A general method for analyzing moderately large deflections of a non-uniform beam: an infinite Bernoulli–Euler–von Kármán beam on a nonlinear elastic foundation

• Published in: Acta mechanica Vol. 225; no. 7; pp. 1967 - 1984
• Main Author: Jang, T. S.
• Format: Journal Article
• Language: English
• Published: Vienna Springer Vienna 01.07.2014; Springer; Springer Nature B.V
• Subjects: Beams (structural); Classical and Continuum Physics; Control; Convergence; Deflection; Differential equations; Dynamical Systems; Elastic foundations; Engineering; Engineering Thermodynamics; Equivalence; Foundation construction; Foundations; Heat and Mass Transfer; Integral equations; Materials elasticity; Mathematical analysis; Mechanics; Nonlinear differential equations; Nonlinear equations; Nonlinearity; Nonuniformity; Rigidity; Solid Mechanics; Theoretical and Applied Mechanics; Vibration; Base Point; Nonlinear Differential Equation; Geometrical Nonlinearity; Steady State Error; Elastic Foundation
• ISSN: 0001-5970; 1619-6937
• DOI: 10.1007/s00707-013-1077-x

The present paper concerns a semi-analytical procedure for moderately large deflections of an infinite non-uniform static beam resting on a nonlinear elastic foundation. To construct the procedure, we first derive a nonlinear differential equation of a Bernoulli–Euler–von Kármán “non-uniform” beam on a “nonlinear” elastic foundation, where geometrical nonlinearities due to moderately large deflection and beam non-uniformity are effectively taken into account. The nonlinear differential equation is transformed into an equivalent system of nonlinear integral equations by a canonical representation. Based on the equivalent system, we propose a method for the moderately large deflection analysis as a general approach to an infinite non-uniform beam having a variable flexural rigidity and a variable axial rigidity. The method proposed here is a functional iterative procedure, not only fairly simple but straightforward to apply. Here, a parameter, called a base point of the method, is also newly introduced, which controls its convergence rate. An illustrative example is presented to investigate the validity of the method, which shows that just a few iterations are only demanded for a reasonable solution.