Analytical treatment of equilibrium configurations of cantilever under terminal loads using Jacobi elliptical functions

The paper provides an exact analytical solution for the equilibrium configurations of a cantilever rod subject to inclined force and tip moment acting on its free end. The solution is given in terms of Jacobi’s elliptical functions and illustrated by several numerical examples and several graphical presentations of shapes of deformed cantilevers. Possible forms of the underlying elastica of a cantilever are discussed in detail, and various simple formulas are given for calculating the characteristic dimensions of the elastica. For the case when a cantilever is subject only to applied force, three load conditions are discussed: the follower load problem, the load determination problem, and the conservative load problem. For all cases, either a formula or an effective procedure for determining the solution is provided. In particular, a new efficient procedure is given to determine all possible equilibrium shapes in the case of the conservative load problem.