Numerical Stability Criteria for Localized Post-buckling Solutions in a Strut-on-Foundation Model

[+] Author and Article Information
M. Khurram Wadee

Department of Engineering, School of Engineering, Computer Science and Mathematics, University of Exeter, North Park Road, Exeter EX4 4QF, UK

Ciprian D. Coman

Department of Mathematics, University of Leicester, University Road, Leicester LE1 7RH, UK

Andrew P. Bassom

Department of Mathematical Sciences, School of Engineering, Computer Science and Mathematics, University of Exeter, North Park Road, Exeter EX4 4QE, UK

J. Appl. Mech 71(3), 334-341 (Jun 22, 2004) (8 pages) doi:10.1115/1.1757486 History: Received September 25, 2002; Revised September 22, 2003; Online June 22, 2004
Copyright © 2004 by ASME
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An elastic strut resting on an elastic foundation acted on by a compressive axial load
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The von Kármán analogy between the post-buckling response of a cylindrical shell element (panel) and the up-down-up response of a strut-on-foundation model. If the panel is thin, the rings have very little bending stiffness and act as thin arches for normal compressive loads (see the top right diagram). The load-deflection curve for such a structure has the well-known shape shown in the bottom graph.
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Variation of Δ (Eq. (23)) with load P for the model with only a quadratic destabilizing nonlinearity (c2=0). Sample values: Δ(1.0)=−9.6×10−7 and Δ(0.0)=−7.5×10−10.
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The variation of foundation force F=y−y2+c2y3 against lateral deflection y for various values of the coefficient c2
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Initial bifurcation diagrams depicting the post-buckling behavior of primary localized solutions of the restabilizing strut model with (a) c2=0.24; (b) c2=0.3, and (c) c2=0.4. Numerical (AUTO97 ) SOLUTIONS FOR PEAK-CENTRED (ϕ0=0) and trough-centered (ϕ0=π) orbits are shown against end-shortening, E , with solid and dashed lines, respectively. Discrete points show solutions obtained using the nonperiodic Rayleigh-Ritz procedure.
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Comparison of buckling solutions obtained using AUTO97 (solid line) and the Rayleigh-Ritz method (discrete points) for the fold points A–F identified on the ϕ0=π branches in Fig. 5
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Variation of the determinant of the Hessian of the energy function, Δ, with end-shortening for various values of c2: (a) 0.24; (b) 0.3; (c) 0.4. The peak-centered branch is labeled Δp and the trough-centered branch is labeled Δt. The fold point on the corresponding bifurcation diagram in Fig. 5 is identified with a large dot.




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