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ay ax

ax3 ay2

3U2 I a2

ax2 ay3

For a circular plate, denoting — — a and spl its slope at the edge, the bending moment is

In (z, x) and (z, y) sections, the radial thicknesses tx, ty of the elliptic cylinder to link with the plates, can be defined by dimensionless coefficients yx, yy such as a2 a tx bx - ax ty by - ay

where t is the thickness of the two plates.

In order to close the two separated plates by a cylinder linked at their edge, let us consider at first the axisymmetric case of a circular cylinder in setting y= jx = yy = tcyl/t, where tcyl is the radial thickness of the cylinder. Let b = bx = by be its external radius and acyl = (a+b)/2 its mean radius. Without changing the frame notations, let us move the (z, r) axes of a plate frame in parallel directions so the origin of the coordinates are now at the median point of the neutral surface of the cylinder. The flexure is a solution of the differential equation [30b]

where the rigidity of the cylinder Dcyl and the coefficient ¡5 are defined by D Etcyl ¡4 3(1 —v2)

In order to determine the thickness tcyl and the length I of the cylinder, first we notice that the cylinder is bent by opposite moments M at its edges z = ±^/2, so the flexure r(z) must be even. Then, the solution of (5.56) is r = Ci sinh ¡5zsin¡z + C2 cosh ¡5zcos ¡z + f (z) (5.58)

where the odd terms, i.e. C3 and C4 coefficients, have been cancelled, and f (z) = q/4fi4D = constant is the particular solution due to loading q. Assuming hereafter the case of very short cylinders, the extensional deformations due to loading q can be neglected in comparison to the effect of the bending moments M and —M applied to the extremities. We have seen with (5.53) that the reactions Vx{±ax, 0}, Vy{0, ±ay} only depend on the loading intensity q whatever the boundaries at the edge (clamped, semi-clamped, or simply supported). In a first-order approximation, we will also neglect the deformation of the cylinder in the z direction.

Since linked to the plates, the extremities of the cylinder cannot appreciably move in the radial direction r, but can rotate. The bending moment is defined by M = Dd2r/dz2. Therefore, the boundary conditions for the outer cylinder are as follows

Denoting co = ¡1/2, we find after substitution,

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