Modified formulas for maximum deflection of a cantilever under transverse loading

Abstract

Modern problems of aerospace industry require consideration of rods experiencing large deflections. The example of such a problem is development of large scale deployable umbrellatype antennas where rods are structural elements. Development of modern analytic methods in the field of solid mechanics allows to model rod bend shapes and to find expressions for maximum deflection. In addition, the analytic methods make it possible to find a full system of solution branches and all possible equilibrium shapes without significant time-consuming for numerical simulations. Wherein relatively simple methods for determining bending shapes in case of large deflections have significant importance for applied use. Namely, they can be used for preliminary design of complex rod constructions. The paper presents the method for obtaining of modified analytic formulas that enable to determine large deflections of a thin elastic cantilever under transverse loading. The method uses a rod’s arc-length saving condition which is important for applied use. The modified formulas allow to achieve accuracy comparable with exact nonlinear solutions given in terms of elliptic integrals and functions. That fact expands the loading range where the linear theory can be used. The authors considered the following cases: concentrated transverse loading on the free end and combined loading (uniformly distributed loading and concentrated transverse loading on the free end). The comparison with experimental data proved accuracy of the proposed method. In addition, the authors obtained approximate formulas based on the modified formulas. The approximate formulas can be use for engineering applications.

Keywords

• geometrical nonlinearity
• large deformations
• cantilever
• euler elastic