Comments on “quartz-window design for gas-pressurized photomultiplier-tube installations”

NUCLEAR

INSTRUMENTS

AND

i29 (I975) 3o9-3IO;

METHODS

AUTHOR'S

© NORTH-HOLLAND

PUBLISHING

CO.

REPLY to

Comments

on " Q u a r t z - w i n d o w

design for gas-pressurized photomnltipller-tube

installations"

B. M. B A I L E Y

Laboratory for Nuclear Science, Massachusetts Institute o f Technology, Cambridge, Massachusetts 02139, U.S.A. Received 19 A u g u s t 1975

Bargmann and Muratori (see preceding article) are to be commended for observing that the author's fig. 3 shows some constant G=contours which are inconsistent with the necessary boundary conditions. This error is not " d u e to incorrect use" of the finite-element method (FEM) of analysis. Rather, it is due to incorrect graphical representation of some results. A corrected

plotting is presented here as fig. 3a, along with more detailed results f r o m a finer mesh in the region of greatest interest (near the window edge) in fig. 3b. These contours have not been faired into smooth curves, but are constructed simply as point-to-point line connections so as not to imply either more or less about the detail of the results.

-300

8.0

-~7.6 g

-

2

-

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I

6.8 0

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I

0.4

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l

I

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I

1.2

1.6

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I 2.0

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I 2.4

I

t

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2.8

X (inches) Fig. 3a. Corrected plot o f c o n s t a n t az c o n t o u r s f r o m fig. 3 o f a u t h o r s ' original p r e s e n t a t i o n (stress values in psi). Applied pressure = 275~psi. 7.8

7.6

~

'

$

0

~

'-150 - 250 -50 3OO

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7.4

7.2

7.0-

2.0

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2.2

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2.4 2.6 X(inches)

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2.8

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Fig. 3b. C o n s t a n t az c o n t o u r s n e a r w i n d o w edge obtained with finer m e s h (6 elements t h r o u g h w i n d o w x 40 elements radially f r o m axis o f s y m m e t r y Y).

309

310

B.M. BAILEY 8.0 °°

ZS---

I--I

co c

500

>" 7o2-

~ 6°6

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0.8

0.4

o

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1.2 1.6 X(inches)

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~7

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2.0

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-3500

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2.8

Fig. 3c. Circumferential stress (o0) c o n t o u r s obtained f r o m 6 x 24-element mesh.

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°f 7.2

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-2000 -2500 -3000

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Fig. 3d. Meridional stress (cr4~) c o n t o u r s obtained f r o m 6 x 24-element mesh.

It is indeed true that in simple, uniform shells, a~ is the principal stress of least concern. But the structure we study here is not a uniform shell, and it is important to understand that as such, it does not behave like one. Figs. 3c and 3d are plots of constant Go and o-o, the circumferential, or hoop, stresses and meridional stresses, respectively. The author regrets that readers might infer a lack of concern with these principal stresses. But the fact emerges that, with their being entirely compressive stresses and of the order of only a few percent of the compressive strength of quartz, they appear to pose no problem in this application. However, study of figs. 3c and 3d shows that this shell geometry, even for the idealized edge conditions assumed for the model, produces results rather different from what Bargmann and Muratori calculate for ao, G4, at the top and at the edge of the shell. The computer program used for the author's study calculates results identically similar to those of theory for ae, ae and a=, when the geometry is that of a uniform shell. But when the edge thickness

differs from that at the top of the shell even by so small an amount as in this case, simple shell behavior no longer obtains. It is from such a plot as fig. 3b that one can infer the existence of possibly severe tensile strains at the edge of the shell. One may estimate from that figure the gradient of o& to be in the range of 7700 psi/inch, near the edge. Thus one might expect a tensile stress of 750 psi or more at the edge, near the midplane, and this is in the range of ten percent of the nominal tensile strength of the material. The author agrees with Bargmann and Muratori concerning the desirability of realistic consideration of shear and bending moments occurring between the window and seat. He feels, however, that FEM, rather than being " t o o sophisticated for this application", is probably the least sophisticated of the analytical methods available. Above all, it seems the simplest (relatively) way to study the behavior of even simply supported shells when their geometry departs from that of uniform thickness.