Closed-Form Solution of Large Deflected Cantilever Beam a gainst Follower Loading Using Complex Analysis

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.

Large Deflection Analysis of Prismatic Cantilever Beam Comparatively by Using Combing Method and Iterative DQM

“…However, some of these studies weren't provide proper accuracy for the solution of problem. Also, some of them ensured very high accuracy but algorithms of these studies caused very high calculation time or difficulty for using a daily solution of the large deflection problem [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22]. Navaee and Elling investigated equilibrium configuration of a prismatic cantilever beam for different loading conditions using the Elliptic Integral Method.…”