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Research Papers

Nudging Axially Compressed Cylindrical Panels Toward Imperfection Insensitivity

[+] Author and Article Information
B. S. Cox

Department of Mechanical Engineering,
Materials and Structures Research Centre,
University of Bath,
Bath, BA2 7AY, UK
e-mail: B.S.Cox@bath.ac.uk

R. M. J. Groh

Bristol Composites Institute (ACCIS),
University of Bristol,
Bristol, BS8 1TR, UK
e-mail: Rainer.Groh@bristol.ac.uk

A. Pirrera

Bristol Composites Institute (ACCIS),
University of Bristol,
Bristol, BS8 1TR, UK
e-mail: Alberto.Pirrera@bristol.ac.uk

1Corresponding author.

Contributed by the Applied Mechanics Division of ASME for publication in the Journal of Applied Mechanics. Manuscript received January 15, 2019; final manuscript received March 21, 2019; published online April 19, 2019. Assoc. Editor: Pedro Reis.

J. Appl. Mech 86(7), 071010 (Apr 19, 2019) (13 pages) Paper No: JAM-19-1025; doi: 10.1115/1.4043284 History: Received January 15, 2019; Accepted March 21, 2019

Curved shell structures are known for their excellent load-carrying capability and are commonly used in thin-walled constructions. Although theoretically able to withstand greater buckling loads than flat structures, shell structures are notoriously sensitive to imperfections owing to their postbuckling behavior often being governed by subcritical bifurcations. Thus, shell structures often buckle at significantly lower loads than those predicted numerically and the ensuing dynamic snap to another equilibrium can lead to permanent damage. Furthermore, the strong sensitivity to initial imperfections, as well as their stochastic nature, limits the predictive capability of current stability analyses. Our objective here is to convert the subcritical nature of the buckling event to a supercritical one, thereby improving the reliability of numerical predictions and mitigating the possibility of catastrophic failure. We explore the elastically nonlinear postbuckling response of axially compressed cylindrical panels using numerical continuation techniques. These analyses show that axially compressed panels exhibit a highly nonlinear and complex postbuckling behavior with many entangled postbuckled equilibrium curves. We unveil isolated regions of stable equilibria in otherwise unstable postbuckled regimes, which often possess greater load-carrying capacity. By modifying the initial geometry of the panel in a targeted—rather than stochastic—and imperceptible manner, the postbuckling behavior of these shells can be tailored without a significant increase in mass. These findings provide new insight into the buckling and postbuckling behavior of shell structures and opportunities for modifying and controlling their postbuckling response for enhanced efficiency and functionality.

