Large deflection and postbuckling behavior of Timoshenko beam–columns with semi-rigid connections including shear and axial effects

The nonlinear large deflection-small strain analysis and postbuckling behavior of Timoshenko beam–columns of symmetrical cross section with semi-rigid connections subjected to conservative and non-conservative end loads (forces and moments) including the combined effects of shear, axial and bending deformations, axial load eccentricities, lateral bracing and out-of-plumbness are developed in a simplified manner. A new set of stability functions based on the “modified shear equation” that includes the effects of shear deformations and the shear component of the applied axial forces is derived. Also, an expression for the axial displacement δ b caused by the “bowing” of the beam–column subjected to end forces and moments with generalized end conditions is derived in a classic manner. The proposed method and corresponding nonlinear equations, although approximate, can be used in the tension and compression stability and nonlinear large deflection-small strain elastic analyses of Timoshenko beam–columns with rigid, semi-rigid, and simple connections. Analytical studies indicate that shear deformations increase the longitudinal and transverse deflections and reduce the buckling axial load capacities of beam–columns. The effects of shear deformations must be considered in the analysis of beam–columns with relatively low effective shear areas (like in short laced columns, columns with batten plates and with open webs) or low shear stiffness (like elastomeric bearings and short columns made of laminated composites with low shear modulus G when compared to their elastic modulus E making the shear stiffness G A s of the same order of magnitude as E I / h 2 ). The shear effects are also of great importance in the tension and compression 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.