# Buckling of Corrugated Ring under Uniform External Pressure

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

_{0}.

## 2. A Statement of the Problem

## 3. Equivalent Bending Stiffness Approach

- (a)
- the applicability of the equivalent bending stiffness approach to the stability problems;
- (b)
- estimation of the error caused by neglecting the curvature of the basic circular ring in determining equivalent bending stiffness;
- (c)
- estimation of the accuracy of the buckling pressure obtained with the equivalent bending stiffness approach and the possibility of its refinement.

## 4. Asymptotic Homogenization Method

## 5. Using the Imperfection Method

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Data Availability

## Abbreviations

ODE | Ordinary Differential Equation |

FEA | finite element analysis |

## References

- Dayyani, I.; Shaw, A.D.; Flores, E.I.S.; Friswell, M.I. The mechanics of composite corrugated structures: A review with applications in morphing aircraft. Comp. Struct.
**2015**, 133, 358–380. [Google Scholar] [CrossRef] [Green Version] - Adriaenssens, S.; Dejong, M. Form finding of corrugated shell structures for seismic design and validation using non-linear pushover analysis. Eng. Struct.
**2019**, 181, 362–373. [Google Scholar] - Malek, S.; Williams, C.J.K. The equilibrium of corrugated plates and shells. Nexus Netw. J.
**2017**, 19, 619–627. [Google Scholar] [CrossRef] [Green Version] - Manko, Z.; Beben, D. Research on steel shell of a road bridge made of corrugated plates during backfilling. J. Bridge Eng.
**2005**, 10, 592–603. [Google Scholar] [CrossRef] - Wang, K.; Zhou, M.; Hassanein, M.F.; Zhong, J.; Ding, H.; An, L. Study on elastic global shear buckling of curved girders with corrugated steel webs: Theoretical analysis and FE modelling. Appl. Sci.
**2018**, 8, 2457. [Google Scholar] [CrossRef] [Green Version] - Semenyuk, N.P. Stability of corrugated arches under external pressure. Int. Appl. Mech.
**2013**, 49, 211–219. [Google Scholar] [CrossRef] - Ross, C.; Humphries, M. The buckling of corrugated circular cylinders under uniform external pressure. Thin Walled Struct.
**1993**, 17, 259–271. [Google Scholar] [CrossRef] - Semenyuk, N.P.; Zhukova, N.B.; Ostapchuk, V.V. Stability of corrugated composite noncircular cylindrical shells under external pressure. Int. Appl. Mech.
**2007**, 43, 1380–1389. [Google Scholar] [CrossRef] - Semenyuk, N.P.; Babich, I.Y. Stability of longitudinally corrugated cylindrical shells under uniform surface pressure. Int. Appl. Mech.
**2007**, 43, 1236–1247. [Google Scholar] [CrossRef] - Semenyuk, N.P.; Zhukova, N.B.; Neskhodovskaya, N.A. Stability of orthotropic corrugated cylindrical shells under axial compression. Mech. Comp. Mater.
**2002**, 38, 243–250. [Google Scholar] [CrossRef] - Norman, A.D.; Seffen, K.A.; Guest, S.D. Multistable corrugated shells. Proc. R. Soc. A
**2008**, 464, 1653–1672. [Google Scholar] [CrossRef] - Wang, X.J.; Wang, Z.M.; Wang, N. Finite element analysis for the stiffness and the buckling of corrugated tubes in heat exchanger. Adv. Mater. Res.
**2012**, 468–471, 1675–1680. [Google Scholar] [CrossRef] - Wennberg, D.; Wennhage, P.; Stichel, S. Orthotropic models of corrugated sheets in finite element analysis. Int. Sci. Res. Netw. Mech. Eng.
**2011**, 2011, 979532. [Google Scholar] [CrossRef] [Green Version] - Sowiński, K. Buckling of shells with special shapes with corrugated middle surfaces–FEM study. Eng. Struct.
**2019**, 179, 310–320. [Google Scholar] [CrossRef] - Dolgikh, D.V.; Kiselev, V.V. Corrugation of a flexible ring under external hydrostatic compression. J. Appl. Math. Mech.
**2010**, 74, 204–213. [Google Scholar] [CrossRef] - Andreeva, L.E. Elastic Elements of Instruments; Israel Program for Scientific Translations: Jerusalem, Israel, 1966. [Google Scholar]
- Seydel, E.B. Schubknickversuche Mit Wellblechtafeln; Jahrbuch d. Deutsch; Versuchsanstallt für Luftfahrt. E.V.: Munich/Berlin, Germany, 1931; Volume 4, pp. 233–235. [Google Scholar]
- Briassoulis, D. Equivalent orthotropic properties of corrugated sheets. Comput. Struct.
