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Research Papers

Probabilistic and Interval Analyses Contrasted in Impact Buckling of a Clamped Column

[+] Author and Article Information
Isaac Elishakoff

Professor
ASME Fellow
Dept. of Ocean and Mechanical Engineering
Florida Atlantic University
Boca Raton, FL 33431
e-mail: elishako@fau.edu

Wim Verhaeghe

Ph.D Student
Dept. of Mechanical Engineering
KU Leuven
B3001 Heverlee, Belgium
e-mail: wim.verhaeghe@mech.kuleuven.be

David Moens

Professor
Dept. of Applied Engineering
Lessius Mechelen, Campus De Nayer
B2860 St-Katelijne-Waver, Belgium
and
Associate Professor
Dept. of Mechanical Engineering
KU Leuven
B3001 Heverlee, Belgium
e-mail: david.moens@mech.kuleuven.be

Manuscript received November 28, 2011; final manuscript received June 29, 2012; accepted manuscript posted July 6, 2012; published online November 19, 2012. Assoc. Editor: Vikram Deshpande.

J. Appl. Mech 80(1), 011022 (Nov 19, 2012) (8 pages) Paper No: JAM-11-1452; doi: 10.1115/1.4007084 History: Received November 28, 2011; Revised June 29, 2012; Accepted July 06, 2012

In this study we contrast two competing methodologies for the impact buckling of a column that is clamped at both ends. The initial imperfection is postulated to be co-configurational with the fundamental mode shape of the column without the axial loading. A solution is also furnished for the case when the initial imperfection is proportional to the Filonenko-Borodich “cosinusoidal polynomial”. Probabilistic and interval analyses are conducted for each case; these are contrasted on some representative numerical data.

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References

Lindberg, H., 1965, “Impact Buckling of a Thin Bar,” ASME J. Appl. Mech., 32, pp. 312–322.
Elishakoff, I., 1978, “Axial Impact Buckling of a Column With Random Initial Imperfections.” ASME J. Appl. Mech., 45(2), pp. 361–365. [CrossRef]
Elishakoff, I., 1978, “Impact Buckling of a Thin Bar Via the Monte Carlo Method,” ASME J. Appl. Mech., 45(3), pp. 586–590. [CrossRef]
Ben-Haim, Y. and Elishakoff, I., 1990, Convex Models of Uncertainty in Applied Mechanics. Elsevier, New York.
Qiu, Z. and Wang, X., 2006, “Interval Analysis Method and Convex Models for Impulsive Response of Structures With Uncertain-But-Bounded External Loads,” Acta Mech. Sin., 22(3), pp. 265–276. [CrossRef]
Qiu, Z., Ma, L., and Wang, X., 2006, “Ellipsoidal Bound Convex Model for the Non-Linear Buckling of a Column With Uncertain Initial Imperfection,” Int. J. Non-Linear Mech., 41(8), pp. 919–925. [CrossRef]
Hu, J. and Qiu, Z., 2010, “Non-Probabilistic Convex Models and Interval Analysis Method for Dynamic Response of a Beam With Bounded Uncertainty,” Appl. Math. Model., 34(3), pp. 725–734. [CrossRef]
Lindberg, H., 1991, “Dynamic Response And Buckling Failure Measures For Structures With Bounded and Random Imperfections,” ASME J. Appl. Mech., 58(4), pp. 1092–1095. [CrossRef]
Filonenko-Borodich, M., 1946, “On a Certain System of Functions and its Applications in Theory of Elasticity,” Prikl. Mat. Mekh., 10, pp. 193–208 (in Russian).
Lindberg, H., and Florence, A., 1987, Dynamic Pulse Buckling: Theory and Experiment, Martinus Nijhoff, Dordrecht, The Netherlands.
Rao, S., 2007, Vibration of Continuous Systems, Wiley, New York.
Vilenkin, N., 1952, “On Some Nearly Periodic Systems of Functions,” Prikl. Mat. Mekh., 16(3), pp. 812–814 (in Russian).
Hoff, N., 1951, “The Dynamics of the Buckling of Elastic Columns,” ASME J. Appl. Mech., 18, pp. 68–71.

Figures

Grahic Jump Location
Fig. 1

Maximum normalized deflection versus normalized time at ξ = 0.25 (dashed lines) and ξ = 0.5 (solid lines). The load ratio is α = 2. The green lines correspond to the solution with the normal mode; the red lines correspond to the solution with the one term Filonenko-Borodich (F-B) approximation.

Grahic Jump Location
Fig. 2

Maximum normalized deflection versus normalized time at ξ = 0.5 for various values of the load ratio. The green lines correspond to the solution with the normal mode; the red lines correspond to the solution with the one term Filonenko-Borodich (F-B) approximation.

Grahic Jump Location
Fig. 3

Maximum integral (span-averaged) deflection versus normalized time for various values of the load ratio. The green lines correspond to the solution with the normal mode; the red lines correspond to the solution with the one term Filonenko-Borodich (F-B) approximation.

Grahic Jump Location
Fig. 4

Reliability as a function of time (α = 0.5)

Grahic Jump Location
Fig. 5

Reliability as a function of time (α = 1)

Grahic Jump Location
Fig. 6

Reliability as a function of time (α = 2)

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