EXPERIMENTAL DETERMINATION OF STRESSES

IN THE CROSS SECTION OF A CYLINDRICAL SPECIMEN

UNDER RADIAL COMPRESSION

Yu. S. BOGOMOLOV, L. M. SEDOKOV

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

The most important mechanical characteristic of a material is its tensile strength. Determining this characteristic for brittle materials—cast iron, bronze, stone, concrete, brittle plastics, and others—is very difficult. These difficulties increase the scatter of test results for one and the same material under identical conditions, which leads to substantial uncertainty in the strength characteristic of the material being tested.

In order to eliminate the considerable scatter of test results, existing standards provide for the use of specimens of complex configurations with a predetermined fracture location, as well as other measures that introduce their own distortions and errors. Thus, for example, when specimens with a predetermined fracture location in the form of “figure-eights” are used, the test results will not correspond to the linear stress state of the metal in the fracture zone when the test is carried out using specimens with a constant cross section.

The technique of fabrication and the testing conditions for specimens in axial tension according to existing standards are of a certain complexity. GOST 4649-50 for plastics provides for specimens of complex shape with rectangular heads, which are connected to the working part of the specimen by fillets. Special clamps are required to install such specimens in the grips of the testing machine.

In addition, possible eccentricity in the application of the tensile force introduces distortion. For plastic materials this is not of substantial importance, since the plastic deformation preceding fracture reduces the eccentricity of the acting force.

All this makes it necessary to determine the tensile strength of materials by indirect methods. However, the existing indirect methods for determining this characteristic in turn have shortcomings, are poorly substantiated theoretically, and the results obtained by these methods depend significantly on the testing conditions. Therefore, further investigations are needed to develop methods for reliably determining the tensile strength of brittle materials.

In the strength-of-materials laboratory of the Tomsk Polytechnic Institute, work was carried out on determining the tensile strength of cast iron \[1, 2\] and brittle plastics by compressing cylindrical specimens along the radius \[3\]. This method is used in testing concrete \[4\].

Fig. 1. Fracture of a cylindrical specimen of a brittle material under radial compression.

The essence of the proposed method for determining the tensile strength of brittle materials is that a cylindrical specimen is subjected to radial compression on an ordinary testing machine, with the breaking load recorded. In this case the specimen fails along a diametral section; the specimen, as it were, splits into two halves (Fig. 1). Often, along the generatrices to which the force is applied, triangular prisms are chipped out.

The strength limit of the material in this type of mechanical testing can be determined from the formula

\[ \sigma_{\mathrm{r}} = A \frac{P_{\max}}{dl}, \]

Here \(P_{\max}\) is the maximum force during the test, \(d, l\) are the diameter and length of the specimen, and \(A\) is a dimensionless coefficient.

The value of the dimensionless coefficient is subject to careful theoretical and experimental determination.

It may be assumed that failure of a cylindrical specimen under radial compression occurs as a result of the action of natural wedges, which appear along the generators from local overstresses near the point of application of the force.

Taking the angle at the vertex of the wedges that form to be equal to \(90^\circ\), one can obtain \(A\) equal to 0.5. As the experiments carried out on cast iron and brittle plastics [2, 3] have shown, this coefficient ranges from 0.7 to 0.9. The increase in the coefficient \(A\) is probably associated with a change in the angle at the vertex of the wedges and with a reduction of the diametral section due to preliminary chipping out of the wedges. Friction along the wedge faces may also have had an effect here.

The value of the dimensionless coefficient \(A\) can be obtained on the basis of the solution of the problem of the stress distribution in the Hertz disk by the methods of the theory of elasticity. It is known [5] that the tensile stresses on areas coinciding with the plane of action of the force under radial compression of a disk are determined by the formula

\[ \sigma_x = \frac{2}{\pi}\frac{P}{dl} = \text{const}. \]

Consequently, extending the action of Hooke’s law under radial compression of the specimen to brittle materials up to failure, we obtain

\[ A = \frac{2}{\pi} \approx 0.64. \]

All this confirms the necessity of investigations to identify the parameters that determine the variation of the dimensionless coefficient \(A\).

