Exact tangent stiffness for imperfect beam-column members

The exact tangent and secant stiffness matrices of an initially curved beam-column element under predominant axial force and end moments are derived. The stability function of solving the differential equilibrium equation for the beam-column element is further extended to allow for the important effect of member initial imperfection. The accuracy of this developed element makes the convergence rate for equilibrium and resistance against divergence batter than that by the conventional cubic element or other currently available elements. The use of a single element per member is adequate enough for the extreme case of a column with both ends tired, in which even two cubic elements cannot generate an accurate result. As a single element can sufficiently model a member, the second-order analysis, the nonlinear integrated design and analysis, and the advanced analysis become simple, reliable, and easy to use for practical design. The present element can also be used as a benchmark element for second-order elastic analysis of 2D and 3D frames and represents the ultimate solution for the imperfect element under Timoshenko's beam-column theory.