Many references about buckling, vibration and optimization of panels and shells. Introduction

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Buckling Analysis of Shell Structures

Predicting the buckling response of thin shells in structural simulations is difficult because most models do not include the physical characteristics of the problem that initiate instabilities. For shell structures, the character of the buckling and load levels that lead to instability are governed by the nonuniformities or imperfections in either the structure or loading. A methodology was developed to accurately predict the buckling response of the thin shell structures by incorporating either the measured imperfections in the structure and loading or accurate statistical approximations to the imperfections. This analysis approach has been applied successfully to a variety of buckling problems. One application is the analysis of dynamic pulse buckling of impulsively loaded thin cylindrical aluminum shells. The cosine distributed external impulsive loads drive the shell inward, producing compressive circumferential stresses and pulse buckling on the loaded side of the shell. These buckles produce strain concentrations that govern the eventual fracture of the structure. Thus, the buckling response needs to be correctly modeled to predict failure. A cross section of a dynamically pulse buckled thin cylindrical shell and the calculated response are shown in Figure 1. In the calculation, we used the measured imperfections in the cylinder shape to initiate the buckling. The buckling response can clearly be seen on the loaded (front) side of the cylinder and is accurately reproduced in the calculation. These modeling techniques have also been applied to other structural applications, including analysis of crash energy management structures for vehicles, static axial collapse of cylinders, and dynamic buckling of thick shells. One example application for thick shells is the buckling that occurs in explosively formed penetrators (EFPs), as shown in Figure 2. Understanding and modeling the processes that lead to dynamic plastic buckling in EFP liners allows the designer to control the buckling process to gain enhanced aerostability.


* S.W. Kirkpatrick and B.S. Holmes, "The Effect of Initial Imperfections on Dynamic Buckling of Shells," ASCE Journal of Engineering Mechanics, Vol. 115, No. 5, pp. 1075-1093, May, 1989.

* Florence, A.L., Gefken, P.R., and Kirkpatrick, S.W., "Dynamic Plastic Buckling of Copper Cylindrical Shells," International Journal of Solids and Structures, Vol. 27, No. 1, pp. 89-103, 1991.

D. Larom, C.T. Herakovich and J. Aboudi (Department of Civil Engineering, University of Virginia, Charlottesville, VA 22903, U.S.A.), “Dynamic response of pulse loaded laminated composite cylinders”, International Journal of Impact Engineering, Vol. 11, No. 2, 1991, pp. 233-248,


ABSTRACT: A finite difference solution is presented for elastic wave propagation in laminated composite hollow tubes under axial plane strain, subjected to a radially symmetric pressure pulse at the inner surface. The solution is applied to two-layered cross-ply and angle-ply AS4/3501-6 graphite/epoxy cylinders, and to homogeneous monoclinic cylinders. The effects of pulse time duration, stacking sequence, ply angle and stiffnesses on the resulting displacement, stresses (radial, hoop, axial and shear) and dynamic stress concentration factor are studied.

W. Gu (1), W. Tang (1) and T. Liu (2)

(1) Department of Naval Architecture and Ocean Engineering, Shanghai Jiao Tong University, Shanghai 200030, China

(2) Department of Naval Architecture and Ocean Engineering, Huazhong University of Science and Technology, Wuhan 430074, China

“Dynamic Pulse Buckling of Cylindrical Shells Subjected to External Impulsive Loading”, Journal of Pressure Vessel Technology, Vol. 118, No. 1, pp. 33-37, February 1996, doi:10.1115/1.2842159

ABSTRACT: The dynamic plastic buckling of cylindrical shells (rings) subjected to general initial impulsive velocity and subjected to impulsive loading is studied based on the energy criterion. A simple analysis method is given and the formulas of critical mode numbers, n cr, and critical impulsive velocity, v cr, are obtained.

Olivier Thomas (1), Eric Luminais (1) and Cyril Touzé (2)

(1) Structural Mechanics and Coupled Systems Laboratory, CNAM, 2 rue Conte, 75003 Paris, France

(2) ENSTA-UME, Chemin de la Huniere, 91761 Palaiseau Cedex, France

“Non-linear modal interactions in free-edge thin spherical shells: measurements of a 1:1:2 internal resonance”, in Computational Fluid and Solid Mechanics, K.J. Bathe (Editor), Paper no. 120-221, 2005

ABSTRACT: This study is devoted to the experimental validation of a theoretical model of large amplitude vibrations of thin spherical shells described in Thomas et al.[1]. A specific mode coupling due to a 1:1:2 internal resonance between an axisymmetric mode and two companion asymmetric modes is especially addressed. The structure is forced with a sinusoidal signal of frequency close to the natural frequency of the axisymmetric mode. The experimental setup, which allows precise measurements of the vibration amplitudes of the three involved modes, is presented. Experimental resonance curves showing the amplitude of the modes as functions of the driving frequency are compared to the theoretical ones. A good qualitative agreement is obtained with the predictions given in the model. The quantitative discrepancies are discussed and an improvement of the model is proposed.

