N

where the second term Ai does not represent the total cone angle of the optical surface of the shell but is a much smaller quantity which can be generated by flexure. Avoiding x-zones with infinite or too large thickness, the general relation expressing the flexure must write

where the constant a2 and sign before it are set such as W(x) is with monotonic sign over the range [-ft, ยก3] and never approach too closely to zero. This implies a shell in extension condition only - or retraction condition only - all over its length.

10.3.2 Truncated Conical Shell Geometry and Cylindrical Flexure

The flexure of a conical shell by an axisymmetric linearly varying load, whose origin is at the cone vertex, is a homothetic cone. A cylindrical shell uniformly loaded can be considered as a particular case where the cone vertex is at infinity. The elaboration of a linear load function for the extension - or retraction - of a cone is a great practical difficulty; therefore we only consider hereafter the case of a conical shell deformed by a uniform load, q = constant. Except for a small axial reaction on its larger ring face, if no other force is applied to the conical shell, then the radial shearing force is null, Qx = 0. Returning to the dimensioned quantities, from (10.36), we have tw/a2 = q/E = constant.

This latter relation allows taking into account the conical geometry. Since with (10.15), the three components of the shear strains are null, a truncated conical shell can be assumed as constituted of separated element rings that are continuously distributed along the x-axis. In the x-variation of the mid-thickness radius a, let us consider that each element ring provides the same amount of flexure w. This is achieved if t( x) q 1

Let us denote (Lj), (Lo) the straight segment lines forming the inner and outer surfaces of a conical shell, (Lm) that of the mid-thickness surface, and i, o the associated low-angle slopes (Fig. 10.13). In a frame r, x where x = x/a0, the equation of the segment lines (Lj) and (Lo) are respectively rt = a0(1 - ix) - 110, ro = a<)(1 -ox) +110, (10.64)

which allow defining the mid-thickness line a(x) and the thickness t(x) as a = a0

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