Page 338 - Mechanics of Microelectromechanical Systems

5. Static response of MEMS 325
The energy method which is utilized here as an alternative tool of
calculating the critical load states that the strain energy stored in a deformed
member is equal to the external work performed by the loads. In the case of
the small-curvature beam of Fig. 5.58, only the bending effects have to be
accounted for. As a consequence, the strain energy stored in the beam
through bending is expressed as:

The bending moment is produced by the axial force and is equal to:

By substituting Eqs. (5.170) and (5.172) into Eq. (5.171), the strain energy
can be calculated as:

The work in this case is produced by the force F traveling over a distance
about the x-axis, namely:

The travel by the force F can be calculated as:

By taking the x-derivative of of Eq. (5.170) and by substituting it into
Eq. (5.175), the work of Eq. (5.174) becomes:

By considering the statement of the energy principle, namely:

it can be found that the critical force is equal to the critical force
corresponding to a straight pinned-pinned beam.