References listed at the end of the paper:

[1] Thomas O, Touzé C, Chaigne A. Non-linear vibrations of free-edge thin spherical shells: modal interaction rules and 1:1:2 internal resonance. International Journal of Solids and Structures 2004, accepted for publication.

[2]  Amabili M, Pellegrini M, Tommesani M. Experiments on large-amplitude vibrations of a circular cylindrical panel. J 
Sound Vib 2003;260(3):537–547. 

[3]  Amabili M. Theory and experiments for large-amplitude 
vibrations of empty and fluid-filled circular cylindrical shells with imperfections. J Sound Vib 2003;262(4):921– 975. 

[4]  Evensen HA, Evan-Iwanowsky RM. Dynamic response and stability of shallow spherical shells subject to time-dependant loading. 1967;AIAA Journal 5(5):969–976. 

[5] Yasuda K, Kushida G. Nonlinear forced oscillations of a shallow spherical shell. Bull JSME 1984;27(232):2233– 2240.

[6] Thomas O, Touzé C, Chaigne A. Asymmetric non-linear forced vibrations of free-edge circular plates, Part II: experiments. J Sound Vib 2003;265(5):1075–1101.

[7] Touzé C, Thomas O, Chaigne A. Hardening/softening behaviour in non-linear oscillations of structural systems using non-linear normal modes. J Sound Vib 2004;273(1– 2):77–101.

G.W. Jones, S.J. Chapman, and D.J. Allwright (OCIAM, Mathematical Institute, University of Oxford), “Axisymmetric buckling of a spherical shell embedded in an elastic medium under uniaxial stress at infinity”, Preprint no. 07/2008, OxMOS: New Frontiers in the Mathematics of Solids Mathematical Institute University of Oxford June, 2008.

ABSTRACT: The problem of a thin spherical linearly-elastic shell, perfectly bonded to an infinite linearly-elastic medium is considered. A constant axisymmetric stress field is applied at infinity in the matrix, and the displacement and stress fields in the shell and matrix are evaluated by means of harmonic potential functions. In order to examine the stability of this solution, the buckling problem of a shell which experiences this deformation is considered. Using Koiter's nonlinear shallow shell theory, restricting buckling patterns to those which are axisymmetric, and using the Rayleigh–Ritz method by expanding the buckling patterns in an infinite series of Legendre functions, an eigenvalue problem for the coefficients in the infinite series is determined. This system is truncated and solved numerically in order to analyse the behaviour of the shell as it undergoes buckling, and to identify the critical buckling stress in two cases — namely where the shell is hollow and the stress at infinity is either uniaxial or radial.

References listed at the end of the paper:

[1] G. W. Jones, Static Elastic Properties of Composite Materials Containing Microspheres (D.Phil. thesis, University of Oxford 2007).

[2] P. Seide, The stability under axial compression and lateral pressure of circular-cylindrical shells with a soft elastic core, J. Aerosp. Sci., 29 (1962) 851–862.

[3] M. J. Forrestal and G. Herrmann, Buckling of a long cylindrical shell surrounded by an elastic medium, Int. J. Solids Struct., 1 (1965) 297–309.

[4] V. V. Vlasov, Stability of composite shells with an elastic core, Mech. Compos. Mater., 12 (1976) 499–502.

[5] S.-L. Fok, Analysis of the buckling of long cylindrical shells embedded in an elastic medium using the energy method, J. Strain Anal., 37 (2002) 375–383.

[6] O. Lourie, D. M. Cox, and H. D. Wagner, Buckling and collapse of embedded carbon nanotubes, Phys. Rev. Lett., 81 (1998) 1638–1641.

[7] C. Q. Ru, Axially compressed buckling of a doublewalled carbon nanotube embedded in an elastic medium, J. Mech. Phys. Solids, 49 (2001) 1265–1279.

[8] Y. F. Luo and J. G. Teng, Stability analysis of shells of revolution on nonlinear elastic foundations, Comput. Struct., 69 (1998) 499–511.

[9] S.-L. Fok and D. J. Allwright, Buckling of a spherical shell embedded in an elastic medium loaded by a far-field hydrostatic pressure, J. Strain Anal., 36 (2001) 535–544.

