D2 y x dyi

I dr2

r dr

Since w(0) can be set to zero or any arbitrary value for the first element, we notice that the constant C5 for the displacement w in (6.31a) can be derived separately after solving C1j...i4 from the four latter equations.

6.3.2 Various Boundaries and Constant Thickness Plain Shells

Given a plain shell - or cup-like shell - of mean curvature <R>, normal thickness t = constant and elastic constants E, v, its rigidity D and characteristic length I are determined by (6.3g) and (6.10b). The functions Yi(r/£) in equation set (6.31) allow deriving the normal displacement w(r) as a function of five boundary conditions for solving the Ci unknowns.

articulated and movable edge built-in and movable edge articulated and immovable edge built-in and immovable edge articulated and movable edge built-in and movable edge articulated and immovable edge built-in and immovable edge

Fig. 6.3 Compared normal flexures w of a constant thickness f/2 menicus under four differing boundary conditions. The meniscus is in Zerodur with <R> = 6 m, 2rm = 1.5 m, t = 42.5 mm. Uniform load q = 105 Pa

At the center of the shell, we set w{0} = 0, dw/dr\r=0 = 0, £« {0} = 0, and at the edge radius r = rmax, we must introduce two other conditions among the following dw/dr = 0, Mr = 0, N = 0, £« = 0, where, from (6.30 f), we also have ett = u/r — w/<R> whatever r.

Typical edge boundary conditions for a shell can be defined at rm following sets.

• Articulated and movable edge:

• Articulated and immovable edge:

Mr = 0 and Nr = 0, dw/dr= 0 and Nr = 0, Mr = 0 and ett = 0, dw/dr = 0 and ett = 0.

The results from calculation of a 1.5-m aperture Zerodur meniscus at f/2, with thickness t = constant = 42.5 mm and uniform load q = 105 Pa, show that for the above four cases the normal displacements w(r) substantially differ (Fig. 6.3).

6.3.3 Some Quantities Involved in a Variable Thickness Shell

A basic geometry for defining a variable thickness shell is a central cup element surrounded by successive ring elements linked together.

Let us consider a shell made of N ring elements with a thickness distribution {h,...tn...,tN} slowly varying from center to edge (Fig. 6.4).

To element number n, expanding in the region rn—1 < r < rn, we associate a thickness tn, a rigidity Dn and a characteristic length £„; the two latter quantities are

Fig. 6.4 Thickness distribution of a shallow shell

- or fast meniscus mirror

- made of N continuously linked elements

defined by (6.3g) and (6.17). Some proportionality relations between these quantities are

All elements n e N are with same quantities E, v, <R> and submitted to the same load q.

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