MEASUREMENT OF STRESSES AND STRAINS ON MODELS OF BOURDON TUBES

G. I. TYZHNOV, V. V. INZHEVITOV

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

1. Strict requirements are imposed on pressure-gauge elastic elements—tubular springs. In all cases it is desirable to have a spring possessing the required sensitivity (compliance) and the greatest strength. These requirements are contradictory. The sensitivity of a spring, for example, increases as the wall is made thinner; however, the stresses increase and the strength of the spring decreases. At the same time an increase in creep and hysteresis will be observed. Conversely, in striving to increase the strength of the spring, we encounter the need to reduce its sensitivity. In order not to reduce the sensitivity of springs and to provide sufficient strength, they are beginning to be made from special alloys that possess better elastic properties than brass L-62, from which most springs are currently manufactured.

Thus, the principal characteristics of a spring—sensitivity, strength, and creep—depend only on its geometrical dimensions and the properties of the material.

The principal theoretical investigations [1, 2] are devoted mainly to the study of the displacements of the end of the spring; moreover, the formulas obtained are well confirmed by experiments, the setup of which presents no difficulties. The problem associated with studying stresses in Bourdon tubes is more difficult, and the approximate solution obtained recently by L. E. Andreeva [3] requires verification.

At present a situation has arisen in which the further development of the theory of the stressed state of Bourdon tubes is hindered by the complete absence of experimental data [3]. This can be explained in part by the fact that springs have very small dimensions and therefore the experimental determination of stresses is associated with certain difficulties.

Springs are manufactured, in the main, with three cross-sectional shapes: elliptical, flat-oval, and segmental.

The purpose of the present work was a comparative study of the deformations and stresses of Bourdon tubes of two cross-sectional shapes currently in use—flat-oval and segmental. The investigations were carried out on springs of the same nominal pressure, 25 kg/cm². The dimensions of the flat-oval spring section: large

major axis \(2a = 19.6\) mm; minor axis \(2b = 3.34\) mm, wall thickness \(h = 1.2\) mm; radius of the central axis \(R = 53\) mm. Dimensions of the spring of segmental cross section: \(2a = 19.7\) mm; \(2b = 3.6\) mm; \(h = 1.2\) mm; \(R = 53\) mm (Figs. 3, 4).

The stresses were studied by several methods.

1. To determine the trajectories of the principal stresses, we applied the method of brittle lacquer coatings [4]. The carbon disulfide recommended as a solvent is toxic, and therefore we dissolved the lacquer in sulfuric ether. The cleaned surfaces of the springs were coated several times with lacquer using a soft brush, with each layer being dried. Final drying was carried out in a thermostat for two hours at \(70^\circ\). Then both springs were simultaneously loaded by a smoothly increasing pressure. In the regions of the greatest tensile stresses the lacquer cracks. The sensitivity of the lacquer decreases during the experiment [5, 6]. Therefore we did not determine the magnitude of the stresses by this method, but limited ourselves only to identifying the trajectories of the principal stresses. The photograph also shows the influence of the end sealing on the direction of the principal stresses (Fig. 1). If we disregard the soldered part of the spring, i.e., the part over which the cracks

Fig. 1.

did not propagate, we shall see that the influence of the end effect (curvature of the cracks) extended over a length not exceeding one third of the spring width. The data obtained were subsequently used in making the models and attaching the gauges.

1. For the study of critical parts of machines and structures, strain-gauge models are being used increasingly often [6, 7]. The model is made, with linear scale maintained, from a material having a low modulus of elasticity. The load on the model must imitate the actual load. Deformations and stresses are measured on the models. The recalculation of the model-test data to the part is performed according to the formulas of the theory of modeling.

Usually models are made at a reduced scale. In the present case, on the contrary, the models were made at an enlarged scale of \(10:1\).

The models of both springs were made of organic glass by gluing together individual sections, which had first been given a wedge shape in order to obtain double curvature. The methods for making the models are described in sufficient detail in the literature.

To simplify fabrication of the models, we made them in the form of short segments (with length approximately equal to the width), and sealed the open ends with special rubber cuffs, which did not restrain the deformation of the end sections. The cuffs were connected to one another by through-bolts, which took up the entire longitudinal load produced by the pressure, thereby relieving the models of it. Thus, the longitudinal stresses $\sigma_1$ measured on such models will differ from the actual ones. However, as calculations show, this error is small and in the present case may be neglected.

Before final assembly of the models, gauges were glued to the inner surface, while the corresponding paired compensating gauges were glued to separate organic-glass plates, which were then placed inside the models. The gauges were glued to the outer surface after assembly of the models.

To fasten the wires, at the places where they connected to the gauge leads, we glued them to the model with BF-2 adhesive. This fastening proved simple and reliable.

The models were loaded with air at a pressure of 400 mm Hg. The pressure value was chosen so that the deformations would be sufficient for measurement, while the stresses would not exceed the proportional limit of the organic glass.

The tests showed the reliability both of the models and of the cuff seals. The appearance of the models and springs is shown in Fig. 2.

Fig. 2.

