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Fig. 10.8 Thickness distributions T(x) of three cylinders generating a x2 flexure mode. Radial reactions at edges in equilibrium with uniform load, Fp = F—p = qL/2. Poisson's ratio v = 1/4. Middle thickness ratio t (0)/a = 1/10. Length parameters p = 0.45, 0.90, 1.35. Mirror aspect ratios L/a = 1.092 p
Fig. 10.8 Thickness distributions T(x) of three cylinders generating a x2 flexure mode. Radial reactions at edges in equilibrium with uniform load, Fp = F—p = qL/2. Poisson's ratio v = 1/4. Middle thickness ratio t (0)/a = 1/10. Length parameters p = 0.45, 0.90, 1.35. Mirror aspect ratios L/a = 1.092 p
An axisymmetric tubular shell with free edges is by itself in a static equilibrium when a uniform load q is applied all over its surface. Returning to (10.19) and using (10.14), we may introduce the expression of the shearing force Qx. Thus we obtain dM =  dX(DdW) = Q (10.35)
and from (10.22), the general differential equation of the flexure of a cylinder may be expressed in a function of the shearing force,
dx a2
Using the dimensionless quantities T and W for the thickness and the flexure, equations (10.25) lead to
dx C
If q is the only external force applied to the cylinder, then the shearing force Qx is null,
so the first lefthand term in the differential equation vanishes. This result is a straightforward inverse proportional law,
^ If an axisymmetric cylindrical shell is uniformly loaded all over one of its surfaces, and if no other external forces are applied, then the thickness T and flexure W are reciprocal functions.
This law is of fundamental importance for the aspherization of Xray mirrors because the boundary conditions at both edges vanish. Therefore only sliding contacts must be set at the end faces to prevent from pressure leak.
1 ^ Singleterm parabolic flexure: Let us consider a parabolic flexure where the central section plane of the cylinder cannot extend or retract. This case corresponds to a dimensionless flexure represented by a single quadratic term,
W = C2 Ew = X2, X = x/a. From (10.38), the dimensionless thickness is simply
Hence the thin shell theory provides a valuable solution that is with infinite thickness at the plane of central section (Fig. 10.9).
2 ^ Twoterm parabolic flexure: In the latter case of a singleterm flexure, although the infinite thickness at the symmetry plane may be rendered finite by an optical tolerancing criterion, this difficulty can be easily circumvented by considering that a radial strain may appear all along the cylinder.
Hence if a parabolic flexure is generated from the origin to the edge x = ±ft, then in order to avoid singular poles at the cylinder edges  which, from the inverse proportional law, entails local infinite thicknesses , it is necessary to
ÏV = X 
/ 
\ 
c2 
= 0.015 <7  

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