Size dependent analysis of tapered FG micro-bridge based on a 3D beam theory

In the current study, an analytical solution based on a modied couple stress theory for a nonlinear model describing the couple 3D motion of a functionally graded tapered micro-bridge is presented. The small scale eects and the nonlinearity arising from the mid-plane stretching are taken into consideration. Governing equations of motion are derived utilizing modied couple stress theory and applying the Hamilton principle. Dynamic and static analyses to determine the eects of lateral distributed forces and mid-plane stretching are investigated. Towards this aim, analytical the Homotopy-pade technique is employed to capture the nonlinear natural frequencies in high amplitude vibrations of tapered micro-bridges with dierent types of geometry and material composition. The obtained results of frequencies propose that there is good agreement between the present analytical results and the numerical ones, as opposed to the well-known multiple-scale method. Furthermore, comparing the results in 2D and 3D analyses shows that in 2D analysis, the stiness and natural frequency of the micro-beam is underestimated and it is observed that increasing the tapered ratio has dierent impacts on natural frequencies for micro-beams with dierent slender ratios.