APPLICATION OF STRAIN GAUGING TO DETERMINE CONTACT STRESSES IN A ROLLED JOINT

D. K. NECHAEV

(Presented by the scientific seminar of the Department of Theoretical and Applied Mechanics)

The load-bearing capacity of a rolled joint, produced by cold expansion of a tube in the socket of a tube sheet, is determined by the magnitude of the residual radial compressive stress (pressure) that arises on the contact surface of the joint. Analytical determination of the contact pressure is associated with solving a problem for a complex stress state beyond the elastic range and can be carried out only with a number of simplifying assumptions. To confirm the results of theoretical studies [1], the author conducted experiments to determine the contact stress in the joint under consideration, using strain gauging.

The experiments were carried out on specimens of “tube–disk” joints, in which the tube and disk were made of low-carbon steel. Seamless tubes with diameters of 76 and 83 mm were used for the specimens. The disks, reproducing a tube sheet, had a diameter of 240 mm and a thickness of 25 mm. Radial expansion of the tubes (rolling) was performed with standard self-feeding rolling tools manually. As the criterion for the quality of the completed joint, the degree of rolling was adopted, defined as the difference between the total increase in the internal diameter of the tube and the magnitude of the clearance between the tube and the wall of the hole before rolling of the tube.

In the completed joint specimens, the degree of rolling, expressed as a percentage relative to the diameter of the hole in the disk, lay within the range 1.5–1.7%.

The contact pressure in the joint was determined from the elastic circumferential (tangential) strains.

Using the equations of elasticity theory, it can be shown that for a circular finite disk with a central hole, expanded within the elastic range by a uniformly distributed radial pressure applied on the inner circle (plane stress scheme), the circumferential strain is determined by the equation

\[ \varepsilon_t = b + K\left(\frac{r_{\mathrm{in}}}{r}\right)^2, \tag{1} \]

\[ \left. \begin{aligned} b &= \frac{p(1-\nu)} {E\left[\left(\dfrac{R_{\mathrm{н}}}{r_{\mathrm{п}}}\right)^2-1\right]},\\[6pt] K &= \frac{p(1+\nu)} {E\left[1-\left(\dfrac{r_{\mathrm{п}}}{R_{\mathrm{н}}}\right)^2\right]} . \end{aligned} \right\} \tag{2} \]

In these expressions, \(E\) is the modulus of normal elasticity; \(\nu\) is Poisson’s ratio; \(r_{\mathrm{п}}\) is the radius of the hole in the disk after removal of the tube; \(R_{\mathrm{н}}\) is the outside radius of the disk; \(r\) is the current radius of the element; \(p\) is the pressure on the inner contour of the disk.

From equations (2) one can obtain a formula for determining the contact pressure

\[ P=\frac{E\left[1-\left(\dfrac{r_{\mathrm{п}}}{R_{\mathrm{н}}}\right)^2\right]K}{(1+\nu)} . \tag{3} \]

The elastic recovery of the disk, after the tube is pushed out of the joint, will relieve the deformation determined by equation (1), since it will be equivalent to the removal of the expanding (rolling) pressure.

The latter makes it possible, using the plot of the function \(\varepsilon_t=f\left(\dfrac{r_{\mathrm{п}}}{r}\right)^2\), to determine the value of \(\kappa\) as the angular coefficient of straight line (I) and to calculate the contact pressure \(p\).

Circumferential deformations were measured by means of wire strain gauges with a base of \(5.5\) mm and a resistance of \(50\) ohms. The arrangement of the strain gauges bonded to the surface of the disk after rolling the tube is shown in Fig. 1. Each of the six gauges located on the surface of the disk was connected in turn into a bridge circuit (Fig. 2). The working gauge \(D_{\mathrm{p}}\) and the compensating gauge \(D_{\mathrm{k}}\), shown in this circuit, form a half-bridge that was assembled on the disk; the other half-bridge is formed by the resistances \(R\) and \(R_i\), the latter of which was varied when adjusting the bridge. A high-sensitivity galvanometer with a scale of \(300—0—300\) was connected into the measuring diagonal of the bridge, where the value of one division was \(6.8\cdot10^{-8}\) A. The strain-gauge circuit was calibrated by tensile stresses that varied within the range from \(1\) to \(7\) kg/mm\(^2\), as a result of which a relationship was obtained between the relative deformation and the galvanometer reading.

All measuring bridges, before removal of the tube, were balanced to the zero reading of the galvanometer. After the tube was pushed out of the joint, which was carried out on a universal testing machine ZDM 10 t/91, all circuits of the working strain gauges were connected in turn and the galvanometer readings were recorded. The latter made it possible, using the calibration curve, to obtain the values of the circumferential deformations.

The experiments were conducted on four specimens, and the values of the relieved circumferential deformations according to the readings of the resistance strain gauges are presented in Fig. 3. The experimental straight line \(\varepsilon_t=f\left(\dfrac{r_{\mathrm{п}}}{r}\right)^2\) makes it possible to determine the coefficients \(b\) and \(\kappa\) in equation (1) and to calculate the contact pressure \(p\), for example, by formula (3).

The values of the average contact pressure—\(p\)—and of the push-out load (shear force)—\(N\)—are given in the table. The recorded

Fig. 1. Arrangement of strain gauges on the “tube–disk” specimen.

Fig. 2. Circuit diagram for connecting the strain gauges.

load is the minimum required to initiate movement of the tube relative to the disk.

The average values of the contact pressure (stress) in the joint under consideration, found by the method described above, proved to be somewhat greater (by 15–20%) than the residual radial stresses

Fig. 3. Values of circumferential strains from resistance strain-gage readings.

calculated analytically for the tested joints [1]. At the same time, the stresses on the contact surface of the joint, determined from the elastic recovery of the disk, differ little in magnitude from the stresses obtained experimentally by other investigators [2], [3] for analogous joints.

Results of pull-out tests

Thus, the experiments confirm that the approximate graphical-analytical method proposed by us for determining residual stresses in a rolled joint [1] is acceptable for practical strength calculation of these joints.

LITERATURE

1. Nechaev D. K. Investigation of the strength of rolled joints of steel tubes. Izvestiya Sibirskogo otdeleniya Akademii nauk SSSR, No. 11, 1958.
2. Glikman L. A. and Stepanov V. A. Residual stresses during rolling. Soviet Boiler and Turbine Engineering, No. 5, 1948.
3. Siebel E. und Kopf E. Beanspruchung in gelochten Platten, “VDI,” Forschungsheft 369, Bd. 5, 1934.