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.
Highlights
Many modern structures are formed by beam elements
Harmonic loads can be of high frequency, when 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
It is imperative that the partial differential equations (PDE) representing the governing equations of a given in-plane structure subjected to harmonic forces has to be solved exactly
Summary
Many modern structures are formed by beam elements. These skeleton like structures are subjected to static and dynamic loads. It is imperative that the PDE representing the governing equations of a given in-plane structure subjected to harmonic forces has to be solved exactly This can be achieved by using Laplace transform [Abu-Hilal (2003), Foda and Albassam (2006), Loudini et al (2006), Saeid (2011)] and/or Green functions [Abu-Hilal (2003), Tang (2008), Davar and Rahmani (2009)]. When considering the capability to solve framed structures with the features of concentrated masses and springs and rigid bodies attached to the members ends, only Seeid (2001) can be highlighted These features are not taken into account by Antes et al (2004), which deals with harmonic loads applied to Timoshenko frames. Since the rigid bodies may or may not have mass, they are alternatively named rigid offsets
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