# Influence of Initial Imperfections on PMMA Spherical Shell

Authors: Haitao Zhu; Di Liu; Han Bao; Zhongjun Ding; Yi Zhang
Journal: Advances in Materials Science and Engineering
 (2022)

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## Abstract

This paper focused on the influence of initial imperfection on the critical load of PMMA pressure spherical shell by introducing buckling modes as the initial defect. Four types of pressure spherical shells including complete spherical shell, spherical shell with single penetration, spherical shell with double penetrations, and hyperspherical shell were studied. The distribution of initial imperfections was given through the modal configuration of linear eigenvalue analysis, and then the buckling critical load of the spherical shell was analyzed by using the consistent modal defect method and N-order modal defect method. Results confirmed that: Initial imperfections have a great different influence on pressure spherical shells with varying penetration. It has the greatest influence on the critical load of a spherical shell with double penetration at a 5 mm defect. With the increase of defect amplitude, the critical load of the spherical shell decreases. The analysis results also show that the first-order mode is usually not the worst geometric defect configuration of PMMA pressure spherical shell. And the worst mode may appear in higher-order mode due to the influence of penetration, initial defect amplitude, and different thickness-diameter ratio.


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## Body

## 1. Introduction

Compared with metal materials, PMMA is widely used in submersible observation windows, pressure-resistant structures, and other equipment because of their excellent properties such as high strength, high-light transmittance, and lightweight. The traditional manned cabin of the submersible is mostly made of high-strength titanium alloy, supplemented by PMMA as an observation window. However, there are still many blind spots in the structure with an observation window as a viewport, and the field of vision is limited, which cannot provide panoramic visibility. With the development of submersible technology, people have been unable to meet the limitations brought by the narrow viewport, so they study all PMMA pressure-resistant structures.The first to propose an all-acrylic hull was Professor Auguste Piccard, but the technical conditions could not be realized at that time. Until the U.S. Navy launched a spherical PMMA submersible named NEMO in 1970, meaning that the problem of all PMMA hulls has been solved technically [1]. The realization of this technology is attributed to Professor Stachiw. J.D.’s extensive theoretical and experimental research on PMMA hull in the 1960s [2–4]. Through continuous research on the PMMA scale model for short-term, long-term, and cyclic pressure tests, the diving depth of the submersible has reached 8000 ft [5, 6]. At the same time, Das also made great contributions to improving the cyclic fatigue life of NEMO [7]. Based on Dr. Stachiw. J.D.’s experimental research, it finally contributed to the formulation of the ASME Safety Standard for Pressure Vessels for Human Occupancy (ASME PVHO-1) [8].However, ASME standards are mostly aimed at the observation window of the submersible, which is not suitable for PMMA spherical shells. Moreover, it is difficult to obtain the analytical formula for a midthick shell with initial imperfections or penetrations. Therefore, it is necessary to develop new criteria to evaluate the critical pressure prediction of PMMA spherical shells.With the development of nonlinear finite element theory and computer science, increasingly scholars use finite elements to predict the buckling behavior of structures. Wunderlich et al. [9] combined a one-dimensional transfer matrix with finite elements to analyze the linear analysis of elastic-plastic rotating shells in 1982. This method can be used to analyze the buckling behavior and defect sensitivity of the dished head under external pressure. Subsequently, in 2002 [10], a new design rule was formulated for the spherical shell under external pressure, taking into account the relevant details such as boundary conditions, material properties, and defects. Grigolyuk and Lopanitsyn [11] proved the high sensitivity of the dome to deviation from the ideal shape by using the Ritz method based on the Marguerre equation. Marcinowski [12] considered all singular characteristics of nonlinear elastic stability, analyzed the buckling problem of a spherical shell with a slender ratio by the finite element method, and verified its correctness. Hong and Teng [13] proposed