Exact solutions for the buckling and postbuckling of shear-deformable beams

The buckling and postbuckling of beams is revisited taking into account both the influence of axial compressibility and shear deformation. A theory based on Reissner’s geometrically exact relations for the plane deformation of beams is adopted, in which the stress resultants depend linearly on the generalized strain measures. The equilibrium equation is derived in a general form that holds for the statically determinate and indeterminate combinations of boundary conditions representing the four fundamental buckling cases. The eigenvalue problem is recovered by consistent linearization of the governing equations, the critical loads at which the trivial solution bifurcates are determined, and the influence of shear on the buckling behavior is investigated. By a series of transformations, the equilibrium equation is rearranged such that it allows a representation of the solution in terms of elliptic integrals. Additionally, closed-form relations are provided for the displacement of the axis, from which buckled shapes are eventually obtained. Even for slender beams, for which shear deformation can usually be neglected, both the buckling and the postbuckling behavior turn out to be affected by shear not only quantitatively, but also qualitatively.