1. Introduction
Toroidal shells have several advantages over traditional spherical and cylindrical shells [
1], including favorable steerability, passability, and stability [
2,
3]. The reason for good passability is that two paths can be chosen to reach a location in toroidal shells. Toroidal shells are widely applied in ocean, nuclear, and civil industries. In ocean engineering, toroidal shells are considered to be a promising pressure structure for deep sea space stations [
2,
3]. However, toroidal shells have a low buckling load and are difficult to fabricate.
Buckling, which has been extensively studied, is the main type of failure affecting toroidal shells subjected to external pressure. Błachut [
4] performed experiments to determine the collapse load and collapsed shape of toroidal shells. The experimental results compared well with their numerical results. Furthermore, studies have investigated the effect of the elliptical section of toroidal shells on their buckling by using the finite element method [
5,
6]. Zingoni proposed an approximate bending solution to solve the axisymmetric bending of elliptic toroidal shells [
7]. He obtained the eigenvalues of part of the toroidal vessel by using the Galerkin’s scheme to calculate stability equations [
8]. Studies proposed analytical algorithms for examining the strength of ribbed toroid shells and performed a nonlinear analysis to analyze the buckling of such shells [
1,
9,
10]. Civalek used the discrete singular convolution method to analyze the buckling of CNT-reinforced laminated non-rectangular plates [
11]. Moradi-Dastjerdi analyzed the thermal and mechanical buckling of an active multidisciplinary sandwich plate by developing a mesh-free solution based on third order shear deformation theory [
12].
Zhang et al. [
13] designed and manufactured segmented toroidal shells composed of cylindrical shells. Furthermore, they investigated the effect of the number of segments on buckling load and imperfection sensitivity. However, it was recognized that, unlike cylindrical shells, spherical shells were more suitable for using as junction elements because the highly symmetrical structure of a spherical shell facilitated hole opening. In addition, surface stress was evenly distributed on spherical shells [
14,
15,
16]. The surface of a spherical shell experienced less stress compared with other structures under the same external pressure in [
17].
The combination design idea mentioned above by Zhang et al. [
13] has also been used in replacing cylindrical shells with multiple intersecting spherical shells. Liang et al. [
18] combined the interior penalty function and the Davidon–Fletcher–Powell method to optimize the design of multiple intersecting spheres. Subsequently, Zhang and Gou examined the effect of material and geometric parameters on the buckling of multiple intersecting spherical shells [
19,
20]. Liu et al. proposed an approximate analytical model for buckling analysis of common spherical–cylindrical–spherical composite structures by using the generalized Galerkin method [
21]. Sobhani et al. investigated the vibration of porous nano-enriched polymer composite coupled spheroidal–cylindrical shells, the wave frequency responses of the nanocomposite linked hemispherical–conical shell, the Circumferential vibration analysis of nano-porous sandwich assembled spherical–cylindrical-conical shells, and the Free vibration of porous graphene oxide powder nano-composite assembled paraboloid–cylindrical shells [
22,
23,
24,
25]. Rezaiee and Masoodi analyzed the buckling of plates and shell structures by an efficient triangular shell element which had six nodes with thirty degrees of freedom [
26]. Zingoni presented a linear–elastic theoretical formulation for determination of the state of stress in large liquid-filled multi-segmented spherical shells [
27]. However, toroidal shells composed of spherical shells, termed sphere-segmented toroidal shells, have rarely been described in literature. The effects of imperfection, ellipticity, and completeness on the buckling characteristics of such structures remain unknown.
This study investigated the buckling characteristics of sphere-segmented toroidal shells under external pressure. The sphere-segmented toroidal shell could overcome the low buckling load and difficult manufacturing of the traditional toroidal shell. The advantages of the proposed experimental methodology were as follows: a pressure pump was used to increase the pressure of the chamber in hydrostatic testing, simulating the pressure under different depths and the reliability of the result could be guaranteed. On the other hand, optical scanning obtained the numerical model with initial geometrical imperfection. The accuracy of the buckling loads of the experimental model could be improved in numerical analysis. The remainder of this paper is organized as follows.
Section 2 introduces the geometric properties of toroidal shells, their fabrication, and related experiments.
Section 3 presents a comparison and analysis of the experimental results and the results of a finite-element analysis. In addition, the effects of ellipticity and completeness on buckling are discussed. Conclusions are presented in
Section 4. This study provides valuable references for the design, manufacture, and analysis of atypical toroidal shells.
Author Contributions
Conceptualization, J.Z.; methodology, C.D. and J.Z.; software, J.Z. and Y.Z.; validation, F.W.; formal analysis, C.D. and J.Z.; investigation, C.D. and J.Z.; data curation, C.D. and F.W.; writing—original draft preparation, C.D.; writing—review and editing, J.Z. All authors have read and agreed to the published version of the manuscript.
