ABSTRACT
This chapter presents the derivation of the equation of equilibrium of a beam with bending stiffness under an axial load by combining earlier results for strings and bars, including uniform and non-uniform cases. The linear (non-linear) boundary conditions at the free end of a cantilever beam are used for the comparison of linear and non-linear buckling, including the lowest-order non-linear approximation and non-linear corrections of all orders. The chapter illustrates that both for the cantilever beam and other types of support the critical buckling load is not changed in the lowest order non-linear approximation because of the onset of buckling at a linear level. It also shows that non-linear effects do change the shape of the buckled elastic beam by the addition of harmonics to the fundamental buckled mode. A beam under traction load does not buckle in the elastic case; that is, it remains straight unless transverse loads are applied.
