Q16Micro and Smart System Technology

Elastic Deformation and Stress Analysis of Beams and Plates

In MEMS design, microstructures such as cantilever beams, bridges (doubly-clamped beams), and diaphragms/plates are the fundamental mechanical elements used in sensors and actuators, and their elastic deformation under applied load must be predicted accurately to design devices with the desired sensitivity, resonant frequency, and mechanical strength margin.

For a cantilever beam of length L, width w, and thickness t, made of material with Young's modulus E, subjected to a point force F at its free end, the classical Euler-Bernoulli beam theory gives the tip deflection delta as shown above, where I is the second moment of area of the beam's rectangular cross-section. This formula, together with the corresponding bending-stress formula (stress at the beam surface is proportional to the bending moment times the distance from the neutral axis, divided by I), allows the designer to compute both the expected deflection (important for actuator displacement or capacitive-sensor gap change) and the peak stress (important for verifying the structure remains within the material's elastic limit and does not fracture) for a cantilever structure under a given applied force.

For a doubly-clamped beam (bridge structure, fixed at both ends) under a central point load, or under a distributed load (such as pressure), the corresponding deflection and stress formulas differ from the cantilever case due to the different boundary conditions (zero deflection and zero slope at both ends rather than only at one end), generally giving a stiffer structure (smaller deflection for the same load) than an equivalent cantilever, and are similarly derived from Euler-Bernoulli beam theory (or, for beams that are not thin relative to their length, from more complete Timoshenko beam theory that also accounts for shear deformation).

For thin plates and diaphragms (such as the pressure-sensor diaphragm discussed elsewhere in this examination), Kirchhoff-Love plate theory extends the beam-bending analysis to two dimensions, relating the plate's deflection to the applied transverse pressure load through a fourth-order partial differential equation (the plate bending equation) involving the plate's flexural rigidity D = Et^3/(12(1-nu^2)), where nu is Poisson's ratio - for a circular clamped diaphragm under uniform pressure, this yields a closed-form maximum central deflection and maximum edge stress formula analogous in spirit to, but algebraically more involved than, the simple cantilever beam formulas given above, since the two-dimensional plate geometry and clamped-edge boundary condition around the full circumference must be accounted for.

In practice, for MEMS structures with complex geometry, non-uniform loading, or significant residual stress (from the fabrication process itself), finite element analysis (FEA) software is used to numerically solve the full elasticity equations for the actual structure geometry, providing detailed stress and deflection maps across the entire structure rather than relying solely on the simplified analytical formulas above - however, the analytical beam and plate formulas remain essential for quick first-order design estimates, for verifying FEA results by comparison against known closed-form solutions in simplified limiting cases, and for building intuitive understanding of how key design parameters (length, thickness, material modulus) influence the resulting mechanical sensitivity and strength of a given MEMS structure.