Page 67 - Mechanics of Microelectromechanical Systems

54 Chapter 1
Solution:
For the fixed-free beam of Fig. 1.30 (a), the linear stiffness that
corresponds to the free end’s deflection was shown to be:

in Example l.l, or in the direct stiffness derivation of subsection 5.2.1. For
the second configuration, the one sketched in Fig. 1.30 (b), where the end
point 1 is confined to move vertically by always having zero slope, this
point’s deflection and slope can be expressed in terms of loading and of the
compliances corresponding to a fixed-free beam as:

Because the slope at that point is zero, it follows from the second Eq. (1.211)
that:

and, as a consequence, by substituting Eq. (1.212) into the first Eq. (1.211),
the sought stiffness is:

For a constant cross-section fixed-free beam, the compliances of Eq. (1.213)
are simply:

By substituting now Eqs. (1.214) into Eq. (1.213), the stiffness of the beam
sketched in Fig. 1.30 (b) becomes: