Review Articles

Dynamic Elastic-Plastic Buckling of Structural Elements: A Review

[+] Author and Article Information
D. Karagiozova1

 Institute of Mechanics, Bulgarian Academy of Sciences, Acad. Georgi Bonchev Street, Block 4, Sofia 1113, Bulgariad.karagiozova@imbm.bas.bg

Marcílio Alves

Group of Solid Mechanics and Structural Impact, Department of Mechatronics and Mechanical Systems Engineering, University of São Paulo, Avenida Professor Mello Moraes 2231, 05508-030 São Paulo, Brazilmaralves@usp.br


Corresponding author.

Appl. Mech. Rev 61(4), 040803 (Jul 08, 2008) (26 pages) doi:10.1115/1.2939481 History: Published July 08, 2008

Structural elements, which deform inelastically, are often used in energy-absorbing devices due to their simple design and the high efficiency achieved by several buckling deformation mechanisms. The application of light ductile materials in transportation systems and increased loading intensity requires studies on the influence of the rate of loading and material characteristics on dynamic buckling behavior. The present review article is focused on summarizing the state of the art related to the inelastic dynamic stability and postbuckling behavior of various basic structural members. In particular, studies on the dynamic response of axially loaded idealized elastic-plastic models, rods, shells with circular and square cross sections, and long tubes are discussed with consideration given to the influence of the geometric and material characteristics as well as the loading conditions on the buckling phenomena observed in these structural elements. The findings from the theoretical and experimental investigations on the phenomenon of dynamic inelastic buckling reported in this review article are based on 118 references.

Copyright © 2008 by American Society of Mechanical Engineers
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Figure 1

Discrete models for elastic buckling: (a) one degree-of-freedom model (1), (b) two degrees-of-freedom model (3), and (c) types of response of the two degrees-of-freedom model (3)

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Figure 2

(a) One degree-of-freedom model for plastic buckling (static) and (b) comparison between the critical elastic and plastic loads (4)

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Figure 3

(a) Two degree-of-freedom elastic-plastic buckling model and (b) comparison between the critical dynamic elastic and plastic loads (5)

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Figure 4

(a) Structural elements representing “Type I” and “Type II” structures, respectively (11) and (b) load-deflection characteristics for the two types of structures

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Figure 5

(a) A model for “Type II” structures, (b) stress loci for the plastic hinge during dynamic and quasi-static response (16), and (c) velocity time history for Type II structure (14)

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Figure 6

Representative results for “Type II” structures; circles are used for the experimental results (14); (a) aluminum alloy: — numerical predictions (8), – – – Ref. 14 when using Eq. 2; (b) mild steel: – – – Ref. 13, — Ref. 14 when using Eq. 2, –⋅–⋅– strain-rate independent model (16), –⋅⋅–⋅⋅– strain-rate dependent model (16)

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Figure 7

Dynamic buckling of rods at high impact velocity. (a) Orthogonal impact (25) and (b) from top to bottom: helical, sinusoidal, and spiral deformation modes (26)

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Figure 8

Permanent plastic strain profile for a 6in. long rod with a diameter of 14 in. Ref. 26

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Figure 9

(a) Axial loading of an elastic plastic column, (b) material model, and (c) stress distributions at different time instances (41)

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Figure 10

Strain distribution along a rod due to different loading conditions (21) (t1=127μs, t2=205μs, and t3=251μs). (a) Impact on a rigid target, V0=128m∕s; - - - experimental results (25) and (b) impact against an identical mass, V0=130m∕s; - - -experimental estimate (31)

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Figure 35

Impact velocity ranges for the different modes of collapse depending on the loading conditions. (a) Constant impact energy, T0=5kJ; L1=550mm, L2=650mm and (b) constant impact masses, G2=200kg, G2=500kg; L=650mm(115).

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Figure 36

(a) Collapse modes versus impact velocity, T-6 and (b) collapse modes for different tempers (117)

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Figure 24

Dynamic effects on the load-deflection curve of brass square tubes subjected to axial impact due to strain hardening (101). (a) Stress-strain material response and (b) squashing force-displacement curves under static and dynamic loading.

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Figure 25

Stress wave speeds in a plate (nx=1, ny=0), A: (σxx<0, σyy=σxx∕2), B: (σxx<0, σyy=0) (102). (a) Variation of the stress wave speeds with the hardening parameter λ, σxy=0.15σ0 and (b) variation of the stress wave speeds with the shear stress σxy, λ=0.003.

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Figure 26

Buckling profiles and plastic strain distributions at the midplane of a square tube wall (z=b∕2, y=b∕4) for various times; Al6063-T5. (a) and (b) V0=64.62m∕s; (c) and (d) V0=91.53m∕s; Ref. 104 is used for the experimental results and Ref. 102 for the numerical simulations

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Figure 27

Dynamic effect on the buckling shape and the crushing distance. (a) Experimental results (104), (b) FE simulations with the same impact velocities (103), and (c) numerically obtained force-displacement histories (103).

