Deformation and Stability of an Elastica under a Point Force and Constrained by a Flat Surface
Date Issued
2010
Date
2010
Author(s)
Wu, Hsu-Hao
Abstract
This thesis is divided into two parts. In the first part we study the deformation and stability of a pinned elastica under a point force moving quasi-statically from one end to the other. The elastica is constrained by a rigid plane wall containing the two ends. Three types of equilibrium configurations can be found; they are non-contact, one-point contact, and one-line contact on the side. A vibration method is adopted to determine the stability of the calculated deformations. In order to take into account the variation of the contact region between the elastica and the plane wall during vibration, an Eulerian version of the governing equations is adopted. It is found that all the point-contact deformations are unstable. On the other hand, there are two different mechanisms a line-contact deformation becomes unstable; one through a secondary buckling and the other through a limit-point bifurcation. In the secondary buckling, the length of the line-contact segment and the axial force satisfy the Euler buckling criteria for a pinned-clamped column. On the other hand, when a line-contact deformation becomes unstable via a limit-point bifurcation, the axial force does not exceed the Euler buckling load. The theoretical predictions are confirmed by experimental observations.
In the second part of the thesis, we adopt a small deformation analysis and replace the rigid surface with an elastic foundation. We study the deformation and stability of a pinned buckled beam under a point force. The buckled beam is constrained by a tensionless elastic foundation, which is flat before deformation. From static analysis, we found a total of five different deformation patterns; they are (1) non-contact, (2) full contact; (3) one-sided contact; (4) isolated contact in the middle, and (5) two-sided contact. For a specified set of parameters, there may coexist multiple equilibria. In order to predict the response of the buckled beam-foundation system as the point force moves from one end to the other, we have to determine the stability of these equilibrium configurations. In order to achieve this, a vibration method is adopted to calculate the natural frequencies of the system, taking into account the slight variation of the contact range between the buckled beam and the tensionless foundation during vibration. It is concluded that among all five deformation patterns, deformations (1), (2), (3), and (4) may become stable for certain loading parameters. In the extreme case when the foundation is rigid, on the other hand, only two types of solutions are stable; i.e., deformations (1) and (3).
Subjects
Elastica
Point force
Constraining wall
Vibration and stability
Buckled beam
Tensionless elastic foundation
Type
thesis
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