Postbuckling behavior of variable-arc-length elastica connected with a rotational spring joint including the effect of configurational force

Abstract

This paper aims to study the behavior of a variable-arc-length (VAL) elastica subjected to the end loading, where a rotational spring joint is placed within the span length of the elastica. One end of the elastica is rested on the pinned support while the other end is placed into the sleeve support. The length of the elastica can be fed into the system through sleeve support by the end thrust where the effect of configurational force has been considered. A rotational spring joint is located within the span length of the elastica. From the equilibrium equations, moment–curvature expression, geometric relations, and boundary conditions, the closed-form solution in terms of elliptic integral of the first and second kinds can be demonstrated. The results obtained from elliptic integral method are validated with those from the shooting method and they are in excellent agreement. In order to interpret the behavior of the elastica, load–deflection curves and equilibrium shapes are established. Interesting features of the results are demonstrated. Particularly, when the stiffness of the spring joint becomes zero, the secondary buckling and the multiple equilibrium shapes can be captured in which the stable equilibrium shapes can be evaluated by using the vibration analysis. For a low value of the stiffness of the spring joint, the elastica has a possibility to exhibit the hardening behavior. When the stiffness of the spring joint becomes large, the elastica shows the softening behavior and its shape is identical to a single portion of VAL elastica.

Appendices

Appendix 1: Secondary buckling

In order to compute the secondary buckling, the following free body diagrams should be drawn. Regarding to the small deflection theory, the end moment at the remote end becomes a very small value so that the configurational force can be neglected. In Fig. 8a, the reference frame is attached to the hinged joint.

Fig. 8

Free body diagram of the elastica during buckling: a right portion; b left portion

The bending moment of free body diagram in Fig. 8a can be expressed by

$$\bar{M} = - \bar{P}\bar{y} + \bar{M}_{B} \bar{x}.$$

(37)

By considering a small deflection, the moment–curvature relationship can be written by

$$\frac{{d^{2} \bar{y}}}{{d\bar{x}^{2} }} = \bar{M}.$$

(38)

By substituting Eq. (37) into Eq. (38), the following equation can be obtained

$$\frac{{d^{2} \bar{y}}}{{d\bar{x}^{2} }} + \bar{P}\bar{y} = \bar{M}_{B} \bar{x}.$$

(39)

From Eq. (39), the general solution takes the following form

$$\bar{y} = A\cos \lambda \bar{x} + B\sin \lambda \bar{x} + \frac{{\bar{M}_{B} }}{{\bar{P}}}\bar{x},$$

(40)

where $\lambda^{2} = \bar{P}$.

From Fig. 8b and considering the small deflection, the angle $\theta$ may be expressed by

$$\theta = \frac{{\bar{M}_{B} }}{{\bar{P}}}.$$

(41)

Hence, the vertical displacement from the pinned support to the hinged joint $\bar{\delta }$ is

$$\bar{\delta } = \alpha \frac{{\bar{M}_{B} }}{{\bar{P}}}.$$

(42)

Subsequently, the boundary conditions for this problem can be established by

$$\bar{y}\left( 0 \right) = 0,\bar{y}^{\prime}\left( {1 + \alpha } \right) = 0,\bar{y}\left( {1 + \alpha } \right) = \alpha \frac{{\bar{M}_{B} }}{{\bar{P}}}.$$

(43a-c)

After applying the boundary conditions in Eqs. (43a–c), the characteristic equation for solving the secondary buckling load can be written by

$$- \frac{1}{{\sqrt {\bar{P}} }}\tan \left( {\sqrt {\bar{P}} \left( {1 + \alpha } \right)} \right) + 1 = 0.$$

(44)

The buckling load can be calculated by using Eq. (44) with the Newton–Raphson method.

Appendix 2: Stability evaluation by using the vibration analysis

In the case of $\bar{\beta } = 0$, there are multiple equilibrium shapes for a given value of total arc-length such as shapes #1, #2, and #3 (see Fig. 6). It is interesting to investigate the stability of each equilibrium shape. In order to evaluate the stability of each shape, the vibration analysis is employed to observe the natural frequency of the fundamental mode of the elastica. The stable shape of the elastica can be judged by the presence of positive value of the natural frequency ($\bar{\omega }^{2} > 0$). While the unstable shape of the elastica can be found by negative value of the natural frequency ($\bar{\omega }^{2} < 0$). The equations of motion (neglected the rotary inertia) of the elastica under some small vibration can be obtained from dynamic equilibrium of the elastica segment shown in Fig. 9.

