Second-order stiffness matrix and load vector of an imperfect beam-column with generalized end conditions on a two-parameter elastic foundation

The stiffness matrix and load vector for an imperfect Euler–Bernoulli beam-column with generalized end conditions subjected to axial and transverse loads are presented. The proposed method includes the effects of initial imperfections (i.e., out-of-straightness, out-of-plumbness, and axial load eccentricities at both ends), a two-parameter elastic foundation, partially restrained sidesway and rotational semirigid connections at both ends, and transverse and end axial loads (tension or compression) on the stiffness matrix and load vector. The proposed method is capable of solving the second-order response and lateral stability, and capturing the phenomenon of deflection reversals in 2D framed structures by using a single segment per element. The effects of shear deformations and torsion along the member are not included in the present research. Three comprehensive examples are provided to show the effectiveness and validity of the proposed matrix method.