**2. Problem description**

Figure 1 shows an inextensible elastic strip with the right end fully clamped at a point B. On the left hand side there is a straight channel with an opening at point A. The distance between points A and B is *L*. Part of the strip is allowed to slide without friction and clearance inside the channel. A longitudinal pushing force *AF* is applied at the left end of the strip inside the channel causing it to buckle in the domain of interest between points A and B. An *xy*-coordinate system is fixed at point A. A point H fixed at position *x L* / 2 and *y h* prevents the elastica from deforming freely after the elastica contacts point H.

The elastic strip is assumed to be straight and stress-free when *AF* =0. The effect of gravity is ignored. The strip is uniform in all mechanical properties along its length. The length and the shape of the elastica in the domain of interest vary as the pushing force *AF* increases. The boundary condition at point A may be called "partially clamped," by which we mean that the strip is allowed to slide freely through the opening A, while the lateral displacement and slope at A are fixed. The dashed and solid curves in Figure 1 represent two typical stages of the elastica deformation when *AF* increases beyond the buckling load. The dashed curve is a symmetric deformation pattern before the elastica contacts the point constraint. The solid curve represents an asymmetric deformation after the elastica contacts the point H. Other deformation patterns may also exist, which will be discussed later.

**Figure 1.** An elastica constrained by a space-fixed point at H. The dashed and solid curves represent typical symmetric and asymmetric deformations, respectively.

#### **3. Load-deflection relation**

44 Advances in Computational Stability Analysis

vibration.

**2. Problem description** 

becomes useless in the case of constrained elastica.

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,

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

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

Figure 1 shows an inextensible elastic strip with the right end fully clamped at a point B. On the left hand side there is a straight channel with an opening at point A. The distance between points A and B is *L*. Part of the strip is allowed to slide without friction and clearance inside the channel. A longitudinal pushing force *AF* is applied at the left end of the strip inside the channel causing it to buckle in the domain of interest between points A and B. An *xy*-coordinate system is fixed at point A. A point H fixed at position *x L* / 2 and

The elastic strip is assumed to be straight and stress-free when *AF* =0. The effect of gravity is ignored. The strip is uniform in all mechanical properties along its length. The length and the shape of the elastica in the domain of interest vary as the pushing force *AF* increases. The boundary condition at point A may be called "partially clamped," by which we mean that the strip is allowed to slide freely through the opening A, while the lateral displacement and slope at A are fixed. The dashed and solid curves in Figure 1 represent two typical stages of the elastica deformation when *AF* increases beyond the buckling load. The dashed curve is a symmetric deformation pattern before the elastica contacts the point constraint.

*y h* prevents the elastica from deforming freely after the elastica contacts point H.

mid-span is analyzed. In Section 6, several conclusions are summarized.

The equilibrium equation at any point (*x*,*y*) of the buckled strip between points A and B, as shown in Figure 1, can be written as

$$EI\frac{d\Theta}{ds} = -\mathcal{Q}\_A \mathbf{x} - F\_A \mathbf{y} + \mathcal{M}\_A \tag{1}$$

*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 parameters (with asterisks):

$$\begin{aligned} \text{(\(\text{s}^\*, \text{x}^\*, y^\*) = \frac{\text{(s, x, y)}}{L}, \text{ (Q.A}^\*, F\_A) = \frac{L^2}{4\pi^2 EI} \text{(Q}\_{A'} F\_A), \text{ } M\_A^\* = \frac{L}{4\pi^2 EI} M\_A \\\\ t^\* = \frac{1}{L^2} \sqrt{\frac{EI}{\mu}} \ t\_{\prime\prime} \text{ } \text{ } \text{\(\text{o}^\* = L^2 \sqrt{\frac{\mu}{EI}}\)} \text{ } \text{\(\text{o}^\* = L^2\text{)}} \end{aligned}$$

 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 obtain the dimensionless equilibrium equation

$$\frac{d\theta}{ds} = 4\pi^2 \left( M\_A - Q\_A x - F\_A y \right) \tag{2}$$

Vibration Method in Stability Analysis of Planar Constrained Elastica 47

(2.03268, 0.0023129) is solid and the upper part is dashed. At the point separating the solid and dashed parts, a symmetry-breaking bifurcation occurs and the elastica evolves to a pair of asymmetric deformations 4(a) and 4(b). As the pushing force continues to increase, it is natural to envision that the elastica may evolve to an "M" shape, i.e., there exist two inflection points in each half of the span, called deformation (3). The load-deflection curve corresponding to deformation (3) starts at ( *AF* , *l* )=(3.97314, 0.0026985) and continues

At point ( *AF* , *l* )=(2.03268, 0.0023129), the symmetric deformation (2) described previously may bifurcate to a pair of asymmetric deformations 4(a) and 4(b). Both of (4a) and (4b) have one inflection point on each of the half span separated by the point constraint. Both (4a) and (4b) start at ( *AF* , *l* )=(2.03268, 0.0023129), while (4a) ends at (2.03545, 0.0028996) and (4b) ends at (2.02600 0.0028996). The slopes of curves (4a) and (4b) are positive and negative throughout, respectively. Stability analysis later indicates that deformation (4b) is unstable while (4a) is stable. As *l* increases further, the pair of deformations (4a) and (4b) evolve to a pair of asymmetric deformations (5a) and (5b). Deformation (5a) has two inflection points on the left span and one inflection point on the right span. On the other hand, deformation (5b) has two inflection points on the right span and one inflection point on the left span. The slope of load-deflection curve of (5b) is negative throughout. On the other hand, the curve of (5a) is of convex shape with the top being at point ( *AF* , *l* ) =(2.03554, 0.0031711), which is very close to the end of curve of deformation (4a). Deformation (5a) is stable first until the curve reaches the top at ( *AF* , *l* ) =(2.03554, 0.0031711). At this critical point, the elastica will jump to another configuration. The load-deflection curves near the symmetry-breaking

