Page 42 - Mechanics of Microelectromechanical Systems

1. Stiffness basics 29
where the stiffness terms are:

and

Note:
In Example 1.1, the cross-stiffness of Eq. (1.96) has been derived by
inversion of a compliance matrix. The minus sign in front of (and which is
denoted by in Eq. (1.96)) in Eq. (1.16) will be explained in Example
1.15.
In a similar fashion, is calculated by means of Eqs. (1.87), (1.88),
(1.91) and (1.93), as:

where the new stiffness term is:

It should be remarked that and are direct-bending stiffnesses,
and this particular subscript notation is utilized instead of using the already-
introduced subscripts l and r of Eqs. (1.7) in order to emphasize the point
where these stiffnesses are calculated (point 1 here), as well as the acting
load (force or moment and the resulting elastic deformation
(deflection or slope/rotation A similar rationale has been applied in the
notation used for the cross-bending stiffness which has been
symbolized by the subscript c in Eqs. (1.7).
Note:
The stiffnesses corresponding to bending about the z-axis are similar to
the ones corresponding to bending about the y-axis, and they are not derived
here. They can easily be obtained from the stiffnesses already formulated by
switching the y and z subscripts, or by using z instead of y, when applicable.

5.2.2 Axial Loading

The axial force at the beam’s free end can be expressed by means of
Castigliano’s first theorem as: