Page 60 - Mechanics of Microelectromechanical Systems

1. Stiffness basics 47

By comparing Eqs. (1.182) and (1.183), it follows that the equivalent axial
rigidity of the composite bar can be calculated as:

or, in the case where more than two components make up the bar, in the
form:

This result is predictable, as the two portions have identical lengths and act
as springs in parallel whose resultant stiffness is therefore the sum of the
individual stiffnesses.
Torsion of composite cross-section bars is very similar to axial loading,
and therefore the torsional rigidity of a composite bar is given by an equation
similar to Eq. (1.184), namely:

or, if the cross-section is composed of more than two different members, the
equivalent rigidity in torsion is:

Note:
Caution should be exercised when a sandwiched member is subject to
mixed loading, such as bending and axial, for instance, because the material
or geometric properties that result from the different rigidity equivalence
operations might be non-consistent. An example will be studied next where
the thickness resulting from bending-related equivalence is not always equal
to the thickness generated through axial-related equivalence.

Example 1.12
A sandwiched cantilever of the type shown in Fig. 1.25 (a) is formed of
two different members that have equal widths and completely
overlap over their length. Apply the bending-related equivalence, as well as
the axial-related one, and find the two corresponding equivalent thicknesses
by considering that is Young’s modulus of the equivalent material. Also
consider that and