A

where r is the distance from the origin of the surface to the x, y projection of a current point of the surface (Fig. 6.1).

Denoting rm the maximum radius of this surface - corresponding to an optics clear aperture 2rm in diameter - the assumption of shallowness is expressed by the limitation of the slope maximum value

Restraining hereafter to axisymmetric loading cases, let q be a uniform load applied in the normal direction to the midsurface and qr a load applied in its meridian in-plane direction (see Fig. 6.1), the differential equations of the equilibrium of bending moments Mr, Mt, shearing force Qr, and tensions Nr, Nt are (1, and 3a)1

1 In all generality, Reissner (1) originally elaborated the elasticity theory for a non-axisymmetric loading - involving q, qr, and qt - and derived the complete equilibrium equation set corresponding to this case.

At the midsurface of curvature 1/<R> the radial and tangential strains err and ett are functions of the radial and tangential tensions and of the displacements w and u in the normal and meridian line directions, respectively. These quantities are related by

The bending moments are

dr2 r dr r dr dr2

where the rigidity is

The membrane type flexure due to shearing force Qr can be neglected in (6.3a) if the shell thickness is thin, i.e. t/rm C 1. Thus, all concentric levels of the shell have the same flexure in the normal direction as that of the midsurface. Assuming that the load qr acting along the meridian lines are derivable from a load potential Q, we set qr = -dQ/dr, (6.4)

where (6.3a) can be satisfied by using a stress function F for representing the tensions as follows

1 dF d2F

r dr dr2

From the Laplacian V2 • = d2 ■/dr2 + (1/r) d ■ /dr, we obtain

0 0