In the paper, entitled:
A Closed Form Solution for the Pull-in Voltage of the Micro Bridge
(Link to PDF: https://pdfs.semanticscholar.org/0d31/33707b1243f6b4e3344c4fa19b831b010b8b.pdf)
... the following equation appears:
I really don't know how to solve this for $\eta_{PI}$, even if all constants are known... do you have any idea ?
EDIT: In LateX:
$\eta_{ P I } = \frac{\int_{ 0 } ^ { L } \left[ \frac { b \phi } { \left( g - \eta _ { P I } \phi \right) ^ { 2 } } + \frac { 0.265 b ^ { 0.25 } \phi } { \left( g - \eta _ { P I } \phi \right) ^ { 1.25 } } + \frac { 0.53 h ^ { 0.5 } \phi } { \left( g - \eta _ { P I } \phi \right) ^ { 1.5 } } \right] dx}{\int _ { 0 } ^ { L } \left[ \frac { 2 b \phi ^ { 2 } } { \left( g - \eta _ { P I } \phi \right) ^ { 3 } } + \frac { 0.33125 b ^ { 0.25 } \phi ^ { 2 } } { \left( g - \eta _ { P I } \phi \right) ^ { 225 } } + \frac { 0.795 h ^ { 0.5 } \phi ^ { 2 } } { \left( g - \eta _ { P I } \phi \right) ^ { 25 } } \right] d x}$
A typical $\phi (x) =a \sin (x)+b \cos (x)+c \sinh (x)+d \cosh (x)$
Update: Because this seems to be a very difficult task to solve analytically, I have also posted a question on Mathematica SE to see if it can be solved numerically. If you are interested in getting the Mathematica code to test it, please have a look here: https://mathematica.stackexchange.com/questions/183262/how-to-solve-this-equation-numerically-or-analytically?