[10] W. T. Koiter, The nonlinear buckling problem of a complete spherical shell under uniform external pressure, Proc. Kon. Ned. Akad. B. Phys., 72 (1969) 40–123.

[11] A. E. H. Love, A Treatise on the Mathematical Theory of Elasticity (Dover, New York 1944).

[12] F. Zhaohua and R. D. Cook, Beam elements on two-parameter elastic foundations, J. Eng. Mech. ASCE, 109 (1983) 1390–1402.

[13] F. I. Niordson, Shell Theory (North-Holland, Amsterdam 1985).

[14] J. N. Goodier, Concentration of stress around spherical and cylindrical inclusions and flaws, Trans. ASME, 55 (1933) 39–44.

[15] S. H. Liu and E. B. Nauman, On the micromechanics of composites containing spherical inclusions, J. Mater. Sci., 25 (1990) 2071–2076.

[16] M. Bilgen and M. F. Insana, Elastostatics of a spherical inclusion in homogeneous biological media, Phys. Med. Biol., 43 (1998) 1–20.

[17] S. Mazzullo, Stress field around an N-layered spherical inclusion, In Proceedings of the European Mechanics Colloquium 214, ‘Mechanical Behaviour of Composites and Laminates’, Kupari, Yugoslavia, 1986, ed. W. A. Green and M. Mi ́cunovi ́c (1987), pages 245–253.

[18] M. Rahman and T. Michelitsch, A general procedure for solving boundary-value problems of elastostatics for a spherical geometry based on Love’s approach, Q. Jl Mech. Appl. Math., 60 (2007) 139–160.

[19] H. L. Langhaar, Energy Methods in Applied Mechanics, (Wiley, New York 1962).

[20] N. N. Lebedev, Special Functions and their Applications, (Dover, New York 1972).

[21] A. I. Lur’e, Three-dimensional Problems of the Theory of Elasticity, (Interscience, New York 1964).

[22] M. K. Wadee, G. W. Hunt, and A. I. M. Whiting, Asymptotic and Rayleigh–Ritz routes to localized buckling solutions in an elastic instability problem, Proc. R. Soc. Lon. Ser. A, 453 (1997) 2085–2107.

[23] C. D. Coman, Inhomogeneities and localised buckling patterns, IMA J. Appl. Math., 71 (2006) 133–152.

G. W. Jones, S. J. Chapman and D. J. Allwright, “Asymptotic Analysis of a Buckling Problem for an Embedded Spherical Shell”, SIAM J. Appl. Math. Vol. 70, 2009, pp. 901-922, DOI:10.1137/080735114

ABSTRACT: The axisymmetric buckling of a spherical shell embedded in an elastic medium with uniaxial compression at infinity is examined in the limit of small shell thickness ratio. An asymptotic method is developed by considering the paradigm problem of a beam attached to a Winkler substrate of variable stiffness, which in the small aspect ratio limit displays the same behavior as the shell. The asymptotic method is then applied to the Euler–Lagrange equations corresponding to shell buckling. The system is analyzed in two distinguished limits, displaying good agreement with the full numerical results.

Qasim Hussain Shah, Hasan M. Abid and Adib B. Rosli, “Cylindrical shell buckling under axial compression load”, In: Advanced Topics in Mechanical Behavior of Materials. IIUM Press, Kuala Lumpur, pp. 3-7. ISBN 9789674181741, 2011

(no abstract given)

M. Hilburger (NASA Langley Research Center, Hampton, Virginia, USA, email: ), “Developments in Shell Buckling Analysis, Design and Testing”

ABSTRACT: High-performance aerospace shell structures are inherently thin-walled because of weight and performance considerations and are often subjected to destabilizing loads. Thus, buckling is an important and often critical consideration in the design of these structures and reliable, validated design criteria for thin-walled shells are needed, especially for shells made of advanced composite materials. Shell-buckling design criteria have a history steeped in empiricism. From approximately 1930 to 1967, many shell-buckling experiments were conducted on metallic shells. Typically, the experiments yielded buckling loads that were substantially lower than the corresponding analytical predictions, which were based on simplified linear bifurcation analyses of geometrically perfect shells with nominal dimensions and idealized support conditions. The primary source of discrepancy between corresponding analytical predictions and experimental results is attributed to small deviations from the idealized geometry of a shell, known as initial geometric imperfections. Empirical design factors, known as "knockdown" factors, were determined from these test data and were to be used in conjunction with linear bifurcation analyses for simply supported shells to adjust or "knockdown" the unconservative analytical prediction. This approach to shell design remains prominent in industry practice, as evidenced by the extensive use of the NASA space vehicle design recommendations. Recent advancements in digital computers, high-fidelity structural analysis tools and testing technologies are enabling the development of a new shell buckling design philosophy, namely, analysis-based knockdown factors. Key enabling technology developments and their implementation in ongoing NASA Shell Buckling Knockdown Factor development activities are presented in this lecture. In addition, the development of a refined shell-buckling preliminary-design criteria that is based on high-fidelity nonlinear finite-element analyses that include the effects of a manufacturing-process-specific geometric imperfection signature is presented.

