Setting as usual the origin of the flexure at r = 0, one obtains after a last integration

This allows derivation of the total sag Az = z{a} of the flexure of a constant thickness plate supported at it's center as

which is negative for g < 0.

• Plate with parabolic flexure: Because of the discontinuity arising in the above case at r = 0, where the slope of the flexure becomes infinite - as does the shearing force -, an efficient means of minimizing the flexure of a plate in gravity is to determine the VTD for which this effect is cancelled. As shown by Lemaitre [41] in comparing four distributions, a VTD that provides a purely parabolic flexure allows minimizing the center-edge sag Az.

Assuming a parabolic flexure z ^ r2, the first left-hand term in (8.62) vanishes; after substitution of t(r) by fT(r) the remaining terms are dT3 d2z v dT3 dz p fa ^ „ + - r T dr = 0. dr dr2 r dr dr r r

For further simplifications, the parabolic flexure may be introduced as

0 0

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