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The objective of this study is to apply Gram-Schmidt orthogonalization procedure for generating displacement functions. This procedure allows us to obtain numerically stable functions to be used in vibration analysis of novel composite beams via the Ritz method. Within this study, the theoretical modeling based on third-order shear deformation beam theory is derived to deal with free and forced vibration of functionally graded graphene nanoplatelet-reinforced composite beams subjected to multiple moving loads. To solve the problems of the beams with various end supports, the Ritz method-based Gram-Schmidt polynomial series is adopted to obtain the accurate results. In forced vibration due to the applied multiple moving loads, the time integration technique of Newmark is also employed to find out time history of dynamic deflection of such beams associated with several parametric studies such as patterns of reinforcement, weight fraction of graphene nanoplatelets, and the number of moving loads. According to the numerical results, it is found that the composite beams with low content of graphene nanoplatelets can result in enhancing performance of dynamic bending resistance. Increasing number of moving loads on the beams yields the increases of dynamic deflection and critical velocity.

Gram-Schmidt-Ritz method for dynamic response of FG-GPLRC beams under multiple moving loads