Analytical solution of the cantilevered elastica subjected to a normal uniformly distributed follower load

We report the full analytical solution of the large deformations of a cantilevered elastica loaded by a uniformly distributed follower pressure. We consider an unshearable, inextensible and linear elastic rod. We obtain a spatial nonlinear differential equation in the curvatures, analogous to the undamped Duffing oscillator with a constant driving force. We solve such differential equation, obtaining the curvature, although in implicit form, for arbitrarily large values of the load. We are then able to obtain the rotations owing to a change of variables from the curvilinear abscissa to the curvature. This step is somewhat mandatory due to the implicit nature of the solution. Finally, with the same change of variables, it is possible to obtain a closed-form solution for the deformation in Cartesian coordinates. The solutions show that the rod deforms into drop-like shapes. The number of drops is equal to the number of spatial periods of the solution, which goes with q*1/3, with q* a dimensionless load normalised to the bending stiffness. Interestingly, we find that for q* >= 3094:2, the number of drop-like shapes does not increase, but remains three. (C) 2020 Elsevier Ltd. All rights reserved.