ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN ANN ARBOR TECHNICAL REPORT NO. 6 ON ELASTIC ELLIPSOIDAL SHELLS OF REVOLUTION P. M. Naghdi C. Nevin De Silva Project 2027 OFFICE OF ORDNANCE RESEARCH, U.S,. ARMY CONTRACT DA-20-018-ORD-12099 PROJECT NO. TB 20001(234), DA PROJECT 599-ol-oo4 August, 1954

ABSTRACT By means of a more recent method of asymptotic integration due to Langer, a solution is obtained which is valid at the apex of the shell and involves Kelvin functions. This solution reduces in the limit to the known theory of shallow Spherical shells. Specifically, the stress distribution is obtained for ellipsoidal shells under a uniform load distributed over a small area about the apex. ii

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN TECHNICAL REPORT NO. 6 ON ELASTIC ELLIPSOIDAL SHELLS OF REVDLUTION INTRODUCTION In a previous paper [1], it was shown that the resulting differential equations for small deformation of thin elastic shells of revolution, as given by Reissner [2], may be combined into a single second-order complex differential equation. This differential equation is valid for shells of revolution of uniform thickness and a large class of nonuniform thickness. In the present paper, the solution of the complex differential equation mentioned is obtained by means of a more recent method of asymptotic integration due to Langer [31. This solution is valid at the apex of the shell (where a second order pole is present in the differential equation) and involve Bessel functions of complex argument. Specifically, shells closed at the apex 0 = 0 and subjected to uniform distributed load over a small region about the apex is treated. Also included is the reduction of the solution to the known results for shallow spherical shells [41* THE BASIC EQUATIONS With the use of cylindrical coordinates r, G. z, the parametric equations of the middle surface of a shell of revolution, as shown in Figure 1, are (1) r = r (e) W z = z (e) and (2) r' = ac cos, z'= a sin 0 where a = [(r')2 + (Z,)2]1/2 ) is the inclination of the tangent to the meridian of the shell and the primes denote differentiation with respect to ~. 1

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN To make the paper self-contained, we record here the relevant equations of the small deflection theory of elastic shells of revolution with axisymmetric loading, as given in [2]. rV = - r a Pvd dNg = r'H + z'V; a Q = -z'H + r'V. CoN = (rH)' + r a PH rNS = (rH) cos ~ + (rV) sin (3) rQ = -(rH) sin ~ + (rV) cos D rt M = C [i3' + v r 1] u = ( N) w = (N - N) - r' d where Nt, N,T and Q are the stress resultants; MS and Mg are the stress couples (Figure 1); H and V denote the "horizontal" and "vertical" stress resultants related to Q and Ng; u and w are the displacements in the radial-and axial direo tions, respectively; P is the negative change in 0 due to deformation; pt and pt are the components of load intensity in the r and z directions; h is the thickness of the shell; D = Eh3/12(l,-v2) and E and v are Young's modulus and Poisson's ratio, respectively. The components of stress, due to the stress couples (bending) and due to stress resultants Ng and Ng (membrane), as well as the shearing stress f'r are defined in the usual manner by -6M _ 6M9 agb - (Iab)max'b (cOb)max N_ Ne 3Q - m = - h = ___2h where the subscripts b and m refer to bending and membrane stresses, respectiv 2

