6.4 Edge Cylinder Link and Boundary Conditions

Let us join the outer contour of the meniscus shell element N to one end of a cylinder N+1 by a continuous link. We then obtain a vase shell configuration where the set up of the cylinder dimensions and boundary conditions at its opposite end shall allow analysis of three manifolds.

6.4.1 Three Geometrical Configurations and Boundaries

The shallow shell analysis will be investigated for the three following geometries: a meniscus shell (Fig. 6.4), a vase shell, and a closed shell (Fig. 6.5).

Each geometry can be characterized by the axial and radial thicknesses of the outer cylinder and a specific couple of boundary conditions. For instance, a vase shell with an outer cylinder of extremely thin radial thickness is an equivalent geometry for a meniscus shell. In practice, only the boundary conditions following below can be easily achieved. We call "external end" of the cylinder the opposite end to that which is linked to the meniscus shell.

• Meniscus shell boundaries and thin cylinder external end boundaries: The axial thickness of the outer cylinder is, for instance, set to tz,N+1 = tz,N and is given a negligible radial thickness compared to tN. Its external end is simply supported without restraint on the radial displacement. In other terms, the boundary conditions (6.32a) for an articulated and movable external end are

where the subscript E holds for "external end." These conditions correspond to (6.32a) (see also Fig. 6.3).

• Vase shell and cylinder external end boundaries: In this design we always set tz,N+i > tz,N which increases the self-rigidity for a perimeter supporting of the mirror [12, 13]. The boundaries for the external end of the outer cylinder are similar to the above case, i.e. (6.38a).

• Closed shell and cylinder external end boundaries: This design is made of two vase shells oppositely linked together at their cylinder rear surfaces [11]. The boundaries at this junction are a built-in and movable surface,

The junction allows analysis by only considering the first vase shell - with variable thickness - whilst the second complementary vase shell must have an appropriate mean thickness if this latter is a constant. The above conditions correspond to (6.32b) (see also Fig. 6.3).

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