Page 47 - Mechanics of Microelectromechanical Systems
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34 Chapter 1
beams of small curvature. Thin curved beams have their width w (the cross-
sectional dimension which is in the plane of the curved beam) generally less
than 10% of the radius R, as mentioned by Den Hartog [9] for instance.
During planar deformation of thin beams, the loading mainly consists of the
bending moment that acts perpendicularly to the beam’s plane, whereas for
thick beams, the effects of normal loading and shearing forces have to also
be taken into account.
Figure 1.20 Loads and displacements for a curved beam of relatively-small curvature
The aim here is to define the six displacement components (three
translations and three rotations) of the free-fixed curved beam of Fig. 1.20 in
terms of the six load components, the geometry of the beam, namely its
radius R, angle constant cross-section (the eccentricity e included), and the
material parameters. Various compliances will be defined by this approach,
similar to the stiffnesses which have been obtained previously for a straight
beam. Formally, the load and displacement components are separated into in-
plane and out-of-the-plane.
5.3.1 Thick Curved Beams
For a thick curved beam, the neutral axis (the axis where the stresses normal
to the cross-section are zero) is offset from the geometric symmetry axis, by
the quantity e – the eccentricity, as shown in Fig. 1.21. The eccentricity is
dependent on the shape of the cross-section, and it can generally be
calculated (see Young and Budynas [4], for instance) by means of the
equation:
For a rectangular cross-section, as the one sketched in Fig. 1.21, the
eccentricity is:
