Selection of Pressure for Pressure Testing Cylindrical Shells with Openings
Abstract Generated abstract
This paper examines how to select hydrostatic proof-test pressure for cylindrical boiler-drum shells with openings, where plastic deformation near stress concentrators can strengthen the material and create beneficial residual stresses. It derives limits based on elastic and plastic stress-concentration factors, the Bauschinger effect during unloading, shell-shape deviations, and stress distributions around openings, and compares these estimates with hardness-measurement results from full-scale drum tests. The study argues that current test pressures may be insufficient to induce useful plastic deformation in thicker or accurately shaped shells, and recommends increasing test pressure to about 225 atm for 92 mm walls and 250 atm for 115 mm walls, while remaining below the pressure that would cause plastic deformation during unloading.
Full Text
SELECTION OF THE PRESSURE TESTING PRESSURE FOR CYLINDRICAL SHELLS WITH OPENINGS
L. A. POLINOVSKII
(Presented by the seminar of the Laboratory of Plastic Deformations at the Department of Strength of Materials)
Many parts, including shells, are subjected to preliminary plastic deformation under loads higher than the service loads. In this process the material is strengthened, and in zones of stress concentration, after unloading, residual stresses act which reduce the stress concentration under working loads. Of great importance is the selection of optimal regimes of preliminary plastic deformation. The pressure corresponding to the onset of plastic deformation at the contour of an opening for a thin-walled shell is
\[ P_{\mathrm{t}} = 2\,\frac{\sigma_{\mathrm{t}}\cdot t}{\alpha_{\sigma} D_{\mathrm{cp}}}, \]
where \(\sigma_{\mathrm{t}}\) is the yield strength of the shell material;
\(t\) and \(D_{\mathrm{cp}}\) are the wall thickness and the diameter of the middle surface of the shell;
\(\alpha_{\sigma}\) is the stress-concentration factor in the elastic region.
After unloading, residual stresses act in places of nonuniform plastic deformation; their magnitude is determined as the difference between the working and unloading stresses [1]. During unloading the material may deform plastically. Owing to the Bauschinger effect, this may lead to a decrease in the yield strength in the directions of action of the working loads. In order that unloading not be accompanied by plastic deformation, the condition
\[ \sigma_{\mathrm{res}} = \sigma - \sigma_{\mathrm{p}} \leq -\beta \sigma_{\mathrm{t}}, \tag{1} \]
must be satisfied at the contour of the opening, where \(\sigma_{\mathrm{res}}\) are the residual and
\(\sigma_{\mathrm{p}}\) the unloading stresses, and
\(\sigma\) are the stresses under plastic deformation;
\(\beta\) is a coefficient accounting for the Bauschinger effect; usually, for a tension–compression deformation cycle, \(\beta \geq 0.8\).
Let us introduce the notation:
\[ \sigma = \alpha \sigma_{\mathrm{cp}}, \qquad \sigma_{\mathrm{p}} = \alpha \sigma_{\mathrm{cp}}, \]
where \(\alpha\) is the stress-concentration factor under plastic deformation;
\(\sigma_{\mathrm{cp}}\) are the greatest principal stresses in the absence of concentration.
Then, in accordance with the condition of elastic unloading [1]:
\[ \alpha \geqslant \frac{\alpha_\sigma}{1+\beta}. \]
Let us denote the allowable value of the stress-concentration factor in the plastic region by
\[ \alpha_{\mathrm{a}} = 0.55\alpha_\sigma . \tag{2} \]
For a small ratio of the diameter of the opening to the diameter of the middle surface of a thin-walled shell, its curvature may be neglected and it may be treated as a flat plate with an opening under biaxial tension. The stress distribution in the critical section for this case is [2]:
\[ \sigma = \sigma_{\mathrm{g}}\left[1+0.75\left(\frac{a^2}{r^2}+\frac{a^4}{r^4}\right)\right], \tag{3} \]
where \(a\) is the radius of the opening;
\(r\) is the distance to its center.
The concentration factor is
\[ \alpha_\sigma = 2.5. \]
Then, according to (2):
\[ \alpha_{\mathrm{a}} = 1.39. \tag{4} \]
Ovality and local deviations from the shell shape have a major effect on stress concentration in the elastic region; because of them, as shown by a strain-gage investigation of boiler drums, \(\alpha_\sigma\) varies from 2.5 to 3.5 [3]. Since local deviations from the shape may be of different character, an analytical estimate of this factor is difficult.
