Full Forward Solution of Large Deflection, End Loaded Cantilever Beams Using Elliptic Integrals

Abstract

This paper presents an algorithm to solve for all solutions to the forward problem for large deflections of inextensible end loaded Euler beams, a problem often encountered in compliant mechanism design and analysis. The forward problem is characterized by known end moment and end force (magnitude and direction), and the horizontal, vertical, and rotational deflections of the end of the beam must be found. Previous solutions have relied on the use of numerical solvers, which normally result in finding a single solution, but are unable to find all possible solutions for a given loading condition. The algorithm presented here works by reformulating the problem to have a single unknown, the end angle of the beam. Using this reformulation, a search vector of possible end angles can be used to find all solutions within desired bounds for the rotation of the end of the beam. The results were compared to nonlinear finite element modeling for verification. The results show that the vast majority of possible load conditions result in multiple (at least two) solutions, with larger end forces generally leading to more solutions. This finding suggests that such solutions may be used to design novel multi-stable compliant mechanisms, including the possibility of metamaterials with variable volume.