Qualitative Convergence in the Discrete Approximation of the Euler Problem*

摘要

There are two mathematically rigorous ways to derive Euler's differential equation of the elastica. The first is to start from integral rules and use variational principles, whereas the second is to regard the continuous rod as a limit of a discrete sequence of elastically connected rigid elements when the length of the elements decreases to zero. Discrete models of the Euler buckling problem are investigated. The global numbersof solutions of the boundary-value problem is expressed as a function of the number of elements in the discrete model,s=s(n), at constant loadingP. The functionss(n) are described by the characteristic parametersn1andn2, and graphs ofn1(P) andn2(P) are plotted. Observations related to these diagrams reveal interesting features in the behavior of the discrete model, from the point of view of both theory and application.