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.

Copyright © 2014 by ASME
Topics: Stress , Buckling , Deflection
Your Session has timed out. Please sign back in to continue.


Lakes, R. S., 2001, “Extreme Damping in Compliant Composites With a Negative Stiffness Phase,” Philos. Mag. Lett., 81(2), pp. 95–100. [CrossRef]
Dong, L., and Lakes, R. S., 2012, “Advanced Damper With Negative Structural Stiffness Elements,” Smart Mater. Struct., 21, p. 075026. [CrossRef]
Dong, L., and Lakes, R. S., 2013, “Advanced Damper With High Stiffness and High Hysteresis Damping Based on Negative Structural Stiffness,” Int. J. Solids Struct., 50, pp. 2416–2423. [CrossRef]
Kalathur, H., and Lakes, R. S., 2013, “Column Dampers With Negative Stiffness: High Damping at Small Amplitude,” Smart Mater. Struct., 22, p. 084013. [CrossRef]
Jaglinski, T., Stone, D. S., and Lakes, R. S., 2005, “Internal Friction Study of a Composite With a Negative Stiffness Constituent,” J. Mater. Res., 20(9), pp. 2523–2533. [CrossRef]
Vinogradov, A. M., 1987, “Buckling of Viscoelastic Beam Columns,” AIAA J., 26(3), pp. 479–483. [CrossRef]
Timoshenko, S., and Gere, J. M., 1963, Theory of Elastic Stability, McGraw-Hill, New York.
Budiansky, B., 1974, “Theory of Buckling and Post-Buckling Behavior of Elastic Structures,” Adv. Appl. Mech., 14, pp. 1–65. [CrossRef]
Bolotin, V., 1964, The Dynamic Stability of Elastic Systems, Holden-Day, San Francisco, CA.
American Wood Council, 2012, ASD/LRFD Manual for Engineered Wood Construction, Section M10, Mechanical Connections, American Wood Council, Leesburg, VA, p. 74.


Grahic Jump Location
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.

Grahic Jump Location
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.

Grahic Jump Location
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.

Grahic Jump Location
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.

Grahic Jump Location
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.

Grahic Jump Location
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.

Grahic Jump Location
Fig. 7

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



Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In