a semianalytical finite element method for rotating a shell with a defect under asymmetric load and verified the correctness through a large number of numerical analyses. Xu [14] studied the postbuckling and defect sensitivity of different types of cylinders by using Hui’s postbuckling method and compared it with ABAQUS simulation results. It is verified that Hui’s method significantly increased the valid region compared with the general stability theory. Based on Koiter’s research in the 1960s, Hutchinson [15] established a new spherical shell buckling theory. Numerical results of axisymmetric large deflection behavior of the ideal sphere are presented, and the role of axisymmetric defects in reducing buckling pressure is widely analyzed. Stein Montalvo et al. [16] conducted dynamic pressure buckling tests on defective spherical shells made of viscoelastic materials, proving that the lower critical load can be determined by material characteristics. An empirical model is introduced to show that reducing critical load caused by viscoelastic creep deformation is in the same way as reducing critical load affected by increasing defects in the elastic shell. In addition, the redesign of the NEMO submersible is also based on the fundamentals of finite element simulation [7].Wang Bo and Hao Peng have made a lot of contributions to the recent advances in imperfections. In the research of accurately predicting the lower-bound buckling load under the influence of geometric defects, Bo and Xiangtao [17] found that the improved worst multiple perturbation load approach (WMPLA) can provide an improved knockdown factor(KDF) by examples in the open literature. Subsequently, Wang et al. [18] found a new lower-bound curve that approximately produces a series of improved KDFs based on decades collection of test data. In another study, Yang et al. [19] proposed a novel bi-stage random field parameter estimation method with limited samples which achieves the best performance in terms of accuracy, efficiency, and stability. In addition, they used the incomplete reduced stiffness method to analyze the imperfection sensitivity of cylindrical shells [20] and developed a finite element numerical procedure for predicting buckling loads [21]. The above works have made an outstanding contribution to the buckling load prediction of cylindrical shell structures with geometric imperfection.For a stability-endangered shell structure, in order to obtain accurate realistic buckling load, imperfections must be included in finite element simulation analysis. So as to make the simulated imperfection more “realistic,” the only way seems to be to model them stochastically [22, 23], which finally corresponds to the measurement results of a large number of similar shell applications [24]. Several geometrical imperfections modeling methods can be used to predict the buckling load of PMMA spherical shells. The consistent modal defect method is convenient to calculate and can quickly estimate structural stability [25]. The improved consistent modal defect method can obtain a lower buckling load in the high-order modal than in the low-order modal [26]. The worst geometric imperfection is generally obtained from the buckling experience, and this defect mode is considered to be the worst, similar to the lowest eigenmode, and also includes the nonlinear eigenmodes [27–29]. Although this mode is “almost the worst,” the probability of occurrence in practice is not too high. The stochastic approach [30] or N-order modal defect method [31] considers the defect configuration of high-order modes. The displacement of the first n-order buckling modal is used to simulate the distribution of N kinds of geometric imperfections, calculate the corresponding critical load, and take the minimum value to evaluate the stability of the structure [32, 33]. The influence of higher-order modes on the stability of the shell structure is considered, which is more accurate and comprehensive and can provide a reference for structural stability design in engineering.In this study, the numerical analysis based on the linear buckling modal imperfection (LBMI) method is used to evaluate the influence of geometric defects on the buckling of complete spherical shell and spherical shell with penetration in detail. In addition, the consistent modal defect method is compared with the N-order modal defect method to analyze the possible order of the worst mode and predict the nonlinear buckling load of the PMMA pressure spherical shell. Finally, the influence of different types of penetrations on the buckling behavior of spherical shells is analyzed by N-order modal defect method and linear buckling method.