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References

Reis, P. M., 2015, “A Perspective on the Revival of Structural (In)stability With Novel Opportunities for Function: From Buckliphobia to Buckliphilia,” ASME J. Appl. Mech., 82(11), p. 111001. [CrossRef]
Hu, N., and Burgueño, R., 2015, “Buckling-Induced Smart Applications: Recent Advances and Trends,” Smart Mater. Struct. 24(6), pp. 1. [CrossRef]
Groh, R. M. J., Avitabile, D., and Pirrera, A., 2018, “Generalised Path-Following for Well-Behaved Nonlinear Structures,” Comput. Methods Appl. Mech. Eng., 331, pp. 394–426. [CrossRef]
Pirrera, A., Avitabile, D., and Weaver, P. M., 2010, “Bistable Plates for Morphing Structures: A Refined Analytical Approach With High-Order Polynomials,” Int. J. Solids Struct., 47(25–26), pp. 3412–3425. [CrossRef]
Arena, G., Groh, R. M. J., Brinkmeyer, A., Theunissen, R., Weaver, P. M., and Pirrera, A., 2017, “Adaptive Compliant Structures for Flow Regulation,” Proc. R. Soc. Lond. A: Math. Phys. Eng. Sci., 473(2204), p. 20170334. [CrossRef]
Andò, B., Baglio, S., Trigona, C., Dumas, N., Latorre, L., and Nouet, P., 2010, “Nonlinear Mechanism in Mems Devices for Energy Harvesting Applications,” J. Micromech. Microeng., 20(12), p. 125020. [CrossRef]
Pellegrini, S. P., Tolou, N., Schenk, M., and Herder, J. L., 2012, “Bistable Vibration Energy Harvesters: A Review,” J. Intel. Material Syst. Struct., 24(11), pp. 1303–1312. [CrossRef]
Alkharabsheh, S. A., and Younis, M. I., 2013, “Statics and Dynamics of Mems Arches Under Axial Forces,” J. Vib. Acoust., 135(021007), pp. 1–7.
Arbocz, J., and Starnes, J. H., Jr., 2002, “Future Directions and Challenges in Shell Stability Analysis,” Thin-Wall. Struct., 40(9), pp. 729–754. [CrossRef]
Koiter, W. T., 1945, “The Stability of Elastic Equilibrium,” Ph.D. thesis, Techische Hooge School, Delft.
Arbocz, J., and Babcock, C. D., 1969, “The Effects of general imperfections on the Buckling of Cylindrical Shells,” ASME J. Appl. Mech., 36(1), pp. 28–38. [CrossRef]
Song, Y. C., Teng, J. G., and Rotter, J. M., 2004, “Imperfection Sensitivity of Thin Elastic Cylindrical Shells Subject to Partial Axial Compression,” Int. J. Solids Struct., 41(24–25), pp. 7155–7180. [CrossRef]
Jiménez, F. L., Marthelot, J., Lee, A., Hutchinson, J. W., and Reis, P. M., 2017, “Technical Brief: Knockdown Factor for the Buckling of Spherical Shells Containing Large-Amplitude Geometric Defects,” ASME J. Appl. Mech. 84(3), p. 034501. [CrossRef]
Mang, H. A., Schranz, C., and Mackenzie-Helnwein, P., 2006, “Conversion From Imperfection-Sensitive into Imperfection-Insensitive Elastic Structures. I: Theory,” Comput. Methods Appl. Mech. Eng., 195(13–16), pp. 1422–1457. [CrossRef]
Schranz, C., Krenn, B., and Mang, H. A., 2006, “Conversion From Imperfection-Sensitive into Imperfection-Insensitive Elastic Structures. II: Numerical Investigation,” Comput. Methods Appl. Mech. Eng., 195(13–16), pp. 1458–1479. [CrossRef]
Ning, X., and Pellegrino, S., 2015, “Imperfection-Insensitive Axially Loaded Thin Cylindrical Shells,” Int. J. Solids Struct., 62, pp. 39–51. [CrossRef]
Ning, X., and Pellegrino, S., 2017, “Experiments on Imperfection Insensitive Axially Loaded Cylindrical Shells,” Int. J. Solids Struct., 115–116, pp. 73–86. [CrossRef]
Burgueño, R., Hu, N., Heeringa, A., and Lajnef, N., 2014, “Tailoring the Elastic Postbuckling Response of Thin-Walled Cylindrical Composite Shells Under Axial Compression,” Thin-Wall. Struct., 84, pp. 14–25. [CrossRef]
Hu, N., Burgueño, R., and Lajnef, N., 2014, “Structural Optimization and Form-Finding of Cylindrical Shells for Targeted Elastic Postbuckling Response,” Proceedings of the ASME 2014 Conference on Smart Materials, Adaptive Structures and Intelligent Systems, Newport, Rhode Island, American Society of Mechanical Engineers (ASME); Paper No.: SMASIS2014- 7446, p. V001T03A008.