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN It was shown in [1] that -the differential equations in P and (rH), resulting from (3) and the equations of equilibrium and compatibility, as derived by Reissner [2], may be combined in the normal form: 25 l 2 3C2E ()r~= 1/2 2 i W W>3w I W(F + ik G) f( ) (5) W' + [2 i3 () +() (g)] W(F + ik G) f() provided that k, given byl (6) k ( - 22v (+Xf - L22 - 2I2 2 + e —- 2v- J is constant. The various quantities occurring in (5) and (6) are defined by W = () /2(r)1/2( + ik I); i = = (rH), m = [12(1-v2)] 1/2 y2M 2,2f( ) Eh2 r2ho A = Q -l' [ 0 +(r'/: r' ht = LF(fW L(r/c) + 3 - it r' h 8 = 2 [ f(j f- + 2v.. +.... h,h r h (r/c) h F = 2ki2 m (rV-) cot 0 -1 zf rt (z'/o4Eh).Z:., G f]- 2 [z'r'.G = [hF (].r. + V (r/CEh) r _;+ v rt] PH P (r/ofEh) 1/ and (k + i-2)V, V fW) (8) 1 = _r /(r/)+i [(L/a? i]r 2 (r/a) 4 L(r/ca) \r 5 (r/a)' h._ h h( _ (h' 2 (r/a) h 2 h 4 \h This restriction, although obtained in an entirely different manner, is similar to that given previously by Meissner [15].

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN (8) = E (rV), PH = r c P where ho is the value of h at some reference section (say ~ = and r2, the length of the normal intercepted between the generating curve of the middle surface and the axis of rotation, is given by r2 = r/sin 0. In what follows, the other principal radius of curvature, that is the radius of curvature of the generating curve, will be denoted by rl. For ellipsoidal shells of revolution, the equation of the middle surface in rectangular Cartesian coordinates is specified by 2 + + 2 a2 c2 a and c being the semi-major axes of the ellipsoid. Selecting the independent variable e as 0, it follows from the geometry of the middle surface and (2) that a = rl. Thus, the radii of curvature are c2 a (9) r, = ac = all + o2 0]3/2 r2 [1 + p2 cos2 0]1/2 and a sin (10) r =. [1-+ P2 CoS2 0]1/2 2 2 (11) p2 = 2 a a2 From (7) and (9), we have c4 Ia -5/2 (12) 2C12 = y() -), f (f ) = (1 + p2 cos2 0) and when the thickness h is uniform, C + 2 3/2 (13) = O X = - () (1 + p cos2 0) Since we are mainly concerned with shells of uniform thickness, in the remainder of this paper ho will be replaced by h, unless otherwise stated. As remarked previouslyr [1], for ellipsoidal shells r1 is not constant it follows that when h is uniform, k is a function of 0. However, with a view towards approximating k by a constant so that the condition for the validity of equation (5) is fulfilled, we note that restriction of (c/a) to 0 (1) is consistent with v << 212 and by (6),

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN [1 ->or k r 1. With this approximation, equation (5) is valid and the coefficient functions A and A read: = (1 + 2 + 22 C 0; 2) (14) A = cot XT c (1 + p2 Cos2 0)2 where A1 Il [ p2 sin. 2 + 5 V- 0os~t 2l (\2\rsin2 + pa Cos2 1 16 1 + p2 os2 2 + p2 COS2 0 2 \a/ (1 + p cos2 0)2 1 and the transformation for W in (7) is (15) W = (a [sin 0 ( p2 cos2 /2 ( + i ) SOLUTION OF THE DIFFERENTIAL EQUATION BY ASYMPTOTIC INTEGRATION Inspection of the coefficient function A as given by (14) reveals the presence of a pole of second order (at 0 = O) in the differential equation (5). Consequently, the solution of (5) by the classical method of asymptotic integration fails to yield a solution valid at the apex of the shell. In order to obtain a solution of (5) which is valid at 0' =".D, recourse will be made to a more recent technique of asymptotic integration, due to Langer [5]. According to Langer, it becomes necessary to modify (5) into a new normal form. For this purposel we introduce a new independent variable t as follows: (16) t = sin d 0 = 2 (t Then (5) may be written as2 d(2W t dW + 4F 2 L +. = R(L-t2)l/4 (17) d4t2 -2t2 dt:)t t = Rt. 2If it is desired to cover the entire region of the shell, ide., O C 0 <, | instead of transformation (16) t = sin 0/2q should be used. This entails replacing the factor 4 by 4q2 in (17) where q > 1.