## 2. Finite Element Model Preparation

The models used in this research were spherical shells with an outer diameter of 1860 mm and thickness of 130 mm, which materials were PMMA. There are four kinds of models: complete spherical shell, spherical shell with single penetration, spherical shell with double penetrations, and hyperspherical shell. The shell with penetration is equipped with an aluminum hatch. The opening angle of the shell with penetration is 50°, and the angle of the hyperspherical shell is 100°, as shown in Figure1. The material properties of PM MA and Al hatch are E1 = 3000 MPa, u1 = 0.3, E2 = 70000 MPa, and u2 = 0.33. E1 and E2 are the elastic moduli, and u1 and u2 are the Poisson’s ratios.Figure 1
Penetration and boundary conditions of PM MA spherical hull model: (a) complete shell, (b) single penetration, (c) double penetration, and (d) hyperspherical shell.
(a)(b)(c)(d)Linear and nonlinear solvers of Abaqus-2019 were used to evaluate the buckling behavior of the pressure hull numerically. A linear perturbation solver was used to calculate the buckling loads and their corresponding mode shapes. Static-Riks solver in general was used to obtain the defect analysis curves of the PMMA spherical shell.The element type was a rotating shell with linear order, C3D8. Hex structured structure was used to divide the grid, and the grid size is 60 mm, so the mesh convergence study led to the whole spherical shell has 4952 elements, one penetration shell has 3696 elements, the double penetration shell has 3136 elements and the hyperspherical shell has 3672 elements.The boundary conditions of the spherical shell are shown in Figure1. The pressure shell is theoretically unconstrained, but to prevent its rigid displacement during the calculation process, three points on the equator are selected to limit the displacement in the six directions of the model. The loading condition was uniform pressure on the outer surface. The same boundary conditions and loading conditions are adopted for single penetration, double penetration, and hyperspherical shell models.


## 3. Buckling Characteristics Analysis of PMMA Sphere under External Pressure

In consideration of the safety of pressure shells, modal imperfections are often introduced as the worst defects. It is specified in the European standard that the shape of geometric imperfections should be the worst defect, that is, the imperfection that causes the fastest reduction of the buckling load of the shell [34]. It is suggested to use modal imperfections to analyze buckling characteristics of the shell when the shape of the worst imperfection is unknown.Based on buckling mode, the modal defect method needs to carry out an eigenvalue buckling analysis on the ideal spherical shell under compression to obtain several order buckling modal eigenvalues and modal displacements. The characteristic buckling critical load is obtained by multiplying modal eigenvalue with external load, which can be verified by classical theory. The modal displacement is written into node data as a result file, and the modal defect method can introduce initial imperfection by modifying the node displacement.The distribution of initial imperfections is based on the buckling modal displacement. In this section, two modal defect methods were used for analysis. In the first-order modal defect method, the first-order modal displacement of the buckling eigenvalue of the spherical shell is taken as the distribution form of the initial imperfection, and the critical buckling load corresponding to the defect amplitude can be obtained only by analyzing once. The N-order modal defect method is to take the first n-order modal displacement as the initial geometric imperfection distribution mode. Introduce then kinds of defect distribution forms into the spherical shell, respectively, and then calculate the n modal defect distributions under each defect amplitude to obtain n critical values under one defect amplitude. Finally, select the minimum value to evaluate the structural stability under the defect amplitude. The first-order modal method is simple and effective, while the n-order modal method is more comprehensive.In this research, the first-order modal defect method and the N-order modal defect method are presented in detail. The two methods are used to analyze the geometrical imperfections modes of complete PM MA spherical shell, spherical shell with single penetration, spherical shell with double penetration, and hyperspherical shell, and then compare the prediction results.


