Abstract

This paper gives exact, closed-form expressions for the deflection, under its own weight, moments, and shears, in a thick, horizontally oriented, circular mirror on a double-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. It is found that 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. Numerical results are presented and interpreted in detail.

© 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. Nat. Bur. Stand. (U.S.) 71C, 1 (1967).
  5. A. F. Kirstein, R. M. Woolley, J. Res. Nat. Bur. Stand. (U.S.) 72C, 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. L. A. Selke, Appl. Opt. 9, 149 (1970).
    [CrossRef] [PubMed]
  11. S. Timoshenko, Theory of Plates and Shells (McGraw-Hill Book Company, Inc., New York, 1959), p. 71.
  12. A. E. H. Love, A Treatise on the Mathematical Theory of Elasticity (Dover Publications, New York, 1944), p. 481.

1970 (1)

1968 (3)

1967 (1)

A. F. Kirstein, R. M. Woolley, J. Res. Nat. Bur. Stand. (U.S.) 71C, 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. Nat. Bur. Stand. (U.S.) 72C, 21 (1968).

A. F. Kirstein, R. M. Woolley, J. Res. Nat. Bur. Stand. (U.S.) 71C, 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]

Selke, L. A.

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. Nat. Bur. Stand. (U.S.) 72C, 21 (1968).

A. F. Kirstein, R. M. Woolley, J. Res. Nat. Bur. Stand. (U.S.) 71C, 1 (1967).

Appl. Opt. (3)

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. Nat. Bur. Stand. (U.S.) (2)

A. F. Kirstein, R. M. Woolley, J. Res. Nat. Bur. Stand. (U.S.) 71C, 1 (1967).

A. F. Kirstein, R. M. Woolley, J. Res. Nat. Bur. Stand. (U.S.) 72C, 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 (4)

Fig. 1
Fig. 1

Circular mirror on a double-ring support.

Fig. 2
Fig. 2

Deflection vs radius of a 2.54-cm-thick mirror supported at r0 = 5.08 cm, r1 = 10.16 cm.

Fig. 3
Fig. 3

Deflection vs radius of a 7.61-cm-thick mirror supported at r0 = 5.08 cm, r1 = 10.16 cm.

Fig. 4
Fig. 4

Deflection vs radius of a 7.61-cm-thick mirror supported at r0 = 10.30 cm, r1 = 10.40 cm.

Equations (30)

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

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