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

Volume 109　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　1960

INFLUENCE OF THE MECHANICAL PROPERTIES OF THE TUBE MATERIAL AND TUBE SHEET ON THE QUALITY OF THE ROLLED JOINT

D. K. NECHAEV

(Submitted by the Department of Applied Mechanics, TPI)

The effectiveness of a rolled joint, used for fastening thin-walled tubes in the sockets of tube sheets of pressure vessels, depends on the mechanical properties of the tube and sheet materials.

The strength of a rolled joint for steel parts is ensured when the condition [4] is satisfied

\[ N \ll F_k f \sigma_r^{o}, \tag{1} \]

where \(N\) is the axial force that the joint can carry;
\(F_k\) is the nominal contact area;
\(f\) is the coefficient of friction (adhesion) during longitudinal displacement of the parts;
\(\sigma_r^{o}\) is the residual radial stress on the contact surface of the joint.

In expression (1), the variable quantity depending on the rolling process is the residual radial stress, which we have adopted as the criterion in evaluating the quality of the joint under consideration.

To obtain a joint of maximum strength, the materials of the tube and the sheet (plate) must be plastically deformed so that the greatest possible residual radial stress arises on the contact surface of the joint. Studies [4] show that this can be achieved if the pressure on the inner contour of the tube is brought to the value \(1.155\,\sigma_T\), where \(\sigma_T\) is the yield strength of the tube material. At the same time, the pressure on the outer contour of the tube must not exceed the value \(1.155\,\sigma'_T\), where \(\sigma'_T\) is the yield strength of the sheet material.

At the initial stage of the rolling process, only the tube undergoes plastic expansion. This case of radial-plastic deformation of the tube over a short length (less than the diameter) may be regarded as the plastic deformation of a flat ring under the action of internal pressure (plane stress state).

In the theory of plasticity it has been shown [3] that the plastic equilibrium of a ring (disk) under plane-stress conditions is possible if the ratio of the outer radius \(r_{\mathrm{н}}\) to the inner radius \(r_{\mathrm{в}}\) satisfies the condition \(1 < \dfrac{r_{\mathrm{н}}}{r_{\mathrm{в}}} < 2.963\). In this case, in a plane ring subjected to a uniform pressure \(p\) on the inner surface \(r = r_{\mathrm{в}}\), the maximum value of the radial stress (pressure) likewise occurs at the inner surface of the ring and cannot exceed the value

\[ \sigma_{r\max} = -p = \frac{2}{\sqrt{3}}\,\sigma_{\mathrm{T}} = 1.155\,\sigma_{\mathrm{T}}. \]

Thus, it may be assumed that plastic deformation of a tube by an internal radial pressure of \(1.155\,\sigma_{\mathrm{T}}\) over its entire cross section is possible only at a certain thickness of its wall. Thus, if the ratio \(\dfrac{r_{\mathrm{н}}}{r_{\mathrm{в}}}\) of the outer radius of the tube to the inner radius exceeds 2.963, purely plastic equilibrium is impossible, i.e., the outer part of the tube will remain elastically deformed.

Fig. 1. Variation of the radial stress on the outer contour of a plastically deformed tube as a function of its wall thickness.

The dependences obtained for the plastic deformation of a plane ring [3] were used by us to determine the stresses in the wall of a tube during its radial-plastic expansion in the hole of a tube sheet. In our notation these dependences have the form:

\[ \frac{r^2}{r_{\mathrm{н}}^2} = \frac{\sqrt{3}}{2}\, e^{\sqrt{3}\left(\Theta-\frac{\pi}{6}\right)} \cdot \frac{1}{\cos\Theta}; \tag{2} \]

\[ \sigma_r = -\frac{2}{\sqrt{3}}\, \sigma_{\mathrm{T}} \sin\left(\Theta-\frac{\pi}{6}\right), \tag{3} \]

where \(r\) is the current radius of an arbitrary element;
\(\sigma_r\) is the radial stress;
\(\Theta\) is an auxiliary parameter (angle) that makes it possible to solve equations (2), (3) simultaneously.

The change in radial stress (pressure) through the thickness of the tube wall during its plastic expansion in the tube-sheet seat by the limiting pressure \(1.155\,\sigma_T\) is shown by the curves in the diagram (Fig. 1). It follows from this diagram that the magnitude of the pressure on the contact surface of the joint is determined by the tube-wall thickness. Consequently, by specifying the pressure on the outer surface of the tube, we can always, with the aid of the radial-stress curve, find the corresponding inner radius of the tube, and conversely. For example, when expanding a tube in a steel plate whose material has a yield point 30% lower than that of the tube material \((\sigma'_T = 0.7\,\sigma_T)\), the limiting pressure on the outer contour of the tube must not exceed 70% of \(1.155\,\sigma_T\). The curve in Fig. 1 shows that this is feasible under the condition

\[ \frac{r}{r_n} = \frac{r_b}{r_n} \approx 0.7. \]

The graphical relationship obtained between the radial stress (pressure) on the outer contour of the plastically deformed tube and its radial dimensions makes it possible to consider the question of the influence of the mechanical properties of the tube and tube-sheet materials on the magnitude of the residual radial stress in the joint.

Case I. The yield point of the tube-sheet material is higher than the yield point of the tube material

With identical mechanical properties of the tube and tube-sheet (plate) materials, the best joint is achieved if a pressure equal to \(1.155\,\sigma_T\) is applied to the inner contour of the tube. The residual radial stress on the contact surface of the joint in this case can be found from the diagram (Fig. 2) as the difference between the elastic-plastic and elastic stresses [4]. The latter is determined from the solution of a fictitious problem of the theory of elasticity [2].

