Deflection and stability of elastically restrained nonuniform beam

A semiexact solution approach is presented to study the static deflection and stability of a general, elastically end-restrained, nonuniform beam under combined loads. The governing equation is a nonhomogeneous fourth-order ordinary differential equation with variable coefficients. The Green's function for the static deflection and the characteristic equation for the stability problem are concisely expressed in terms of the four fundamental solutions of the governing equation. These fundamental solutions can be obtained approximately through a simple and efficient numerical algorithm. Of the four typical boundary conditions, when placed under the same distributed axial loads, the critical buckling load of the beam with clamped-clamped boundary condition increases most rapidly, and the load of the beam with clamped-free boundary condition increases most slowly. Under the same boundary condition and among the axial loads discussed, the critical buckling load of the beam subjected to linearly distributed follower forces g₀[l - (x/l)] increases more rapidly than that of the beam subjected to the other types of distributed follower forces, as the breadth taper ratio is increased.