Theoretical modeling of boundary conditions in microfabricated beams

The authors report detailed modeling of step-up boundary conditions in surface micromachined beams and investigate their effects on the onset of buckling in doubly supported beams. Both cantilever and doubly supported beams are considered. Finite element analysis is used to accurately model the mechanical behavior of the step-up boundary conditions. An extended beam model which uses equivalent torsional and axial stiffnesses in conjunction with a simply supported boundary condition is developed to account for the finite stiffness of a step-up boundary condition. The finite element results are used to calculate the values of the equivalent stiffnesses of the extended beam model for practical geometries of step-up boundary conditions. The extended beam model is used to calculate buckling loads for doubly supported beams. Equivalent stiffness values for torsional and axial springs are for geometries of practical importance to microelectromechanical systems