### 3.1. Geometrical Imperfection Modes for Ideal PMMA Spherical Shell without Penetration

Linear buckling analysis is a typical problem of solving eigenvalues and gives the instability modes of mode shapes, buckling load, and response to the solution of the first stability problem by solving eigenvalues. These mode shapes can be used as initial geometric imperfections to predict the critical instability force of the pressure hull, but the analysis results are frequently higher than the actual value.Table1 shows the buckling modes of the complete spherical shell buckling analysis using the first-order modal defect method. Select 10 groups from 5 mm to 50 mm as defect amplitude. According to the stress nephogram of the initial deflection, the critical state, and the buckling state of the complete spherical shell under compression listed in the table, it can be seen that the initial deflection is the same under this method. However, with the increase of initial defect amplitude, the deformation degree increases. The critical state is the same, and the buckling state is different in degree according to the magnitude of the defect amplitude.Table 1
Buckling modes of the complete spherical shell using the first-order modal defect method.Figure2(a) shows the first 10 modal load curves under 1–10 mm imperfections obtained by N-order modal defect method. Each curve represents different defect amplitudes. The lowest point on each curve is the buckling load under the worst of the first ten modes under the corresponding defect amplitude. As shown in the figure, the worst mode does not appear in the first mode, indicating that the first mode is not the worst mode of the spherical shell.Figure 2
Modal load curve of the complete spherical shell using N-order modal method: (a) defect amplitude 1–10 mm, (b) defect amplitude 5–50 mm, and (c) defect amplitude 15–50 mm (worst mode point).
(a)(b)(c)In the first ten modal load curves, the first four critical loads are at low points, most of which are 2nd-order modes, and the individual lowest points appear in the 3rd and 4th-order modes. Therefore, under this model, we should focus on the critical load corresponding to the first four modes. The figure also shows the maximum value of the critical load corresponding to the 9th mode. In fact, the difference between the eigenvalues of 3–9 modes is very small, but the critical load has a large deviation. There is no obvious difference between the critical loads corresponding to orders 3–8, but the buckling load changes greatly in the 9th order. It shows that even if the modal eigenvalues are similar, the buckling critical load may deviate greatly due to different buckling structures.Figure2(b) shows the first 20 modal load curves under 5–50 mm imperfections of the complete spherical shell. Similarly, the worst mode is not the first mode. Figure 2(c) shows the first 7 modes of 15–50 mm defect amplitude in Figure 2(b). Since the curves in Figure 2(b) are too dense to be seen clearly, it is partially enlarged to observe the worst-order mode.The variation trend of buckling load in the two figures is basically the same. The mode of the worst buckling load under each defect amplitude has been marked in the figure. For the selected initial imperfections, the mode of the worst buckling load appears in the first few modes, but the worst mode is rarely the first mode. It illustrates that for the buckling analysis of PMMA pressure spherical shell with initial imperfections, the first-order mode is not the worst mode.Eigenvalue buckling analysis is very useful as a preliminary evaluation of nonlinear buckling analysis. It can predict the approximate location of the critical instability force. Therefore, there is a basis for the magnitude of the applied force in nonlinear buckling analysis.


### 3.2. Geometrical Imperfection Modes for PMMA with Penetration

Theoretically, the pressure-resistant structure of the submersible is not a complete spherical shell, at least one perforation is required for the entry of the observer and equipment. The analysis of the complete spherical shell is only a comparison to judge whether the existence of perforation has an impact on its structural stability.Figure3 shows the first-order and worst-order modal load curves of four types of a spherical shell, respectively. The blue curve represents the buckling load of 1–20 modes under 5 mm imperfection, the black curve represents the buckling load of each defect under the first mode, and the red curve is the buckling load of each defect under the worst mode.Figure 3
Comparison between first-order modal method and N-order modal method in buckling load of four types of spherical shells (a) Complete spherical shell (b) Spherical shell with single penetration (c) Spherical shell with double penetration (d) Hyperspherical shell.
(a)(b)(c)(d)Figure3(a) is the modal load curve of the complete spherical shell. The worst mode is the 2nd-order mode. Compared with the first-order modal load curve, the gap gradually increases with the increase of defect amplitude. The maximum value of the gap is at a defect amplitude of 30 mm, the maximum modal load is reduced by 8.5%, then the gap is gradually narrowed. Figure 3(b) is the modal load curve of the spherical shell with single penetration. The first mode of the spherical shell is the worst mode under the defect of 5–35 mm, so there is only one curve. Then the defect amplitude increases and the worst mode changes. Figure 3(c) is the modal load curve of the spherical shell with double penetration. The worst mode under defect amplitude of 5–35 mm is the 2nd-order mode, but there is little difference between the buckling load of the worst mode and the first mode, the maximum difference is 2.7%. Figure 3(d) is the modal load curve of the hyperspherical shell. The worst mode under 10–35 mm defect amplitude is the 4th-order mode. Compared with the first-order mode, the maximum buckling load reduction of the worst-order mode is 10.5%. Thus, the worst modes of PM MA pressure spherical shells with different penetrations are different.There is little difference between the buckling load of single penetration shell, hyperspherical shell, and complete spherical shell, which is much larger than the gap between double penetration shell and complete spherical shell. It is proved that for the spherical shell with penetration, the size of the penetration in a certain range has been found to not affect the short-term critical pressure. However, double penetrations have a great influence on the critical load of the spherical shell when the initial imperfection is less than 15 mm. And then the critical load of double penetration spherical shell is higher than that of a complete spherical shell and single penetration spherical shell as the defect amplitude exceeds 15 mm.Table2 lists the worst modes of four kinds of spherical shells under different defect amplitudes. With the increase of defect amplitude, the worst modes of four spherical shells will also change. However, for different models, there are stable worst modes when the defect amplitude changes in a certain range.Table 2
The worst mode of the spherical shell under different defect amplitudes.
The worst buckling modeDefect5101520253035404550Complete spherical shell2233333176Single penetration11111112186Double penetration2222222111Hyperspherical shell244444431212Since the modal displacement is introduced as the initial imperfection in the modal defect method, the larger the modal displacement, the larger the initial defect distribution. For the spherical shell, when the initial imperfection changes in a certain range, the stress of the model is basically the same, the only difference is that the critical load changes. According to Table1, no matter whether the initial imperfection introduced into the spherical shell is large or small, the initial deformation position is the same, but the deformation degree increases with the increase of defect amplitude. This also leads to more instability phenomena in the final buckling state when the initial defect amplitude is large, and the worst mode order changes accordingly. That is to say, the increase of the initial defect amplitude makes the spherical shell more prone to instability.In Table2, the worst mode of a complete spherical shell within 10 mm of initial imperfection appears in the 2nd-order, and then the worst mode becomes 3rd-order when the defect amplitude in the stage of 15–35 mm. 1st-order is the worst mode of single penetration spherical shell during defect amplitude 5–35 mm, while double penetration spherical shell is 2nd-order. The worst mode of the hyperspherical shell is 4th-order in the range of 10–35 mm. When the defect amplitude is more than 40 mm, the worst-mode order of the four kinds of spherical shells changes greatly. The abnormal phenomenon of the worst mode after exceeding 40 mm indicates that there are special points in the sensitivity of the spherical shell to large defect amplitude. If this situation exists in the design, it should be paid attention to.


