Abstract

This paper shows how exact, closed-form expressions can be determined for the deflections under its own weight of a thick, horizontally oriented, circular mirror on a ring support. A theory developed by Reissner for thick plates that includes shear deformations is used, and the results are reduced to those of classical plate theory. An optimum support radius is obtained, for which the center deflection is equal to the deflection of the outer edge. For mirrors having thickness-to-diameter ratios greater than approximately one-tenth, shearing deformations can contribute significantly to the total deflection and hence should not be neglected.

© 1970 Optical Society of America

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References

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  1. A. Couder, Bull. Astron. 7, 201 (1932).
  2. G. Schwesinger, J. Opt. Soc. Amer. 44, 417 (1954).
    [CrossRef]
  3. W. A. Bassali, Proc. Cambridge Phil. Soc. 53, 728 (1957).
    [CrossRef]
  4. A. F. Kirstein, R. M. Woolley, J. Res. Natl. Bur. Std. C71, 1 (1967).
  5. A. F. Kirstein, R. M. Woolley, J. Res. Natl. Bur. Std. C72, 21 (1968).
  6. T. V. Prevenslik, Appl. Opt. 7, 2123 (1968).
    [CrossRef] [PubMed]
  7. A. J. Malvick, E. T. Pearson, Appl. Opt. 7, 1207 (1968).
    [CrossRef] [PubMed]
  8. E. Reissner, J. Appl. Mech. 12, A-69 (1945).
  9. E. Reissner, Quart. Appl. Math. 5, 55 (1947).
  10. S. Timoshenko, Theory of Plates and Shells (McGraw-Hill Book Company, Inc., New York, 1959), p. 71.
  11. A. E. H. Love, A Treatise on the Mathematical Theory of Elasticity (Dover Publications, New York, 1944), p. 481.

1968 (3)

1967 (1)

A. F. Kirstein, R. M. Woolley, J. Res. Natl. Bur. Std. C71, 1 (1967).

1957 (1)

W. A. Bassali, Proc. Cambridge Phil. Soc. 53, 728 (1957).
[CrossRef]

1954 (1)

G. Schwesinger, J. Opt. Soc. Amer. 44, 417 (1954).
[CrossRef]

1947 (1)

E. Reissner, Quart. Appl. Math. 5, 55 (1947).

1945 (1)

E. Reissner, J. Appl. Mech. 12, A-69 (1945).

1932 (1)

A. Couder, Bull. Astron. 7, 201 (1932).

Bassali, W. A.

W. A. Bassali, Proc. Cambridge Phil. Soc. 53, 728 (1957).
[CrossRef]

Couder, A.

A. Couder, Bull. Astron. 7, 201 (1932).

Kirstein, A. F.

A. F. Kirstein, R. M. Woolley, J. Res. Natl. Bur. Std. C72, 21 (1968).

A. F. Kirstein, R. M. Woolley, J. Res. Natl. Bur. Std. C71, 1 (1967).

Love, A. E. H.

A. E. H. Love, A Treatise on the Mathematical Theory of Elasticity (Dover Publications, New York, 1944), p. 481.

Malvick, A. J.

Pearson, E. T.

Prevenslik, T. V.

Reissner, E.

E. Reissner, Quart. Appl. Math. 5, 55 (1947).

E. Reissner, J. Appl. Mech. 12, A-69 (1945).

Schwesinger, G.

G. Schwesinger, J. Opt. Soc. Amer. 44, 417 (1954).
[CrossRef]

Timoshenko, S.

S. Timoshenko, Theory of Plates and Shells (McGraw-Hill Book Company, Inc., New York, 1959), p. 71.

Woolley, R. M.

