Dear Werner,

Probably the best plug-in soluctions for circular/rectangular membranes are
from "Roark's Formulas for Stress & Strain" by Warren C. Young.  Chapter 10
in the 6th edition covers both circular and rectangular conditions.
However, note that this will cover small deflections only, as will any
plug-in solution.  The reason for this is that when small deflection
assumptions are made, the equations can be linearized and the solution is
closed-form.  However, for larger deflections you run into two types of
geometric nonlinearities and possibly material nonlinearities.  First,
stress stiffening occurs in membranes.  Stress stiffening comes from the
fact that the membrane is fixed on all sides which causes the membrane to
change stiffness with deflection.  This change in stiffness is due to the
axial strain that needs to occur in the membrane to allow it to deflect.
An simple analogy is as follows:  suppose you have a string that people
hold at each end and you deflect it by pushing in the middle of the string.
If the people at the ends of the string pull harder and put more tension on
the string, then you will get a smaller deflection when you push with the
same force as before.  The problem in solving stress stiffening is that you
do not know how stiff the member is until you calculate the deflection but
you cannot calculate the deflection until you know the stiffness.  This
requires a iterative solution and the problem is nonlinear.  Large
deflections cause a similar geometric nonlinearity.  Also, you may have
material nonlinearities (plasticity) that will arise from the large stress
induced by trying to deflect a membrane that much.  The stresses tend to be
much higher because of the constraints involved and large axial strains are
required to accomplish a small lateral displacement.

I hope there is something there that helps.

Larry

> 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.
>  <<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<
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___________________________________________________________

Larry L. Howell
Assistant Professor
Mechanical Engineering Department
Brigham Young University
Provo, UT  84602

Phone:   (801) 378-8037
Fax:     (801) 378-5037