Transfer matrix formulation for dynamic response of Timoshenko beams resting on two-parameter elastic foundation subjected to moving load

Abstract

In this study, the dynamic response of beams resting on two-parameter elastic foundation subjected tomoving load is investigated by using the transfer matrix method (TMM). The Timoshenko beam theory(TBT) which considers shear deformation and rotational inertia is used to model the beam. The twoparameter elastic foundation model is selected as Pasternak foundation that takes into account a shear layerat the end of linear springs of Winkler foundation. The TMM which uses the relation between analyticallyobtained state vectors of each end of the beam is applied to solve the free vibration problem. After performingthe free vibration analysis, the mathematical model is simplified into an equivalent single degree of freedom(SDOF) system by using the exact mode shapes to obtain dynamic responses. The generalized displacementis calculated for each mode by using the Runge-Kutta algorithm. A numerical case study is presented for asimply-supported Timoshenko beam on the Pasternak foundation subjected to a concentrated load. Thenatural frequencies obtained from finite element method (FEM) results of SAP2000 are presented with theresults of TMM for comparison purposes using the Winkler foundation. The effects of shear layer on thenatural frequencies of the model are revealed. The mode shapes are plotted. The proposed approach forcalculating dynamic responses is validated by using the results of FEM for Winkler foundation model. Then,the effects of Winkler springs and shear layer of the foundation model on the dynamic responses arepresented in figures. The effects of modal damping are discussed. Finally, the critical velocities for the modelare calculated for various elastic foundation scenarios and the effects of elastic foundation parameters on thedynamic response of beam model subjected to moving load with high velocity are observed.