**1. Introduction**

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The primary goal of the research in constrained elastica is to understand the behavior of a thin elastic strip under end thrust when it is subject to lateral constraints. It finds applications in a variety of practical problems, such as in compliant foil journal bearing, corrugated fiberboard, deep drilling, structural core sandwich panel, sheet forming, nonwoven fabrics manufacturing, and stent deployment procedure. By assuming small deformation, Feodosyev (1977) included the problem of a buckled beam constrained by a pair of parallel walls as an exercise for a university strength and material course. Vaillette and Adams (1983) derived a critical axial compressive force an infinitely long constrained elastica can support. Adams and Benson (1986) studied the post-buckling behavior of an elastic plate in a rigid channel. Chateau and Nguyen (1991) considered the effect of dry friction on the buckling of a constrained elastica. Adan et al. (1994) showed that when a column with initial imperfection positioned at a distance from a plane wall is subject to compression, contact zones may develop leading to buckling mode transition. Domokos et al. (1997), Holms et al. (1999), and Chai (1998, 2002) investigated the planar buckling patterns of an elastica constrained inside a pair of parallel plane walls. It was observed that both point contact and line contact with the constraint walls are possible. Kuru et al. (2000) studied the buckling behavior of drilling pipes in directional wells. Roman and Pocheau (1999, 2002) used an elastica model to investigate the post-buckling response of bilaterally constrained thin plates subject to a prescribed height reduction. Chen and Li (2007) and Lu and Chen (2008) studied the deformation of a planar elastica inside a circular channel with clearance. Denoel and Detournay (2011) proposed an Eulerian formulation of the constrained elastica problem.

The emphasis of these studies was placed on the static deformations of the constrained elastica. Very often, multiple equilibria under a specified set of loading condition are

© 2012 Chen and Ro, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2012 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

possible. Since only stable equilibrium configurations can exist in practice, there is a need to determine the stability of each of these equilibria in order to predict the behavior of the constrained elastica as the external load varies. For an unconstrained elastica, vibration method is commonly used to determine its stability; see Perkins (1990), Patricio et al. (1998), Santillan, et al. (2006), and Chen and Lin (2008). This conventional method, however, becomes useless in the case of constrained elastica.

Vibration Method in Stability Analysis of Planar Constrained Elastica 45

*x* 

The solid curve represents an asymmetric deformation after the elastica contacts the point

H

C A B

**Figure 1.** An elastica constrained by a space-fixed point at H. The dashed and solid curves represent

*L* 

*h* 

The equilibrium equation at any point (*x*,*y*) of the buckled strip between points A and B, as

*<sup>d</sup> EI Q x F y M ds*

*QA* and *MA* are the internal shear force and bending moment, respectively, provided by the partial clamp at A. (positive when counter-clockwise) is the rotation angle of the strip at point (*x,y*). *EI* is the flexural rigidity of the elastic strip. *s* is the length of the strip measured from point A. For convenience we introduce the following dimensionless

> <sup>2</sup> \* \* <sup>2</sup> ( , ) ( , ), <sup>4</sup> *A A A A <sup>L</sup> Q F QF EI*

 is the mass per unit length of the elastica. t is time and is a circular natural frequency, which will be discussed in the dynamic analysis later. After substituting the above relations into Equation (1), and dropping all the superposed asterisks thereafter for simplicity, we

\*\*\* (, , ) (, , ) , *sxy sxy <sup>L</sup>*

obtain the dimensionless equilibrium equation

 <sup>2</sup> <sup>1</sup> , *EI t t L*

*A A A*

2*L EI* 

(1)

\*

<sup>2</sup> 4 *<sup>A</sup> <sup>A</sup> <sup>L</sup> M M EI*

typical symmetric and asymmetric deformations, respectively.

**3. Load-deflection relation** 

*y* 

*FA*

shown in Figure 1, can be written as

parameters (with asterisks):

H. Other deformation patterns may also exist, which will be discussed later.

*L/2* 

The difficulty of the conventional vibration method arises from the existence of unilateral constraints. A unilateral constraint is capable of exerting compressive force onto the structure, but not tension. Mathematically, this type of constraints can be represented by a set of inequality equations. This poses challenges in determining the critical states of the loaded structure. In order to overcome this difficulty, the conventional stability analysis needs modification. In this chapter, we introduce a vibration method which is capable of determining the stability of a constrained elastica once the equilibrium configuration is known. The key of solving the vibration problem in constrained elastica is to take into account the sliding between the elastica and the space-fixed unilateral constraint during vibration.

In this chapter, we consider the vibration of an elastica constrained by a space-fixed point constraint. This particular constrained elastica problem is used to demonstrate the vibration method which is suitable to analyze the stability of a structure under unilateral constraint. In Section 2, we describe the studied problem in detail. In Section 3, we describe the static load-deflection relation. In Section 4, we introduce the theoretical formulation of the vibration method. In Section 5 an imperfect system when the point constraint is not at the mid-span is analyzed. In Section 6, several conclusions are summarized.
