C2 E

Let us define a mirror length to radius ratio L/a similarly as quantities Li/r0 or L2/r0 in Table 10.1. From the above equation where xmax = L/2, this mirror aspect ratio is

After substitution of W into (10.26), we obtain the second-order equation d2

For the integration, we introduce the cubic power of the reduced thickness, u = T3, (10.32)

so the differential equation writes

Denoting t0 the thickness at the origin, the associated quantities are T(0) =t0/a =

uQ . The quantity u0 provides the starting condition in the integration from the origin x = 0. Assuming an infinitesimally small increment A%, it is well known that 2

hence the differential equation becomes

where C2 is the unknown. Starting the integration from a given value of the reduced thickness T(0) =t0/a = u^3, the first increment u1 is obtained by noticing that u(f) should be continuous and symmetric with respect the central section plane; this entails that u—1 = u1. The iterations are carried out by varying C2 up to obtaining uN(¡) = 0.

Results from iterations provide the reduced thickness T(f) and the constant C2 as functions of the reduced thickness at the origin t0/a and parameter ¡5 that defines the mirror aspect ratio L/a (Table 10.2 and Fig. 10.8).

Whatever v e [0, 1/2] and T(0), there exists a solution ¡5 for which d2T/ df2 = 0 at the origin, i.e. where the cylinder is of constant section in the central region. The application of the radial forces at the simply supported edges requires use of axially thin collars which thus avoids any bending moment. However, even this simple boundary condition is a practical difficulty for a mirror substrate in glass or vitroceram.

Uniform load and free ends - Inverse proportional law: Compared to the case of plane plates, an important particularity of cylinders is that edge reacting forces are not necessarily required to generate a flexure by external loads.

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