Hu, N., and Burgueño, R., 2015b, “Tailoring the Elastic Postbuckling Response of Cylindrical Shells: A Route for Exploiting Instabilities in Materials and Mechanical Systems,” Extreme Mech. Lett., 4, pp. 103–110. [CrossRef]
Hu, N., and Burgueño, R., 2015c, “Elastic Postbuckling Response of Axially-Loaded Cylindrical Shells With Seeded Geometric Imperfection Design,” Thin-Wall. Struct., 96, pp. 256–268. [CrossRef]
Hu, N., and Burgueño, R., 2017, “Harnessing Seeded Geometric Imperfection to Design Cylindrical Shells With Tunable Elastic Postbuckling Behavior,” ASME J. Appl. Mech., 84(1), p. 011003. [CrossRef]
White, S. C., and Weaver, P. M., 2016, “Towards Imperfection Insensitive Buckling Response of Shell Structures: Shells with Plate-Like Post-Buckled Responses,” Aeronaut. J. 120(1224), pp. 233–253. [CrossRef]
Bielski, J., 1995, “A Global Plasticity Formulation Combined With a Semi-Analytical Analysis of Imperfect Shells of Revolution,” Thin-Wall. Struct., 23(1), pp. 399–411. ISSN 0263-8231. [CrossRef]
Lu, Z., Obrecht, H., and Wunderlich, W., 1995, “Imperfection Sensitivity of Elastic and Elastic-Plastic Torispherical Pressure Vessel Heads,” Thin-Wall. Struct., 23(1), pp. 21–39. ISSN 0263-8231. [CrossRef]
Lee, A., Jiménez, F. L., Marthelot, J., Hutchinson, J. W., and Reis, P. M., 2016, “The Geometric Role of Precisely Engineered Imperfections on the Critical Buckling Load of Spherical Elastic Shells,” ASME J. Appl. Mech., 83(11), p. 111005. [CrossRef]
Cox, B. S., Groh, R. M. J., Avitabile, D., and Pirrera, A., 2018, “Modal Nudging in Nonlinear Elasticity: Tailoring the Elastic Post-Buckling Behaviour of Engineering Structures,” J. Mech. Phys. Solids, 116, pp. 135–149. [CrossRef]
Jun, S. M., and Hong, C. S., 1988, “Buckling Behaviour of Laminated Composite Cylindrical Panels under Axial Compression,” Comput. Struct., 29(3), pp. 479–490. [CrossRef]
Ramm, E., 1977, “A Plate/Shell Element for Large Deflections and Rotations,” Formulations and Computational Algorithms in Finite Element Analysis, Formulations and Computational Algorithms in Finite Element Analysis, K. J. Bathe, T. Oden, and W. Wunderlich, eds., MIT Press, Boston, MA.
Riks, E., 1971, “The Application of Newton’s Method to the Problem of Elastic Stability,” ASME J. Appl. Mech., 39(4), pp. 1060–1065. [CrossRef]
Neville, R. M., Groh, R. M. J., Pirrera, A., and Schenk, M., 2018, “Shape Control for Experimental Continuation,” Phys. Rev. Lett. 120(25), p. 254101. [CrossRef] [PubMed]
Groh, R. M. J., and Pirrera, A., 2018, “Orthotropy as a Driver for Complex Stability Phenomena in Cylindrical Shell Structures,” Compos. Struct., 198, pp. 63–72. [CrossRef]
Groh, R. M. J., and Pirrera, A., 2018, “Extreme Mechanics in Laminated Shells: New Insights,” Extreme Mech. Lett., 23, pp. 17–23. [CrossRef]
Thompson, J. M. T., 2015, “Advances in Shell Buckling: Theory and Experiments,” Int. J. Bifur. Chaos, 25(1), pp. 1–25. [CrossRef]
Kocsis, A., and Károlyi, G., 2006, “Conservative Spatial Chaos of Buckled Elastic Linkages,” Chaos, 16(3), p. 033111. [CrossRef] [PubMed]
Gürdal, Z., Tatting, B. F., and Wu, C. K., 2008, “Variable Stiffness Composite Panels: Effects of Stiffness Variation on the In-Plane and Buckling Response,” Compos. Part A Appl. Sci. Manuf., 39(5), pp. 911–922. [CrossRef]
Thompson, J. M. T., and Virgin, L. N., 1988, “Spatial Chaos and Localization Phenomena in Nonlinear Elasticity,” Phys. Lett. A, 126(8–9), pp. 491–496. [CrossRef]
El Naschie, M. S., and Al Athel, S., 1989, “On the Connection between Statical and Dynamical Chaos,” Z. Naturforsch. A, 44(7), pp. 645–650. [CrossRef]

