Exact Second-Order Dynamic Stiffness Matrix for Torsion and Warping of Axially-Loaded Members

In order to establish a three-dimensional dynamic stability analysis of beam-column systems, this paper presents an exact second-order dynamic stiffness matrix of axially-loaded members considering the natural frequency and axial compression effect. First, dynamic equilibrium analysis is developed for torsion and warping of an axially-loaded member. A fourth-order differential equation governing twist angle is derived, where a dimensionless quantity [Formula: see text] associated with the natural frequency is defined for the dynamic effect, and another dimensionless quantity [Formula: see text] indicates the Saint Venant torsion and axial load effect. Then, the exact element dynamic stiffness matrix defining the correlation between the element-end torsional stress resultants (torques and bimoments) and the associated deformations (angles and rates of twist) is obtained. Exact element dynamic torsional stiffness matrices for three configurations of warping restraints at the ends of the element are then developed via the matrix structural analysis. Moreover, for structural three-dimensional analysis, a 14-order exact element second-order dynamic stiffness matrix is derived through analogies between dynamic axial–torsional and axial–flexural problems.