STRESS DISTRIBUTION AROUND OPENINGS IN THIN-WALLED SHELLS

L. A. POLINOVSKII

(Presented by the scientific seminar of the Laboratory of Plastic Deformations)

Thin-walled shells, for which the ratio of wall thickness to the diameter of the median surface is less than 0.07, are widely used in power and chemical machine building, and many of them have side openings for the insertion of branch pipes. The stress state in thin-walled shells may approximately be considered plane [1]. Near stress concentrators in shells, local plastic deformation is often allowed; in studying it, experimental-analytical methods are becoming increasingly important, since they make it possible to obtain the necessary quantitative information while taking account of the real properties of the material. Among other experimental-analytical methods, the method of measuring the hardness of a plastically deformed material [2, 3] is distinguished by its simplicity and universality. It is applicable to studies of stress concentration in small parts and models, and also on actual shells such as the shells of boiler drums with an internal diameter of 1600–1800 mm and a wall thickness of 92–115 mm.

The sequence for investigating the stress-strain state by the hardness-measurement method is as follows:

1. By uniaxial tension of specimens made from the shell material, a calibration curve is constructed in the coordinates: hardness \((H)\)—stress intensity \((\sigma_i)\).

2. In the region under investigation, the hardness is measured. The method of measurement depends on the dimensions of the plastic zone and the conditions of the investigation: on the outer surface of small shells and models, the hardness is measured by Vickers; on the inner surface of large shells, by Brinell. The results of the hardness measurements are processed by constructing graphs of hardness variation with respect to the coordinates; in doing so, the hardness values are smoothed.

3. The field of hardness distribution in the region of the concentrator is converted, by means of the calibration curve, into the corresponding field of stress-intensity distribution.

4. By the joint solution of two differential equations of equilibrium and the expression for \(\sigma\) through the components of the stress tensor \(T_\sigma\), the three unknowns of the tensor characterizing the plane stress state are determined. The differential equations are integrated by the finite-difference method using conditions on the free ...

contour, where one of the principal stresses is equal to zero, and the other is \(\sigma_i\), with the free contour being the trajectory of this principal stress.

The study was carried out on the shell of a boiler drum during its hydrostatic pressure test. Figure 1 gives the calibration graph for the shell material; Fig. 2 shows the field of distribution of hardness and stress intensity around the opening.

Fig. 1. Calibration graph: hardness H—stress intensity \(\sigma_i\).

Fig. 2. Variation of hardness and stress intensity at the opening along the axial section of the shell.

stresses around the opening. The solution was obtained by writing the equilibrium equations in the trajectories of the principal stresses:

\[ \frac{\partial \sigma_1}{\partial S_1}+(\sigma_1-\sigma_2)\frac{\partial \nu}{\partial S_2}=0, \tag{1} \]

\[ \frac{\partial \sigma_2}{\partial S_2}+(\sigma_1-\sigma_2)\frac{\partial \nu}{\partial S_1}=0, \tag{2} \]

where \(\sigma_1, \sigma_2\) are the principal stresses;
\(S_1, S_2\) are coordinates measured along the trajectories of \(\sigma_1\) and \(\sigma_2\);
\(\nu\) is the angle of inclination of the principal stress to the \(X\)-axis.

The third equation was

\[ \sigma_i^2=\sigma_1^2+\sigma_2^2-\sigma_1\sigma_2. \tag{3} \]

The equations were solved while simultaneously constructing a net of trajectories of the principal stresses. Of greatest interest is the stress distribution in the most dangerous, axial section of the shell. The procedure for constructing the net of trajectories of the principal stresses is shown in [4]. After determining the position of the next nodal point from equation (1), \(\nu\) was determined for the point of the next row, and from equation (2), \(\sigma_2\); \(\sigma_1\) was determined from equation (3).

The interpretation was stable owing to the fact that, from equation (3), the larger principal stress was determined as

\[ \sigma_1=\frac{\sigma_2\pm\sqrt{4\sigma_i^2-3\sigma_2^2}}{2}, \]

and the subtrahend under the radical sign was considerably smaller than the minuend.

In studying the stressed state around an opening of diameter \(105\ \mathrm{mm}\) in the shell of a boiler drum with an internal diameter of \(1600\ \mathrm{mm}\) and a wall thickness of \(115\ \mathrm{mm}\) during its hydraulic pressure test under a pressure of \(200\ \mathrm{atm}\), it was found that in the region adjacent to the axial section of the shell, the trajectories of \(\sigma_1\) are close to arcs of circles whose center coincides with the center of the opening, i.e., the deformation is close to axisymmetric. This circumstance made it possible to simplify the research procedure considerably.

Fig. 3. Plots of the principal stresses along the axial section of the shell.

According to the simplified method, it is sufficient to measure the hardness along the trace of the axial section (the \(X\) axis in Fig. 2) and solve equations (2) and (3) jointly. From equation (2), written for the present case,

\[ \frac{\partial \sigma_2}{\partial X}+\frac{(\sigma_1-\sigma_2)}{X}=0, \tag{4} \]

\(\sigma_2\) is determined; \(\sigma_1\) is obtained from equation (3).

As boundary conditions, the conditions on the free contour were used, where \(\sigma_2=0\), \(\sigma_1=\sigma_i\). The variation in stress intensity along the \(X\) axis is also given in Fig. 2. The resulting diagrams of the principal stresses are shown in Fig. 3. The difference between the maximum value of \(\sigma_1\), calculated by the numerical solution of the system of equations (1), (2), (3) (the diagrams are shown in Fig. 3 by dashed lines) in the region adjacent to the \(X\) axis, and that determined by solving equations (4), (3), is 2.5%; the greatest deviations of \(\sigma_2\) at the boundary of the plastic zone are 6.5%, owing to the accumulated error of the adopted simplification. The approximate calculation may be used when the region in which the stresses are determined is limited to a size of 0.8 of the hole diameter.

Conclusions

1. A simplified experimental-analytical method has been developed for studying stress concentration around openings in thin-walled shells subjected to internal pressure.

2. Using the developed method, the stresses in the dangerous section of the shell have been determined.

Literature

1. N. M. Belyaev. Strength of Materials. GIFML, Moscow, 1956.

2. G. A. Smirnov-Alyaev. Strength of Materials under Plastic Deformation. Mashgiz, Moscow—Leningrad, 1961.

3. G. D. Del. Hardness of deformed metal. Izv. AN SSSR, “Metals,” No. 4, Moscow, 1967.

4. 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). Studies in the field of hardness measurement. Moscow—Leningrad, 1967.