General Equation of Shallow Spherical Shells

Combining (6.3d) and (6.3e) with the above equations, a first fundamental relation for the stress function F and normal flexure w is

The shearing force Qr in (6.3c) is derived from the bending moments in Eqs. (6.3f),

A second fundamental relation between the F and w functions is obtained by substitution of the shearing force Qr in (6.3b). Use of (6.5b) leads to2

Let us now assume that the meridian loads are always null, qr = 0, for all loading cases investigated hereafter. From (6.4), this leads to only considering the two following cases for the load potential, q = Q = 0 or q = - 2Q/<R>= constant. (6.9)

Hence, the left-hand terms of the two fundamental relations both vanish, so these relations can be combined into a single equation. Let us define quantities X and t by the following set, where i = \J—T,

XEt / <R >= i/t1, 1 / (XD <R>)= - i/t1, (6.10a)

which gives

X = i^J 12 (1-v2) / Et2, I =y <R> t/y 12 (1-V2). (6.10b)

If qr = 0 and if one of conditions (6.9) on q is satisfied, then, multiplying (6.6) by —X and summing with (6.8) leads to

This is the general equation of shallow spherical shells for a load q = 0 or q = constant. It was first derived by Eric Reissner [2] in 1947. This equation is integrable by writing w — XF = O + W, (6.12)

where the O and W functions are general solutions of

Assuming complex constants Ai to A4, the respective solutions are

2 For a non-axisymmetric load where the tangential component qt is derivable from a load potential, as qt = — T ^, Reissner [1] demonstrated that the two fundamental equations (6.6) and (6.8) still apply.

W = A3 [ W (r/i) - i W2(r/i)] + A4 [W3(r/i) - i W (r/i)], (6.14b)

where the yn functions are zero-order Kelvin functions as follows3

W1 = ber (r/i), w = bei (r/i), w = ker (r/i), W4 = kei (r/i), (6.15)

which are explicitly given in Sect. 6.2.3.

The general solutions for w and F are derived by identification of the real and imaginary parts. Introducing real coefficients C1 to C8 and the characteristic length i, we set

A1 = i (C5 + iC8), A2 = i (C7 + iC); A3 = i (C1 + iC2), A4 = i (C3 + iC), (6.16)

and representing the W(r/i) functions by the concise writing w, W/i = C1W1 + C2W2 + C3W3 + C4W4

+i (C2 W1 - C1W2 + C4 W - C3W4), O/i = C5 + iC8 + (C7 + iC) ln (r/i).

where, from (6.3g) and the second of equations (6.10a), the fourth power of the characteristic length also writes

Returning to the above expressions of W/i and O/i, the identifications of the real and imaginary parts in (6.12) provide the determination of the normal displacement and stress function respectively as w = i [C1W + C2W2 + C3W3 + C4W4 + C5 + C7 ln (r/i)], (6.18a)


F = —— [-C2W1 + C1W2 - C4W3 + C3W4 - C8 - C6ln (r/i)]. (6.18b) < R >

Thus, in the general form corresponding to an axisymmetric load q = 0 or q = constant, each function w and F is represented by eight unknown coefficients where Ci to C4 are common coefficients.

3 In Timoshenko W-K [29], these authors use the following representation for the Wi Kelvin functions : W1 = ber, W2 = -bei, W3 = -(2/n) kei, W4 = -(2/n) ker on p. 560 and in the Tables on pp. 491-494. We adopt hereafter the Reissner representation [25, 26].

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