Abstract

This paper gives exact, closed-form expressions for the deflection under its own weight, moments, and shears, in a thick, solid, horizontally oriented, circular mirror having a central hole on a ring support. A theory, developed by Reissner for thick plates, that includes shear deformations is used, and it is shown how the results for the solid mirror can be reduced to obtain results for cored, or sandwich, mirrors. 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.

© 1971 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. 71C, 1 (1967).
  5. A. F. Kirstein, R. M. Woolley, J. Res. Nat. Bur. Stand. 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. L. A. Selke, Appl. Opt. 9, 1453 (1970).
    [CrossRef] [PubMed]
  12. L. A. Selke, Internat. J. Solids Structures (in press).
  13. W. P. Barnes, Appl. Opt. 8, 1191 (1969).
    [CrossRef] [PubMed]
  14. G. R. Cowper, J. Appl. Mech. 33, 335 (1966).
    [CrossRef]
  15. S. Timoshenko, Theory of Plates and Shells (McGraw-Hill, New York, 1959), p. 61.
  16. A. E. H. Love, A Treatise on the Mathematical Theory of Elasticity (Dover, New York, 1944), p. 481.
  17. C. Chang, I. Ebcioglu, J. Basic Eng.513 (Dec.1961).
    [CrossRef]
  18. J. Penzien, T. Didriksson, Amer. Inst. Aeron. Astron. J. 2, 531 (1964).
  19. S. Kelsey, R. Gellatly, B. Clark, Aircraft Eng. 30, 294 (1958).
    [CrossRef]
  20. A. Leissa, C. Lo, F. Niedenfuhr, Amer. Inst. Aeron. Astron. J. 3, 566 (1965).

1970 (2)

1969 (1)

1968 (3)

1967 (1)

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

1966 (1)

G. R. Cowper, J. Appl. Mech. 33, 335 (1966).
[CrossRef]

1965 (1)

A. Leissa, C. Lo, F. Niedenfuhr, Amer. Inst. Aeron. Astron. J. 3, 566 (1965).

1964 (1)

J. Penzien, T. Didriksson, Amer. Inst. Aeron. Astron. J. 2, 531 (1964).

1961 (1)

C. Chang, I. Ebcioglu, J. Basic Eng.513 (Dec.1961).
[CrossRef]

1958 (1)

S. Kelsey, R. Gellatly, B. Clark, Aircraft Eng. 30, 294 (1958).
[CrossRef]

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).

Barnes, W. P.

Bassali, W. A.

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

Chang, C.

C. Chang, I. Ebcioglu, J. Basic Eng.513 (Dec.1961).
[CrossRef]

Clark, B.

S. Kelsey, R. Gellatly, B. Clark, Aircraft Eng. 30, 294 (1958).
[CrossRef]

Couder, A.

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

Cowper, G. R.

G. R. Cowper, J. Appl. Mech. 33, 335 (1966).
[CrossRef]

Didriksson, T.

J. Penzien, T. Didriksson, Amer. Inst. Aeron. Astron. J. 2, 531 (1964).

Ebcioglu, I.

C. Chang, I. Ebcioglu, J. Basic Eng.513 (Dec.1961).
[CrossRef]

Gellatly, R.

S. Kelsey, R. Gellatly, B. Clark, Aircraft Eng. 30, 294 (1958).
[CrossRef]

Kelsey, S.

S. Kelsey, R. Gellatly, B. Clark, Aircraft Eng. 30, 294 (1958).
[CrossRef]

Kirstein, A. F.

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

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

Leissa, A.

A. Leissa, C. Lo, F. Niedenfuhr, Amer. Inst. Aeron. Astron. J. 3, 566 (1965).

Lo, C.

A. Leissa, C. Lo, F. Niedenfuhr, Amer. Inst. Aeron. Astron. J. 3, 566 (1965).

Love, A. E. H.

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

Malvick, A. J.

Niedenfuhr, F.

A. Leissa, C. Lo, F. Niedenfuhr, Amer. Inst. Aeron. Astron. J. 3, 566 (1965).

Pearson, E. T.

Penzien, J.

J. Penzien, T. Didriksson, Amer. Inst. Aeron. Astron. J. 2, 531 (1964).

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, New York, 1959), p. 61.

Woolley, R. M.

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

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

Aircraft Eng. (1)

S. Kelsey, R. Gellatly, B. Clark, Aircraft Eng. 30, 294 (1958).
[CrossRef]

Amer. Inst. Aeron. Astron. J. (2)

A. Leissa, C. Lo, F. Niedenfuhr, Amer. Inst. Aeron. Astron. J. 3, 566 (1965).

J. Penzien, T. Didriksson, Amer. Inst. Aeron. Astron. J. 2, 531 (1964).

Appl. Opt. (5)

Bull. Astron. (1)

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

J. Appl. Mech. (2)

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

G. R. Cowper, J. Appl. Mech. 33, 335 (1966).
[CrossRef]

J. Basic Eng. (1)

C. Chang, I. Ebcioglu, J. Basic Eng.513 (Dec.1961).
[CrossRef]

J. Opt. Soc. Amer. (1)

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

J. Res. Nat. Bur. Stand. (2)

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

A. F. Kirstein, R. M. Woolley, J. Res. Nat. Bur. Stand. 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 (3)

S. Timoshenko, Theory of Plates and Shells (McGraw-Hill, New York, 1959), p. 61.

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

L. A. Selke, Internat. J. Solids Structures (in press).

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Figures (6)

Fig. 1
Fig. 1

Circular mirror having a central hole on a ring support.