1. The stresses on the inner surface of the models were measured only with strain gauges having a 5 mm base, while on the outer surface they were measured both with gauges and with lever-type strain meters. This increased the reliability of the measurements and made it possible to critically evaluate the results obtained.

The points at which measurements were made are marked by numbers in Figs. 3 and 4. At each point the strains were measured in two directions, corresponding to the directions of the principal stresses. For electrical strain-gauge measurements we used the forty-point VST-3 instrument. The supply voltage during measurements on organic glass was 1.5 V, and during measurements on metals, 6 V. This is due to the fact that organic glass has low thermal conductivity, as a result of which the gauges on it heat up more than on metals.

Stress envelopes

Legend in Fig. 3:

— stress theory [1] without allowance for curvature
- - - stress theory [1] with allowance for curvature
-·-·- experimental envelope
○○○ measurements on models by sensors
▼▼▼ — " — " — by strain gauges
●● — " — " — on parts
⊕ tension
⊖ compression

Outer contour
Inner contour

Fig. 3.

It is known that short-base gauges are very sensitive to transverse strains; therefore, in calculating the strains we used formulas that take this circumstance into account [8]

\[ \left. \begin{aligned} \varepsilon_1 &= \frac{C}{S(1-B)}(n_1 - Bn_2) \\ \varepsilon_2 &= \frac{C}{S(1-B)}(n_2 - Bn_1) \end{aligned} \right\}, \ldots\ldots \tag{1} \]

where the instrument constant \(C = 1.1 \cdot 10^{-5}\), the strain sensitivity of the wire \(S = 2\), and \(n_1\) and \(n_2\) are the numbers of scale divisions of the instrument for the two gauges glued at the given point.

The geometric constant of the sensor grid is

\[ B=\frac{1}{\dfrac{2l}{\pi r}+1}=0.045, \]

where the length of the longitudinal sections of the sensor is \(l=4.98\ \text{mm}\), and the radius of the loops is \(r=0.15\ \text{mm}\).

When formula (1) is used, the need for calibration of the sensors is eliminated.

The principal stresses were calculated by the usual formulas

\[ \begin{aligned} \sigma_1&=\frac{E}{1-\mu^2}(\varepsilon_1+\mu\varepsilon_2),\\ \sigma_2&=\frac{E}{1-\mu^2}(\varepsilon_2+\mu\varepsilon_1). \end{aligned} \tag{2} \]

The modulus of elasticity of the organic glass, determined experimentally, was \(E=30000\ \text{kg}/\text{cm}^2\); Poisson’s ratio \(\mu=0.35\) was taken from the handbook [4].

1. The stresses calculated for the model by formulas (2) must be recalculated for the actual part, using the relation expressing the similarity between the model and the part [7]

\[ \sigma_{\text{part}}=\beta\sigma_{\text{mod.}}, \]

where the scale of force similarity is \(\beta=q_{\text{part}}:q_{\text{mod.}}\).

The magnitude of the air pressure in the model, equal to \(400\ \text{mm Hg}\), corresponds to a pressure of \(0.544\ \text{kg}/\text{cm}^2\). The nominal pressure for both springs is \(25\ \text{kg}/\text{cm}^2\); therefore,

\[ \sigma_{\text{part}}=\frac{25}{0.544}\sigma_{\text{mod.}}=46\sigma_{\text{mod.}}. \]

The measurement results on the models, recalculated for the actual parts, are shown in Figs. 3 and 4.

1. The stresses were also measured directly on the springs. However, their dimensions are small, and therefore for the measurements it is necessary to use sensors with the smallest possible base, on the order of \(1\ \text{mm}\). Because such sensors were not available, we had to limit ourselves to determining the stresses only at two points, in which the stress gradient is minimal: at points 1 and 9 of the flat-oval spring and at points 11 and 15 of the segment-shaped spring (Figs. 3, 4). The strains at these points were measured with specially made sensors with a base of \(3\ \text{mm}\). To make them we used available sensors with a base of \(5\ \text{mm}\), whose ends were bent upward. Such sensors are completely insensitive to transverse deformations. To measure stresses in the longitudinal direction, ordinary sensors with a base of \(30\ \text{mm}\) were used.

The sensors were calibrated on a beam of uniform strength. The strains at the places where the sensors were bonded to the beam were measured directly with a lever strain gauge. As the experiments showed, the measurement error of sensors with a \(3\ \text{mm}\) base was \(7\%\), and with a \(30\ \text{mm}\) base no more than \(0.5\%\). On this basis, at the indicated points of the springs three sensors with a \(3\ \text{mm}\) base were bonded in the transverse direction, and one sensor with a \(30\ \text{mm}\) base in the longitudinal direction. The stresses were measured at a pressure of \(25\ \text{kg}/\text{cm}^2\). The measurement results are marked in Figs. 3 and 4.

In Fig. 3, the solid lines show the stress diagrams for the spring of flat-oval cross section, calculated by the formulas

by L. E. Andreeva [3], without taking curvature into account, and by dash-dotted lines—with curvature taken into account.

