PAMM · Proc. Appl. Math. Mech. 9, 255 – 256 (2009) / DOI 10.1002/pamm.200910101

Modes of transition from imperfection sensitivity to imperfectioninsensitivity

Herbert A. Mang∗1, Gerhard Hoefinger1, and Xin Jia1

1 Vienna University of Technology, Karlsplatz 13/202, 1040 Vienna, Austria

In order to a priori find the mode of conversion of a structure from imperfection sensitivity to insensitivity in the course ofsensitivity analysis of the initial postbuckling path (cf. [2]), terms appearing in the consistently linearized eigenvalue problem(cf. [1]) are studied. Two main classes can be identified (cf. [4]). The first one is characterized by the restriction of theprebuckling deformations to axial deformations. For this class, the quality of the aforementioned conversion is better than theone for the other class, for which the above restriction does not hold.

c© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim

1 Introduction

Analysis of the initial postbuckling behavior of elastic structures is used to assess the imperfection sensitivity of a structure. Incertain cases imperfection sensitivity can be reduced or even converted into imperfection insensitivity by variation of a scalardesign paramter κ. Therefore, a comprehensive analysis of the structure should include an investigation of the possibility ofsuch a conversion. The following considerations show how to a priori determine the quality of this conversion.

A quantitative expression to determine whether a structure is imperfection sensitive or insensitive is deduced from the se-ries expansion for the load level along the secondary (equilibrium) path λ(η) =

∑λiη

i. If the lowest non-vanishing term inthe series has an even exponent and its coefficient λi is positive, imperfection insensitivity is on hand. In general, this term isλ2.

The basis for the classification mentioned above is the so-called consistently linearized eigenvalue problem, introduced in [1].Let KT be the tangent stiffness matrix of the system, (u(λ), λ) be a parametrization of the primary equilibrium path, andsolve [

KT (λ) +(λ∗

j − λ)KT,λ(λ)

]· v∗

j = 0 with KT (λ) := KT (u(λ)) (1)

for pairs (v∗j (λ), λ∗

j (λ)). At the stability limit λ = λC , KT is singular (∃v1 with KT · v1 = 0). Thus, one of the solutions isλ∗

1 = λ and v∗1 = v1. Furthermore,

v∗1,λ = a1v1, v∗

1,λλ = 3(a21 + a∗

1

)v1 +

N∑j=2

v∗Tj · KT,λλ · v1

(λ∗1 − λ∗

j )(v∗Tj · KT,λ · v∗

j )v∗

j (2)

where a1 and a∗1 are quantites which also appear in the series expansion of the secondary path. For details see [2].

2 Classification

We discern two classes of structures. Their distinguishing feature is the presence or absence rsp. of the remarkable orthogo-nality

v∗Tj · KT,λλ · v1 = 0, j = 2 . . .N (3)

which leads to

v∗1,λλ = 3(a2

1 + a∗1)v1 (4)

Because of (3), v∗1(λ) does not change its direction which is the consequence of the restriction of the prebuckling deformations

to axial deformations.

As a consequence, for Class I a1 = 0 implies λ2 > 0, i.e. imperfection insensitivity, whereas for Class II, a1 = 0 ⇒ λ2 = 0,

∗ Corresponding author: e-mail: herbert.mang@tuwien.ac.at, Phone: +43 1 58801 20210, Fax: +43 1 58801 20299

c© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim

256 Short Communications 4: Structural Mechanics

12 4( () ), ( ), a

4

2

2 0

2 0

imperfectionsensitive

imperfectioninsensitive

02

1ahilltopbuckling

Fig. 1 Class I: By variation of κ, λ2 canbe increased such that it takes on a positivevalue. The sign of λ4(λ2 = 0) is not re-stricted.

12 4( () ), ( ), a

4

22 0

2 0

imperfectionsensitive

imperfectioninsensitive

021a

hilltopbuckling

Fig. 2 Class IIa: By variation of κ, λ2 canbe increased such that it takes on a positivevalue; λ4(λ2 = 0) < 0, a1(λ2 = 0) = 0.

12 4( () ), ( ), a

02

imperfectionsensitive

0

2

4

4

2

1a

hilltopbuckling

Fig. 3 Class IIb: By variation of κ, λ2 canbe increased up to the maximum value of 0,for which λ4 = λ4,κ = λ4,κκ = 0. Hence,there is no conversion to imperfection in-sensitivity.

which is the point where the conversion to imperfection insensitivity can occur [4]. The relation between λ2 and a1 can bestudied using the definition KT (η) := KT (u(λ(η))) and the equation

λ2 = − a1

vT1 · KT,λλ · v1

vT1 · KT,ηη · v1 (5)

If a1 = 0, also vT1 · KT,λλ · v1 = 0, and l’Hospital’s rule can be used to compute λ2.

2.1 Class I

Fig. 1 refers to sensitivity analysis starting at hilltop buckling which is characterized by the coincidence of the bifurcationpoint representing the stability limit with the snap-through point. For hilltop buckling, λ2 < 0, λ4 < 0, a1 = −∞ [4]. As aspecial case of conversion from imperfection sensitivity to insensitivity, zero-stiffness postbuckling may be mentioned. It ischaracterized by λi = 0 ∀i [5].

2.2 Class II

In contrast to Fig. 3, Fig. 2 refers to a situation with a conversion from imperfection sensitivity into imperfection insensitivity(Class IIa). λ4(κ) shows that this conversion is of worse quality than the one in Fig. 1. The parameter κ in Fig. 3 is thethickness of a cylindrical panel [3]. Expectedly, an increase of the thickness of the panel will not render an imperfectionsensitive into an imperfection insensitive structure (Class IIb).

Acknowledgements Gerhard Hoefinger and Xin Jia acknowledge financial support by the Austrian Academy of Sciences.

References

[1] P. Helnwein, Zur initialen Abschätzbarkeit von Stabilitätsgrenzen auf nichtlinearen Last-Verschiebungspfaden elastischer Strukturenmittels der Methode der Finiten Elemente [in German; On ab initio assessability of stability limits on nonlinear load-displacement pathsof elastic structures by means of the finite element method]. (Ph.D. Thesis, Vienna University of Technology, Österreichischer Kunst-und Kulturverlag: Vienna, 1997).

[2] H.A. Mang, C. Schranz and P. Mackenzie-Helnwein, Conversion from imperfection-sensitive into imperfection-insensitive elastic struc-tures I: Theory, Comput. Methods Appl. Mech. Engrg. 195, 1422-1457 (2006).

[3] C. Schranz, B. Krenn, H.A. Mang, Conversion from imperfection-sensitive into imperfection-insensitive elastic structures II: Numericalinvestigation, Comput. Methods Appl. Mech. Engrg. 195, 1458-1479 (2006).

[4] H.A. Mang, X. Jia, G. Hoefinger, Hilltop buckling as the Alpha and Omega in sensitivity analysis of the inital postbuckling behavior ofelastic structures, J. Civ. Eng. Manag., 15, 35-46 (2009).

[5] A. Steinboeck, X. Jia, G. Hoefinger, H.A. Mang, Conditions for symmetric, antisymmetric, and zero-stiffness bifurcation in view ofimperfection sensitivity and insensitivity, Comput. Methods Appl. Mech. Engrg. 197, 3623-3636 (2008).

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