It is necessary to properly transform the triaxial stress state to generate an equivalent stress. In general, tensile tests of materials are performed in the uniaxial direction, and the yield stress and tensile strength, which are the results of the test, are also values in the uniaxial direction; however, it is difficult to compare as the stress exists in a triaxial stress state. Therefore, it is convenient to define the equivalent stress to express the triaxial stress state and compare it with the uniaxial stress. The equivalent stress is also called the von Mises stress, and its expression as a principal stress state is shown in Equation (1). The distortion energy theory was used to calculate the stress. In this theory, the strain energy of a material is composed of volumetric strain energy and shear strain energy. The yield occurs when the shear strain of the material is equal to the shear strain energy of the same material in a simple tensile state. This theory is also called von Mises theory.

In nonlinear analysis, the stiffness matrix is not a constant but changes according to the value of the displacement vector, so the desired solution can be obtained through several calculations. The load acts on a step-by-step basis, dividing at each step into sub-steps and performing the analysis while incrementing the load in the order of the divided sub-loads. For the analysis of the sub-load, the slope (stiffness K) was repeatedly obtained, and the convergence was determined when the error was less than the force criterion. The Newton-Rapson method was used to obtain the slope. In Figure 6 (a), when the first-stage load is applied, it can be observed that the error or force convergence is less than the force criterion after seven stiffness gradients seven times. Subsequently, the sub-steps converged immediately after the 3rd and 4th loads were applied. Figure 6 (b) shows the convergence process in which four loads are sequentially applied and eleven cumulative iterations were performed [8].

Figure 6: 
Solution information of force convergence (a) and displacement convergence (b)

As shown in Figure 7, the outer surface of the threaded part of the lower shell was under compression, as depicted by the red arrow in Figure 7. Additionally, a von Mises stress of 769 MPa occurred near the lowermost thread, which does not exceed the tensile strength of 880 MPa; thus, it was concluded that the structural stability is secured under the operating conditions.

Figure 7: 
The result of von Mises stress of the accumulator

In contrast, when the pressure was 40 MPa, the maximum von Mises stress of the accumulator was computed as 1029 MPa, making it difficult to have structural stability.

A walled pressure vessel subjected to cyclic internal pressure represents a component subjected to mean tensile stress. The mean stress can have substantial influence on the fatigue behavior.

In fatigue life analysis, the S-N curve is widely used for life prediction, especially in high-cycle fatigue situations where the cyclic load is in the elastic region and the failure cycle is high. The S-N curve generated from the fully reversed loading type which is σ_m = 0 and R = −1 . The mean stress, σ_m and the stress ratio, R are given by Equation (2) and Equation (3).

Alternating stresses are usually lower than the yield strength of the material; however, the application of such alternating stresses can cause fatigue failure. To analyze fatigue failure, the Gerber method was utilized. The Gerber method, represented by Equation (4), is commonly used for ductile materials to predict the fatigue failure under a specific force or stress application.

In Equation (4), σ_alt is the absolute maximum value of the alternating stress, S_ult is the ultimate tensile strength, and σ_m is the mean stress of the maximum and minimum peak points of the sinusoidal alternating stress.

The safety factor is the ratio between the design allowable stress and the yield strength or the ultimate strength, which is the design criterion. If the design allowable stress is calculated using the safety factor and compared with the stress acting on the current structure, it is easy to verify the existence of any damage. When the von Mises discriminant is used, the safety factor is calculated by dividing the uniaxial yield strength of the material by the von Mises equivalent stress. The safety factor of the accumulator is shown in Figure 8.

Figure 8: 
Safety factor of the accumulator

Similar to the structural analysis results and failure of the actual model, the fatigue analysis results also showed the lowest fatigue life near the lowermost thread of the lower shell. The results of the fatigue life are illustrated in Figure 9.

Figure 9: 
Fatigue life of the accumulator

Figure 9 shows that a minimum of 160 cycles was found at the lowermost thread. As the design life is set to 10⁶, the minimum lifespan is 160, and it has a low safety factor of 0.695, which model cannot withstand the fatigue life of 10⁶ cycles.