The closest thing you will find to an analytical solution is work being done at MIT by Dr.
Senturia and Elmer Hung. They are using variational calculus to find an approximate
solution. However, FEM must be done to find the basis functions. This technique does offer
a significant decrease in simulation time.

Look at:

E. Hung, "Low-Order Models for Fast Dynamical Simulation of MEMS Microstructurs", 1997
International Conference on Solid-State Sensors and Actuators, June, 1997.

S. Senturia, "CAD Challenges for Microsensors, Microactuators and Microsystems", IEEE
Proceedings, August 1998.

***************************************************

Jim Culver
Principal Engineer
Raytheon
Communication Division
1501 72nd Street
St Petersburg, FL  33710

voice (813) 302-3146
fax   (813) 302-4791
e-mail jwcb@eci-esyst.com

***************************************************



> From mems-mgr@ISI.EDU Sat Aug  1 20:01:45 1998
> Received: from darkstar.isi.edu (sentry) by thor (4.1/SMI-4.1)
>       id AA14708; Sat, 1 Aug 98 20:01:41 EDT
> X-Proxy: callisto.eci-esyst.com protected by CyberGuard
> Received: (from daemon@localhost)
>       by darkstar.isi.edu (8.8.7/8.8.6) id QAA18337
>       for mems-out-list; Sat, 1 Aug 1998 16:46:24 -0700 (PDT)
> Received: (from mems@localhost)
>       by darkstar.isi.edu (8.8.7/8.8.6) id QAA18329;
>       Sat, 1 Aug 1998 16:46:23 -0700 (PDT)
> From: "WJ Karl" <ELP95WJK@sheffield.ac.uk>
> Date:          Thu, 30 Jul 1998 17:06:04 +0100
> Subject:       Membrane deflection
> To: MEMS@ISI.EDU
> X-Url: http://mems.isi.edu/mems.html
> Reply-To: "WJ Karl" <ELP95WJK@sheffield.ac.uk>, mems-cc@ISI.EDU
> Content-Length: 1215
>
>  Dear colleages,
>
>  I am looking for an analytical solution to calculate the deflection
>  and the stress/strain behaviour of a clamped circular/square or
>  rectangular membrane. It seems there is no complete analytical
>  model for this published (especially in the rectangular case) and
>  everybody uses FEM instead.
>
>  I am looking for a plug-in solution able to do the following (yes,
>  I am optimistic ..) :
>
>  + any geometry/aspect ratio (rectangular case) can be calculated
>  + deflections bigger than membrane thickness are valid
>  + deflection and strain/stress values as function of position
>
>  Any hint appreciated. Please ... don't just mention Timoshenko.
>
>  Thanks,
>
>  Werner.
>   <<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<
>  W  J   K  A  R  L
>  MEMS Unit, U of Sheffield         Home 'sweet Home :
>  Mappin Street                     Cobden View Rd. 7o
>  Sheffield S1 3JD, UK              Sheffield S1o 1HQ
>  +44 (0) 114 222 5182/5890         South Yorks., UK
>  Fax (0) 114 272 6391              +44 (0) 114 266 2831
>  elp95wjk@sheffield.ac.uk
>  >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>  What has two legs and three ears ? - A happy Tyson :-)
>
>