A. F. Kirstein, R. M. Woolley, J. Res. Natl. Bur. Std. C72, 21 (1968).

A. F. Kirstein, R. M. Woolley, J. Res. Natl. Bur. Std. C71, 1 (1967).

Appl. Opt. (2)

Bull. Astron. (1)

A. Couder, Bull. Astron. 7, 201 (1932).

J. Appl. Mech. (1)

E. Reissner, J. Appl. Mech. 12, A-69 (1945).

J. Opt. Soc. Amer. (1)

G. Schwesinger, J. Opt. Soc. Amer. 44, 417 (1954).
[CrossRef]

J. Res. Natl. Bur. Std. (2)

A. F. Kirstein, R. M. Woolley, J. Res. Natl. Bur. Std. C71, 1 (1967).

A. F. Kirstein, R. M. Woolley, J. Res. Natl. Bur. Std. C72, 21 (1968).

Proc. Cambridge Phil. Soc. (1)

W. A. Bassali, Proc. Cambridge Phil. Soc. 53, 728 (1957).
[CrossRef]

Quart. Appl. Math. (1)

E. Reissner, Quart. Appl. Math. 5, 55 (1947).

Other (2)

S. Timoshenko, Theory of Plates and Shells (McGraw-Hill Book Company, Inc., New York, 1959), p. 71.

A. E. H. Love, A Treatise on the Mathematical Theory of Elasticity (Dover Publications, New York, 1944), p. 481.

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Figures (6)

Fig. 1
Fig. 1

Circular mirror on a ring support.

Fig. 2
Fig. 2

Stresses acting on an infinitesimal element of disk.

Fig. 3
Fig. 3

Deflection vs radius for a 1-cm thick mirror supported at the optimum radius as determined from the Reissner theory.

Fig. 4
Fig. 4

Deflection vs radius—curves A, 3-cm thick mirror supported at r1 = 10.170 cm; curves B, 7-cm thick mirror supported at r1 = 10.110 cm—support radii determined from Reissner theory.

Fig. 5
Fig. 5

Deflection vs radius—5-cm thick mirror supported at three different radii.

Fig. 6
Fig. 6

Deflection vs radius—curves A, 5-cm thick mirror, curves B, 7-cm thick mirror—both mirrors supported at arbitrary radius of 5 cm.

Tables (1)

Tables Icon

Table I Optimum Support Radius (r2 = 15 cm)

Equations (30)