Funding
This work was supported by the National Natural Science Foundation of China (Grant Nos. 52071160 and 52222111).
Data Availability Statement
The data presented in this study are available on request from the corresponding author. The data are not publicly available due to privacy.
Conflicts of Interest
The authors declare no conflict of interest.
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Figure 1.
Geometric notations (a) and photograph (b) of a sphere-segmented toroidal shell.
Figure 2.
Schematic of the process for manufacturing sphere-segmented toroidal shells.
Figure 3.
Photographs of ultrasonic measurement scene (a), optical scanning scene (b), and hydrostatic tests scene (c).
Figure 4.
Geometric deviation between the scanned models and perfect geometries of two laboratory models sphere-segmented toroidal shells (named ST-1 and ST-2, respectively).
Figure 5.
Pressure versus time curves for ST-1 and ST-2 sphere-segmented toroidal shells under uniform external pressure.
Figure 6.
Failed sphere-segmented toroidal shell and collapse locations.
Figure 7.
Convergence study of ST-1 sphere-segmented toroidal shell.
Figure 8.
Finite-element models of sphere-segmented toroidal shells and relevant information.
Figure 9.
Equilibrium paths, stress distribution, and post-buckling modes of ST-1 and ST-2 sphere-segmented toroidal shells.
Figure 10.
Deformation evolution of ST-1 and ST-2 sphere-segmented toroidal shells; six points correspond to those in
Figure 9.
Figure 11.
Normalized linear and nonlinear buckling loads for sphere-segmented toroidal shells.
Figure 12.
Imperfection sensitivities (a) and equilibrium path (b) of sphere-segmented toroidal shell.
Figure 13.
(a) Geometric notations for sphere-segmented toroidal shells under the influence of ellipticity. (b) Ellipticity parameter definition diagram.
Figure 14.
First-order eigenmode of sphere-segmented toroidal shells with ellipticity (K) of 1.024, 1.066, 1.083, and 1.095.
Figure 15.
Equilibrium paths of sphere-segmented toroidal shells with ellipticity of (K) = 1.024, 1.066, 1.083, and 1.095.
Figure 16.
First-order eigenmode of sphere-segmented toroidal shells with a segment number of 1–6.
Figure 17.
Post-buckling modes of sphere-segmented toroidal shells (
n = 1–6) correspond to
Figure 16.
Table 1.
Geometric properties, buckling load, and experiment-derived collapse pressure of ST-1 and ST-2 sphere-segmented toroidal shells.
Model | | | | St. dev. | | | | |
---|
ST-1 | 0.686 | 0.826 | 0.753 | 0.037 | 19.206 | 6.229 | 5.523 | 1.128 |
ST-2 | 0.688 | 0.818 | 0.746 | 0.031 | 18.452 | 6.292 | 5.525 | 1.139 |
Table 3.
Statistical results related to linear and nonlinear buckling loads (see
Table 2 and
Table 4).
Model | | | | St. dev. |
---|
| 26.549 | 26.758 | 26.645 | 0.067 |
| 9.922 | 10.668 | 10.394 | 0.155 |
Table 5.
Geometrical parameters and buckling loads of sphere-segmented toroidal shells with different ellipticities (K).
K | | | a | b | | | |
---|
1 | 60.00 | 60.00 | 126.35 | 126.35 | 11.505 | 12.615 | 10.398 |
1.024 | 60.79 | 58.42 | 127.87 | 124.84 | 11.584 | 12.548 | 10.412 |
1.047 | 61.50 | 57.01 | 129.22 | 123.44 | 11.582 | 12.666 | 10.540 |
1.066 | 62.10 | 55.80 | 130.41 | 122.3 | 11.089 | 12.548 | 10.334 |
1.083 | 62.59 | 54.85 | 131.40 | 121.37 | 11.504 | 12.507 | 10.497 |
1.095 | 62.96 | 54.07 | 132.12 | 120.61 | 11.682 | 12.669 | 10.446 |
1.104 | 63.21 | 53.57 | 132.59 | 120.14 | 11.681 | 12.600 | 10.565 |
Table 6.
First yield load (Py) and nonlinear buckling load (Δ = 0 and 0.1 t) for sphere-segmented toroidal shells with a segment number (n) of 1–6.
| n = 1 | n = 2 | n = 3 | n = 4 | n = 5 | n = 6 |
---|
| 12.007 | 11.635 | 11.362 | 11.491 | 11.651 | 11.505 |
Δ = 0 | 12.753 | 12.350 | 12.585 | 12.431 | 12.420 | 12.615 |
Δ = 0.1 t | 10.483 | 10.381 | 10.244 | 10.186 | 10.492 | 10.398 |
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