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN where I/2 (18) (-t2)5/4 (/ f(t) [F + i G Expanding the coefficient function 4A1L(t)/l-t2 in (17) by partial fractions, there results (19) 4 (t) = 3/4 + 2 (1 - t2) t2 where (20) 2 4 -J/4 (c/a)4 (1-2t2)2 + 1 -t2 Lt2 (l_t2)2 [1+p2 (1.2t2 j.l2 t2j is bounded in /t/ < 1, i.e., in 0 < 0 <. Now, by means of the transformation (21) W = (1 - t2/ W we obtain from (17) the new normal form (22) = + W = R where (23) 1A.3 = 2 + (t2 + t2(_t2)-2 ( 4 is bounded in /t/ < 1. Since the particular solution of (22) depends on the specific loading, we shall consider in this section only the solution of the homogeneous differential equation associated with (22). As pointed out by Langer, the homogeneous solution of (22) may be written ass (2;) { i= {n1/2 /4 t/4l/ai 4 (24) W2 1 t The numerator of the second term in the coefficient of in (22), when written The numerat1/ [(2 cor of the seond term in the coefficient of in (22) when writtenr of the Bessel functions in the solution of (22) is determined by 1/2 A.

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN where F2)i1 0~/ /4 I3/4 log [(2i3 42)1/2] (25) = Lt2)1j (2i3 p2)1/2 ( and 0(E) denotes a boundedfunction. In (24), H(I)() and Hf)(r) are Hankel functions4 (26a) = (8i3 12)1/2 and 4 (26b) = + dt In view of (24) and (25), it follows that the homogeneous solution of (22) may be represented asymptotically by (27) WH A lta/)2 1/4 H( ) + (2 which is valid in /t/ < 1 and where the constants A and B are complex. It is advantageous to express solution (27) in terms of Kelvin functions, as follows: (28) Wl = [(1)1 /2 A [berl (s) + i beil (s)] + B1 [keri (s)+ikeil slj where (29a) s = (81.2)1/2 f and (29b) Al = Ao + i Al, B = Bo + i Bi By means of (15), (21), and solution (28), we record the expressions for,s,1 B', and.'V, since they are required in the solution of specific problems. = j2 (lt2)[l+p2(l-2t2)2]/2 t2 /iA berl (s) (30a) -A1 beil (s) + B0 keri (s) - BL kei (s)} 4The notation used is that of Watson. See reference[6].

ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN = (L) f(l-t2)[l~+p2(1.-2t2)21 ]2)1 J {Ao bei1 (s) (30a) + AL ber1 (s) + BO kei1 (s) + B1 ker1 (s)} =; + (2A2)1/2 (E ) [1+p2 (N2t) it a (30b) -{/2 ] 1/2 (30b), + (212)I/2- /[l+p2(1-2t2)2 -] _ where again primes denote differentiation with respect to 0, t = sin 0/2, and = [l+a(1-2ta)2] /4 1- (t2)l/2 [l+p2(1-2t2)21] / i f.3st2 8.2t (l.t2)1/ [l+p2(1-2t2) (ll-l2t2) I - ( - [l"pa(1-ta] l (= - [Ao ber, (s) - A, beil (s) + Bo ker, (s) - B, kei, (s)] C/ = d [AO beil (s) + Al ber, (s) + Bo keil (s) + B1 kerl (s)] dCs REDUCTION TO THE THEORY OF SHALLOW SPHERICAL SHELLS In this section, we consider the transition from the solution (30) to the known results for shallow spherical shells, due to E. Reissner [4]. For this purpose, we first examine solution (30) for the case of a spherical shell and then proceed to show its reduction and correspondence to the limiting case of shallow shells. Since for spherical shells c/a = 1, then = rl = r2, p2 = 0, and by (7), f = 1. Hence 212 =(a/h)m, and the quantitiesn and; in (14) and (26) become (31) 0= i = 0/2. The solution (30) is now considerably simplified and reads as follows:' (32a) = [s 1 {nAo berl (s) - A1 bell (s) + Bo kerl (s) - B1 keil(s __ __ _ __ _ __ __ _ __ _ __ __ _ __ _ 8 _____________________