### 3.3. Comparison between Spherical Shells

The defect modal analysis of PMMA ideal spherical shell, single opening spherical shell, double opening spherical shell, and the super spherical shell are carried out, and the prediction results are compared.Table3 shows the comparison between the buckling modes of the consistent defect mode method and the N-order characteristic defect mode method. The introduction of geometric defects in the consistent defect mode method adopts the first-order modal displacement. In addition to the single penetration spherical shell, the amplitude defects of the N-order modal defect method of the other three spherical shells adopt 2nd-order modal displacement under the imperfection of 5 mm, and the single penetration spherical shell adopts the 18th-order modal displacement under the imperfection of 45 mm.Table 3
Comparison between 1st-order andN-order modal methods in buckling shape.


## 3.1. Geometrical Imperfection Modes for Ideal PMMA Spherical Shell without Penetration

Linear buckling analysis is a typical problem of solving eigenvalues and gives the instability modes of mode shapes, buckling load, and response to the solution of the first stability problem by solving eigenvalues. These mode shapes can be used as initial geometric imperfections to predict the critical instability force of the pressure hull, but the analysis results are frequently higher than the actual value.Table1 shows the buckling modes of the complete spherical shell buckling analysis using the first-order modal defect method. Select 10 groups from 5 mm to 50 mm as defect amplitude. According to the stress nephogram of the initial deflection, the critical state, and the buckling state of the complete spherical shell under compression listed in the table, it can be seen that the initial deflection is the same under this method. However, with the increase of initial defect amplitude, the deformation degree increases. The critical state is the same, and the buckling state is different in degree according to the magnitude of the defect amplitude.Table 1
Buckling modes of the complete spherical shell using the first-order modal defect method.Figure2(a) shows the first 10 modal load curves under 1–10 mm imperfections obtained by N-order modal defect method. Each curve represents different defect amplitudes. The lowest point on each curve is the buckling load under the worst of the first ten modes under the corresponding defect amplitude. As shown in the figure, the worst mode does not appear in the first mode, indicating that the first mode is not the worst mode of the spherical shell.Figure 2
Modal load curve of the complete spherical shell using N-order modal method: (a) defect amplitude 1–10 mm, (b) defect amplitude 5–50 mm, and (c) defect amplitude 15–50 mm (worst mode point).
(a)(b)(c)In the first ten modal load curves, the first four critical loads are at low points, most of which are 2nd-order modes, and the individual lowest points appear in the 3rd and 4th-order modes. Therefore, under this model, we should focus on the critical load corresponding to the first four modes. The figure also shows the maximum value of the critical load corresponding to the 9th mode. In fact, the difference between the eigenvalues of 3–9 modes is very small, but the critical load has a large deviation. There is no obvious difference between the critical loads corresponding to orders 3–8, but the buckling load changes greatly in the 9th order. It shows that even if the modal eigenvalues are similar, the buckling critical load may deviate greatly due to different buckling structures.Figure2(b) shows the first 20 modal load curves under 5–50 mm imperfections of the complete spherical shell. Similarly, the worst mode is not the first mode. Figure 2(c) shows the first 7 modes of 15–50 mm defect amplitude in Figure 2(b). Since the curves in Figure 2(b) are too dense to be seen clearly, it is partially enlarged to observe the worst-order mode.The variation trend of buckling load in the two figures is basically the same. The mode of the worst buckling load under each defect amplitude has been marked in the figure. For the selected initial imperfections, the mode of the worst buckling load appears in the first few modes, but the worst mode is rarely the first mode. It illustrates that for the buckling analysis of PMMA pressure spherical shell with initial imperfections, the first-order mode is not the worst mode.Eigenvalue buckling analysis is very useful as a preliminary evaluation of nonlinear buckling analysis. It can predict the approximate location of the critical instability force. Therefore, there is a basis for the magnitude of the applied force in nonlinear buckling analysis.


