Stability and minimum lateral bracing for stepped columns with semi-rigid connections including shear effects: i) theory

A complete column classification and the corresponding stability equations for single stepped columns with sidesway inhibited, partially inhibited, and uninhibited, subjected to concentrated axial loads located at the ends and at the intermediate joint including semi-rigid connections and shear force effects are presented 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 (Ψ) 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. Four different types of divergent instability are possible for a single-stepped column subjected to concentric axial loads: 1) buckling with sidesways between the two ends and intermediate joint totally inhibited; 2) buckling with sidesway between the two ends totally uninhibited; 3) buckling with sidesway between the bottom end and intermediate joint totally inhibited; and 4) buckling with sidesways between the two ends and intermediate joint uninhibited or partially inhibited. The stability analysis of a single-stepped column consists of determining the eigenvalue of a 2x2 matrix for the first three types of buckling just mentioned and of a 3x3 matrix for members buckling with sidesways between the two ends and intermediate joint uninhibited or partially inhibited. The definite criterion on the minimum stiffness of lateral bracings for single-stepped columns is also presented. The proposed method is general and can be extended to multi-stepped columns.A complete column classification and the corresponding stability equations for single stepped columns with sidesway totally inhibited, partially inhibited, and uninhibited subjected to concentrated axial loads located at the ends and at the intermediate joint including semi-rigid connections and shear force effects are presented 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 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. Four different types of divergent instability are possible for a single stepped column subjected to concentric axial loads: 1) buckling with sidesways between the two ends and intermediate joint totally inhibited; 2) buckling with sidesway between the two ends totally uninhibited; 3) buckling with sidesway between the bottom end and intermediate joint totally inhibited; and 4) buckling with sidesways between the two ends and intermediate joint uninhibited or partially inhibited. The stability analysis of a single stepped column consists in determining the eigenvalue of a 2x2 matrix for the first three types of buckling just mentioned and of a 3x3 matrix for members buckling with sidesways between the two ends and intermediate joint uninhibited or partially inhibited. Definite criterion on minimum stiffness of lateral bracings for single stepped columns is also presented. The proposed method is general and can be extended to multi-stepped columns