General exact harmonic analysis of in-plane timoshenko beam structures

Abstract

The exact solution for the problem of damped, steady state response, of in-plane Timoshenko frames subjected to harmonically time varying external forces is here described. The solution is obtained by using the classical dynamic stiffness matrix (DSM), which is non-linear and transcendental in respect to the excitation frequency, and by performing the harmonic analysis using the Laplace transform. As an original contribution, the partial differential coupled governing equations, combining displacements and forces, are directly subjected to Laplace transforms, leading to the member DSM and to the equivalent load vector formulations. Additionally, the members may have rigid bodies attached at any of their ends where, optionally, internal forces can be released. The member matrices are then used to establish the global matrices that represent the dynamic equilibrium of the overall framed structure, preserving close similarity to the finite element method. Several application examples prove the certainty of the proposed method by comparing the model results with the ones available in the literature or with finite element analyses.Keywords exact harmonic analysis; Laplace transform; Timoshenko beam; dynamic stiffness matrix; rigid offsets; end release. General exact harmonic analysis of INTRODUCTIONMany modern structures are formed by beam elements. These skeleton like structures are subjected to static and dynamic loads. Their beam members can be of various sizes, including beams with small length to beam height ratio. For the analysis of these structures, it is important to use a more refined beam theory, where the assumption of the cross section to remain plane is not enforced. Besides, harmonic loads can be of high frequency, when then it is important to keep in the beam model the cooperation of the rotatory inertia to the overall structure response. This motivates the use of the Timoshenko beam theory to obtain the damped steady state response for general plane frames subjected to harmonically time varying external forces. The study reported here concerns with an exact harmonic analysis using the classical Dynamic Stiffness Matrix, DSM. The problem at hand is non-linear and transcendental with respect to the excitation frequency [Howson and Williams (1973)]. The approach used to solve it is by the use of the Laplace transform.Focusing on the calculation of natural frequencies, Howson and Williams (2003) present a formulation based on the classical DSM obtained by a set of decoupled fourth order partial differential equations (PDE) for the unknowns deflection and rotation of the beam cross section. Dias and Alves (2009) also derive the DSM via the same procedure but reaches an improved formulation, argued to be more suitable for the eigenproblem solution. It has been noticed [Schanz and Antes (2002)] that the dynamic analyses of beams can be performed by decoupling deflection and rotation. In the study described here, the original coupled PDEs, combining deflection, rotation, bending moment and shear f...