Research Papers

On Establishing Buckling Knockdowns for Imperfection-Sensitive Shell Structures

[+] Author and Article Information
S. Gerasimidis

Civil and Environmental Engineering Department,
University of Massachusetts,
Amherst, MA 01003

E. Virot

Emergent Complexity in Physical Systems
Ecole Polytechnique Federale de Lausanne,
Lausanne CH 1015, Switzerland;
School of Engineering and Applied Sciences,
Harvard University,
Cambridge, MA 02138

J. W. Hutchinson, S. M. Rubinstein

School of Engineering and Applied Sciences,
Harvard University,
Cambridge, MA 02138

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received May 22, 2018; final manuscript received May 26, 2018; published online June 18, 2018. Editor: Yonggang Huang.

J. Appl. Mech 85(9), 091010 (Jun 18, 2018) (14 pages) Paper No: JAM-18-1303; doi: 10.1115/1.4040455 History: Received May 22, 2018; Revised May 26, 2018

This paper investigates issues that have arisen in recent efforts to revise long-standing knockdown factors for elastic shell buckling, which are widely regarded as being overly conservative for well-constructed shells. In particular, this paper focuses on cylindrical shells under axial compression with emphasis on the role of local geometric dimple imperfections and the use of lateral force probes as surrogate imperfections. Local and global buckling loads are identified and related for the two kinds of imperfections. Buckling loads are computed for four sets of relevant boundary conditions revealing a strong dependence of the global buckling load on overall end-rotation constraint when local buckling precedes global buckling. A reasonably complete picture emerges, which should be useful for informing decisions on establishing knockdown factors. Experiments are performed using a lateral probe to study the stability landscape for a cylindrical shell with overall end rotation constrained in the first set of tests and then unconstrained in the second set of tests. The nonlinear buckling behavior of spherical shells under external pressure is also examined for both types of imperfections. The buckling behavior of spherical shells is different in a number of important respects from that of the cylindrical shells, particularly regarding the interplay between local and global buckling and the post-buckling load-carrying capacity. These behavioral differences have bearing on efforts to revise buckling design rules. The present study raises questions about the perspicacity of using probe force imperfections as surrogates for geometric dimple imperfections.

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Fig. 3

Further details of the behavior of the reference shell subject to case A conditions. Representative curves of axial load versus end shortening are plotted for several levels of probing force imperfection in (a). The associated curves of radial displacement at the probe location versus end shortening are shown in (b). The behavior in the three zones is discussed in the text.

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Fig. 2

Local and global buckling loads as a function of the amplitude of the probing force imperfection for the reference shell subject to loading case A. The lower branch of the curves is the local buckling load and the upper branch is the global buckling load. The plots include the results of Haynie et al. [10]. (a) The results as F/FC versus PR/2πD. (b) The same results as F/FC versus δ/t. The local and global buckling behaviors defining the three zones are discussed in the text.

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Fig. 1

(a) Geometry and conventions for a lateral probe force imperfection. (b) Dimensionless relation between the probing force, P, and the inward radial deflection at the probe, δ, with no end shortening for case A. The values of P chosen for the simulations of Haynie et al. [10] are denoted by solid dots on the curve.

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Fig. 4

The roles of the four sets of boundary conditions on local (lower curves) and global (upper curves) buckling of the reference shell. (a) Local and global buckling loads as a function of the probing force imperfection amplitude PR/2πD. (b) Local and global buckling loads as a function of the probing force imperfection amplitude δ/t. Cases A and C for which overall upper end rotation is suppressed are essentially identical. Cases B and D for which overall upper end rotation is unconstrained are also essentially identical. Overall end-rotation constraint has relatively little effect on local buckling but significantly effects global buckling in zones 2 and 3.

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Fig. 5

The influence of the shell length for the case of probing force imperfections for case A in (a) and case D in (b). The parameters characterizing the three shells are the same as for the reference shell except for their length L. The reference shell has L/R=3.44.

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Fig. 6

Full equilibrium solution generated using the Riks method compared to the results using automatic stabilization. (a) Load versus end shortening and (b) normalized inward displacement to the shell, wP/t, at the probe force location versus end shortening. Reference shell with case A boundary conditions and PR/2πD=0.570.

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Fig. 7

(a) Schematics of the experimental setup. (b) Prescribed end shortening with no end rotation (case A). (c) Prescribed end shortening with end rotation allowed (case B). The rotation is enabled by two steel plates with a steel marble located at the center of the top end of the shell, as shown in the schematic on the right.