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M r = h / 2 h / 2 z σ r r d z , H r θ = h / 2 h / 2 z σ r θ d z , V r = h / 2 h / 2 σ r z d z , V θ = h / 2 h / 2 σ θ z d z .
w = 1 D { E o + F ln ( r ) + B 4 [ r 2 ln ( r ) r 2 ] + A 4 r 2 r 4 W 64 π r 2 2 } .
( r V r ) / r = r ( W / π r 2 2 ) ,
V r k 2 [ 2 V r ( 1 / r 2 ) V r ] = D ( 2 w / r ) ,
V θ k 2 [ 2 V θ ( 1 / r 2 ) V θ ] = 0 ,
M r = D ( 2 w r 2 + ν r w r ) + 2 k 2 V r r D ( 1 + ν C n ν C s ) W π r 2 2 ,
H r θ = k 2 [ r ( / r ) ( V θ / r ) ] ,
β r = ( w / r ) + ( V r / C s ) ,
β θ = V θ / C s .
V r = [ ( B / r ) + ( r W / 2 π r 2 2 ) ] ,
V θ = D o k K 1 ( r k ) C k I 1 ( r k ) ,
M r = ( 1 ν ) F r 2 B [ ( 1 + ν ) ln ( r ) 2 + ( 1 ν ) 4 2 k 2 r 2 ] ( 1 + ν ) A 2 + W π r 2 2 [ r 2 ( 3 + ν ) 16 + k 2 ( D ( 1 + ν ) C n D ν C s ) ] ,
β θ = ( D o / k C s ) K 1 ( r / k ) ( C / k C s ) I 1 ( r / k ) ,
β r = 1 D [ F r + B r 4 ( 2 ln ( r ) 1 + 4 D r 2 C s ) + A r 2 r W 2 π r 2 2 ( r 2 8 + D C s ) ] ,
H r θ = [ D o K 2 ( r / k ) + C I 2 ( r / k ) ] .
r = 0 r = r 2 r = r 1 ( a ) w I is finite ( i ) V r I I = 0 ( l ) w I = 0 ( b ) w I / r = 0 ( j ) M r I I = 0 ( m ) w I I = 0 ( c ) V r I = 0 ( k ) H r θ I I = 0 ( n ) β r I = β r I I ( d ) M r I is finite ( o ) M r I = M r I I ( e ) β r I = 0 ( p ) V r I = V ¯ r + V r I I ( f ) V θ I is finite ( q ) H r θ I = H r θ I I ( g ) H r θ I is finite ( r ) β θ I = 0 ( h ) β θ I = 0 ( s ) β θ I I = 0.
B I = F I = D o I = C I = D o I I = C I I = 0 ,
B I I = W / 2 π ,
F I I = ( W / 8 π ) [ r 1 2 ( 4 D / C s ) ] ,
A I I = W ( 1 + ν ) 2 π [ ( 1 + ν ) ln ( r 2 ) + ( 1 + 3 ν ) 4 + 2 D ( 1 + ν ) r 2 2 ( 1 C s 2 C n ) + ( 1 ν ) 2 r 1 2 r 2 2 ] ,
A I = W 2 ( 1 + ν ) π [ ( 1 + ν ) ln ( r 1 r 2 ) + ( 1 + 3 ν ) 4 + 2 D ( 1 + ν ) r 2 2 ( 1 C s 2 C n ) + ( 1 ν ) 2 r 1 2 r 2 2 ] ,
E o I = W r 1 2 8 π ( 1 + ν ) [ ( 1 + ν ) ln ( r 1 r 2 ) ( 1 + 3 ν ) 4 2 D ( 1 + ν ) r 2 2 ( 1 C s 2 C n ) r 1 2 r 2 2 ( 3 5 ν ) 8 ] ,
E o I I = W 8 π { D C s [ 4 ln ( r 1 ) 2 r 1 2 r 2 2 ] + 4 D r 1 2 C n r 2 2 + r 1 2 [ ln ( r 2 r 1 2 ) + 3 + ν 4 ( 1 + ν ) ] ( 3 5 ν ) 8 ( 1 + ν ) r 1 4 r 2 2 } .
w I = W ( 1 + ν ) 8 π D { ( r 2 r 1 2 ) [ 2 D ( 1 + ν ) r 2 2 ( 1 C s 2 C n ) + ( 1 + ν ) ln ( r 1 r 2 ) + ( 1 + 3 ν ) 4 ] + 1 8 [ ( 3 5 ν ) r 1 4 r 2 2 + 4 ( 1 ν ) r 2 r 1 2 r 2 2 ( 1 + ν ) r 4 r 2 2 ] } ,
w I I = W 8 π D { D C s [ 4 ln ( r 1 r ) + 2 ( r 2 r 1 2 ) r 2 2 ] + 4 D ( r 1 2 r 2 ) C n r 2 2 + r 1 2 ln ( r 2 r r 1 2 ) + r 2 ln ( r r 2 ) ( 3 5 ν ) r 1 4 8 ( 1 + ν ) r 2 2 r 4 8 r 2 2 ( r 2 r 1 2 ) ( 3 + ν ) 4 ( 1 + ν ) + r 2 r 1 2 ( 1 ν ) 2 r 2 2 ( 1 + ν ) } .
C n = 5 6 ( E h / ν ) ,
C s = 5 6 G h .
w I = W ( r 2 r 2 2 ) 64 π r 2 2 D [ ( 5 + ν ) ( 1 + ν ) r 2 2 r 2 ] + W ( r 2 r 2 2 ) 8 π D [ 2 5 h 2 ( 1 ν 2 ) ] .
w shear = W ( r 2 r 2 2 ) 8 π D [ 2 5 h 2 ( 1 + ν / 8 + ν 2 / 8 ) ( 1 ν 2 ) ] .
4 r 1 2 [ 2 ( 1 + ν ) ln ( r 1 / r 2 ) ( 3 + ν ) ] + ( 7 + 3 ν ) r 2 2 ( 16 D / C s ) × ( 1 + ν ) [ 1 + 2 ln ( r 1 / r 2 ) ] + [ 32 D ( 1 + ν ) / C n ] = 0.

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