Analytical Approximate Solution for Large Post-Buckling Behavior of a Fixed-Pinned Beam Subjected to Terminal Force with Shear Force Effect

Based on Euler–Bernoulli beam theory, this paper investigates large post-buckling deformation of a slender elastic beam with fixed-pinned end. Owing to the asymmetric boundary conditions, it is difficult to establish analytic solution. Based on the Maclaurin series expansion and orthogonal Chebyshev polynomials, the governing differential equation with both sinusoidal nonlinearity and cosinusoidal nonlinearity can be reduced to a polynomial equation, and the geometry condition with sinusoidal nonlinearity can also be simplified to be a cubic polynomial integral equation. The admissible lateral displacement function to satisfy the fixed-pinned boundary conditions is derived in an elegant way. The analytical approximations are obtained with the harmonic balance method. Two approximate formulae for axial load and lateral load are established for small as well as large angle of rotation at the pinned end. These approximate solutions show excellent agreement with those of the shooting method for a large range of the rotation angle at the pinned end. Moreover, due to brevity of expressions, the present analytical approximate solutions are convenient to investigate effects of various parameters on the large post-buckling response of fixed-pinned beams.