Page 71 - Mechanics of Microelectromechanical Systems

58 Chapter 1
Equation (1.224) shows that the stiffness of the two-beam structure can be
found by summing up the stiffnesses given in Eq. (1.215), and therefore it
can be considered that the two beams act as two springs in parallel.
In addition, the slope at point 2 is calculated here. This slope can be
found in a similar fashion, by following the two-step procedure outlined
previously. The new reactions have to be determined by applying Eqs.
(1.220) and (1.221). The only difference is that a dummy moment has to
be added in the bending moment of the second Eq. (1.222). It can be shown
that the two reactions of Eq. (1.223) are the same, whereas the bending
moment reaction becomes:

With this addition, the slope at point 2 is calculated as:

and its value is zero since the dummy moment is also zero. This result
confirms the physical intuition that the system should deform under the
action of the horizontal force such that the rigid segment 2-3 translates
horizontally.

7. PLATES AND SHELLS

Plates and shells are structural components that can use their elastic
deformation in MEMS applications for devices such as valves, pumps,
switches etc. By design, plates and shells have their thickness much smaller
than the in-plane dimensions. Den Hartog [7] and Reddy [5], among others,
mention that the thickness in these members is less than 1/10 of the smallest
in-plane dimension, and moreover, if the thickness is less than 1/20 of the
smallest planar dimension, the member is considered thin.
The plate is generally a flat member and it is the two-dimensional
correspondent of the straight beam, whereas the shell is curved and is similar
to a curved beam. In their thin variant, both members can accommodate
bending and axial (stretching) loads. For small deformations, where the
deflections are less than the thickness, the bending is preponderant, and the
load-deformation relationship is linear. For large deformations, 5 times larger
than the thickness or more, the membrane behavior becomes paramount, and
the bending effects can be neglected. In such cases, the load-deformation law
is non-linear. The transition cases, where the maximum deformation is
comparable to the thickness, approximate equations can be formulated to
accommodate both the bending and the membrane effects. For a circular thin
plate of radius R and thickness t, clamped on its edge and acted upon by a