Figure 4 shows experimental stress diagrams for a tube of segment-shaped cross section. Squares mark the points at which the stresses were obtained by means of the polarization-optical method (see the article in the present collection).

Outer contour  Inner contour

Fig. 4.

1. As is known, the travel of the end of a tubular spring depends on the magnitude of the relative deformation of the section (flattening), i.e., on the ratio \(\Delta b : b\), where \(b\) is the dimension of the minor semi-axis of the section, and \(\Delta b\) is its change. These deformations are measured by means of an ordinary dial indicator at several places of the middle section of the model. The results of the measurements are given in Table 1; the numbers of the measurement points are coordinated with Figs. 3 and 4.

Since the magnitude of the relative deformation of the section is variable, it may be assumed that the travel of the end of the spring is determined by its average value. Table 1 gives the corresponding data on the magnitude of the relative deformation.

Table 1

The mean value of the relative deformation was obtained by planimetering the area of the variation \(\Delta b : b\) across the width of the spring. For the springs studied, of flat-oval and segment-shaped cross section, the displacement of the end at a pressure of \(25\ \mathrm{kg/cm^2}\) is, respectively, \(5.8\ \mathrm{mm}\) and \(5.2\ \mathrm{mm}\). Their ratio is \(5.8 : 5.2 = 1.11\), while the ratio of the mean values of the relative deformations, according to Table 1, is \(0.0128 : 0.0116 = 1.10\). Since the difference between the ratios is less than \(1\%\), the assumption that the displacement of the spring end is determined by the mean value of the relative deformation of the cross section is, at least for these springs, confirmed.

Conclusions

1. A procedure has been developed for the experimental determination of stresses on models of Bourdon tubular springs; it has been tested on two cross-sectional shapes: flat-oval and segment-shaped. The results of measurements on models and on actual parts agree well. At most points, the results of measurements on the models with the aid of strain gauges give underestimated values in comparison with the transducers. This should be attributed to the fact that their base length is greater than that of the transducers. A drawback of the experiment was the accidental breakage of one of the transducers at point 16 of the flat-oval-section model; the magnitude of the stresses at this point can therefore be judged only approximately.

The method of strain-gauge models should be regarded as very promising for studying stresses in small parts, for which the usual measurement methods are inapplicable.

1. The brittle-coating method makes it possible, by simple means, to obtain pictures of the distribution of the principal stresses. Unfortunately, coatings of the recommended compositions do not possess stable properties; this complicates the investigations and can often devalue the results obtained.

2. From a comparison of the theoretical [3] and experimental curves for the flat-oval spring, it is seen that both the qualitative and quantitative agreement of the stresses \(\sigma_2\) is satisfactory. If a correction for curvature is introduced into the formulas, the agreement improves appreciably, especially at the dangerous points at the ends of the major axis. The agreement of the stresses \(\sigma_1\) is considerably worse, which may be explained by two reasons. First, the accuracy of measuring small stresses is considerably reduced; and, second, it must be taken into account that the magnitude of the stresses \(\sigma_1\) was measured on a model which, with respect to these stresses, does not completely reproduce the operating conditions of the part.

3. The experiments showed that the largest of the principal stresses is \(\sigma_2\). The stress \(\sigma_1\) is “passive,” and it may approximately be taken equal to \(\sigma_1 \approx \mu\sigma_2\). This circumstance makes it possible to conclude that preliminary investigations can be carried out on flat models of cross sections.

4. The deformations of cross sections are also well studied on models. If the assumption that the magnitude of the displacement of the spring end depends on the mean relative deformation of the cross section is confirmed further for springs of other pressures as well, this will make it possible to create a simpler theory of Bourdon springs than the existing ones.

5. In a comparative evaluation of springs by stresses, the flat-oval-section spring should be regarded as the better one, since it has a more favorable stress distribution than the segment-shaped-section spring. There is no doubt that in the places of the greatest—

with greater curvature, a plastic hinge arises in a segment-shaped spring. This is also confirmed by practice: springs of plano-oval cross section, all other conditions being equal, give a smaller residual deformation (extension) than springs of segment-shaped cross section.

References

1. V. I. Feodos'ev. Elastic Elements of Precision Instrument Engineering. Oborongiz, 1949.

2. W. Wuest. Theorie der Hochdruckröhrenfeder. Ingenieur—Archiv XXIV B, 1956.

3. L. E. Andreeva. Elastic Elements of Instruments. Mashgiz, 1962.

4. Mechanical Engineer’s Handbook, vol. 3. Mashgiz, 1956.

5. G. I. Tyzhnov. On the question of determining stresses by the method of brittle coatings. Zavodskaya laboratoriya, No. 4, 1960.

6. Stresses and Strains in Machine Parts and Assemblies. Collection edited by N. I. Prigorovskii. Mashgiz, 1961.

7. N. I. Prigorovskii, A. K. Preis. Investigation of Stresses and Rigidity of Machine Parts on Photoelastic Models. Publishing House of the Academy of Sciences of the USSR, Moscow, 1958.

8. Measurement of Stresses and Forces in Machine Parts, edited by N. I. Prigorovskii. Mashgiz, 1955.