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8 tk a2

where Sr is a dimensionless maximum stress (Fig. 3.6).

For identical dimensions, loads, and flexures, the configuration VTD 3 provides the smallest radial maximum stress. For these configurations the tangential maximum stress St is smaller than Sr.

### 3.3.3 Hybrid Configurations

With the CTD configuration such as displayed by Fig. 3.3, the Sphe 3 mode is purely achieved up to r < a, i.e. the optical clear aperture of the mirror is limited to the inner plate or meniscus which is clamped into the ring. The ring allows one to generate the

bending moment Mr {a} by applying an external pair of ring forces, but its flexure is not purely of the fourth degree. However, from the latter results, we can obtain also an exact Sphe 3 flexure of the zone outer to r = a.

Let us consider a configuration composed of two concentric zones. The first zone is a CTD such as defined by Eq. (3.26) and applying for 0 < r < a; the second zone extending on a < r < b is a VTD 3 defined by Eq. (3.31) where a is replaced by b. After this substitution, the bending moment of the outer zone is represented by

If the clamping of the inner VTD 3 with the outer CTD is done at r = a with the same thicknesses, we then set the same uniform loadings q in intensity and sign. These two conditions are written by

T40(a)|VTD3 = T40(a)|CTD and q|VTD3 = q|CTD, (3.34)

the first equality providing the radial location of the link between the two distributions as, from (3.26a) to (3.31a),

Therefore, the radius a of the rigidity junction, where is also acting in the ring-force reaction, is

After substitution of this ratio in the bending moment of the VTD 3, we obtain Mr {a} = (3 + v)qa2/16 which is identical to Mr {a} of the CTD [see (3.26b)], so that the condition (3.35) is satisfied.

^ A mirror with two concentric zones provides a Sphe 3 deformation mode z = A4o r4 on all its surface r e [0, b ], if a uniform load q applied to all its surface is in reaction with a ring force at the rigidity junction r — a. The thickness t — T40 to and the geometric parameters are represented by

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