**1986**, 23, 129–138. [Google Scholar] [CrossRef] - Donnell, L.H. The flexibility of corrugated pipes under longitudinal forces and bending. Trans. ASME
**1932**, 54, 69–75. [Google Scholar] - Ye, Z.; Berdichevsky, V.L.; Yu, W. An equivalent classical plate model of corrugated structures. Int. J. Solids Struct.
**2014**, 51, 2073–2083. [Google Scholar] [CrossRef] [Green Version] - Xia, Y.; Friswell, M.I.; Flores, E.I.S. Equivalent models of corrugated panels. Int. J. Solids Struct.
**2012**, 49, 1453–1462. [Google Scholar] [CrossRef] [Green Version] - Nguyen-Minh, N.; Tran-Van, N.; Bui-Xuan, T.; Nguyen-Thoi, T. Static analysis of corrugated panels using homogenization models and a cell-based smoothed Mindlin plate element. Front. Struct. Civ. Eng.
**2019**, 13, 251–272. [Google Scholar] [CrossRef] - Kolpakov, A.G.; Rakin, S.I. Calculation of the effective stiffness of the corrugated plate by solving the problem on the plate cross-section. J. Appl. Mech. Tech. Phys.
**2016**, 57, 757–767. [Google Scholar] [CrossRef] - Kolpakov, A.A.; Kolpakov, A.G. Discussion of the effective stiffnesses in: Ye, Berdichevsky, and Yu [Int. J. Solids Struct. 51 (2014) 2073–2083]. Int. J. Solids Struct.
**2019**, 174–175, 145–146. [Google Scholar] [CrossRef] - Gimena, L.; Gimena, F.; Gonzaga, P. Structural analysis of a curved beam element defined in global coordinates. Eng. Struct.
**2008**, 30, 3355–3364. [Google Scholar] [CrossRef] - Mercuri, V.; Balduzzi, G.; Asprone, D.; Auricchio, F. Structural analysis of non-prismatic beams: Critical issues, accurate stress recovery, and analytical definition of the Finite Element (FE) stiffness matrix. Eng. Struct.
**2020**, 213, 110252. [Google Scholar] [CrossRef] - Andrianov, I.V.; Diskovsky, A.A.; Kholod, E.G. Homogenization method in the theory of corrugated plates. Tech. Mech.
**1998**, 18, 123–133. [Google Scholar] - Syerko, E.; Diskovsky, A.A.; Andrianov, I.V.; Comas-Cardona, S.; Binetruy, C. Corrugated beams mechanical behavior modeling by the homo–genization method. Int. J. Solids Struct.
**2013**, 50, 928–936. [Google Scholar] [CrossRef] [Green Version] - Andrianov, I.V.; Awrejcewicz, J.; Diskovsky, A.A. Design optimization of FGM beam in stability problem. Eng. Comput.
**2019**, 36, 248–270. [Google Scholar] [CrossRef] - Andrianov, I.I.; Awrejcewicz, J.; Diskovsky, A.A. The optimal design of a functionally graded corrugated cylindrical shell under axisymmetric loading. Int. J. Nonlinear Sci. Numer. Simul.
**2019**, 20, 387–398. [Google Scholar] [CrossRef] - Andrianov, I.V.; Diskovsky, A.A.; Syerko, E. Optimal design of a circular diaphragm using the homogenization approach. Math. Mech. Solids
**2017**, 22, 283–303. [Google Scholar] [CrossRef] - Feodosiev, V.I. Advanced Stress and Stability Analysis, Worked Examples; Springer: Berlin, Germany, 2005. [Google Scholar]
- Bensoussan, A.; Lions, J.-L.; Papanicolaou, G. Asymptotic Analysis for Periodic Structures; North-Holland: Amsterdam, The Netherlands, 1978. [Google Scholar]
- Andrianov, I.V.; Awrejcewicz, J.; Manevitch, L.I. Asymptotical Mechanics of Thin-Walled Structures: A Handbook; Springer: Berlin, Germany, 2004. [Google Scholar]
- Manevich, L.I.; Andrianov, I.V.; Oshmyan, V.G. Mechanics of Periodically Heterogeneous Structures; Springer: Berlin, Germany, 2002. [Google Scholar]
- Ziegler, H. Principles of Structural Stability, 2nd ed.; Springer: Basel, Switzerland, 1977. [Google Scholar]
- Timoshenko, S.P.; Gere, J.M. Theory of Elastic Stability, 2nd ed.; McGraw-Hill: New York, NY, USA, 1961. [Google Scholar]
- Panovko, Y.a.G.; Gubanova, I.I. Stability and Oscillations of Elastic Systems Paradoxes, Fallacies, and New Concepts; Consultants Bureau: New York, NY, USA, 1965. [Google Scholar]
- Grigolyuk, E.I.; Kabanov, V.V. Stability of Shells; Nauka: Moscow, Russia, 1978. (In Russian) [Google Scholar]
- Vol’mir, A.S. Stability of Deformable Systems; Foreign Technology Division, Air Force Systems Command; Wright-Patterson Air Force Base: Dayton, OH, USA, 1967. [Google Scholar]
- Thielemann, W.; Esslinger, M. Beul- und Nachbeulverhalten isotropic Zylinder unter Äussendruck. Stahlbau
**1967**, 36, 161–174. [Google Scholar] - Myachenkov, V.I. The stability of cylindrical shells under axisymmetric transverse pressure. Sov. Appl. Mech.
**1970**, 6, 19–23. [Google Scholar] [CrossRef] - Zevin, A.A. Estimates of eigenvalue of self-adjont boundary-value problems with periodic coefficients. Ukr. Math. J.
**1998**, 50, 719–722. [Google Scholar] [CrossRef] - Dell’Isola, F.; Romano, A. On the derivation of thermomechanical balance equations for continuous systems with a nonmaterial interface. Int. J. Eng. Sci.
**1987**, 25, 1459–1468. [Google Scholar] [CrossRef] [Green Version] - Dell’Isola, F. Linear growth of a liquid droplet divided from its vapour by a “Soap Buble”–like fluid Interface. Int. J. Eng. Sci.
**1989**, 27, 1053–1067. [Google Scholar] [CrossRef] [Green Version]