The theoretical load-deflection curves shown in Figure 2 give us a mental picture how the elastica evolves as the pushing force *AF* increases. First of all, the elastica remains still when *AF* is smaller than 1, the Euler buckling load. As soon as *AF* reaches 1, the elastica jumps to symmetric deformation (2) in contact with the point constraint. As *AF* increases, a symmetry-breaking bifurcation occurs and the elastica evolves to asymmetric deformation (4a) first and smoothly to (5a). As *AF* continues to increase up to a certain value, a second jump occurs. Following this jump the elastica will eventually settle to a self-contact configuration. This self-contact configuration requires a length increment *l* over 8, which is well beyond the range of Figure 2. The above scenario has been verified experimentally in Chen and Ro (2010). In the next section we describe the vibration method used to determine

As mentioned above, the deformation patterns discussed in Section 3 may not necessarily be stable. If the deformation is unstable, then it can not be realized in practice. In order to study the vibration and stability properties of the elastica, we first derive the equations of motion

of a small element *ds* supported by the point constraint, as shown in Figure 3.

beyond the range of Figure 2. The slope of this curve is slightly negative.

point are magnified and shown in the inset of Figure 2.

the stability of the static deformations.

**4. Vibration and stability analyses** 

**4.1. Lagrangian and Eulerian descriptions** 

The method of static analysis can be found in Chen and Ro (2010). In this section we introduce several deformation patterns of the constrained elastica. All the physical quantities described henceforth are dimensionless.

The length of the elastica being pushed in through the opening is *l l* 1 , where *l* is the dimensionless length of the elastica between points A to B. Figure 2 shows the relation between the edge thrust *AF* and the length increment *l* . The height of the point constraint *h* is 0.03. The dashed and solid curves in this load-deflection diagram represent unstable and stable configurations, respectively. The method used in determining the stability of the static deformation will be described in detail in Section 4.

**Figure 2.** Load-deflection curves for *h*=0.03. The solid and dashed curves represent stable and unstable deformations, respectively.

The symmetric deformation before contact occurs is called deformation (1), whose locus starts at ( *AF* , *l* ) = (1,0) and ends at (0.99668, 0.0022188). The slope of this load-deflection curve is slightly negative. After the middle point C of the elastica touches the point constraint H, the deformation pattern initially remains symmetric, called deformation (2). The load-deflection curve of deformation (2) starts at ( *AF* , *l* )=(0.99668,0.0022188) and ends at (3.97314, 0.0026985). It is noted that the lower part of this load-deflection curve up to (2.03268, 0.0023129) is solid and the upper part is dashed. At the point separating the solid and dashed parts, a symmetry-breaking bifurcation occurs and the elastica evolves to a pair of asymmetric deformations 4(a) and 4(b). As the pushing force continues to increase, it is natural to envision that the elastica may evolve to an "M" shape, i.e., there exist two inflection points in each half of the span, called deformation (3). The load-deflection curve corresponding to deformation (3) starts at ( *AF* , *l* )=(3.97314, 0.0026985) and continues beyond the range of Figure 2. The slope of this curve is slightly negative.

At point ( *AF* , *l* )=(2.03268, 0.0023129), the symmetric deformation (2) described previously may bifurcate to a pair of asymmetric deformations 4(a) and 4(b). Both of (4a) and (4b) have one inflection point on each of the half span separated by the point constraint. Both (4a) and (4b) start at ( *AF* , *l* )=(2.03268, 0.0023129), while (4a) ends at (2.03545, 0.0028996) and (4b) ends at (2.02600 0.0028996). The slopes of curves (4a) and (4b) are positive and negative throughout, respectively. Stability analysis later indicates that deformation (4b) is unstable while (4a) is stable. As *l* increases further, the pair of deformations (4a) and (4b) evolve to a pair of asymmetric deformations (5a) and (5b). Deformation (5a) has two inflection points on the left span and one inflection point on the right span. On the other hand, deformation (5b) has two inflection points on the right span and one inflection point on the left span. The slope of load-deflection curve of (5b) is negative throughout. On the other hand, the curve of (5a) is of convex shape with the top being at point ( *AF* , *l* ) =(2.03554, 0.0031711), which is very close to the end of curve of deformation (4a). Deformation (5a) is stable first until the curve reaches the top at ( *AF* , *l* ) =(2.03554, 0.0031711). At this critical point, the elastica will jump to another configuration. The load-deflection curves near the symmetry-breaking point are magnified and shown in the inset of Figure 2.

The theoretical load-deflection curves shown in Figure 2 give us a mental picture how the elastica evolves as the pushing force *AF* increases. First of all, the elastica remains still when *AF* is smaller than 1, the Euler buckling load. As soon as *AF* reaches 1, the elastica jumps to symmetric deformation (2) in contact with the point constraint. As *AF* increases, a symmetry-breaking bifurcation occurs and the elastica evolves to asymmetric deformation (4a) first and smoothly to (5a). As *AF* continues to increase up to a certain value, a second jump occurs. Following this jump the elastica will eventually settle to a self-contact configuration. This self-contact configuration requires a length increment *l* over 8, which is well beyond the range of Figure 2. The above scenario has been verified experimentally in Chen and Ro (2010). In the next section we describe the vibration method used to determine the stability of the static deformations.
