Direct Derivation of the Invariant Imbedding Equations for Beams from a Variational Principle.

by D. W. Alspaugh, Robert E. Kalaba

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A demonstration of the application of invariant imbedding techniques to a problem involving the equilibrium configuration of a beam. The equilibrium configuration of a beam supporting a distributed load, free at one end and clamped at the other, is characterized by a minimum of potential energy. Using the traditional reasoning leads to the formulation of an unstable two-point boundary-value problem for a fourth-order Euler equation. This Memorandum shows that the solution of the minimization problem can be characterized by an initial-value problem. Relationships between the set of invariant imbedding equations and the Euler equations are described. An analytic solution to a simple problem is given to demonstrate the technique. 19 pp. Refs. (KB)

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