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Research Papers

The Postbuckling Behavior of Planar Elastica Constrained by a Deformable Wall

[+] Author and Article Information
Shmuel Katz

Faculty of Mechanical Engineering,
Technion—Israel Institute of Technology,
Haifa 32000, Israel
e-mail: katzshm@technion.ac.il

Sefi Givli

Faculty of Mechanical Engineering,
Technion—Israel Institute of Technology,
Haifa 32000, Israel
e-mail: givli@technion.ac.il

1Corresponding author.

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received January 2, 2017; final manuscript received February 16, 2017; published online March 8, 2017. Assoc. Editor: Yihui Zhang.

J. Appl. Mech 84(5), 051001 (Mar 08, 2017) (15 pages) Paper No: JAM-17-1004; doi: 10.1115/1.4036018 History: Received January 02, 2017; Revised February 16, 2017

Attributed to its significance in a wide range of practical applications, the post-buckling behavior of a beam with lateral constraints has drawn much attention in the last few decades. Despite the fact that, in reality, the lateral constraints are often flexible or deformable, vast majority of studies have considered fixed and rigid lateral constraints. In this paper, we make a step toward bridging this gap by studying the post-buckling behavior of a planar beam that is laterally constrained by a deformable wall. Unfortunately, the interaction with a compliant wall prevents derivation of closed-form analytical solutions. Nevertheless, careful examination of the governing equations of a simplified model reveals general properties of the solution, and let us identify the key features that govern the behavior. Specifically, we construct universal “solution maps” that do not depend on the mode number and enable simple and easy prediction of the contact conditions and of the mode-switching force (the force at which the system undergoes instantaneous transition from one equilibrium configuration (or mode) to another). The predictions of the mathematical model are validated against finite element (FE) simulations.

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Figures

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Fig. 1

Schematic illustration of the system, consisting of a clamped–clamped beam that is laterally constrained by a rigid wall at the bottom and a WF located at distance D0 above the beam

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Fig. 2

As Δ is increased, several configurations might take place: (a) a region (length) of contact with the WF, and point contacts with the rigid wall. (b) Line-contact regions with the rigid wall. (c) Once the line-contact configuration becomes unstable, the beam switches to a new configuration with two additional folds. (d) Line-contact segments with the rigid wall for the four-folds configuration. On the right, results of FE simulations, where vertical displacement is denoted by different colors.

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Fig. 3

Transition from point-contact to line-contact configuration. (a) Domains of point-contact and line-contact configurations. (b) Same as (a) but for a larger range of stiffness. The solid line, indicating transition from point-contact to line-contact configuration, starts at (3π, 5/2) at the left end, and approaches ζ/n=2 for large values of β1/4L0/n. The second solid line shows the so-called run away of the beam deflection, where the magnitude of the deflection becomes extremely large. Lines of constant γ are illustrated by dashed lines. (c) H̃2 at the onset of transition from point-contact to line-contact configuration (red line in (a) and (b)). Linear approximation is given by Eq. (2.22). (Color images can be viewed online).

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Fig. 4

In line-contact configuration, H̃1 and H̃2 are directly related to γ through Π1

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Fig. 5

Mode switching is governed by the length of the longest line segment, c. Three cases are of special interest: (i) the lengths of all line-contact segments, except one, are very small; (ii) all line-contact segments have the same length; and (iii) the symmetric case where the beam is comprised of n identical and symmetric parts. (a) Case (ii) and case (iii) are identical for n=1. This not the case for n≥2, as illustrated in (b) for n=2.

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Fig. 6

Switching from mode n to mode n+1 for the symmetric case. (a) Mode switching for n≥2 and for n=1 are illustrated by a solid and a broken lines, respectively. (b) Same as (a) but for a larger range of stiffness. The solid straight line shows the so-called run away of the beam deflection, where the magnitude of the deflection becomes extremely large. Lines of constant γ are illustrated by dashed lines. (c) H̃2 at the onset of mode switching. Linear approximation is given by Eq.(2.28).

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Fig. 7

Summary of predicted behavior, showing domains of point-contact configuration, line-contact configuration, and mode switching associated with the symmetric case. Results of the FE simulations are shown with arrows in (a). Filled circles indicate transitions from point-contact to line-contact configuration based on the FE results. Occurrence of mode switching, based on FE results, is indicated by the departure of the arrows from the original vertical branch.

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Fig. 8

Force–displacement response. A comparison between results of the theoretical model and FE analysis for: (a) β1/4L0=17.8, (b) β1/4L0=8.4, and (c) ϕ=2. Results of the mathematical model are shown with continuous lines, where each line corresponds to a different mode. Dashed and solid lines distinguish between point-contact and line-contact configurations, respectively. Results of the FE simulations are illustrated with filled diamonds, where transition from point-contact to line-contact configuration is indicated with a circle.

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