## 3.2. Geometrical Imperfection Modes for PMMA with Penetration

Theoretically, the pressure-resistant structure of the submersible is not a complete spherical shell, at least one perforation is required for the entry of the observer and equipment. The analysis of the complete spherical shell is only a comparison to judge whether the existence of perforation has an impact on its structural stability.Figure3 shows the first-order and worst-order modal load curves of four types of a spherical shell, respectively. The blue curve represents the buckling load of 1–20 modes under 5 mm imperfection, the black curve represents the buckling load of each defect under the first mode, and the red curve is the buckling load of each defect under the worst mode.Figure 3
Comparison between first-order modal method and N-order modal method in buckling load of four types of spherical shells (a) Complete spherical shell (b) Spherical shell with single penetration (c) Spherical shell with double penetration (d) Hyperspherical shell.
(a)(b)(c)(d)Figure3(a) is the modal load curve of the complete spherical shell. The worst mode is the 2nd-order mode. Compared with the first-order modal load curve, the gap gradually increases with the increase of defect amplitude. The maximum value of the gap is at a defect amplitude of 30 mm, the maximum modal load is reduced by 8.5%, then the gap is gradually narrowed. Figure 3(b) is the modal load curve of the spherical shell with single penetration. The first mode of the spherical shell is the worst mode under the defect of 5–35 mm, so there is only one curve. Then the defect amplitude increases and the worst mode changes. Figure 3(c) is the modal load curve of the spherical shell with double penetration. The worst mode under defect amplitude of 5–35 mm is the 2nd-order mode, but there is little difference between the buckling load of the worst mode and the first mode, the maximum difference is 2.7%. Figure 3(d) is the modal load curve of the hyperspherical shell. The worst mode under 10–35 mm defect amplitude is the 4th-order mode. Compared with the first-order mode, the maximum buckling load reduction of the worst-order mode is 10.5%. Thus, the worst modes of PM MA pressure spherical shells with different penetrations are different.There is little difference between the buckling load of single penetration shell, hyperspherical shell, and complete spherical shell, which is much larger than the gap between double penetration shell and complete spherical shell. It is proved that for the spherical shell with penetration, the size of the penetration in a certain range has been found to not affect the short-term critical pressure. However, double penetrations have a great influence on the critical load of the spherical shell when the initial imperfection is less than 15 mm. And then the critical load of double penetration spherical shell is higher than that of a complete spherical shell and single penetration spherical shell as the defect amplitude exceeds 15 mm.Table2 lists the worst modes of four kinds of spherical shells under different defect amplitudes. With the increase of defect amplitude, the worst modes of four spherical shells will also change. However, for different models, there are stable worst modes when the defect amplitude changes in a certain range.Table 2
The worst mode of the spherical shell under different defect amplitudes.
The worst buckling modeDefect5101520253035404550Complete spherical shell2233333176Single penetration11111112186Double penetration2222222111Hyperspherical shell244444431212Since the modal displacement is introduced as the initial imperfection in the modal defect method, the larger the modal displacement, the larger the initial defect distribution. For the spherical shell, when the initial imperfection changes in a certain range, the stress of the model is basically the same, the only difference is that the critical load changes. According to Table1, no matter whether the initial imperfection introduced into the spherical shell is large or small, the initial deformation position is the same, but the deformation degree increases with the increase of defect amplitude. This also leads to more instability phenomena in the final buckling state when the initial defect amplitude is large, and the worst mode order changes accordingly. That is to say, the increase of the initial defect amplitude makes the spherical shell more prone to instability.In Table2, the worst mode of a complete spherical shell within 10 mm of initial imperfection appears in the 2nd-order, and then the worst mode becomes 3rd-order when the defect amplitude in the stage of 15–35 mm. 1st-order is the worst mode of single penetration spherical shell during defect amplitude 5–35 mm, while double penetration spherical shell is 2nd-order. The worst mode of the hyperspherical shell is 4th-order in the range of 10–35 mm. When the defect amplitude is more than 40 mm, the worst-mode order of the four kinds of spherical shells changes greatly. The abnormal phenomenon of the worst mode after exceeding 40 mm indicates that there are special points in the sensitivity of the spherical shell to large defect amplitude. If this situation exists in the design, it should be paid attention to.


