Methodology for Studying Residual Stresses in Tubular Pressure-Gauge Springs

G. I. Tyzhnov

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

The occurrence of residual stresses in tubular pressure-gauge springs is a consequence of plastic deformations of the metal during the bending operation through which the springs pass in the manufacturing process. The magnitude and character of the residual stresses affect the proportional limit, fatigue strength, relaxation, and corrosion resistance of the springs [1]. The presence of residual stresses is confirmed by a simple experiment involving the cutting of annular strips from the wall of the spring (Fig. 1).

Fig. 1.

To study residual stresses in pressure-gauge springs, with certain modifications, a known method of investi-

for studying residual circumferential stresses on ring specimens [2, 3]. This method consists in the following: from the surface of a previously cut ring, layers of metal are successively removed by etching in acid or by mechanical machining; after each such operation, the change in the diameter of the remaining ring and the thickness of the removed layer are measured. Knowing the magnitude of the change in diameter and the layer thickness, and assuming that the residual stresses were uniformly distributed over the cross section of the layer and along the length of the ring, one can calculate the magnitude of the stress that the removed layer had. The total stress in the \(n\)-th layer is determined by the sum

\[ \sigma_n = \sigma_{1n} + \sigma_{2n} + \sigma_{3n}, \tag{1} \]

where \(\sigma_{1n}\) is the stress in the \(n\)-th layer, relieved by cutting the ring;
\(\sigma_{2n}\) is the stress remaining in this layer before its removal;
\(\sigma_{3n}\) is the stress caused by deformation of the ring as a result of removing the preceding stressed \(n-1\) layers.

Usually, to measure the change in the diameter of rings, optimeters or special devices with resistance gauges are used [2, 4]. It turned out that for such thin-walled parts as manometric springs, these methods cannot be applied because of the low rigidity and the usually existing deviations from the correct ring shape. Therefore, for measuring the diameter of springs in the study of residual stresses, another method was proposed, according to which the change in the distance \(L\) between the fused ends of glass pointers glued with BF-2 adhesive to the surface of the spring is taken approximately as the magnitude of the change in diameter (Fig. 2). Measurements were made on an IZA-2 comparator and on a UIM-21 measuring microscope with a reading accuracy of \(\pm 0.001\) mm. Taking into account the sighting error, the total error in the experiments proved to be \(\pm 0.002\) mm. Let us estimate the systematic error of the proposed method. To do this we introduce the generally accepted assumptions: 1) after removal of the layer, the ring shape of the deformed spring is preserved; 2) the spring undergoes only bending deformation, i.e., the length of the neutral layer does not change. Suppose that after removal of a layer the diameter of the spring \(D_0\) increased to the value \(D_1\), and the distance between the ends of the pointers \(L_0\)—to \(L_1\) (Fig. 3). On the basis of the adopted assumptions, the length of the spring between the two pointers is

\[ l = \varphi_0 \frac{D_0}{2} = \varphi_1 \frac{D_1}{2}, \tag{2} \]

where \(\varphi_0\) and \(\varphi_1\) are the central angles between the pointers before and after etching.

According to Fig. 3,

\[ \frac{\Delta L}{2} = A_1O_1 \cos \frac{\Delta\varphi}{2} - A_0O_0. \tag{3} \]

Bearing in mind that

\[ \frac{\Delta\varphi}{2} = \frac{\varphi_0 - \varphi_1}{2}, \]

and taking \(\varphi_0 = \pi\), by jointly solving equations (2) and (3) we obtain

\[ D_1 \sin \frac{\pi D_0}{2D_1} = D_0 \pm 2b\left(1 - \sin \frac{\pi D_0}{2D_1}\right) \pm \Delta L, \tag{4} \]

where \(b\) is the length of the pointer; the positive signs are taken in the case when,

Fig. 2.

Fig. 3.

if the pointer is glued on the outside, and the diameter of the spring increased after removal of the layer. Equation (4) has no exact solution. The results of an approximate graphical solution for \(D_0 = 100\) mm and \(b = 10\) mm are given in the table. Usually \(\Delta L < 0.5\) mm; therefore, taking \(\Delta D = \Delta L\), we admit an error of no more than 0.6%.

