Second-order Stiffness Matrix and Loading Vector of a Tapered Rectangular Timoshenko Beam-column With Semirigid Connections

The second-order stiffness matrix and loading vector of a linearly tapered Timoshenko beam-column with rectangular cross-section, constant width and generalized end conditions subject to a constant axial load is derived. The proposed model includes the simultaneous effects of: (1) axial load (tension or compression) applied at both ends; (2) transverse load; (3) generalized end conditions; (4) bending and shear deformation along the member; (5) shear stress correction factor induced by the tapering of the cross section; and (6) the shear force induced by the applied axial load as the member deflects laterally using two different approaches, those by Engesser and Haringx which are proportional to the total slope (dy/dx) and to the angle of rotation (theta) along the span of the member, respectively. The governing differential equations are solved by the power series method. The concept of fixity factor of tapered Timoshenko beam-columns with semirigid end connections is introduced. The proposed method and corresponding equation are capable of determining the static and stability behavior of elastic framed structures made of tapered beam-columns with semi-rigid connections using a single segment per element. Three comprehensive examples are presented that show the accuracy and validity of the proposed method and the calculated results are compared with those obtained using ABAQUS.