The stresses at points of the disk located in the vicinity of the point of application of the compressive force differ substantially from the theoretical values. The tensile stresses exceed the theoretical ones by a factor of 5 to 13, while the compressive stresses are 3–4 times smaller than the theoretical ones.

2. The signs of the principal stresses in the contact zone do not change, i.e., \(\sigma_x\) remains positive and \(\sigma_y\) negative.

3. In the contact zone the principal stresses have extreme values at a distance of approximately 0.03 of the diameter from the point of load application: the tensile stresses are maximal, and the compressive stresses are minimal.

4. A substantial difference between the actual and theoretical stresses is observed at a distance of no more than one tenth of the diameter from the point of load application. Consequently, the boundary of the contact zone may be regarded as the family of points lying in the region bounded by a circle of radius about one tenth of the disk diameter.

References

1. H. Hertz. Über die Berührung fester elastischer Körper, 1881. Über die Berührung fester elastischer Körper und über die Härte, 1882.

2. A. N. Dinnik. Impact and Compression of Elastic Bodies. 1909.

3. N. M. Belyaev. Works on the Theory of Elasticity and Plasticity. GosINTI, Moscow, 1957.

4. Yu. S. Bogomolov, L. M. Sedokov. Experimental determination of stresses in the cross section of a cylindrical specimen under radial compression. TPI Collection, No. 1963.

5. L. M. Sedokov, Yu. S. Bogomolov. Manual for laboratory work on electrical strain-gauge measurement in the study of the course Strength of Materials. Tomsk, 1962.