Slope-deflection equations for stability and second-order analysis of Timoshenho beam–column structures with semi-rigid connections

A new set of slope-deflection equations for Timoshenko beam–columns of symmetrical cross section with semi-rigid connections that include the combined effects of shear and bending deformations, and second-order axial load effects are developed in a classical manner. The proposed method that also includes the effects of the shear component of the applied axial force as the member deflects laterally (Haringx Model) has the following advantages: (1) it can be utilized in the stability and second-order analyses of framed structures made up of Timoshenko beam–columns with rigid, semi-rigid, and simple end connections; (2) the effects of semi-rigid connections are condensed into the slope-deflection equations for tension or compression axial loads without introducing additional degrees of freedom and equations; (3) it is more accurate than any other method available and capable of capturing the phenomena of buckling under axial tension forces; and (4) it is powerful, practical, versatile and easy to teach. Analytical studies indicate that shear deformations increase the lateral deflections and reduce the critical axial loads of framed structures made of members with low shear stiffness. The effects of shear deformations must be considered in the analysis of beam–columns with relatively low effective shear areas (like laced columns, columns with batten plates or with perforated cover plates, and columns with open webs) or with low shear stiffness (like elastomeric bearing and short columns made of laminated composites with low shear modulus G when compared to their elastic modulus E ) making the shear stiffness GA s of the same order of magnitude as EI / L 2 . The shear effects are also of great importance in the static, stability and dynamic behavior of laminated elastomeric bearings used for seismic isolation of buildings. Four comprehensive examples are included that show the effectiveness of the proposed method and equations.