Fig. 2
Fig. 2

Deflection vs radius: curves A, 3-cm-thick mirror supported at r1 = 10.829 cm; curves B, 7-cm-thick mirror supported at r1 = 10.859 cm; support radii determined from Reissner theory.

Fig. 3
Fig. 3

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

Fig. 4
Fig. 4

Deflection vs radius: sandwich mirror having upper and lower plate thickness of 0.65 cm and core height of 5.70 cm, for three different effective core shear moduli; support radii determined from Reissner theory.

Fig. 5
Fig. 5

Deflection vs radius: sandwich mirror having upper and lower plate thickness of 0.65 cm and core height of 5.70 cm and having different effective core shear moduli; all mirrors supported at arbitrary radius of 10 cm.

Fig. 6
Fig. 6

Deflection vs support radius: sandwich mirror having upper and lower plate thickness of 0.65 cm and core height of 5.70 cm, for an effective core shear modulus of 0.57 × 1011 dynes cm−2.

Tables (2)

Tables Icon

Table I Optimum Support Radius-Solid Mirrora

Tables Icon

Table II Optimum Support Radius-Sandwich Mirrora

Equations (25)

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V r = [ - B r + r W 2 π ( r 2 2 - r 0 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 0 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 0 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 - r 0 2 ) } .
r = r 0 , r = r 2 , r = r 1 , ( a ) V r 0 = 0 , ( d ) V r II = 0 , ( g ) w I = 0 , ( b ) M r 0 = 0 , ( e ) M r II = 0 , ( h ) w II = 0 , ( c ) H r θ 0 = 0 , ( f ) H r θ II = 0 , ( i ) β r I = β r II , ( j ) V r I = V ¯ r + V r II , ( k ) M r I = M r II , ( l ) H r θ I = H r θ II , ( m ) β θ I = 0 , ( n ) β θ II = 0.
D 0 I = D 0 II = C I = C II = 0 ,
B I = r 0 2 W 2 π ( r 2 2 - r 0 2 ) ,
B II = r 2 2 W 2 π ( r 2 2 - r 0 2 ) ,
A II = W 2 π ( r 2 2 - r 0 2 ) ( 1 + ν ) [ 2 D ( 1 + ν ) ( 1 C s - 2 C n ) + ( r 2 2 + r 0 2 ) ( 1 + 3 ν ) 4 + ( 1 - ν ) r 1 2 2 - r 2 4 ( 1 + ν ) ln ( r 2 ) ( r 2 2 - r 0 2 ) + r 0 4 ( 1 + ν ) ln ( r 0 ) ( r 2 2 - r 0 2 ) + r 0 2 ( 1 + ν ) ln ( r 1 ) ] .
A I = A II + [ B II - B I ] ln ( r 1 ) ,
F I = r 0 2 B I ( 1 - ν ) [ ( 1 + ν ) ln ( r 0 ) 2 + ( 1 - ν ) 4 - 2 k 2 r 0 2 ] + ( 1 + ν ) r 0 2 2 ( 1 - ν ) A I - W r 0 2 π ( 1 - ν ) ( r 2 2 - r 0 2 ) { r 0 2 ( 3 + ν ) 16 + k 2 - [ D ( 1 + ν ) C n - D ν C s ] } ,
F II = r 2 2 B II ( 1 - ν ) [ ( 1 + ν ) ln ( r 2 ) 2 + ( 1 - ν ) 4 - 2 k 2 r 2 2 ] + ( 1 + ν ) r 2 2 2 ( 1 - ν ) A II - W r 2 2 π ( 1 - ν ) ( r 2 2 - r 0 2 ) { r 2 2 ( 3 + ν ) 16 + k 2 - [ D ( 1 + ν ) C n - D ν C s ] } ,
E 0 I = - F I ln ( r 1 ) - B I 4 [ r 1 2 ln ( r 1 ) - r 1 2 ] - A I r 1 2 4 + r 1 4 W 64 π ( r 2 2 - r 0 2 ) ,
E 0 II = - F II ln ( r 1 ) - B II 4 [ r 1 2 ln ( r 1 ) - r 1 2 ] - A II r 1 2 4 + r 1 4 W 64 π ( r 2 2 - r 0 2 ) .
w I = 1 D { E I + F I ln ( r ) + B I 4 r 2 [ ln ( r ) - 1 ] + A I r 2 4 - r 4 W 64 π ( r 2 2 - r 0 2 ) } ,
w II = 1 D { E II + F II ln ( r ) + B II r 2 4 [ ln ( r ) - 1 ] + A II r 2 4 - r 4 W 64 π ( r 2 2 - r 0 2 ) } .
C n = ( 5 E h ) / ( 6 ν ) ,
C s = ( 5 G h ) / 6.
D = t ( h 1 + t ) 2 E 2 ( 1 - ν 2 ) ,
C n = ,
C s = h 1 G c .
W = ( 2 ρ p t + ρ c h 1 ) π ( r 2 2 - r 0 2 ) .

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