(No references given. This is not a paper, just an abstract.)

Harik, V. M. ; Gates, T. S. ; Nemeth, M. P., “Applicability of the Continuum-shell Theories to the Mechanics of Carbon Nanotubes”, Institute For Computer Applications In Science And Engineering Hampton VA, Accession Number : ADA401873, Handle / proxy Url , April 2002

ABSTRACT: Validity of the assumptions relating the applicability of continuum shell theories to the global mechanical behavior of carbon nanotubes is examined. The present study focuses on providing a basis that can be used to qualitatively assess the appropriateness of continuum-shell models for nanotubes. To address the effect of nanotube structure on their deformation, all nanotube geometries are divided into four major classes that require distinct models. Criteria for the applicability of continuum models are presented. The key parameters that control the buckling strains and deformation modes of these classes of nanotubes are determined. In an analogy with continuum mechanics, mechanical laws of geometric similitude are presented. A parametric map is constructed for a variety of nanotube geometries as a guide for the applicability of different models. The continuum assumptions made in representing a nanotube as a homogeneous thin shell are analyzed to identify possible limitations of applying shell theories and using their bifurcation-buckling equations at the nano-scale.

Vasyl Harik (Nanodesigns Consulting, P.O. Box 5303, Wilmington, DE, 19808-5303, USA), “Trends in recent publicaiotns on nanoscale mechanics”, Chapter in Trends in Nanoscale Mechanics, Vasyl Harik, editor, Springer, 2014, pp 213-222, DOI: 10.1007/978-94-017-9263-9_9

ABSTRACT: This part of the edited volume highlights trends in recent publications by providing examples of important research papers in different areas of nanoscale mechanics. Research papers on novel applications of carbon nanotubes, nanocomposites, nanodevices, quantum anti-dots, and other nanostructures are noted.

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2. V. Holovatsky, O. Voitsekhivska, I. Bernik, Effect of magnetic field on electron spectrum in spherical nano-structures. Condens. Matter Phys. 17(1), 13702:1–8 (2014)

3. C.M. Wang, A.N. Roy Chowdhury, S.J.A. Koh, Y.Y. Zhang, Molecular dynamics simulation and continuum shell model for buckling analysis of carbon nanotubes. in Modeling of Carbon Nanotubes, Graphene and their Composites, ed. by K.I. Tserpes, N. Silvestre. Springer Ser. Mater. Sci. 188, 239 (2014)

4. K. Moth-Poulsen, T. Bjornholm, Molecular electronics with single molecules in solid-state devices. Nat. Nanotechnol. 4, 551–556 (2009)

5. H.-E. Schaefer, Carbon nanostructures—Tubes, graphene, fullerenes, wave-particle duality, nanoscience (Springer, Berlin, 2010)

6. X. Xiao, T. Li, Z. Peng, H. Jin, Q. Zhong, Q. Hu, B. Yao, Q. Zhang, Q. Luo, C. Zhang, L. Gong, J. Chen, Y. Gogotsi, J. Zhou, Freestanding functionalized carbon nanotube-based electrode for solid-state asymmetric supercapacitors. Nano Energy 6, 1–9 (2014)

7. P. Egberts, Z. Ye, X.-Z. Liu, Y. Dong, A. Martini, R.W. Carpick, Environmental dependence of atomic-scale friction at graphite surface steps. Phys. Rev. B 88, 035409/1-0 (2013)

8. X. Li, W. Qi, D. Mei, M.L. Sushko, I. Aksay, J. Liu, Functionalized graphene sheets as molecular templates for controlled nucleation and self-assembly of metal oxide-graphene nanocomposites. Adv. Mater. 24, 5136–5141 (2012)

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11. V.M. Harik, Mechanics of Carbon Nanotubes (Nanodesigns Press, Newark, Delaware, 2011)

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13. A.A. Pelegri (Mina), S.D. Tse, B.H. Kear, in Multifunctional Graphene Composites for Lightning Strike Protection: Structural Mechanics and System Integration (Rutgers University, Rutgers, 2012). A.A. Pelegri, X. Huang, Nanoindentation on soft film/hard substrate and hard film/soft substrate material systems with finite element analysis. Composites Sci. Techn. 68(1), 147–155 (2008)

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