Optimal design of cantilevered elastica for minimum tip deflection under self-weight
The optimal distribution of material to minimize the vertical deflection of the free end of a horizontal cantilever is determined. The beam is only subjected to its own weight. Large deflections are considered, and the structure is modeled as an inextensible elastica. A minimum-area constraint is included, and is active in a region near the tip. After the problem is formulated, numerical results are obtained with the use of a shooting method. The moment of inertia is assumed to be proportional to the area or its square or cube. The results depend on this relationship, the minimum-area constraint, and a nondimensional parameter depending on the beam's density, length, and modulus of elasticity. In the numerical results presented, if the minimum area is 1/20 of the area of the uniform beam, the tip deflection for the optimal design is 78-89% smaller than that for the uniform beam. An experiment is conducted and the data are in close agreement with the numerical results.
cantilever elastica self-weight large deflections optimal beam minimum deflection