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Research Papers

Buckling Mode Jump at Very Close Load Values in Unattached Flat-End Columns: Theory and Experiment

[+] Author and Article Information
R. S. Lakes

e-mail: lakes@engr.wisc.edu

W. J. Drugan

e-mail: drugan@engr.wisc.edu
Engineering Physics Department,
Engineering Mechanics Program,
University of Wisconsin-Madison,
Madison, WI 53706

Manuscript received June 10, 2013; final manuscript received July 19, 2013; accepted manuscript posted July 31, 2013; published online September 23, 2013. Editor: Yonggang Huang.

J. Appl. Mech 81(4), 041010 (Sep 23, 2013) (8 pages) Paper No: JAM-13-1234; doi: 10.1115/1.4025149 History: Received June 10, 2013; Revised July 19, 2013; Accepted July 31, 2013

Buckling of compressed flat-end columns loaded by unattached flat platens is shown, theoretically and experimentally, to occur first at the critical load and associated mode shape of a built-in column, followed extremely closely by a second critical load and different mode shape characterized by column end tilt. The theoretical critical load for secondary or end tilt buckling for a column geometry tested is shown to be only 0.13% greater than the critical load for primary buckling, in which the ends are in full contact with the compression platens. The experimental value is consistent with this theoretical one. Interestingly, under displacement control, the first buckling instability is characterized by a smoothly increasing applied load, whereas the closely following second instability causes an abrupt and large load drop (and hence exhibits incremental negative stiffness). The end tilt buckling gives rise to large hysteresis that can be useful in structural damping but that is nonconservative and potentially catastrophic in the context of design of structural support columns.

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Copyright © 2014 by ASME
Topics: Stress , Buckling , Deflection
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References

Figures

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Fig. 1

Geometry of linear elastic springs demonstrating negative stiffness in a lumped system. (Adapted from Jaglinski et al. [5].) Displacement u is applied at point A at left, producing a force. (a) The springs are initially unstretched. (b) The springs are deformed to a new equilibrium configuration that exhibits negative stiffness. (c) Snap-through to a new stable configuration.

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Fig. 2

Polymer column. (a) Column secured between flat platens solely by contact under compression. (b) Column deflection at first buckling, identical to that of a clamped-ended column. (c) Column deflection at second buckling, exhibiting end tilt. (d) Column end tilt in close-up.

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Fig. 3

(a) The force-displacement relationship of a PMMA column with d = 6.453 mm and  = 197.6 mm from zero load through both buckling events. (b) Curve fit of the full experimental data set but only up to just prior to snap (second buckling) instability. A linear fit to the initial part of the experimental data was used to determine the first buckling threshold. The second buckling threshold was taken to be the point just prior to the first abrupt load drop. On the left, a portion of the raw experimental data for snap-through associated with tilt of column ends is shown. (c) Lateral deflection versus axial displacement of a column with d = 6.453 mm and  = 150.4 mm.

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Fig. 4

Load thresholds for buckling of PMMA columns for several aspect ratios; comparison of theory and experiment. Glued columns are considered as built-in.

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Fig. 5

Hysteresis at 1 Hz of a PMMA column with d/ = 0.033. The displacement amplitude is 0.53 mm and the column length is 197.6 mm. (a) Ends unattached. (b) Ends glued.

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Fig. 6

(a) Curved end of column, modeled as a portion of a circle having radius R. (b) Displacement of forces action line due to rotation of curved column ends.

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Fig. 7

The two columns with different end conditions analyzed to predict the second buckling load

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