RE: [ATM] Break Strength of Glass

["Donald Good" <donald.good@comcast.net>]

A 36" disk is 1018 sq in and at 5 psi, the total load is a little over 5000
lbs.  Also, a 5.5mm (.22") sag on a 36" mirror is close to f11 (see
http://www.atmsite.org/contrib/Prewitt/sagitta/).  The formula (from
"Formulas for Stress, Strain, and Structural Matrices" by Walter D. Pilkey)
for the deformation of a uniformly loaded disk is:
w=(pR^4/64D)[(5+v)/(1+v)-(6+2v)/(1+v)a^2+a^4)]
where:
R=disk radius
a=r/R (fractional radius)
p=uniform pressure
D=Eh^3/[12(1-v^2)]
h=plate thickness
v=Poisonn's ratio
E=Young's modulus

a=0 at center and the central deformation (sag) becomes
w=(pR^4/64D)[(5+v)/(1+v)]

In cgs units (to avoid the pound/poundal confusion in English units)
R=91.44cm (36")
p=351.5 g/sq.cm (5 psi)
h=.95cm (3/8")
For plate glass: (Corning type 0080)
v=.24
E=700,000,000 g/sq.cm

w=30.6cm
I'm sure that the glass would shatter/explode before that.

To get w=.55cm, p=(64Dw/R^4)[(1+v)/(5+v)]=6.32 g/sq.cm (.09 psi)
Which is a total load of 91.6 pounds.

I see the problem of the vacuum technique as the normal variation in
temperature and atmospheric pressure will cause the focal length to
continually change.  

Normal daily barometric change is .02" - .10" Hg or .01 - .05 psi.  Weather
(high - low) changes are on the order of 1" Hg or .5 psi.  Daily barometric
changes can be greater than the vacuum pressure of .09 psi.  Weather changes
are about 5 times greater.

The ideal gas law is PV=nRT or P/T=nR/V as a function of volume.  Assuming 1
atmosphere (14.7 psi) and 50 deg F (510 deg Rankine) gives
P/T=14.7/510=.0288 which is constant for a constant volume (the vacuum
chamber behind the mirror).  It is also the change in pressure for a 1 deg F
(or R) change.  A 4 degree change (well within an evening cooldown) is also
greater than the .09 psi vacuum.

So some sort of continual vacuum regulation would be necessary to keep a
constant focus.

Finally, I believe that the shape of the mirror surface would be a sphere
because the glass disk would act approximately like a balloon skin.

Clear skies
Don


-----Original Message-----
From: atm-bounces@atmlist.net [mailto:atm-bounces@atmlist.net] On Behalf Of
Bob Walker
Sent: Friday, August 27, 2004 11:39 PM
To: ATM Group
Subject: [ATM] Break Strength of Glass

Here's one for all of you structural engineers out there.  I'm thinking of
vacuum flexing a large, thin mirror just for the heck of it.  I did a PLOP
analysis and figured that with a 36" x 3/8" disc of plate glass, about 5 psi
of vacuum would produce about 5 1/5 mm of sag (and sagitta, too), resulting
in an f/10 mirror.  Probably nothing but an ant-burner, but what else am I
going to do with 3/8" glass?  The question, which has resisted all my
efforts to research it via Google, is: can such a piece of glass bend that
much without breaking?  When I built my bridge, I had no trouble finding
tables and equations for beam strength of any species of wood you care to
name, but no dice for glass. As I recall, even Schmidt himself broke a plate
or two with his vacuum pan, but still I'd rather not.

 *  Best regards, Bob
*
         *           *   *

            *         *


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