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Fig. 8

(a) Landscape of stability of the shell without end rotation (case A). (b) Landscape of the same shell with overall end rotation allowed (case B). The squares, circles and triangles are, respectively, the ridge, valley, and projection of the ridge on the opposite face of the valley. The gray area indicates the range of axial loads where a valley is found in the landscape (values estimated by linear extrapolation).

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Fig. 10

Comparison between buckling theory and experiments for cylindrical shells under axial compression with an axisymmetric dimple imperfection (5) located at the shell midsection and subject to case A end conditions (adapted from Ref. [23]). (a) Global buckling load as dependent on imperfection amplitude for ℓy=1.05ℓC. (b) Global buckling load as dependent on the dimple wavelength for a fixed imperfection amplitude, δ/t=0.363. The calculations were performed for a shell with R/t=200, L/R=2.8 and ν=0.4, values very close to those measured for the test specimens as discussed in the original paper.

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Fig. 9

(a) Projection of the landscape of stability in the plane containing the probing displacement with the axial force as the horizontal axis, case A. (b) The same for case B. (c) Projection in the plane containing the probing force with the axial force as the vertical axis, case A. (d) The same for case B.

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Fig. 16

Nonlinear response of the spherical shell with P held fixed with subsequent application of external pressure p for six levels of surrogate probing force imperfections characterized by (δ/t,PR/2πD). (a) p/pC versus pole deflection Δwpole0/t. (b) normalized volume change ΔV/ΔVC versus Δwpole0/t. The results can be used to generate behavior for both prescribed pressure and prescribed volume change, as will be discussed in the text. These have been computed with R/t=200 and ν=0.3, but the results in (a) are independent of R/t. The results in (b) have some dependence on R/t.

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Fig. 11

Shapes of the geometric dimple imperfection (ℓ̂y=0.55ℓC, ℓ̂x/ℓ̂y=2) and the probing force imperfection at the same imperfection amplitude δ

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Fig. 12

Local and global buckling for the reference cylindrical shell with the local dimple imperfection (4.4) (ℓ̂y=0.55ℓC and ℓ̂x=2ℓ̂y) compared with corresponding results from Sec. 2 for the probing force imperfections. Results for the four sets of end conditions (3) are presented: (a) Case A and C, which are nearly identical. (b) Case B and D, which are nearly identical.

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Fig. 13

Nonlinear buckling behavior for the reference cylindrical shell with the local dimple imperfection (7) with ℓ̂y=0.55ℓC and ℓ̂x=2ℓ̂y subject to case A end conditions

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Fig. 14

Buckling imperfection-sensitivity curves for geometric dimple imperfections compared to Koiter's [16] worst case imperfection, the axisymmetric sinusoidal imperfection with ν=0.3. The lowest buckling loads are plotted for localized dimple imperfections (7) having aspect ratios ranging from ℓ̂x/ℓ̂y=2 to ∞ (the axisymmetric limit). For all the dimple imperfections, ℓ̂y=0.55ℓC. The curves for the dimple imperfections have been computed for the reference shell but they are essentially independent of R/t. Koiter's curve is independent of R/t.

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Fig. 17

(a) Buckling imperfection-sensitivity for special shells under prescribed external pressure for both probing force imperfections and geometric dimple imperfections with B=1.5. These theoretical curves have been computed with R/t=200 and ν=0.3 but they are essentially independent of R/t. (b) Imperfection-sensitivity prediction for geometric dimple imperfections minimized over B and comparison with the experiments on clamped hemispherical shells with precisely manufactured dimple imperfections of Lee et al. [27].

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Fig. 15

Dimensionless relation between the pole probing force, P, and the inward deflection at the pole, δ=wpole0, computed with p=0 for R/t=200 and ν=0.3. Over the range plotted the curve would be essentially identical had it been computed under a constraint of no volume change. These results are independent of R/t but they do depend on ν.

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Fig. 18

The maximum pressure prior to buckling, pmax, and the pressure in the stable dimple buckled state, pbuckled, as a function of the probing force imperfection amplitude, δ/t(=wpole0/t) for prescribed volume change. These results have been computed with ν=0.3. The curve for pmax is essentially independent of R/t. Curves for three values of R/t are shown for the pressure in the buckled state. The solid dot indicates the largest value of the imperfection for which snap buckling occurs under prescribed volume change for the specific value of R/t.



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