For this purpose, an experimental study was made of the distribution of stresses at various points of the cross section of a Hertz disk over a wide range of variation of the compressive force. The experiment was carried out on an organic-glass disk 40 cm in diameter and 4.5 cm thick. Along a radius of 20 cm the stresses were measured at 10 points, i.e., the gauges were bonded at intervals of 2 cm. To estimate the plane-stress state at each point under investigation, two gauges were placed along the directions of the principal stresses \(\sigma_x\) and \(\sigma_y\). Strain gauges with a 5 mm base were bonded to the disk in the direction of the vertical and horizontal diameters in such a way that on one side of the disk their direction corresponded to \(\sigma_x\), and on the other to \(\sigma_y\). The strain gauges were bonded with dichloroethane adhesive, which consists of 100 parts by weight of dichloroethane and 3–5 parts of organic-glass shavings. The surface of the disk to which the strain gauges are bonded is carefully cleaned with fine emery paper, the disks being placed perpendicular to the axis of the gauge. Degreasing is carried out with acetone and alcohol.

During bonding, the surface of the gauge and the surface of the disk at the points under investigation are coated with dichloroethane adhesive. After drying for 3–5 minutes, the gauge to be bonded and the disk are again coated, and the gauge is immediately placed on the disk. After orienting the axis of the gauge in the required direction, the gauge is covered with triacetate film. Drops of adhesive are squeezed out from under the gauge with a rubber roller or by hand, and it is pressed down with a weight of 400–500 grams. Under the load, the bonded gauge is kept at room temperature for 24 hours.

The readings of the transducers were measured using the high-stability 40-point strain-gauge installation VST-3 of the Ural Polytechnic Institute.

When calculating stresses from the readings of strain gauges having a small base, under a plane stress state difficulties arise owing to distortion of the magnitude of the deformation of the part under investigation. This distortion of the measured strain value is explained by the mechanical and electrical imperfections of loop-type wire gauges. In addition, it is necessary to take into account the dependence of the ratio of the principal stresses on the coordinates of the points at which the stresses are determined.

Thus, A. I. Gugenberger [6], in investigating certain cases of the plane stress state, shows that the coefficient of distortion of strain by loop gauges varies very substantially and lies within the limits from 0.35 to 2.7.

The accuracy of calculations according to the method and formulas of G. A. Mitichenko [7] depends on the accuracy with which the characteristics of the gauges, the strain-sensitivity coefficient of the wire, the modulus of elasticity of the material, and the coefficient of transverse deformation are determined, which in practical application gives a considerable uncertainty in the final results.

Preliminary calibration of the gauges under a linear stress state according to a method developed at the Department of Strength of Materials of the Tomsk Polytechnic Institute makes it possible to take into account the errors introduced into the measurements by the mechanical and electrical imperfections of the wire gauges. The essence of the proposed method consists in the simultaneous calibration of two gauges under linear tension (compression). The calibration was carried out on a beam of equal resistance. One of the gauges is bonded along the tensile (compressive) stresses, and the other perpendicular to the first. The gauges were calibrated in such a way that one time they were located on the stretched fibers, and the other time on the compressed fibers, for which purpose the calibration beam with the bonded gauges was turned over. Figure 2 gives a calibration graph for gauges with a base of 5 mm. A control calibration of two gauges on a plate subjected to direct axial tension practically confirmed the results obtained in calibration on the beam of equal resistance.

If the readings of the recording instrument connected to gauge 1, the direction of whose grid axis coincides with the direction of the acting stresses under axial tension (compression), are denoted by \(N_1\), then its operation is determined by the angle \(\alpha\). Denoting by \(N_2\) the readings of the recording instrument connected to gauge 2, whose grid axis is perpendicular to the acting stresses, we determine the operation of this gauge by the angle \(\beta\).