Figures

Grahic Jump Location
Fig. 1

Perfect and imperfect bifurcation branching responses: (a) subcritical and (b) supercritical [27]

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Fig. 2

Axially compressed cylindrical panel with geometric dimensions and imposed boundary conditions: (a) Pinned-Pinned-Pinned-Pinned (PPPP) edges, where rotations are free at each edge and (b) Clamped-Clamped-Pinned-Pinned (CCPP) edges, where rotations are constrained on the curved sides

Grahic Jump Location
Fig. 3

Initial load-displacement response of (a) PPPP and (b) CCPP panels solved using an in-house matlab-based nonlinear finite element analysis. Validation is provided with equivalent abaqus models (dashed black lines). (See online version for color.)

Grahic Jump Location
Fig. 4

The load-displacement response for the Pinned-Pinned-Pinned-Pinned (PPPP) model: (a) a single equilibrium path starting from the unloaded state, which exhibits multiple instabilities and entanglement, (b) an identical plot to (a) with the stable solutions highlighted, (c) a close-up view of the nonlinear response in the vicinity of the first critical point, (d) the stable solutions in the close-up view (c), (e) a close-up view of the deep postbuckled region, and (f) the stable regions within this deep postbuckled region. (See online version for color.)

Grahic Jump Location
Fig. 5

The load-displacement response for the Clamped-Clamped-Pinned-Pinned (CCPP) model: (a) a single equilibrium path starting from the unloaded state, which exhibits multiple instabilities and entanglement, (b) an identical plot to (a) with the stable solutions highlighted, (c) a close-up view of the nonlinear response in the vicinity of the first critical point, (d) the stable solutions in the close-up view (c), (e) a close-up view of the deep postbuckled region, and (f) the stable regions within this deep postbuckled region. (See online version for color.)

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Fig. 6

Stable secondary paths for the PPPP load case, which branch from the fundamental path depicted previously in Fig. 4. (a) A total of 18 branching paths are presented. There exist many other paths, but to the authors’ knowledge, the paths shown are the only bifurcated paths that present stable solutions, and hence a potential opportunity for modal nudging. In (b), 13 equilibrium paths are presented, the majority of which have similar responses. The gray curves correspond to the original baseline response of the PPPP panel previously shown in Fig. 4. (See online version for color.)

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Fig. 7

Stable secondary paths for the CCPP load case, which branch from the fundamental path depicted previously in Fig. 5 (a) illustrates three bifurcation (branching) paths in with stable (blue) and unstable (red) regions highlighted and (b) highlights these stable regions further. There exist many other paths, but to the authors’ knowledge, the paths shown are the only bifurcated paths that present stable solutions, and hence a potential opportunity for modal nudging. The gray curves correspond to the original baseline response of the CCPP panel previously shown in Fig. 5.

Grahic Jump Location
Fig. 8

Highlighted stable regions for nudge-state selection: (a) PPPP, states I–VI, and (b) CCPP, states VII–VIII. States I, II, VII, and VIII are stable regions on the fundamental paths, and states III–VI are stable regions on branching paths. (See online version for color.)

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Fig. 9

Mode shapes used in nudging. Each figure represents the actual mode shape from the selected stable regions and therefore the absolute deformation before normalization for the nudging procedure. States I–VI are each selected from the PPPP response and the final two states VII and VIII are selected from the CCPP response.

