Need equation for deflection of fabric over slotted with vac vacuum 3

[Rocketman41]

Conditions:
Impervious fabric pre-tensioned at (T pli) and over a vacuum slot (L in) wide
NEED:
Def(h) as a function of Slot width (L), Vacuum (V ), and pre-tension (P).
Note: I am familiar with the beam equations below but, none of them apply. see diagram below.

** h=5PL^4/(384*EI)
** R=(h^2+(L/2)^2)/2h
Angle of deflection the fabric (beam) makes = pressure1*slotwidth^3/(24*E*I)

 

[SWComposites]

you need membrane deflection equations, not beam equations; the fabric does not have bending stiffness.

 

[Rocketman41]

Additional information re: my posting on fabric deflection.
I am a retired mechanical engineer. I'm writing a technical article that requires the equation.
I have searched Google and many other web sites but have not been able the find an appropriate equation for the conditions explained. I think it will need to be developed from scratch.
I will certainly give credit in the published article to the person that developes the required equation.

Thanks in advance

 

[3DDave]

The radius is likely to be the same as for a uniform cylinder that has the given pressure difference and the related tension.

As the radius of a uniform cylinder goes up the tension for a given pressure difference goes up so to keep deflection lower the tension in the fabric needs to be higher.

The end conditions that seal off the slot will affect the longitudinal strain as well.

If one needs to add fabric strain I see over 250 pages of research into measuring that:

https://core.ac.uk/download/pdf/33796985.pdf

 

[rb1957]

one method with some (possibly questionable) logic is to equate the change in length of an arc to the change in length due to membrane stress from pressure ...

original length is W
arc length is 2*R*theta, where sin(theta) = W/(2R)
so change in length = 2R*theta-W (assuming the material doesn't side into the gap, yes?)

membrane stress = pR/t
membrane strain = pR/(tE) ... E = ?
change in length = pRW/(tE)

Not sure how to account for your pre-tension.

"Hoffen wir mal, dass alles gut geht !"
General Paulus, Nov 1942, outside Stalingrad after the launch of Operation Uranus.

 

[SWComposites]

searching for “ membrane deflection under pressure” brings up lots of what look like potentially useful info.

 

[zeusfaber]

Recognising that this is a slot, are we not looking at a catenary equation (once you get away from the ends of the slot)?

A

 

[rb1957]

it'll form some sort of curve. Assuming circular is an assumption. Assuming catenary is another, where you'd model the pressure as weight.

"Hoffen wir mal, dass alles gut geht !"
General Paulus, Nov 1942, outside Stalingrad after the launch of Operation Uranus.

 

[3DDave]

It is a catenary when the load is in one direction - such as gravity. Pressure follows the surface and is normal to the surface so the curvature is constant - a radius.

 

[LittleInch]

You also need some sort of radius on the edge of the slot or your fabric won't last very long....

not sure what the ratio needs to be for the slot length / width needs to be for the equations to be an infinite slot length, but my guess is at least 5.

otherwise the end effect will be complex to say the least. You also need to radius the end of the slot.

It always helps to know why you're doing this and what sort of deflection you are aiming for.
Is the material operating int he elastic zone as a spring constant type of thing? Or does the spring constant vary with increased deflection?
Is the membrane fixed at the edges of the slot or is there stretch further way back to some anchor point?

Remember - More details = better answers
Also: If you get a response it's polite to respond to it.

 

[Wrenchbender]

Found this piece by a government lab. Only gives the equations, no derivations, but you might search further for those.

https://engineeringlibrary.org/reference/membranes-air-force-stress-manual

(Also remember doing something similar as a student for vibrating drum heads - time dependent wave eqns, but its been too long and i'm a lot dumber now.)