where a is an unknown. Thus, after substitution and using p = r/a, we obtain

After derivation and division by p, the researched thickness is a solution of the general equation

a where V2 is a Laplacian with respect to the radial variable p. However, it is preferable to solve the thickness T(p) from the integro-differential form (8.70). For this,

one remarks that for p ^ 0, the second term is with a negligible variation which entails T ^ (- ln p)1/3. In the region of the edge, from first and second-order variations of 1 - p, we find the asymptotic form

Therefore, at some distance from its center, the solution for the plate VTD is a quasi-conical thickness distribution.

The asymptotic representation (8.72) allows the numerical integration from the edge of the plate which also must satisfy the comparison condition (8.64) of the normalized thickness for equal mass plates, that is /Q pT(p) dp = 1/2, and then determines t from (8.63). The results from integration of (8.70), which is not Poisson's ratio dependent, are a = 0.022316... ~ 1/45 (8.73)

and a quasi-conical T(p) distribution given in Table 8.4 and shown by Fig. 8.18.

From (8.73) and (8.65), the substitutions of a and ยก5 into (8.69) provide the dimensioned form of the parabolic flexure as

from which the center-edge sag is

Table 8.4 T(p) values of the normalized thickness distribution from (8.70) of a plate with parabolic flexure in the gravity field when supported at its center (Lemaitre [41])


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