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Figure 28

Influence of the material model on the buckling shape of square tubes; Al6063-T5, V0=100m∕s and G=0.2kg. (a) and (b) Intermediate, final buckling shapes and strain distribution along the shell at (x=b∕4, y=0) when the strain-rate effects are not taken into account and (c) and (d) same as for (a) and (b) but the strain-rate effects are included in the material model (68)

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Figure 29

Deformation maps for long tubes (106). (a) and (b) Square and circular tubes subjected to static axial loading, respectively and (c) and (d) as in (a) and (b) but for axial impact.

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Figure 30

Influence of the impact velocity on the dynamic buckling transition of aluminum alloy circular tubes (110). (a) Buckling transition for different tube lengths (b) L=360mm, (c) L=500mm, and (d) L=650mm.

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Figure 31

Typical stages of deformation of square tubes under axial impact, showing the transition from progressive buckling to global bending, b=80mm, AA6060-T6, (112). (a) L=960mm, h=3.5mm, and V0=20m∕s; (b) L=878mm, h=4.5mm, and V0=13m∕s; and (c) L=1600mm, h=3.5mm, and V0=20m∕s.

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Figure 32

Collapse mode maps depending on the impact velocity, showing the transition from progressive buckling to global bending, b=80mm, AA6060-T6 (112). (a) Quasistatic loading, (b), V0=13m∕s, and (c) V0=20m∕s.

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Figure 33

(a) Influence of the impact velocity on the critical tube length for buckling transition of circular tubes made from different materials (numerical results) (113); (b) L=630mm, without strain-rate effects; (c) as in (b) but with strain-rate effects included in the material model; and (d) new transition velocity when the strain-rate effect is considered (114)

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Figure 11

Effect of the hardening modulus on the initial buckling shape of a rod for G∕m=5 and V0=50m∕s(21). (a) and (b) Axial wave propagation history for λ=0.0284 and λ=0.01, respectively and (c) corresponding transverse profile of the rod at the initiation of buckling.

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Figure 12

Influence of the material strain-rate sensitivity on the flexural wavelength versus step loading intensity (46)

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Figure 13

Types of buckling. (a) Classical static progressive buckling with axisymmetric folds, mixed modes, and asymmetric folds for quasi-static compression (63); (b) dynamic plastic buckling at impact with V0=108m∕s(64); and (c) dynamic progressive buckling with mixed mode at V0=78.15m∕s, 2R=30mm, h=0.7mm, and attached mass of 81g(68)

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Figure 14

Speeds of the stress waves in a circular shell made of an elastic-plastic material with a linear strain hardening (SRS, strain rate sensitive; SRI, strain rate insensitive) (86). (a) von Mises yield criterion and (b) Tresca yield criterion.

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Figure 15

Regions for different buckling modes in circular shells (89)

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Figure 16

Influence of the material model on the buckling shape of cylindrical shells, A6063-T5, T0=2.2kJ, V0=75m∕s. (a) Undeformed shell wall; (b) buckling initiation, t0=0.217ms; (c) final shape for material approximation A2; (d) and (e) same as in (b) and (c) but for material approximation A3; (f) and (g) distribution of the plastic strains corresponding to cases (b) and (d); and (h) approximations of the actual material properties (86)

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Figure 17

Final buckling shapes of an aluminum alloy shells, Eh=542.6MPa and σ0=285MPa. (a) and (b) Kinematic strain hardening, V0=65m∕s and 100m∕s, respectively and (c) and (d) same as (a) and (b) but with isotropic strain hardening (84).

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Figure 18

Buckling shapes of steel circular shells, T0=2.3kJ and V0=10m∕s. (a) Undeformed, (b) and (c) initial and final buckling shapes (92), and (d) specimen wrinkled under quasi-static axial compression (D∕t=28.97)(93).

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Figure 19

Influence of the experimental setup on the final buckling shape of a shell (84). (a) and (b) Impact against a rigid wall at V0=65m∕s, (c) and (d) impact by a mass at V0=65m∕s, and (e) and (f) and (g) and (h) as in (c) and (d) and (e) and (f), respectively, but for V0=80m∕s

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Figure 20

Deformation modes of square tubes made of aluminum alloy AA6060. (a) Temper T4 and (b) Temper T6 (97)

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Figure 21

(a) Engineering stress-strain curves for aluminum alloys AA6060-T4 and AA6060-T6 and (b) ratio between the calculated and the experimental mean crushing force (97)

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Figure 22

Comparison between the static, ———, and dynamic, – – –, force-displacement curves for axially impacted square cross-section shells, L=310mm, b=80mm, and h=2.5mm(98)

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Figure 23

Ratio between the dynamic and static mean loads versus axial displacement, L=310mm, b=80mm, and h=2.5mm; (a) effect of temper and (b) effect of impact velocity (98)

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Figure 34

Counterintuitive response for circular tubes with 2R=50.2mm, h=2.19mm, σ0=240MPa, Eh=460MPa, V0=10.4m∕s, and G=209kg(115)

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Figure 37

Collapse modes of square tubes. (a) V0=13m∕s, L=1120mm, h=2.5mm, (112); (b) V0=13m∕s, L=1520mm, h=2mm, (112); and (c) Quasistatic, L=1679mm, h=2mm(117).



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