Page 39 - Mechanics of Microelectromechanical Systems

26 Chapter 1
By considering the following equations that define the cross-sectional
amounts of interest:

it can be shown that the ratio of Eq. (1.84) only depends on the length 1 and
thickness t, and for a Poisson ratio of (for a polysilicon material)
and a value of (Young and Budynas [4]), the slope calculated with
shearing effects taken into account is up to 30% smaller than the slope
determined without considering shearing, as shown in Fig. 1.17.

Figure 1.17 Plot of the ratio of slopes at midspan – according to Eq. (1.84)

5.2 Stiffnesses of Constant Cross-Section Straight Beam
Using Castigliano’s First Theorem

Castigliano’s first theorem, as introduced in this chapter, enables
calculation of the stiffnesses that connect a force/moment to the
corresponding linear/angular displacement. A fixed-free straight beam of
constant cross-section is considered here, loaded as shown in Fig. 1.18.
Bending about the y-axis is produced by and Bending about the z-
direction is generated by and Axial deformation is created by the
force and torsion is caused by the moment

5.2.1 Bending About the y-Axis

We shall assume here that the beam is relatively long (length is at least 5
times larger than the maximum cross-sectional dimension), and that plane
sections that are perpendicular to the beam’s midsurface (neutral fiber)