The author determined the residual stresses \(\sigma_i\) in the surface layers of pressure-gauge springs. Before etching in nitric acid, the corresponding part of the spring surface was coated with a layer of lacquer to protect it from etching. Removal of the stressed surface layer caused a change in the diameter of the spring. If a stretched layer is removed from the inner surface of the spring, then the latter will evidently increase its diameter. Knowing the magnitude of the change in diameter, it is possible to determine the magnitude of the stress that the removed layer had. Removal of the stressed layer is equivalent to the application to the remaining section of a centrally tensile force \(\Delta P\) and a bending moment \(M_0 = \Delta P \cdot e\) (Fig. 4) [2]. Leaving aside the action of the tensile force, since it does not cause a change in the curvature of the spring, we take into account only the bending moment \(M_0\), which can be found from the known formula of strength of materials

\[ \Delta D = \frac{2M_0 R^2}{EI}, \tag{5} \]

where \(\Delta D\) is the change in the diameter of the spring due to the moments \(M_0\) applied at its ends; \(R\) is the radius of the spring; \(EI\) is the stiffness. Assuming that the stresses \(\sigma_i\) were uniformly distributed over the layer, we may write

Fig. 4.

\[ M_0=\Delta P\cdot e=\sigma_l\cdot \Delta h\cdot k\cdot e. \]

After substituting this expression into (5), we obtain

\[ \sigma_l=\pm \frac{\Delta D E I}{2R^2 k e \Delta h}. \tag{6} \]

Here \(\Delta D\) is the change in the diameter of the spring after removal of the layer, \(k\) is the width and \(\Delta h\) the thickness of the layer, \(EI\) is the stiffness of the spring section, and \(e\) is the distance from the center of gravity of the section to the middle of the removed layer; the plus sign refers to etching from the inside, and the minus sign to etching from the outside.

The bending deformation of thin-walled curved tubes is considerably greater than that calculated according to the ordinary theory of bending [5], i.e., the actual stiffness of the section proves to be less than the calculated value. In addition, it must be taken into account that the stiffness of tubes depends on the direction of deformation: in bending it is less than in straightening. This little-known and, at first glance, strange behavior of curved tubes is due to the peculiar character of the deformation of the section [6, 7]. Since this phenomenon has still not received a theoretical evaluation, in order to increase the accuracy of determining residual stresses it is better to determine the actual stiffness of the section experimentally. For this purpose the spring, in a simple device, was loaded with concentrated diametrically applied forces, after which the change in its diameter was measured. The actual stiffness of the section was calculated by the formula corresponding to this method of loading a curved bar of small curvature,

\[ EI=\frac{\pi P R^3}{2\Delta D_p}. \tag{7} \]

Now, substituting the value of the stiffness found into formula (6), we obtain

\[ \sigma_l=\pm \frac{\pi \Delta D P R}{4\Delta D_p k e \Delta h}, \tag{8} \]

where \(P\) is the magnitude of the concentrated force, and \(\Delta D_p\) is the change in the diameter of the spring due to this force. The remaining notation is as before. A study of the accuracy of the proposed method for determining residual stresses in surface layers, within the assumptions adopted above, showed that the total maximum error will not exceed 10%. The principal source of errors is the operation of measuring the change in diameter (4%). Experiments showed that the magnitude of the residual stresses in the surface layers of brass pressure-gauge springs made of alloy L-62 depends on the method of manufacturing the springs and on subsequent heat treatment. In springs that had not undergone heat treatment, the magnitude of the residual stresses in surface layers \(0.03\)--\(0.05\) mm thick can reach \(+20\ \mathrm{kg/mm^2}\).

The distribution of residual stresses through the wall thickness of a pressure-gauge spring was also studied on strips cut from the spring by the method of successive removal of metal layers. In conducting such experiments and in the subsequent calculations there are certain differences from the known methods for determining residual stresses in closed rings. The criterion for the correctness of determining the residual stresses in this case is equality of the positive and negative parts of the area of the stress diagram, since, from the equilibrium condition, the moment and resultant of the residual stresses of the mentally cut-off part must be equal to zero.

The special feature of the method for determining residual stresses in pressure-gauge springs is that the strip being cut out is only a part of the cross section and therefore, generally speaking, for it the moment and the resultant force need not be equal to zero. This situation can be explained by Fig. 5. Suppose, for example, that the strip cut from the spring, with section \(AA_1B_1B\), is nonuniformly stretched (owing to compression of the remaining part of the section) and that the residual-stress diagram has the form of a trapezoid.

Fig. 5.

After being cut out, the strip undergoes a double deformation: it shortens—owing to release of the uniformly distributed component \(\sigma_0\)—and changes its radius of curvature—owing to release of the moment of the residual stresses. Consequently, in this case the magnitude of the residual stresses cannot be determined from the bending deformation alone, because in that case the component \(\sigma_0\) is lost. To study the distribution of residual stresses in a part of the spring section, first, on one batch of identical springs, the magnitude of the surface residual stresses \(\sigma_a\) was determined (by the method described above), while on another batch of springs annular strips were cut out, the changes in their diameters after cutting them from the spring were measured, and the component \(\sigma_1\) in equation (1) was calculated; then, by successive etching, \(\sigma_{2n}\) and \(\sigma_{3n}\) were determined, after which the stress-distribution diagram was constructed. This diagram must have equality of the areas of positive and negative stresses. The constructed diagram was shifted until its origin coincided with the value \(\sigma_a\) determined earlier and, in this way, the complete diagram of residual stresses over the region \(AA_1B_1B\) of the section of the pressure-gauge spring was obtained (Fig. 5). Consequently, the total stress in some part of the section of a curvilinear member will be

\[ \sigma_n = \sigma_0 + \sigma_{1n} + \sigma_{2n} + \sigma_{3n}. \tag{9} \]

