Second-order slope–deflection equations for imperfect beam–column structures with semi-rigid connections

A new set of second-order slope–deflection equations for Euler–Bernoulli beam–columns of symmetric cross-section including the effects of initial imperfections (i.e., initial curvature, out-of-plumbness and axial load eccentricities) and semi-rigid connections is developed in a classical manner. The proposed method has the following advantages: (1) it can be utilized in the stability and second-order analysis of framed structures made of Euler–Bernoulli imperfect beam–columns with rigid, semi-rigid, and simple connections subject to axial and transverse loads; (2) the effects of semi-rigid connections and member imperfections are condensed into the slope–deflection equations for tension and compression axial loads; (3) it is more accurate than any other method available in the technical literature and capable of capturing not only the first-order and second-order elastic responses of frames made of imperfect beam–columns but also the phenomenon of reversals of deflections along the members as the axial loads are increased; and (4) it is powerful, practical, versatile and easy to apply. Analytical studies indicate that the initial imperfections (a) act as if they were additional transverse loads proportional to the bending stiffness and magnitudes of the imperfections of the corresponding beam–column; and (b) are detrimental to structures, increasing the lateral deflections, moments, and shears, and also reducing the critical axial loads of beam–columns and framed structures. This is particularly critical in structures made of beam–columns with initial crookedness and low stiffness connections subjected to high compressive axial loads. In addition, the effects of initial curvature are amplified by the compressive axial loads applied at the ends of the member. Four comprehensive examples are included that show the effectiveness of the proposed method.