Continuity Conditions of a Shell Element Ring

From equation set (6.14), we have seen that the dimensionless radial variable x = r/in, as defined by the characteristic length in of ring number n, must be the argument of the four functions r Let us denote Q,„ the five unknown coefficients of the element ring n whose associated thickness and rigidity are tn and Dn. From (6.31a), the coefficient set C5,n is straightforwardly derived from C1n to C4,n once known. By use of (6.3g), (6.17), and (6.33), the four remaining continuity conditions - corresponding to equation set (6.31b-e) - may be written at the junction of radius rn between rings n and n + 1 as [10, 15]

---► X Chn — = X Ci n+1 —x = invariant for n — n + 1, (6.34a)

Nr - il (- X C+1,n'd1 + X Ci-1 n^Wr) = inv., (6.34c)

After solving the aboveN-set coefficients Q,n for i = 1, ...4, theN-set for C5,n is derived from continuity of the normal displacements by use of (6.31a) where the origin of the normal displacement w must be fixed; for instance, one may set w1 (0) = 0. Given two boundary conditions at the outer contour of the outmost element, solving all Ci,n determines all {wn, un} displacements.

If the central element, n = 1, is not a ring but a plain meniscus (r0 = 0), then

Since ^i(0) = ber (0) = 1 and y2(0) = bei (0) = 0, the set up of the center of the first element as origin of the displacements is achieved if C5j1 = -C1j1. Thus, only the two unknowns C1;1 and C2j1 must be considered.

The outermost ring element optically used is numbered n = N and extends from rN-1 to rN. This element may either receive the mirror external reaction at its edge or be linked to a special outer ring or cylinder.

For instance, for a shell whose thickness variation is made of N = 3 elements, and where the central element n = 1 is a cup, the continuity conditions lead to 2 + 4 + 4 unknowns. Taking also into account the boundary conditions for the external contour of the outmost element n = 3 leads to introducing 2 supplementary unknowns. These latter unknowns, hereafter chosen as the radial bending moment Mr,4 and radial tension Nr,4, require two additional columns in the matrix to solve. Therefore, denoting w = dw/dr, the square matrix and associated column of unknowns are radius r1