Evaluation of the optimum pre-stressing pressure and wall thickness determination of thick-walled spherical vessels under internal pressure

doi:10.1016/j.jfranklin.2006.02.013

Journal of the Franklin Institute

Volume 344, Issue 5, August 2007, Pages 439-451

https://doi.org/10.1016/j.jfranklin.2006.02.013 Get rights and content

Abstract

In the present study, in the first part for a spherical vessel with known dimensions and working pressure, two methods of hoop and equivalent stress optimization across the wall thickness are employed to determine the best autofrettage pressure. In the next part for a predefined working pressure the minimum wall thickness of the vessel is calculated using two other design criteria i.e. (A) optimizing the hoop stress, and (B) assuming a suitable percent for the penetration of yielding within the wall thickness. Finally, the optimum thickness and the necessary strengthening pressure are extracted and different plots are introduced for different types of structural materials under different internal pressures.

Introduction

The ever increasing demands for axisymmetrical pressure vessels have high applications in industries such as chemical, nuclear, fluid transmitting plant, power plant and on the other hand in the military equipments, have turned the attention of many designers to this particular area of engineering. The progressive, hardly appropriate scarcity of some materials and also the high cost of their production have caused researchers not to confine their design to the customary elastic regime and have attracted their attention to the elastic–plastic approach which offers a more efficient use of such materials. In brief, increasing the strength to weight ratio and extending the fatigue life are some of the main objectives of optimal design for the thick-walled sphere. The optimization can be carried out primarily by generating a residual stress field in the sphere wall, known as the autofrettage process. The study of this technique has been the subject of numerous researches for many years, such as those given by Majzoobi [1], Manning [2], Chen [3], [4], Franklin [5], and has been discussed by Kargarnovin [6] and Ruilin Zhu [7] for elasto-perfectly plastic materials. For the linear work hardening material, however, little progress has been made, as the analysis is a more complicated compared to the above-mentioned behavior.

Most of the earlier solutions for residual stresses are based on the assumption of elastic unloading and only a few researchers have considered a criterion known as the reverse yielding point [8]. Hence, it has been noticed that the inclusion of plastic behavior on the unloading phase has been an untouched area of work to be considered in our study in this paper. To do this, a closed form solution of residual stresses in autofrettaged spheres is obtained. In the next step, a simple, accurate and reasonable analytical equation for determining the best autofrettage pressure with known dimensions of the sphere and the working pressure is derived. Moreover, for the cases in which the working pressure and inner radius are known, the minimum acceptable thickness is determined.

In the present research for the thick-wall sphere the following basic assumptions are made: the material is assumed to be elastic-linear work hardening which yields according to Tresca/von-Mises criterions and hardens isotropically, and the Bauschinger effect is ignored.

Section snippets

Elasto-plastic loading

Consider a thick-walled sphere with inner radius a, and outer radius b, which is subjected to the inner pressure P_i (see Fig. 1). The following non-dimensional parameters are used: $β = \frac{b}{a}, ρ = \frac{r}{a}, ρ_{c} = \frac{c}{a}, ρ_{c}^{'} = \frac{c^{'}}{a}, P = \frac{P_{i}}{σ_{0}}, p_{w} = \frac{P_{w}}{σ_{0}}, S_{r} = \frac{σ_{r}}{σ_{0}}, S_{θ} = \frac{σ_{θ}}{σ_{0}}, S = \frac{σ_{e}}{σ_{0}} = S_{θ} - S_{r}, \in_{r} = \frac{E ε_{r}}{σ_{0}}, \in_{θ} = \frac{E ε_{θ}}{σ_{0}},$ where P_w, P_i, $σ_{0}$ , $σ_{e}$ , $σ_{r}$ , $σ_{θ}$ and E are the working pressure, the autofrettage pressure, the initial yielding stress, the equivalent stress, the radial stress, the hoop stress and Young^’s modulus, respectively. Moreover, c and c′ are

Elastic–plastic unloading

When the pressure is removed from the sphere under consideration, if it causes plastic flow over part of the sphere, some residual stresses result in. In order to calculate this induced residual stresses, it is necessary to superpose the obtained stresses in loading phase due to the internal pressure P and stresses caused during unloading phase i.e. −P.

It is clear that the equilibrium and compatibility relations during unloading are similar to those given in Eqs. (2), (3) in which the

Determination of the residual stresses

Depending on the geometrical factor β and the amount of removed pressure from the vessel, different deformation modes are produced on the sphere.

If $P > P^{*}$ then during unloading, yielding will occur otherwise, the unloading is entirely elastic. In order to analyze the effect of these changes, at first the value of the two aforementioned pressures, i.e. P and P^*, have to be equated. In order to get to this state, the value of P from Eq. (8) is set equal to the value of P^*, obtained from Eq. (17).

Equivalent stress and hoop stress optimization

Now, suppose that the dimensions of the sphere and also the working pressure are known. The optimum autofrettage pressure is an internal pressure which is applied to the sphere before it is being put into operation. The main task of this pressure is to initiate yielding in the inner surface of the sphere and beside; it optimizes the distribution of the hoop stress, or equivalent stress, throughout the wall thickness.

Thickness optimization

In Fig. 3, Fig. 4 it can be seen that the value of the equivalent stress in radius $ρ_{c, opt}$ is less than the value of the initial yield stress. This means that in this case the vessel is capable to withstand higher level of working pressure p_w. On the other hand, if the value of p_w is prescribed, then the optimum thickness and initial strengthening pressure also can be obtained simultaneously. The design criterion is to consume less material in the production of the vessel. Where, by introducing

Conclusion

In this paper based on the theory of elasto-plasticity for optimal design of thick-walled spherical vessels, the governing equations were derived and solved. Based on the obtained results, it is concluded that

(1)
Autofrettage considerably reduces stress level in the wall of a sphere.
(2)
The method presented in this paper is very simple, applicable, accurate and reasonable compared to other existing methods.
(3)
The autofrettage pressure must be always greater than the working pressure.
(4)
Refered to Fig. 5,

Reference (9)

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P.C.T. Chen, Stress and deformation analysis of autofrettaged high pressure vessels, ASME Special Publication, vol....
P.C.T. Chen
The Bauschinger and hardening effect on residual stresses in an autofrettaged thick-walled cylinder
J. Pressure Vessel Technol.
(1986)

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