Development of analytical vibration solutions for microstructured beam model to calibrate length scale coefficient in nonlocal Timoshenko beams

The present study takes an analytical approach for solving the free vibration problem of a microstructured beam model, in which transverse displacement springs are added to allow for the transverse shear deformation effect in addition to the rotational springs. The exact vibration frequencies for the discrete microstructured beam model with simply supported ends are obtained via matrix decomposition. In addition, a general solution technique involving the use of Pade approximants for the continualization procedure is proposed in order to obtain the continuous equivalent system for the discrete microstructured beam model. The analytical vibration solutions of the equivalent continuous system are obtained and their accuracy is assessed by using the exact solutions. It is found that the solutions of the equivalent continuous system have a first order accuracy when compared with the exact solutions of their discrete counterpart. The length scale coefficient in the nonlocal Timoshenko beam model is calibrated by using the analytical solutions. Two nonlocal Timoshenko beam models, i.e., the Wang model (without the length scale effect in the shear stress strain relation) and the Reddy model, are evaluated based on their ability to capture the nonlocal effect. (C) 2013 AIP Publishing LLC.