Fig. 9

Dynamic equilibrium of the elastica segment

From Fig. 9, the equations of motion can be written in non-dimensional form by

$$\bar{F}_{x}^{\prime } = - \frac{{\partial^{2} \bar{x}}}{{\partial \bar{t}^{2} }},$$

(45)

$$\bar{F}_{y}^{\prime } = - \frac{{\partial^{2} \bar{y}}}{{\partial \bar{t}^{2} }},$$

(46)

$$\bar{M}^{\prime} = \bar{F}_{y} \cos \theta - \bar{F}_{x} \sin \theta ,$$

(47)

$$\theta^{\prime} = \bar{M},$$

(48)

$$\bar{x}^{\prime} = \cos \theta ,$$

(49)

$$\bar{y}^{\prime} = \sin \theta .$$

(50)

The non-dimensional terms of displacements $\left( {\bar{x} \, \text{and} \, \bar{y}} \right)$, horizontal and vertical forces ($\bar{F}_{x}$ and $\bar{F}_{y}$), and moment $\bar{M}$ have already been introduced in Sect. 3. It should be noted that the horizontal force in static equilibrium $\bar{F}_{x}$ consists of the force $\bar{P}$ and the configurational force $\bar{M}_{B}^{2} /2EI$. In this section, the non-dimensional of time $\bar{t}$ is defined by $\bar{t} = \left( {t/L^{2} } \right)\sqrt {EI/\mu }$ in which $\mu$ is mass distribution of the elastica. Under a small vibration, the displacement $\left( {\bar{x} \, \text{and} \, \bar{y}} \right)$, slope $\theta$, force $\bar{F}$, and moment $\bar{M}$ are perturbed from its equilibrium position by

$$\begin{aligned} \bar{x} & = \bar{x}_{e} + \bar{x}_{d} \sin \bar{\omega }\bar{t},\quad \bar{y} = \bar{y}_{e} + \bar{y}_{d} \sin \bar{\omega }\bar{t},\quad \theta = \theta_{e} + \theta_{d} \sin \bar{\omega }\bar{t}, \\ \bar{F}_{x} & = \bar{F}_{xe} + \bar{F}_{xd} \sin \bar{\omega }\bar{t},\quad \bar{F}_{y} = \bar{F}_{ye} + \bar{F}_{yd} \sin \bar{\omega }\bar{t},\quad \bar{M} = \bar{M}_{e} + \bar{M}_{d} \sin \bar{\omega }\bar{t}, \\ \end{aligned}$$

(51a-f)

where $\left( \cdot \right)_{e}$ and $\left( \cdot \right)_{d}$ are referred to the static and dynamic states, respectively. The natural frequency is presented in non-dimensional term by $\bar{\omega } = \omega L^{2} \sqrt {\mu /EI}$. By substituting Eq. (51) into Eqs. (45)–(50) and using the linearization process, a system of differential equations in terms of dynamic parameters for the vibration of the elastica about its equilibrium position can be expressed by

$$\bar{F}_{xd}^{\prime } = \bar{\omega }^{2} \bar{x}_{d} ,$$

(52)

$$\bar{F}_{yd}^{\prime } = \bar{\omega }^{2} \bar{y}_{d} ,$$

(53)

$$\bar{M}_{d}^{\prime } = - \left( {\bar{F}_{ye} \theta_{d} + \bar{F}_{xd} } \right)\sin \theta_{e} + \left( { - \bar{F}_{xe} \theta_{d} + \bar{F}_{yd} } \right)\cos \theta_{e} ,$$

(54)

$$\theta_{d}^{\prime } = \bar{M}_{d} ,$$

(55)

$$\bar{x}_{d}^{\prime } = - \theta_{d} \sin \theta_{e} ,$$

(56)

$$\bar{y}_{d}^{\prime } = \theta_{d} \cos \theta_{e} .$$

(57)

In Eqs. (52)–(57), there are 5 unknown parameters for a prescribed value of the vertical disturbance at the hinge joint $\bar{y}_{d} \left( {\bar{s} = \alpha } \right) = \delta_{yd}$ where they are $\bar{F}_{xd} \left( {\bar{s} = 0} \right)$, $\bar{F}_{yd} \left( {\bar{s} = 0} \right)$, $\theta_{d} \left( {\bar{s} = 0} \right)$, $\Delta \theta_{d} \left( {\bar{s} = \alpha } \right)$ and $\bar{\omega }^{2}$. Hence, the five constraint conditions have been presented

$$\bar{x}_{d} \left( {\bar{s} = \bar{s}_{t} } \right) = 0,\quad \bar{y}_{d} \left( {\bar{s} = \bar{s}_{t} } \right) = 0,\quad \theta_{d} \left( {\bar{s} = \bar{s}_{t} } \right) = 0,\quad \bar{M}_{d} \left( {\bar{s} = \alpha } \right) = 0,\quad \bar{y}_{d} \left( {\bar{s} = \alpha } \right) - \delta_{yd} = 0.$$

(58a-e)

After minimization of Eq. (58), the five dynamic parameters can be obtained.