Fig. 2. Distribution of radial residual stresses in the joint of the tube and plate.

If the yield point of the steel plate material is higher than that of the tube material, the stress distribution in the plate will be

less than in the tube [1]. But the stress curve within the limits of the tube-wall thickness remains the same (the upper curve in Fig. 2), and the residual radial stress on the contact surface of the joint can likewise be found by the method noted above.

In our case these stresses were determined graphically from the segments between the curve showing the change in radial stresses in the joint in the first phase of the rolling process and the curve of elastic stresses during unloading. The curves of residual radial stresses obtained in this way for plates having yield strengths of 125, 150, and 200%, in comparison with the yield strength of the tube material, are presented in Fig. 3.

Fig. 3. Curves of residual radial stresses on the contact surface of the joint for different ratios of the mechanical properties of the tube and plate (tube sheet) materials.

Case II. The yield strength of the tube-sheet material is lower than the yield strength of the tube material

If the sheet material is weaker than the tube material, then the limiting radial pressure that can be produced by rolling on the outer contour of the tube is determined by the yield strength of the tube-sheet material. On the other hand, this pressure will depend on the thickness of the tube wall.

The obtained curves of radial stresses for tubes with different wall thicknesses (Fig. 1) make it possible to determine the residual stress on the contact surface of the joint by the method indicated above.

by the method. These stresses are found graphically, as the distance between the curve of radial stresses corresponding to the plastic expansion of the tube and the curve of elastic unloading. The latter begins on the plastic curve at the point corresponding to the inner radius of the tube, and the residual radial stress is found at \(\frac{r}{r_n}=1\). The results for determining the residual radial stresses are represented by curves designated 40, 50, 60, 70, and 80% (Fig. 3).

For the case considered above \((\sigma'_T = 0.7\,\sigma_T)\), the residual radial stresses are determined on this diagram by the rectilinear segment of the curve designated 70%. If the tube has a wall thickness greater than its value determined with the aid of the stress curve (Fig. 1), then the pressure on the outer contour of the tube during its plastic expansion does not reach the limiting value for the plate, whose yield strength is \(0.7\,\sigma_T\). In this case, the condition for obtaining a high-quality joint is determined no longer by the limiting stress of the plate material \(\sigma'_T\), but by the limiting stress of the tube material \(\sigma_T\). Correspondingly, there is a change in the character of the residual-stress curve for values of \(\frac{r_n}{r_v}\) greater than 1.42 (Fig. 3).

The curve designated 40% (Fig. 3) shows that, in accordance with this theory, for the tube sizes used in boiler construction \(\left(1.1 < \frac{r_n}{r_v} < 1.2\right)\), an expanded tube joint cannot be achieved if the yield strength of the tube material is much higher than that of the tube sheet. In practice this is often confirmed when unexpanded tubes are flared out.

Consideration of the residual-stress curves presented in Fig. 2 permits the conclusion that the use, for tube sheets, of a material with higher mechanical characteristics than the tube material has no particular advantages in comparison with the use, for the tubes and tube sheets, of a material with identical mechanical properties. At the same time, if the strength of the tube-sheet material is considerably lower than that of the tube material, better results can be obtained by using for the tube sheet a material of increased strength (hardness). Thus, for example, with identical mechanical properties of the tube and tube-sheet materials and with the ratio \(\frac{r_n}{r_v}=1.2\), the residual radial stress on the contact surface of the joint is \(\sigma^0_r = 0.175\,\sigma_T\) (Fig. 3); if the yield strength of the tube-sheet material is half that of the tube material \((\sigma'_T = 0.5\sigma_T)\), then the value of the residual stress decreases to \(\sigma^0_r = 0.098\,\sigma_T\). Thus, in the second case we have a reduction of the residual radial stresses by more than 40%.

These conclusions were confirmed by us in the experimental determination of the strength of expanded joints of boiler tubes under axial loads. Thus, tests carried out on specimens of “tube—disk” joints, in which the tube and the disk were made of low-carbon steel, showed that increasing the yield strength of the disk material leads to an increase in the strength of the joi-

…increase. In this case, the increase in strength has a noticeable effect as long as the difference in yield limits of the tube and tube-sheet materials does not exceed 20%; a further increase in the yield limit (hardness) of the tube-sheet material has no substantial effect on the strength of the rolled joint.

Conclusions

The highest efficiency of a rolled joint made of steel parts can be obtained by using, for the tube sheet, a material of somewhat greater strength than the tube material. In this case, it is inadvisable for the difference in the mechanical properties of the materials of the joint elements to exceed 20% in yield limit and 15 HB in hardness.

The recommended combination of mechanical properties of the materials of the joint parts, while ensuring high strength and operational reliability of the rolled joint, protects the tube sheet from increased plastic deformation (work hardening) during tube rolling.

References

1. Glikman L. A., Stepanov V. A. Residual stresses during rolling. Soviet Boiler and Turbine Construction, No. 5, 1948.
2. Ilyushin A. A. Plasticity. Moscow–Leningrad, Gostekhizdat, 1943.
3. Nadai A. Plasticity. ONTI, NKTP, USSR, 1936.
4. Nechaev D. K. Study of the strength of rolled joints of steel tubes. Izvestiya SO AN SSSR, No. 11, 1958.