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Research Papers

A New Geometrically Exact Model for Buckling and Postbuckling Statics and Dynamics of Beams

[+] Author and Article Information
Hamed Farokhi

Department of Mechanical and Construction Engineering,
Northumbria University,
Newcastle upon Tyne NE1 8ST, UK
e-mail: hamed.farokhi@northumbria.ac.uk

Mergen H. Ghayesh

School of Mechanical Engineering,
University of Adelaide,
Adelaide 5005, South Australia, Australia
e-mail: mergen.ghayesh@adelaide.edu.au

1Corresponding author.

Contributed by the Applied Mechanics Division of ASME for publication in the Journal of Applied Mechanics. Manuscript received February 3, 2019; final manuscript received March 7, 2019; published online April 12, 2019. Assoc. Editor: Yihui Zhang.

J. Appl. Mech 86(7), 071001 (Apr 12, 2019) (10 pages) Paper No: JAM-19-1061; doi: 10.1115/1.4043144 History: Received February 03, 2019; Accepted March 08, 2019

In this study, a new geometrically exact nonlinear model is developed for accurate analysis of buckling and postbuckling behavior of beams, for the first time. Three-dimensional nonlinear finite element analysis is conducted to verify the validity of the developed model even at very large postbuckling amplitudes. It is shown that the model commonly used in the literature for buckling analysis significantly underestimates the postbuckling amplitude. The proposed model is developed on the basis of the beam theory of Euler–Bernoulli, along with the assumption of centerline inextensibility, while taking into account the effect of initial imperfection. The Kelvin–Voigt model is utilized to model internal energy dissipation. To ensure accurate predictions in the postbuckling regime, the nonlinear terms in the equation of motion are kept exact with respect to the transverse motion, resulting in a geometrically exact model. It is shown that even a fifth-order truncated nonlinear model does not yield accurate results, highlighting the significant importance of keeping the terms exact with respect to the transverse motion. Using the verified geometrically exact model, the possibility of dynamic buckling is studied in detail. It is shown that dynamic buckling could occur at axial load variation amplitudes as small as 2.3% of the critical static buckling load.

Copyright © 2019 by ASME
Topics: Stress , Buckling
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Figures

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Fig. 1

Schematic of a doubly clamped beam with an initial imperfection under an axial load

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Fig. 2

Static buckling diagrams of a doubly clamped beam with one movable end under an axial load obtained using the geometrically exact model: (a) transverse displacement at midpoint and (b) axial displacement at the movable end, A1 = 0

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Fig. 3

Buckling diagrams of a doubly clamped beam with one movable end under an axial load obtained via different models: (a) transverse displacement at midpoint and (b) magnified version of (a) in the vicinity of the onset of buckling, A1 = 0.0001

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Fig. 4

Buckling diagram of different initially imperfect axially loaded doubly clamped beams obtained via the geometrically exact model developed in this study and the 3D nonlinear finite element analysis: (a) transverse displacement at midpoint and (b) longitudinal displacement at the movable tip

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Fig. 5

Static buckling diagrams of a doubly clamped beam under an axial load obtained via the model developed in this study (solid line) and the existing one in the literature (dashed line): (a) L/h = 20, (b) L/h = 100, and (c) L/h = 500

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Fig. 6

The first transverse natural frequency as a function of the axial load for various imperfection amplitudes

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Fig. 7

Dynamic buckling diagram of the axially loaded doubly clamped beam showing the transverse displacement and oscillation envelope at midpoint. P0 = 0, A1 = 0, ωp/ω1 = 2.00, and ω1 = 22.3733. Stable response: solid line; unstable response: dashed line.

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Fig. 8

Dynamic buckling diagram of the axially loaded doubly clamped beam showing the transverse displacement and oscillation envelope at midpoint. P0 = 0, A1 = 0, ωp/ω1 = 1.96, and ω1 = 22.3733. Stable response: solid line; unstable response: dashed line.

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Fig. 9

Dynamic buckling diagram of the axially loaded doubly clamped beam showing the maximum transverse displacement at midpoint. P0 = 20, A1 = 0, and ω1 = 15.8480.

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Fig. 10

Dynamic buckling diagram of the axially loaded doubly clamped beam showing the transverse displacement and oscillation envelope at midpoint. P1 = 0.05P0, A1 = 0, and ωp = 25.00. Stable response: solid line; unstable response: dashed line.

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Fig. 11

(a) and (b) Time trace and phase-plane of the w motion at x = 0.5 at P0 = 21.67, respectively; (c) and (d) time trace and phase-plane of the w motion at x = 0.5 at P0 = 47.66, respectively. tn indicates the time normalized relative to the oscillation period.

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Fig. 12

Dynamic buckling diagram of the axially loaded doubly clamped beam showing the transverse displacement and oscillation envelope at midpoint. P1 = 0.05P0, A1 = 0, and ωp = 25.00. Stable response: solid line; unstable response: dashed line.

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Fig. 13

Parametric resonance of the axially loaded doubly clamped beam: (a) the transverse displacement at midpoint and (b) the axial displacement at movable tip; P0 = 10, P1 = 1.4, A1 = 0, and ω1 = 19.41. Stable response: solid line; unstable response: dashed line.

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Fig. 14

Effect of P0 on parametric resonance of the axially loaded doubly clamped beam: (a) the transverse displacement at midpoint and (b) the axial displacement at movable tip; P1 = 1.2 and A1 = 0

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Fig. 15

Resonance response of the axially loaded initially imperfect doubly clamped beam: (a) the transverse displacement at midpoint and (b) the axial displacement at movable tip; P0 = 12, P1 = 1.6, A1 = 0.01, and ω1 = 18.69. Stable response: solid line; unstable response: dashed line.

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