Buckling and postbuckling of extensible, shear-deformable beams: Some exact solutions and new insights

This paper presents exact solutions for the buckling loads and postbuckling states of extensible, shear deformable beams. The governing equation for the large-amplitude lateral deformation of beams in compression is expanded in Taylor series up to the cubic nonlinearity. Closed-form solutions in terms of the axial and shear stiffnesses are developed for statically determinate and statically indeterminate beams. Namely, pinned–pinned, cantilevered, clamped–clamped, and clamped–pinned beams are considered with the loaded end is a roller that is able to slide. The postbuckling response under a given axial load is exactly derived. The dependence of the buckling load on the length-to-radius of gyration is discussed. It is shown that the extensibility and the shear deformation significantly affect the buckling loads and the postbuckling response. For conventional materials with positive Poisson’s ratio, the inclusion of the axial and shear deformation results in a meaningful reduction of the buckling load. It is further shown that the buckling load can be enhanced by designing artificial metamaterials materials with an effective negative Poisson’s ratio.