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Fig. 10

Modal nudging the structural response of panel PPPP using states from the baseline response of PPPP: (a) Nudge-I, u¯η=0.760 provides a successful nudge to a critical load of R¯=2.056; (b) Nudge-II, u¯η=0.844 results in a successful nudge to a critical load of R¯=2.135; (c) Nudge-III, u¯η=0.941 provides a successful nudge to an initial critical load of R¯=1.194, but also shows signs of potentially increasing to R¯=1.880; (d) Nudge-IV, with a nudging imperfection of u¯η=0.735, a stable well-behaved response to a critical load of R¯=1.481 is achieved. Modal nudging the structural response of panel PPPP using states from the baseline response in PPPP and CCPP: (e) Nudge-V, u¯η=0.738 provides a successful nudge to a critical load of R¯=2.092; (f) Nudge-VI, u¯η=0.745 results in a successful nudge to a critical load of R¯=2.460; (g) Nudge-VII, with a nudging state from CCPP, and a value of u¯η=0.849 provides a successful nudge to an initial critical load of R¯=1.721; (h) Nudge-VIII, with a nudging state from CCPP, and with a nudging imperfection of u¯η=0.702, a stable well-behaved response to a critical load of R¯=1.626 is achieved. The corresponding deformation mode shape is included for each response. (See online version for color.)

Grahic Jump Location
Fig. 11

Modal nudging the structural response of panel CCPP using states from the baseline response in PPPP: (a) Nudge-I, u¯η=0.760 provides a successful nudge to a critical load of R¯=1.637; (b) Nudge-II, u¯η=0.563 results in a successful nudge to a critical load of R¯=2.085; (c) Nudge-III, a value of u¯η=0.543 provides a successful nudge to an initial critical load of R¯=1.401; (d) Nudge-IV, with a nudging imperfection of u¯η=0.601, a stable well-behaved response to a critical load of R¯=1.496 is achieved. Modal nudging the structural response of panel CCPP using states from the baseline response in PPPP and CCPP: (e) Nudge-V, u¯η=0.546 provides a successful nudge to a critical load of R¯=1.721; (f) Nudge-VI, u¯η=0.488 results in a successful nudge to a critical load of R¯=2.295; (g) Nudge-VII, with a nudging state from CCPP, and a value of u¯η=0.607 provides a successful nudge to an initial critical load of R¯=2.040; (h) Nudge-VIII, with a nudging state from CCPP, and with a nudging imperfection of u¯η=0.562, a stable well-behaved response to a critical load of R¯=2.062 is achieved. The corresponding deformation mode shape is included for each response. (See online version for color.)

Grahic Jump Location
Fig. 12

Modal nudging the structural response of panel PPPP using state II with modifications made to the initial geometry of the panel. (a) The nudged response for each of the modifications made, a similar response is observed for all structural modifications, the nudging parameter η = 16.5 is identical for all responses. (b) A closer view of the variations observed in the structural response: (1) The panel thickness t is increased by 5%; (2) The panel arc-length s is reduced by 5%; (3) The panel radius is reduced by 5%; (4) The panel length L is reduced by 5%; (5) The original nudged response as illustrated in Fig. 10(b); (6) The panel length L is increased by 5%; (7) The panel radius r is increased by 5%; (8) The panel arc-length s is increased by 5%; (9) The panel thickness t is reduced by 5%. (c) Minor instabilities incurred for a reduction of the panel radius (3) and the increase in panel length (6). The other solutions have been removed from inset (c) for clarity. (See online version for color.)

Grahic Jump Location
Fig. 13

(a) Forced pendulum, (b.1) buckled elastica, (b.2) single loop of buckled elastica, and (c) conceptual stroboscopic map (Poincaré phase portrait). The forced pendulum is capable of exhibiting controlled and predictable behavior provided that the applied force F is small. In this scenario, the motion is such that the pendulum swings back and forth in a predictable manner, corresponding to the trapped motion in (c). However, when the applied perturbation F is large, the motion of the pendulum is such that it revolves continuously about its pinned end, corresponding to the untrapped stable regions in (c). At some point between these two scenarios, the motion of the pendulum becomes temporally chaotic, corresponding to the red regions of chaotic orbits in (c). Analogously, assuming an infinitely long elastica—where the arc-length variable s replaces the temporal variable t—shows signs of spatial chaos for small imperfection amplitudes. In (b.2), we see only one loop; however, for an infinitely long elastica, many loops would generate at arbitrary and unpredictable locations. For large imperfections, it is possible to accurately predict the deformation of the structure corresponding to the untrapped stable solution in (c). (See online version for color.)

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