Heavy Elastica soil-supported with lifting load and bending moment applied to an end: a new analytical approach for very large displacements and experimental validation

The Heavy Elastica problem is a classical issue and has been addressed by many authors. The only analytical approach applied is the perturbative method, and the expansion through Taylor's series is the most recurrent technique. This transforms the nonlinear differential problem into a system of nonlinear (trigonometric) algebraic equations to compute the expansion coefficients. The main disadvantage is that the size of the obtained nonlinear systems is equal to the degree of the series expansion of the solution; therefore, to gain a good accuracy, it is necessary to manage the series expansion up to a considerable degree. Possibly for this reason, the use of perturbative techniques has gradually been abandoned and many authors have addressed the problem of Heavy Elastica with numerical methods. In this paper, a new analytical method based on the parabolic curvilinear abscissa mapping is presented. The result is that the problem of the Heavy Elastica takes on the same complexity as the well-known Elastica problem (concentrated forces and moment applied) and is therefore solved. The proposed method is compared with the Runge-Kutta integration approach and finite element results, showing the correctness of the solution suggested and its remarkable computational advantage. A further comparison is carried out by means of a number of experimental tests that agree with the analytical expected values. By expressing the proposed solution in a dimensionless form, Design Charts are given; they provide the results in the whole field of the domain, so that both kinematic and static variables can be deduced without computer aid.