284 W. M. Shepherd For b> chte required functions could then be obtained using the recur rence relation b (1 — z) 2Fi {a, b + 1; c; z) = {(6 — (1 — — (c — a)} 2FX (a,b; c; z) + — 2FX 1; c; z). (14) For b< cti was usually most convenient to compute numerically 2Fx {a, c — 2 ; c; z) and then to use the above recurrence relation. A useful check on the calculations can be obtained by working back to 2Fx (a, o ; c ; z) which, of course, should equal unity.
Stress Systems in an Infinite Sector 285 1—The Sector under Two Equal Inward Normal Forces at Corresponding Points of the Two Edges Using polar co-ordinates r, 0 in the plane of the plate, the sector is upposed bounded by the lines 0 = ± a and to be of uniform thickness Ic. Consider the systems of mean stresses corresponding to the two stress functions* Xi and X2- Corresponding to Xi = rw+2 cos ( + 2) 0, ve have w 00 = (n + 2) ( 1) rn cos (n -b 2) 0, ft rr = — (w + 2) (n -f- 1) rn cos (n -f 2) 0, r0 = (n + 2) (« + 1) rn sin ( + 2) 0.
W. M. Shepherd 286 We then have 00 = (cos mp + i sin mp) {(2 — m2) + 3 (A + iB), rr = (cos mp + isin mp) {(2 — m2) + 3/m} (C + /D), r% — (cos mp + i sin mp) {(2 — m2) + 3/m} (E + /F), where A — sminh ma cos 2a cosh m0 — m sinh ma cosh m0 cos 20 — 2 cosh ma sin 2a cosh m0, B = m sinji ma sinh mO sin 20 — m cosh ma sin 2a cosh m0 — 2 sinh ma cos 2a cosh m0, C = m sinh ma cosh m0 cos 20 — m sinh ma cos 2a cosh m0 — 2 cosh ma sin 2a cosh m0, D == m cosh ma sin 2a cosh m0 — m sinh ma sinh m0 sin 20 — 2 sinh ma cos 2a cosh m0, E = m cosh ma sin 2a sinh m0 — m sinh ma cosh m0 sin 20, F = m sinh ma cos 2a sinh m0 — m sinh ma sinh m0 cos 20.
Stress Systems in an Infinite Sector 287 and let the corresponding stresses be 00m, rrm, /*0W. Then 0 0 = (1 + ra2) (4 + ra2) {cos rap (A0A + B0B) — sin rap (A0B — B0A)}, rrm— (1 + ra2) (4 + rn2) {cos rap (A0C + B0D) — sin rap (A0D — B0C)}, r0m = (1 + ra2) (4 + ra2) {cos rap (A0E + B0F) — sin rap (A0F — B0E)}.
288 W. M. Shepherd The integral (1.3) is, however, uniformly convergent for all values of p if | 6 | < a and for p ^ 0 if 0 = ± a. Therefore lim F (P> 0) = - ^ f" S12LP dm e->±a 2-kc Jo M provided that p ^ 0. It follows that P>0 (1.4) = + I ? < ° Now 00 as defined by (1.2) is a continuous bounded function </> (0, p) of 0 and p throughout the interior of the sector, the neighbourhood of the points p = 0, 0 = ± a being excluded. On the edges of the sector 00 is given by 00 = lim (0, p).
Stress Systems in an Infinite Sector 289 difficulty similar to that encountered in finding 00 again appears. The integral foo (PC + QD) cos rap dm Jo is not convergent when 0 = ± a, but is uniformly convergent when | 0 | < a. As before it is permissible to integrate with respect to p under the integral sign and we obtain I (0, p) = f" rr d9 = - f" {(PC + QD)S2-^£ J o 2tcc J o l ra + (PD - QC) - 1)| dm m Fig. 1 This integral is uniformly convergent at all points of the interior and of the edges of the sector. It may also be written in the form w r 1(0, p) = ± -■ r {(PC + QD — 1) 2nc — 2 Jo ^ m + (PD - QC) (cos m? 1)| dm .
290 W. M. Shepherd 2—The Sector under Two Equal Normal Forces, One Inward and One Outward, at Corresponding Points of the Two Edges Derived from the stress function X3 = r”+2 sin ( + 2) 6, we have 00 = (« + 2) (n + 1) sin ( ft + 2) 0, rr = — (ft + 2) (ft + 1) sin (ft + 2) 0, r 0= — (n + 2) (ft + 1) rn cos ( + 2) 0.
Stress Systems in an Infinite Sector 291 by (1.2), the quantities A, B, C, D, E, F, A0, B0 being given in this case by* :— A — mocsh ma cos 2a sinh m0 — m cosh ma sinh m0 cos 20 — 2 sinh ma sin 2a sinh m0, B — m cosh ma cosh m0 sin 20 — misnh ma sin 2a sinh m0 — 2 cosh ma cos 2a sinh m0, C — cmosh ma sinh mO cos 20 — m cosh ma cos 2a sinh m0 — 2 sinh ma sin 2a sinh m0, D = m sinh ma sin 2a sinh m0 — m cosh ma cosh m0 sin 20 — 2 cosh ma cos 2a sinh m0, E = m sinh ma sin 2a cosh m0 — m cosh ma sinh m0 sin 20, F = m cosh ma cos 2a cosh m0 — m cosh ma cosh m0 cos 20, A0 = — 2 sinh2 ma sin 2a = (1 — cosh 2ma) sin 2a, B0 = m sin 2a — sinh 2ma cos 2a.
292 W. M. Shepherd The stresses 00 and rr are even in 0 and /*0 is odd.
Stress Systems in an Infinite Sector 293 It follows that 06A — cos mp (A — mB) — sin mp (mA + B), 00B = sin mp (A — mB) + cos mp (mA + B), rrA — cos mp (C — mD) — sin mp (mC + D), rrB = sin mp (C — mD) + cos mp (mC + D), rOA = cos mp (E — mF) — sin mp (mE + F), rdB = sin mp (E — mF) + cos mp (mE + F). When 6 = a, E = E0 = 2 cosh2 ma sin 2a = (1 + cosh 2ma) sin 2a, F = F0 = m sin 2a -J- sinh 2ma cos 2a.