Analytical approximations to large hydrothermal buckling deformation of a beam

This paper presents analytical approximate solutions for the geometrically nonlinear, large deformation postbuckling of a linear-elastic and hygrothermal beam with axially nonmovable pinned-pinned ends and subjected to a significant increase in swelling. The solution for the limiting case of a string is also obtained. Analytical approximations to large deformation of the beam are established by combining the Newton's method with the method of harmonic balance. In contrast to the classical method of harmonic balance, the linearization is performed prior to proceeding with harmonic balancing, thus resulting in a set of linear algebraic equations instead of one of nonlinear algebraic equations. We are hence able to establish analytical approximate solutions. These approximate solutions show excellent agreement with the referenced solution obtained by the shooting method, and are valid for a small as well as a large angle of rotation at the end of the beam.