Considering the plane stress state as the sum of two linear ones (Fig. 2), one can graphically determine the values of the stresses \(\sigma_1\) and \(\sigma_2\). Gauge 1, when \(\sigma_2 = 0\), gives a reading corresponding to the segment \(AB\), but the simultaneous action of the stress \(\sigma_2\) causes a decrease in the reading by an amount equal to the segment \(LM = GL = EB\).

Thus, in calculating stresses under a plane stress state, the reading of gauge 1 should be considered as corresponding to the segment \(AE\), and that of gauge 2 to the segment \(DH\). Then the calculation formulas have the form

\[ \sigma_1 = \frac{\operatorname{tg}\alpha}{\operatorname{tg}^2\alpha - \operatorname{tg}^2\beta} \left( N_1 - \frac{\operatorname{tg}\beta}{\operatorname{tg}\alpha} N_2 \right), \]

\[ \sigma_2 = \frac{\operatorname{tg}\alpha}{\operatorname{tg}^2\alpha - \operatorname{tg}^2\beta} \left( N_2 - \frac{\operatorname{tg}\beta}{\operatorname{tg}\alpha} N_1 \right). \tag{1} \]

Having the results of calibration of gauges with a base of 5 mm,

\[ \tan \alpha=\frac{N_1}{\sigma_1}=6.15, \qquad \tan \beta=\frac{N_2}{\sigma_2}=2.4, \]

Fig. 2. Calibration graph for gauges with a base of 5 mm: scheme of a plane stress state as the sum of two linear ones.

we obtain the working formulas

\[ \begin{aligned} \sigma_1 &= 0.192\,N_1 - 0.075\,N_2,\\ \sigma_2 &= 0.192\,N_2 - 0.075\,N_1. \end{aligned} \tag{2} \]

The described method for determining stresses in a plane stress state, with practically any ratio of the principal stresses, made it possible to process the results of experiments on compression of a disk by

radius between the parallel plates of a press, under a load from 200 to 2000 kg. Data on the magnitude of the stresses \(\sigma_x\) arising in the compressed disk along the vertical and horizontal radii are shown in the table for a load of 1200 kg.

Table

Comparison of the distribution of stresses \(\sigma_x\) in a compressed disk, obtained experimentally, with the results of an analytical calculation by the methods of the theory of elasticity

In Figs. 3a and 3b are given diagrams of the distribution of the normal stresses \(\sigma_y\) for the vertical and horizontal radii of the disk, constructed from the experimental data and from the theory of elasticity. Owing to the symmetry of the disk, the diagrams are made for one quarter of the disk, \(OAB\).

The solid line shows the stress diagram determined by the methods of the theory of elasticity. The circles indicate the magnitudes of the stresses calculated by formulas (2). It is seen from Fig. 3 that these points practically lie on the curve constructed from the data of the theory of elasticity. The crosses show the stresses determined under the same experimental conditions from the same strain-gage readings, but calculated in the usual way, with a preliminary calibration of one strain gage. From these points, diagrams of the stress distribution are constructed in the form of a dashed curve. In this case the discrepancies with the diagram made from the data of the theory of elasticity reach 10%.

The experimental data indicate that the experimental determination of the stress distribution over the cross section of a cylinder compressed by radial forces, in the main, confirms the calculation of these stresses by the methods of the theory of elasticity. A significant discrepancy between experiment and calculation occurs in the zone close to the place of application of the load, i.e., for the points farthest from the center of the disk.

This is explained by the fact that, in reality, the radial force is distributed over very small areas, but areas having finite dimensions. As the load increases, the dimensions of these areas also increase, which leads to a further discrepancy between experiment and theoretical calculations. Investigations of a circular disk by the optical method [5] also show a deviation of the experimental data from the calculated data in the zone close to the place of application of the radial force.

