Matrix method for stability and second-order analysis of Timoshenko beam-column structures with semi-rigid connections

The first- and second-order stiffness and load matrices of a beam-column of symmetric cross section with semi-rigid connections including the effects of end axial loads (tension or compression) and shear deformations along the member are derived in a classical manner. Both matrices can be used in the stability, first- and the second-order elastic analyses of framed structures made of Timoshenko beam-columns with rigid, semi-rigid and simple connections of symmetric cross sections. The “modified” stability approach based on Haringx’s model described by Timoshenko and Gere [1] is utilized in all matrices. The model proposed which is an extension of that presented by Aristizabal-Ochoa [2] captures the models of beams and beam-columns based on the theories of Bernoulli–Euler, Timoshenko, and bending and shear. The closed-form second-order stiffness matrix and load vector derived and presented in this paper find great applications in the stability and second-order analyses of structures made of beam-columns with relatively low shear stiffness such as orthotropic composite polymers (FRP) and multilayer elastomeric bearings commonly used in seismic isolation of buildings. The effects of torsional warping along the members are not included. Analytical studies indicate that the buckling load and the stiffness of framed structures are reduced by the shear deformations along the members. In addition, the phenomenon of buckling under axial tension forces in members with relatively low shear stiffness is captured by the proposed equations. Tension buckling must not be ignored in the stability analysis of beam-columns with shear stiffness GAs of the same order of magnitude as EI/h2. The validity of both matrices is verified against available solutions of stability analysis and nonlinear geometric elastic behavior of framed structures with semi-rigid connections using a single segment for each beam and column member without introducing additional degrees of freedom. Four examples are included that demonstrate the simplicity, effectiveness and accuracy of the proposed method and corresponding matrices.

► The stability and second-order response of framed structures are affected by many factors.
► Shear deformations increase the lateral deflections and reduce the critical axial loads.
► They must be considered when beam-columns with relatively low shear stiffness GAsGAs are used.
► End moments and lateral deflections are increased significantly by geometric nonlinear effects.