RE: [ATM] Break Strength of Glass

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

Vladimir is quite right.  I used the wrong value for R.  The following is
reposted with the corrected value for R and calculation.  An extraneous
parenthesis was removed from the first equation for w below.  

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=Poisson'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=45.72 cm (18")
p=351.5 g/sq.cm (5 psi)
h=.95 cm (3/8")
For plate glass: (Corning type 0080)
v=.24
E=700,000,000 g/sq.cm

w=1.9 cm ~ 3/4"
The glass might shatter before that.

To get w=.55 cm (.22"), p=(64Dw/R^4)[(1+v)/(5+v)]=6.32 g/sq.cm (1.44 psi)
Which is a total load of 1465 pounds.

The atmospheric pressure changes I mentioned are of less importance than
stated previously, but might still have some significance to the focal
length, particularly over the course of days.  The temperature changes also
have less significance.

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.

The ideal gas law is PV=nRT or P/T=nR/V as a function of volume.  Assuming a
partial pressure in the vacuum chamber of 13.26 psi (1 atmosphere (14.70
psi) - 1.44 psi) and 50 deg F (510 deg Rankine) gives P/T=13.26/510=.026
which is constant for the constant volume of the vacuum chamber.  It is also
the change in pressure for a 1 deg F (or R) change.  A 10 degree change
(within range of an evening cool down) represents a 1/4 psi change which is
18% of the load psi.

Overall, frequent pressure adjustments may still be necessary.

Vladimir,
I was also able to verify the Pilkey equation with "Roark's Formulas for
Stress and Strain" by Young and Budynas.  The formula in Roark's has a
different form, but is equivalent after initial conditions are substituted
and the formula rearranged.

I should also mention that these equations are based on the Kirchhoff theory
of bending which assumes that the thickness is small compared to the other
two dimensions, i.e. a plate; and that the deflection is small (1/2 or less)
compared to the thickness.  This last part is not quite true.  The
deflection desired (5.5 mm or .22") is 59% of the thickness.  The
assumptions were made to simplify the equations by restricting the solution
to the case where the plate bends only, but there is no stretching like a
balloon.  The general solution for both bending and stretching usually
involves numerical methods, e.g. finite element analysis.

But the Kirchhoff assumption should still be close enough in this case to
give us an approximate solution and an idea of the pressures involved and
the possible effects of temperature and atmospheric pressure.

Clear skies,
Don

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