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TECHNICAL PAPERS

# Post-bauckling and Vibration of Heavy Beam on Horizontal or Inclined Rigid Foundation

[+] Author and Article Information
S. T. Santillan

Department of Mechanical Engineering and Materials Science,  Duke University, Durham, North Carolina 27708-0300

L. N. Virgin

Department of Mechanical Engineering and Materials Science,  Duke University, Durham, North Carolina 27708-0300l.virgin@duke.edu

R. H. Plaut

Department of Civil and Environmental Engineering,  Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061-0105

J. Appl. Mech 73(4), 664-671 (Oct 27, 2005) (8 pages) doi:10.1115/1.2165237 History: Received July 14, 2005; Revised October 27, 2005

## Abstract

A slender, straight beam resting on a flat, rigid foundation does not buckle when subjected to a compressive load, since the load cannot overcome the effect of the beam’s weight. However, it buckles if its ends are moved toward each other. Post-buckling of such a beam is examined, both theoretically and experimentally, for horizontal and inclined foundations. The beam is modeled as an elastica, and equilibrium states with large deflections are computed, including cases in which self-contact occurs. Frequencies and mode shapes for small vibrations about equilibrium are also determined. Agreement between the theoretical and experimental results is very good.

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## Figures

Figure 1

(a) Schematic of heavy inclined strip (b) Photo of experimental system

Figure 2

Coordinate systems for symmetric equilibrium with self-contact

Figure 3

Equilibrium shapes and end-shortening as a function of axial load for horizontal strip with weights (from left to right) w=0, 25, 125, 250, and 343

Figure 4

Midpoint deflection as a function of end-shortening for horizontal strip. Continuous line and 엯, w=32.56; ●, w=9.74.

Figure 5

End-shortening as a function of fundamental frequency for short horizontal strips with w=0, 25, 125, and 250, and mode shapes; equilibrium shapes are shown in gray

Figure 6

End-shortening as a function of fundamental frequency for horizontal strips. Dark curves are (from right to left) for w=8.145, 32.56, and 124.7; light curves are for w=7.030, 28.12, and 107.6. 엯, ●, and ▵, experiment.

Figure 7

End-shortening as a function of lowest four frequencies for horizontal strip with w=32.56. Experiment: 엯 (forced) and ● (free); various lines, theory. Self-contact is indicated by the gray lines: Continuous, theory; dashed, experiment.

Figure 8

First three vibration modes from analysis and experiment for horizontal strip with w=32.56 and δ=0.117; equilibrium shape is dashed

Figure 9

First three vibration modes from analysis and experiment for horizontal strip with w=32.56 and δ=0.821; equilibrium shape is dashed

Figure 10

Equilibrium shapes for the inclined strip with β=0.627. The strip is short-long in (a) and (b), and short in (c) and (d).

Figure 11

End-shortening as a function of fundamental frequency for inclined strips with w=32.56. Solid curve and ●, β=0; dot-dashed curve and 엯, β=0.627; dashed line and ◇, β=0.944; dotted line and ▵, β=π∕2.

Figure 12

First three vibration modes from analysis and experiment for vertical strip with w=32.56 and δ=0.630, along with equilibrium shape (dashed curve)

Figure 13

Midpoint deflection as a function of end-shortening with no foundation, β=0, and w=125

## Errata

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