Coupled lateral–torsional frequencies of asymmetric, three-dimensional frame structures

This paper presents a global analysis approach to the calculation of the natural frequencies of asymmetric, three-dimensional frame structures in which the primary frames run in two orthogonal directions and whose properties may vary through the height of the structure in a step-wise fashion at one or more storey levels. The governing differential equations of a substitute system are formulated using a continuum approach and posed in the form of a simple dynamic member stiffness matrix. Such a formulation allows for the distributed mass and coupled shear–torsion stiffness of the member and thus necessitates the solution of a transcendental eigenvalue problem. The required natural frequencies are finally determined using a stepped cantilever model in conjunction with the Wittrick–Williams algorithm, which ensures that no natural frequencies are missed. When the structure can be represented realistically by a uniform cantilever, solutions can be found easily by hand. A parametric study comprising four, three-dimensional, asymmetric frame structures is given to compare the accuracy of the current approach with that of a full finite element analysis.