Non-linear analysis of beams under electrostatic loads

The aim of this work is that of evaluating the relative contribution of the different non-linearities in the simple case of slender cantilever beams and plates under electrostatic loads. This case not allows analytical solution to be achieved and therefore a numerical approach must be followed, multipurpose commercial software do not feature simultaneous solution of electrostatic and structural problems. In this work a solution algorithm for the coupled electro-mechanical system to be implemented in a finite element commercial software (ANSYS TM) is proposed. The solution follows a Newton iterative method in which the solution of the Linear system is obtained trough the biconjugate gradient stabilized method. This approach is compared with the already proposed relaxation scheme. The 2D case was firstly considered taking into account the contribution of the hinging field on the tip of the beam. In order of evaluate the accuracy of such a model a 3D model has also been developed taking into account the fringing field on the lateral surfaces, the anticlastic curvature of the beam and the lateral effect of the constraint. The results obtained emphasize the coupling between electrical and mechanical solution as an error around 30% is obtained if the mechanical solution is calculated on the base of the undeformed electric field. On the other hand the 2D mode give a suitable model of the structure as an error of the order of than 2.5% with respect to the 3D case has been obtained.