The stress concentration at openings during hydrostatic tests of a boiler-drum shell was investigated by the hardness-measurement method [4]. The stress-concentration factor near the openings where plastic deformation began was 3.0–3.3, i.e., it decreased somewhat in comparison with its greatest value determined on similar shells [3] in the elastic region. The calculated value depends on the tolerance for shell out-of-roundness, and its greatest value at the existing manufacturing accuracy is 3.5, i.e.,
\[ \alpha_{\mathrm{a}} = 1.9. \tag{5} \]
The variation of the stress-concentration factor under large plastic deformations was investigated by the hardness-measurement method during full-scale tests of boiler drums at the experimental facilities of the Promstalkonstruktsiya, TsNIITMASH, and TsKTI institutes. In Fig. 1, the dashed line shows the change in the value of \(\alpha\) as a function of pressure for a shell with an inside diameter of 1600 mm and a wall thickness of 92 mm. The allowable value of the stress-concentration factor \(\alpha_{\mathrm{a}}\) corresponds to a pressure, according to (4) and according to (5), of 225 atm.
The investigation was carried out at the opening with the smallest concentration in the elastic region (\(\alpha_\sigma = 2.5\)), and consequently also in the plastic region. The allowable pressure for case (4) is close to the limiting burst pressure. Taking into account that the influence of the initial out-of-roundness of the shell decreases with the development of plastic deformation, we choose \(P_{\mathrm{a}} = 225\) atm. At present, the hydrostatic test pressure is taken as 25% above the working pressure and is equal to 195 atm, which corresponds to \(\alpha \approx 2.4\). During plastic deformation of the material in the zone near the opening, local strengthening of the material occurs, and after unloading there act
residual stresses, which reduce the magnitude of the stress concentration during operation of the shell. The concentration factor of the operating stresses, taking the residual stresses into account, will be:
\[ \alpha_p = \alpha n - \alpha_\sigma (n - 1), \]
for
\[ \alpha_\sigma > \alpha > \alpha_p, \]
where \(n\) is the ratio of the hydrostatic test pressure to the operating pressure.
The variation of this coefficient for the shell of a boiler drum operating at a pressure of \(155\ \text{atm}\) is also given in Fig. 1. The coefficient
Fig. 1.
of concentration of the operating stresses decreases, with increasing hydrostatic test pressure, from 2.5 to 1.25 when operating after testing at a pressure corresponding to \(\alpha_t\) (the onset of plastic deformation during unloading).
Fig. 2.
In connection with the recent production of drums with an increased wall thickness of up to \(115\ \text{mm}\), the hydrostatic test pressure used does not
causes plastic deformation at openings in shells of regular cylindrical shape. Therefore, it is advisable to increase the hydrostatic test pressure for shells with a wall thickness of 92 mm to 225 atm, and to 250 atm for shells with a wall thickness of 115 mm.
When determining stresses by the hardness-measurement method at different stages of deformation, it was established that the maximum principal stress in the critical section in the zone of plastic deformations varies within 4–8%, and for approximate calculations may be taken as constant. For the recommended pressure \(P_d\), its value will be
\[ \sigma = a_d \sigma_{\mathrm{av}} = a \frac{P_d D_{\mathrm{av}}}{2t}. \]
The distribution of unloading stresses is calculated by formula (3). The plot of the variation of residual stresses in the critical section after hydrostatic tests at the recommended pressure is shown in Fig. 2.
Conclusions
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The test pressure must not exceed the pressure after whose removal the material undergoes plastic deformation at the stress concentrators. It is determined as a function of the values of the stress-concentration factors in the elastic and plastic ranges. The latter, in turn, depend on the tolerance for distortion of the shell shape.
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The advisability has been demonstrated of increasing the hydrostatic test pressure of boiler-drum shells by 15–28%, at which the greatest strengthening of the material occurs and, after unloading, residual stresses act that reduce the stress concentration at openings in the shell.
References
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G. Hencky. On the theory of plastic deformations and the residual stresses caused by them in a material. In: Theory of Plasticity, IL Publishing House, 1948.
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G. N. Savin. Stress Concentration Around Openings, GITTL, 1951.
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Wellinger et al. Stresses in boiler drums. Mitteilungen der VGB, No. 90, 1964.
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G. D. Del’. Determination of the stress state of a material in the plastic region by hardness measurement. Proceedings of the Metrological Institutes of the USSR, issue 91 (151), Standards Publishing House, Moscow–Leningrad, 1967.