**Figure 3.**Projection of external pressure to the basic ring: $p,{p}^{0}$ are the intensities of external pressure acting on the corrugated and basic circular ring; $\theta $ is the angle between the tangents to the corrugated and the basic circlular ring; $ds\mathrm{and}dl$ are the arcs of the corrugated and basic circular ring, $dl=dscos\theta ;P=pds=\frac{pdl}{cos\theta};{P}_{n}=Pcos\theta =pdl;{p}^{0}=p.$

**Figure 4.**Ratio of buckling loads of corrugated (4) and basic circular rings ${p}_{b}^{0}$ on parameter $nh\text{}\left(h=0.05;n=0\dots 100\right)$.

**Figure 5.**The part of the ring lying above the horizontal axis of symmetry: numbers 1 and 2 correspond to the middle lines of the basic circular ring before and after buckling; $\tilde{N},\tilde{M},\tilde{w},\tilde{r}$ are the internal force, moment, deflection, and radius at the points A and C.

**Figure 6.**Projections of displacements on the basic circular ring. Here ${u}^{\prime},{w}^{\prime}$ are the tangential and normal displacements of the corrugated ring; $u,w$ are the projections of these displacements on the basic circular ring; ${u}^{\prime}=\frac{ru+{r}_{\phi}w}{A};$ ${w}^{\prime}=\frac{rw-{r}_{\phi}u}{A}$.

**Figure 7.**Decrease of the buckling pressure of a corrugated ring with an increase in the depth of the corrugation.

**Figure 8.**Comparison of buckling pressures ${\overline{p}}_{b}^{2}\mathrm{and}{\overline{p}}_{b}^{2e}$ for corrugated and circular rings having the same volume of material.

**Figure 9.**Comparison of buckling pressure obtained using the asymptotic homogenization method (36) and equivalent bending stiffness approach (7).

**Figure 10.**Diagrams of loading a corrugated ring with a corrugation (39): (

**a**) $n=16,h=0.01$; (

**b**) $n=8,h=0.02;{\overline{p}}_{b}^{0}=3,{\overline{p}}_{b}^{1}=2.98,{\overline{p}}_{b}^{2}=2.94$ are buckling pressures for the smooth ring and for the corrugated ring obtained by the equivalent bending stiffness approach and asymptotic homogenization method.

**Table 1.**Buckling loads (6) for the circular ring with the equivalent bending stiffness for constant value of nh.

nh = 1.6 | n = 160; h = 0.01 | n = 80; h = 0.02 | n = 40; h = 0.04 | n = 20; h = 0.08 | n = 16; h = 0.1 |
---|---|---|---|---|---|

p_{b} | 0.6767 | 0.6767 | 0.6767 | 0.6766 | 0.6764 |

**Table 2.**Comparison of the results of calculation of the deflection at the point $\phi =0$ for $n=16,h=0.01$; here ${\Delta}_{i}=\frac{{w}_{i-1}-{w}_{1}}{{w}_{i-1}}100,i=1-4$.

Number of Iteration/Pressure | p = −1 | p = −1.5 | p = −2 |
---|---|---|---|

3 | Δ_{3} = 0.8% | Δ_{3} = 1.09% | Δ_{3} = 12% |

4 | Δ_{4} = 0.01% | Δ_{4} = 1.06% | Δ_{4} = 3.18% |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Andrianov, I.I.; Andrianov, I.V.; Diskovsky, A.A.; Ryzhkov, E.V.
Buckling of Corrugated Ring under Uniform External Pressure. *Symmetry* **2020**, *12*, 1250.
https://doi.org/10.3390/sym12081250

**AMA Style**

Andrianov II, Andrianov IV, Diskovsky AA, Ryzhkov EV.
Buckling of Corrugated Ring under Uniform External Pressure. *Symmetry*. 2020; 12(8):1250.
https://doi.org/10.3390/sym12081250

**Chicago/Turabian Style**

Andrianov, Igor I., Igor V. Andrianov, Alexander A. Diskovsky, and Eduard V. Ryzhkov.
2020. "Buckling of Corrugated Ring under Uniform External Pressure" *Symmetry* 12, no. 8: 1250.
https://doi.org/10.3390/sym12081250