Multiple beam systems - How to obtain the PDEs/BCs? How to solve?

The paper considers an N-fold beam system.Commonly, Hamilton’s principle is used to derive the partial differential equations (PDEs) along with its boundary conditions (BCs). An initiatory example (N=2,plane motion), however, already shows that such a procedure results in an untolerable effort. Instead, a clear structured algorithm for the PDEs and the corresponding BCs is derived. In order to demonstrate the procedure (based on Lagrange’s principle), it is restricted to plane motions and eventually expanded to the general case. A single beam with only elastic deflections as well as one with only rigid body motions leads to the basic properties that are eventually combined for the general motion. A solution may be obtained via a Ritz series expansion along with an “order-n-algorithm”. This procedure does not need the knowledge of the BCs, and the question arises wether we need them at all. The answer, however, is twofold: “Yes” for the generation of shape functions, arising from simplified model descriptions and “No” for the over all problem. Combination of both may deliver overwhelming computational time advantages.