The Buckling of Thin Cylindrical Shells under Axial Compression

Kármán, Th. von; Tsien, Hsue-shen

doi:10.2514/2.7056

Cited by 80 publications

(44 citation statements)

References 7 publications

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“…From the first nonlinear studies of the pressurized spherical shell [Karman & Tsien, 1939] and the axially compressed circular cylindrical shell [Karman & Tsien, 1941], the complex post-buckling phenomena of these two problematic archetypal buckling problems have been explored very much in tandem. Ad hoc analyses of increasing complexity were made (see for example the review by Thompson [1960a] for papers on the sphere), and it was shown for both problems that geometrical imperfections in the shape of the shell's middle surface gave rise to a severe imperfection-sensitivity in which the bifurcation of the perfect shell was rounded off, and the failure load of an imperfect shell fell off dramatically with the magnitude of the imperfection.…”

Section: Historical Surveymentioning

confidence: 99%

Shock-Sensitivity in Shell-Like Structures: With Simulations of Spherical Shell Buckling

Thompson

Sieber

2016

Int. J. Bifurcation Chaos

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Under increasing compression, an unbuckled shell is in a metastable state which becomes increasingly precarious as the buckling load is approached. So to induce premature buckling, a lateral disturbance will have to overcome a decreasing energy barrier which reaches zero at buckling. Two archetypal problems that exhibit a severe form of this behavior are the axially-compressed cylindrical shell and the externally pressurized spherical shell. Focusing on the cylinder, a nondestructive technique was recently proposed to estimate the “shock-sensitivity” of a laboratory specimen using a lateral probe to measure the nonlinear load-deflection characteristic. If a symmetry-breaking bifurcation is encountered on the path, computer simulations showed how this can be suppressed by a controlled secondary probe. Here, we extend our understanding by assessing in general terms how a single control can capture remote saddle solutions: in particular, how a symmetric probe could locate an asymmetric solution. Then, more specifically, we analyze the spherical shell with point and ring probes, to test the procedure under challenging conditions to assess its range of applicability. Rather than a bifurcation, the spherical shell offers the challenge of a destabilizing fold (limit point) under the rigid control of the probe.

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Section: Historical Surveymentioning

confidence: 99%

Shock-Sensitivity in Shell-Like Structures: With Simulations of Spherical Shell Buckling

Thompson

Sieber

2016

Int. J. Bifurcation Chaos

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“…Let us note that the original von Kármán strains [15] do not have the quadratic in-plane displacement gradients in the membrane tensor ϕ. The quadratic in-plane displacement gradients are considered in the model for the sake of completness and they are not expected to enhance the accuracy of the original model significantly.…”

Section: Kinematic Relations For a Laminatementioning

confidence: 99%

A model for fast delamination analysis of laminated composite structures

Katajisto

Kere

Lyly

2020

RakMek

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Delamination is one of the major failure mechanisms for composites and traditionally the simulation requires high expertise in fracture mechanics and dedicated knowledge of the Finite Element Analysis (FEA) tool. Yet, the simulation cycle times are high. Geometrically nonlinear analysis approach, which is based on the Reissner-Mindlin-Von K´arm´an type shell facet model, has been implemented into the Elmer FE solver. Altair ESAComp software runs the Elmer Solver in the background. A post-processing capability, which enables the prediction of the delamination onset from the FEA output, has been implemented into the AltairESAComp software. A Virtual Crack Closure Technique (VCCT) speciﬁcally developed for shell elements deﬁning the Strain Energy Release Rate (SERR) related to the diﬀerent delamination modes at the crack front is used. The onset of delamination is predicted using the relevant delamination criteria that utilize the SERR data and material allowables in the form of fracture toughness. The modeling methodology is presented for laminates including initial through-the-width delamination. Examples include delamination in the solid laminate and debonding of the skin laminate in the sandwich structure. Rather coarse FE mesh has proved to yield good results when compared to typical approaches that utilize the standard VCCT or Cohesive Zone Elements.

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“…The limit capacity in postcritical de¯ections is reached by plasticization of the plate, which develops ridge-shaped buckles causing the plate to act approximately as a truss. Using such a truss analogy, simple formulas have been developed for the maximum loads of rectangular plates (von KaÂ rmaÂ n et al, 1932), with the remarkable property that the maximum distributed load is independent of the plate dimensions.…”

Section: Thin-wall Beams Plates and Shellsmentioning

confidence: 99%

Structural Stability

2011

Robust Chaos and Its Applications

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In this chapter, we discuss the concept of structural stability as a criterion for robustness of invariant sets of dynamical systems. In Sec. 4.1, we define and state important properties of this notion, along with its conditions as given in Sec. 4.1.1. An example of a proof of Anosov's theorem on structural stability of diffeomorphisms of a compact C ∞ manifold M without boundary due to Robinson and Verjovsky [Robinson & Verjovsky (1971)] is given and discussed in Sec. 4.1.2. At the end of this chapter, we present a weaker version called Ω-stability and then give a set of exercises and open problems concerning structural stability along with some suggested references. The concept of structural stabilityThe concept of structural stability was introduced by Andronov and Pontryagin in 1937, and it plays an important role in the development of the theory of dynamical systems. Conditions for structural stability of highdimensional systems were formulated by Smale in [Smale (1967)]. These conditions are the following: A system must satisfy both axiom A (recall Definition 6.3) and the strong transversality condition. Mathematically, let C r (R n , R n ) denote the space of C r vector fields of R n into R n . Let Dif f r (R n , R n ) be the subset of C r (R n , R n ) consisting of the C r diffeomorphisms.Definition 4.1. (a) Two elements of C r (R n , R n ) are C r ε-close (k ≤ r), or just C k close, if they, along with their first k derivatives, are within ε as measured in some norm. (b) A dynamical system (vector field or map) is structurally stable if nearby systems have the same qualitative dynamics. 61Robust Chaos and Its Applications Downloaded from www.worldscientific.com by DEAKIN UNIVERSITY on 03/15/15. For personal use only.

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The Buckling of Thin Cylindrical Shells under Axial Compression

Cited by 80 publications

References 7 publications

Shock-Sensitivity in Shell-Like Structures: With Simulations of Spherical Shell Buckling

Shock-Sensitivity in Shell-Like Structures: With Simulations of Spherical Shell Buckling

A model for fast delamination analysis of laminated composite structures

Structural Stability

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