## 3.3. Comparison between Spherical Shells

The defect modal analysis of PMMA ideal spherical shell, single opening spherical shell, double opening spherical shell, and the super spherical shell are carried out, and the prediction results are compared.Table3 shows the comparison between the buckling modes of the consistent defect mode method and the N-order characteristic defect mode method. The introduction of geometric defects in the consistent defect mode method adopts the first-order modal displacement. In addition to the single penetration spherical shell, the amplitude defects of the N-order modal defect method of the other three spherical shells adopt 2nd-order modal displacement under the imperfection of 5 mm, and the single penetration spherical shell adopts the 18th-order modal displacement under the imperfection of 45 mm.Table 3
Comparison between 1st-order andN-order modal methods in buckling shape.


## 4. Conclusion

In this paper, the buckling characteristics of PM MA pressure spherical shell of underwater submersible are studied by finite element method and modal defect method, and the following conclusions are obtained:(1)
For the buckling numerical analysis of PM MA pressure spherical shell considering initial defects, the worst initial geometric imperfection distribution is not necessarily the first-order mode. The worst initial defect modes of pressure spherical shells with different penetrations are different. Of the four types of pressure spherical shells in this paper, only the worst mode of single penetration spherical shell under 5–35 mm defect amplitude is the first order.(2)
With the change of defect amplitude, the worst mode of pressure spherical shell will also change. The worst mode generally appears in the first few modes when the defect amplitude is small, and for large defect amplitude, the worst mode may appear in higher-order mode. The worst modes of complete spherical shells mostly appear in 2nd-order and 3th-order modes, single penetration spherical shells mostly appear in the first-order mode, double penetration spherical shells mostly appear in the 2nd-order mode, and hyperspherical spherical shells mostly appear in the 4th-order mode.(3)
The critical buckling load of the spherical shell decreases with the increase of initial defect amplitude. Compared with the first-order mode, the worst-order mode leads to a faster decrease in the critical load of the spherical shell. The larger defect amplitude leads to the lower stability of the spherical shell. So the spherical shell is more prone to instability, resulting in irregular changes in the worst-order mode of the spherical shell. Therefore, to accurately evaluate the bearing capacity and stability of spherical shells, it is necessary to analyze the critical buckling load of the structure in detail according to the specific structure and specific conditions.



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*Source: 3481368-2022-10-06.xml*