Stability of columns with semi-rigid connections including shear effects using Engesser, Haringx and Euler approaches

A complete column classification and the corresponding stability equations that can be used to evaluate the elastic critical axial loads of prismatic columns with sidesway uninhibited, partially inhibited, and totally inhibited including the simultaneous effects of bending and shear deformations and semi-rigid connections are presented and derived using three different approaches. The first two approaches are those by Engesser and Haringx that include the shear component of the applied axial force proportional to the total slope (dy/dx) and to the angle of rotation of the cross-section (ψψ) along the member, respectively. The third approach is a simplified formulation based on the classical Euler theory that includes the effects of shear deformations but neglects the shear component of the applied axial force along the member. Comparisons among the critical loads obtained using these three approaches and those using the classical Euler solutions for classical column cases are presented. Examples that demonstrate the effectiveness and accuracy of the proposed stability equations and the importance of shear deformations and shear component of the applied axial force are presented in detail. These effects must be taken into account particularly in structures made of columns with relatively low effective shear areas AsAs (like laced columns, columns with batten plates or with perforated cover plates, and columns with open webs) or with low shear stiffness GAs of the same order of magnitude as π2EI/h2π2EI/h2 (like short columns made of laminated composites with low shear modulus GG when compared to their elastic modulus EE). These effects become even more significant when the external supports are not perfectly clamped. The effects of semi-rigid connections and those due to shear forces and deformations are condensed into the proposed characteristic equations.