Equilibrium Configurations of Cantilever under Terminal Loads

TL;DR

This paper derives an exact analytical solution for the equilibrium shapes of a cantilever under various terminal loads using Jacobi elliptical functions, providing detailed formulas and numerical illustrations.

Contribution

It introduces a comprehensive analytical approach to determine cantilever configurations under inclined forces and torques, including multiple load scenarios and characteristic dimension formulas.

Findings

Exact solutions for cantilever shapes under different loads

Numerical examples illustrating deformed shapes

Formulas for characteristic dimensions of elastica

Abstract

The paper provides an exact analytical solution for equilibrium configurations of cantilever rod subject to inclined force and torque acting on its free end. The solution is given in terms of Jacobi elliptical functions and illustrated by several numerical examples and several graphical presentations of shapes of deformed cantilever. Possible forms of cantilever underlying elastica are discussed in details and various simple formulas are given for calculation of characteristic dimensions of elastica. For the case when cantilever is subject only to applied force four load conditions are discussed: follower load problem, load determination problem, conservative load problem and rotational load problem. For all the cases the formulas or effective procedure for solution is given.