Page 28 - Mechanics of Microelectromechanical Systems

1. Stiffness basics 15
Because only normal stresses and strains are produced in this particular case,
the strain energy of the generic Eq. (1.37), in combination with Eq. (1.39),
simplifies, for the more generic case where the area is variable, to:

where it has been taken into account that the elementary volume can be
expressed in terms of the cross-sectional area A and the elementary length dx
as:

4.2 Torsion Loading

MEMS deformable components are vastly conceived to have rectangular
cross-sections because of either microfabrication constraints or design
purposes. Torsion loading produces shearing, and the maximum shear stress,
which is generated by a torque acting on a fixed-free bar of rectangular
cross-section, occurs at the middle of the longer side (w) and is expressed as:

where w and t are the cross-sectional dimensions (w > t) and is a torsional
constant depending on the w/t ratio, as mentioned by Boresi, Schmidt and
Sidebottom [1]. For very thin cross-sections, where w/t > 10, as
indicated by the same source. The rotation angle at the free end of Fig. 1.9
(a) – where a torque can be applied about the x-axis – with respect to the
fixed end, spaced at a distance l, is:

and the corresponding shear strain is:

In Eqs. (1.45) and (1.46), is the torsion moment of inertia, which will be
defined later in this chapter.
The total strain energy stored in the bar that is subject to torsion is: