Abstract
The literature reveals that the non-conservative deflection of an elastic cantilever beam caused by applying follower tip loading was investigated and solved by various numerical methods like: Runge Kutta, iterative shooting, finite element, finite difference, direct iterative and non-iterative numerical methods. This is due to the fact that the Euler–Bernoulli nonlinear differential equation governing the problem contains the “slope at the free end”, this slope however needs special numerical treatment. On the other hand, some of these methods fail to find numerical solutions for extremely large loading conditions. Hence, this paper is aimed to obtain a closed-form solution for solving the large deflection of a cantilever beam opposed to a concentrated point follower load at its free end. This closed-form solution when compared with other conventional numerical approaches is characterized by simplicity, stability and straightforwardness in getting the beam deflection and slopes even for extremely large loading conditions. The closed-form solution is obtained by applying complex analysis along with elliptic-integral approach. Very good results were obtained when the elastica of the beam compared with that of various numerical methods which are used in analyzing similar problem.
Highlights
Many engineering fields use cantilever beams (CBs) in their applications
Many researchers investigated the large deflection (LD) of CB loaded by a follower load (FL)
The large deflected behavior of a CB subjected to a tip point load is numerically analyzed by applying the static and dynamic stability-criteria to both uniform and non-uniform cross-section
Summary
Many engineering fields use cantilever beams (CBs) in their applications. These attractive applications are based on the elastic behavior of the beam in its large deflection (LD) state. The large deflected behavior of a CB subjected to a tip point load is numerically analyzed by applying the static and dynamic stability-criteria to both uniform and non-uniform cross-section. These numerical approaches used the iterative shooting with Runge-Kutta (RK) method or other techniques introduced by Rao B.N. and Rao G.V., 1987-1988. In the three papers of Shvarsman, 2007, 2009 and 2013, a direct numerical method is applied to study the “elastica” of a straight non-uniform CB under a tip-concentrated load In these papers, the load is always assumed following the beam axis with constant inclination angle. 2013 assessed the validity of the non-iterative direct method for considering the static analysis of the elastic curved CB against a tip point follower force (FF)
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