Page 342 - Mechanics of Microelectromechanical Systems

5. Static response of MEMS 329
buckling load of the free-fixed column by using 1/4 instead of 1. Another
important consequence is that the maximum postbuckling deflection of the
guided-fixed column is twice the maximum postbuckling deflection of a free-
fixed column with one quarter length‚ as shown in Figs. 5.61 (b) and (c).
Calculating the maximum deflection of a free-fixed column is relatively
easier and it follows the path described previously when studying the large
deflections of a free-fixed beam under the action of a transverse force.
Figure 5.61 (d) is used to briefly formulate the maximum deflection of a
postbuckled free-fixed column. By using the same reasoning that has been
applied for the beam under the action of a transverse load – Fig. 5.44 – it can
be shown that:

where ds‚ and are indicated in Fig. 5.44 and k is given in Eq. (5.134).
Equation (5.186)‚ coupled to Eq. (5.133)‚ gives the length of beam-column
as:

Equation (5.187) is used to determine the force F (which is embedded in k by
way of Eq. (5.134)) corresponding to a certain value of the tip slope The
maximum tip deflection is found by combining Eqs. (5.186) and (5.131)‚
namely:

Example 5.23
Determine the maximum deflection of a guided-fixed microcolumn as
the one sketched in Fig. 5.61 (a) under the compressive action of a force
knowing and E = 160 GPa.

Solution:
The critical load of a free-fixed microcolumn having the length equal to
1/4 the length of the analyzed microcolumn is determined by means of Eqs.
(5.157) and (5.158) and of Fig. 5.56 (e) – showing that K = 2. The critical
load is found to be equal to Solving for in Eq. (5.187) gives a
value of 100°‚ which is further utilized in Eq. (5.188) to find the maximum
tip deflection of the free-fixed beam. This value‚ as mentioned previously‚ is
half the maximum deflection of a guided-fixed microcolumn having four