PROCEEDINGS
OF THE TOMSK ORDER OF THE OCTOBER REVOLUTION AND ORDER
OF THE RED BANNER OF LABOR
POLYTECHNIC INSTITUTE named after S. M. KIROV

Vol. 228 1974

CALCULATION OF WELDED PRESSURE-GAUGE SPRINGS

V. G. AFONIN, M. P. SHUMSKII

(Presented by the scientific and technical seminar of the Department
of Gyroscopic Instruments and Devices)

In [1] a description is given of the designs and test results of pressure-gauge springs manufactured by welding from two profiled strips.

The use of welding makes it possible to realize a wide variety of shapes and standard sizes of elastic sensing elements, including those with variable wall thickness. It becomes possible to use materials with high elastic properties that cannot be used for manufacturing springs from seamless tubes. It is noted in [2] that the wide introduction into production of welded pressure-gauge springs, which could ensure a significant improvement in the metrological characteristics of elastic sensing elements and a reduction in the cost of their production, is to a considerable extent hindered by the absence of engineering methods for their calculation.

Fig. 1

Thus, consideration of the problems of calculating welded springs is quite timely. The general theory of pressure-gauge springs, set forth in [3], [4], is also applicable to welded springs. Certain features of the calculation are due to the presence of welded seams at the ends of the major axis of the section.

A possible calculation scheme for the section of a welded spring is shown in Fig. 1,a. In approximate calculations it is natural to assume that the tangent to the center line of the section at point A does not change its position during deformation of the section, i.e., that the spring wall is rigidly fixed in the section lying on the axis \(OY\). The magnitude of the stresses at point O

of the section will be determined by the magnitude of the bending moment in the restraint, by the normal force tending to tear apart the welded strips, and by the presence of a sharp transition at point O of the inner contour of the section, which causes stress concentration at this point.

The scheme of the forces acting on the welded seam is shown in Fig. 1,b. It is evident that the spring material in the vicinity of point O is in very unfavorable conditions.

In springs made from seamless tubes, the ratio of the radius \(R_1\) of curvature of the mean line of the section to the wall thickness \(H\) has a substantial effect on the magnitude of the stresses at the ends of the major axis of the section. As \(R_1 \to \dfrac{H}{2}\), the stresses on the inner contour of the section increase sharply.

Fig. 2

In welded springs the mean line of the section at point A (Fig. 1,a) is interrupted; there is no rounding of the mean line at the ends of the major axis; therefore, when designing welded springs there is no need to copy the section shapes adopted in the manufacture of springs from solid-drawn tubes and dictated by their technological features. From the works of V. I. Feodos’ev [5] and L. E. Andreeva [6] it is known that a pressure-gauge spring has the greatest sensitivity when the shape of the mean line of the section is similar to the function \(\omega(x)\), which describes the vertical displacements of points of the mean line of the section. The use of welding makes it possible to realize section shapes close to the ideal one. Provided that a high quality of the welded seam is ensured, it may be expected that welded springs will have better characteristics in comparison with those made from seamless tubes. A definitive solution to this question is possible if experimental data are available on the magnitude of the allowable stresses for welded seams obtained under loading conditions close to those occurring in pressure-gauge springs.

One of the possible section shapes of welded springs is shown in Fig. 2. For the calculation of such springs, formulas and graphs developed for springs of oval, figure-eight, and flat-oval sections may be used. In doing so, the following changes must be made in the calculation procedure:

1. The radius of rounding of the mean line of the section at the ends of the major axis should be taken equal to zero:

\[ R_1 = 0. \]

1. The radius \(R_2\) should be taken from the drawing.

1. Determine the angle $\alpha_0$ from the formula

\[ \alpha_0=\frac{\pi}{2}-2\arctg(b-y_0),\qquad y_0=\frac{Y_0}{A}, \]

and determine the value $Y_0$ from the drawing (Fig. 2).

If the technological rounding at the transition from radius $R_2$ to the welded seam is neglected, then the value $Y_0$ may be taken equal to $\dfrac{H}{2}$, and the radius $R_2$ determined from the formula

\[ R_2=\frac{1}{2}\,\frac{A^2+\left(B+\dfrac{H}{2}\right)^2}{B-\dfrac{H}{2}}. \]

1. When calculating the moment of inertia of the section, it is desirable to take into account the moment of inertia of the seams,

\[ I_{\text{seam}}=\frac{4}{3}LH^3. \]

For thin-walled springs, and also in a simplified calculation, the value $I$ of the seam may be neglected.

The calculation of a welded spring with the cross-sectional form corresponding to Fig. 2, in the case where $y_0=0$, may be performed using the graphs of the coefficients $C_i$ intended for calculating springs of oval, figure-eight, and flat-oval section [4]. When determining the values of the coefficients $C_i$ corresponding to the specified value of the minor semiaxis $B$, the radius $R_1$ should be taken equal to zero. In this case the calculation will be performed without taking into account the moment of inertia of the welded seam.

To determine the stresses, one should use the simplified formula

\[ \sigma_{\text{eqv}}=0.89\,\frac{p}{h^2}\,K\cdot C_2\,\frac{\nu+C_3}{\nu+C}. \]

The stress-concentration coefficient $K$ entering into this formula must be determined experimentally.

LITERATURE

1. I. A. Sychev. Development of welded pressure-gauge springs. Bulletin of NIIteplopribor, 1960.

2. K. B. Arutyunov. The state of development and production of elastic sensing elements in instrument engineering. In: Prospects for the Development of Elastic Sensing Elements, TsINTI of the Electrical-Engineering Industry and Instrument Engineering, 1961.

3. M. P. Shumsky. Calculation of pressure-gauge springs. USSR Higher-Education Publishing House, Instrument Engineering, No. 5, 1964.

4. M. P. Shumsky, O. S. Zinkevich, Yu. P. Minchenkov. A unified engineering method for calculating the principal types of single-turn pressure-gauge springs. TPI Publishing House, vol. 192, 1969.

5. V. I. Feodos’ev. Elastic Elements of Precision Instrument Engineering. Oborongiz, Moscow, 1949.

6. L. E. Andreeva. Elastic Elements of Instruments. Mashgiz, Moscow, 1962.