Without presenting here the derivation of the individual components of this formula, which is carried out on the basis of the usual considerations adopted in strength of materials, we write it in expanded form:

\[ \sigma_n = \sigma_0 \pm \frac{\Delta D_0 \alpha E \left(h - 2y_n\right)}{D_1D_2} \pm \frac{\alpha E y_n^2 \Delta D_n}{3D_n^2 \Delta y_n} \pm \]

\[ \pm \frac{\alpha E}{3} \sum_{i=1}^{n-1} \frac{\Delta D_i y_i}{D_i^2} \mp \alpha E \sum_{i=1}^{n-1} \frac{\Delta D_i}{D_i^2} (y_i - 2y_n). \tag{10} \]

The upper signs correspond to the case of etching layers from the inside, and the lower signs—from the outside; \(\Delta D_0\) is the change in the diameter of the strip after it has been cut out of the body of the spring, \(\Delta D_0 = D_1 - D_2\); \(y_n, y_i\) are coordinates measured to the corresponding layer from the non-etched side of the strip; \(\Delta D_n, \Delta D_i\) are changes in the diameter of the ring obtained after removal of the \(n\)-th and \(i\)-th layers; when the diameter increases, \(\Delta D_n\) and \(\Delta D_i\) are taken with a positive sign, and when it decreases—with a negative sign; \(D_n, D_i\) are the mean diameter of the remaining cross section of the ring after removal of the \(n\)-th or \(i\)-th layer. The coefficient of cylindrical rigidity is

\[ \alpha = \frac{1}{1 - \mu^2}. \]

It is introduced when the thickness of the strip \(h\) is small in comparison with the width \(k\), i.e., \(h \leq 0.2\,k\) [3]. Formula (10) differs from the known formula of N. N. Davidenkov [2] in that it includes the term \(\varepsilon_0\), determined experimentally, neglects the influence of the curvature of the cut-out strip, and replaces integrals by finite sums.

Figure 6 shows diagrams of the residual stresses \(\sigma_t\) over the wall cross section of a pressure-gauge spring made of alloy L-62 after the bending operation.

Visible labels in the figure: \(\sigma\), kg/mm\(^2\); “direction of etching”; 1 mm.

Fig. 6.

The diagrams were constructed by the method described above; the calculation of stresses in each removed layer was performed according to formula (10)

Conclusions

For the study of residual stresses in annular thin-walled parts, a method is proposed for measuring the diameter by means of glued-on glass indicators. The systematic error of this method increases as the change in diameter increases; however, for values of such changes ordinarily encountered in practice, the systematic error does not exceed 0.6%. Much larger errors are introduced by measurement errors and by the measuring instrument.

On the basis of the method for determining residual stresses in closed rings proposed by Academician N. N. Davidenkov, a combined method has been developed for determining residual stresses in a portion of the cross section of a pressure-gauge tube. According to the proposed method, the magnitude of the surface residual stresses is first determined on one batch of springs; annular strips are cut from another batch of identical springs, and the distribution of residual stresses dependent on the moment component is studied on them. The complete diagram of residual stresses is obtained by graphically combining the diagram thus obtained with the magnitude of the surface residual stress obtained earlier.

To facilitate calculations and increase accuracy, the actual stiffness of the spring section is best determined experimentally.

Experiments have demonstrated the presence of residual stresses in pressure-gauge springs; their magnitude may reach \(20\ \mathrm{kg/mm^2}\).

References

1. A. V. Bobylev. Corrosion cracking of brass. Metallurgizdat, Moscow, 1956.

2. N. N. Davidenkov. Measurement of residual stresses in tubes. Zh.T.F., vol. 1, issue 1, 1931.

3. M. A. Babichev. Methods for determining internal stresses in machine parts. Academy of Sciences of the USSR, 1955.

4. P. E. Dyachenko and A. N. Dobychnina. Vestnik mashinostroeniya, No. 10, 1951.

5. S. D. Ponomarev et al. Strength calculations in mechanical engineering. Vol. 1, Mashgiz, 1956.

6. S. P. Timoshenko and D. Lessells. Applied theory of elasticity. GNTI, 1930.

7. V. I. Feodos’ev. Elastic elements of precision instrument engineering. Oborongiz, 1949.