Fig. 3. Distribution diagrams of $\sigma_y$: for (a) the vertical and (b) the horizontal diameter of the compressed disk.
$\circ$ — stresses determined by the proposed calibration of the strain gauges;
$\times$ — stresses determined under the same experimental conditions from the same gauge readings, but from calculated data obtained during preliminary calibration of a single gauge.

(a) Vertical diameter:

• ordinate: $\tilde{\sigma}_y$
• stress, kg/cm$^2$
• coordinate $y$
• points: $A$, $B$, $O$
• diameter: $d = 20\ \text{cm}$

(b) Horizontal diameter:

• ordinate: $\tilde{\sigma}_y$
• stress, kg/cm$^2$
• coordinate $x$
• points: $O$, $B$
• diameter: $d = 20\ \text{cm}$

The character and accuracy of application of the compressive force play a substantial role. Deviation of the line of action of the force from the plane passing through the transverse section of the disk with the bonded gages introduces significant errors. To avoid such errors, in trial experiments with disks 10 cm in diameter and 1 cm thick, small recesses were drilled to transmit the compressive forces by means of balls. However, this had an adverse effect on the readings of the peripheral gages located in the immediate vicinity of the recesses; it served as a stress concentrator and led to local chipping of the disk material under a comparatively small load.

In carrying out the main series of experiments with a disk 40 cm in diameter and 4.5 cm thick, it was necessary to abandon the making of recesses and to set up the disk according to the readings of control gages bonded symmetrically on both sides of the disk, seeking agreement of their readings.

The investigation carried out shows the possibility of determining the actual stresses in a plane stressed state with an arbitrary ratio of the principal stresses from the readings of low-base loop-type gages, which, as is known, introduce a significant distortion into the magnitude of the measured strain. But for this purpose it is necessary, as a result of calibration of two gages, to obtain the characteristics of their sensitivity to longitudinal and transverse strains (the angles \(\alpha\) and \(\beta\) in Fig. 2) and to perform the calculation by formulas (1) and (2).

Using this conclusion in a new series of experiments, when the gages were bonded with a very small spacing, it was established that the deviation of the actual values of the tensile stresses along the vertical diameter ends at a distance of approximately 0.1 diameter from the point of application of the compressive force. This distance is practically independent of the magnitude of the acting force.

The greatest increase in the actual tensile stresses occurred at a distance of approximately 0.03 of the disk diameter from the point of application of the compressive force and exceeds the theoretical value by a factor of 6–10. In the experiments the acting force varied from 200 to 2000 kg, which did not cause a change in the magnitude or the location of the greatest tensile stresses.

LITERATURE

1. A. V. Verkhovskii, V. V. Romanovskaya. Determination of the tensile strength by the method of compression of cylindrical specimens. Zavodskaya laboratoriya, No. 11, 1951.

2. L. M. Sedokov. Static tests of brittle materials. Standartizatsiya, No. 1, 1961.

3. L. M. Sedokov, Yu. S. Bogomolov. Determination of the rupture strength of brittle plastics by compression of cylindrical specimens along the radius. Plasticheskie massy, No. 11, 1962.

4. O. E. Pflaumer. Determination of the tensile strength of concrete, TsBTI. Moscow, 1959.

5. M. M. Frocht. Photoelasticity, Vol. II. State Publishing House of Technical-Theoretical Literature, 1950.

6. A. U. Huggenberger. Über die Querdehnungsempfindlichkeit von Dehnungsmeßstreifen und ihre Beseitigung. Grundlagen und Anwendungen des Dehnungsmeßstreifens. Dr. K. Fink, Verlag Stahleisen M. B. H. Düsseldorf, 1952.

7. G. A. Mitichenko. Measurement of stresses on the internal surfaces of vessels under pressure. In: Measurement of Stresses and Forces in Machine Parts, edited by N. I. Prigorovskii. Mashgiz, 1955.