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN (32a) = [' Lj/ Ao beil (s) + Al berl (s) + BO kei1 (s) + B1 ker1 (s) D =; + ( m)/ [2sinJ d. AO ber, (s) - Al bell (s) (32b) + Bo ker (s) - B ke (s) m)j/ I2 + (~)/2 d L Ao beil (s) + Al berl (s) + Bo keil(s 2 sin s. + B1 kerl (s)} where the arguments is reduced to (32c) s (a /2 and (32d) 2 (0 )-ot According to Reissner [41 a spherical shell is defined as shallow when r/a is small compared to unity. This assumption implies that the theory of shallow shells is valid only for small values of 0; in particular, since r = a sin 0, then for small values of 0, r/a _ 0. While the argument "s" given by (32a) is valid for 0 < 0 < x, for small values of 0 it may be written as s (a/h m)1/2 r/a which is identical with the argument of Kelvin functions employed in [4]. In view of this discussion, it is clear that for shallow shells sin 06 — 9 and. - 0 in solution (32). Thus for shallow spherical shells, using well-known recurrence relations for Kelvin functions, we have = fAo-A1 ber (s) A B0-B B1+ j =d [AO-Al ber (s) - AO+ Al bei (s) + B —-1 ker (s) - kei (s ds 2 2 2 (33) = d. A-A1 bei(s) + An+A1 ber (s) + B0-B1 kei (s) + B0+B1 ker (s). ds 2 2 2 2 To show the correspondence of solution (33) with that of Reissner [4], we recall that the basic dependent variables of his work are a stress function F and the displacement normal to the middle surface of the shell, which for small 0 is the same as the deflection w of the present paper. These dependent variables are related to our 8 and'r by dw m dF (354) 4r Eh -, ~ "

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN If (33) and the solution given in [4] are substituted in (34), it is seen that a one to one correspondence exists between the two solutions, provided the coefficient of 1/s in dF/dr is set equal to zero; the function l/s arises from the solution of the differential equationV2F = O. An equivalent differential equation does not arise in the formulation of shells of revolution employed here, and this is also borne out by Reissner's subsequent work on shallow shells [2, p. 243]. ELLIPSOIDAL SHELL UNDER A UNIFORM NORMAL LOAD DISTRIBUTED OVER A SMALL REGION ABOUT THE APEX We consider an ellipsoidal shell clamped at the edge 0 = t/2 subjected to a_uniform normal load Pn distributed symmetrically in the region O < 0 < 0, 0 being small. In view of the presence of a distributed load, it becomes necessary to obtain a particular solution of (22). Since by the first of (3) (35) rV Pn a2 sin2 0l+P2 cs2 0 < 2 } 2 (l+p2 cos2 2)2 P n a2 sing 20 0 < 1/ 2 (l+p2 cos2 ~) then, as in [4], a suitable particular solution may be obtained approximately by the membrane theory of shells. Thus, for small values of 0, the particular solution is (36) Bp = 0 m Ia2 p2 itn - - 1)E h/ 2 (l+p'cos- ) In the following, the loaded and unloaded regions will be distinguished by subscripts I and II, respectively. The requirement that the quantities M0, Mg, N., Ng, Q, w, and w' remain finite at the apex 0 = 0 demands that (37a) 0 = O; PI' I 7 I, L I r r remain finite where the quantities in the above include both the homogeneous and particular solutions, i.e., PI = (AH) + (P)I etc. 10

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN The boundary conditions at the clamped edge 0 = /2 are (37b) 0 = n/2; II =, II = (rV) where II= (H)II etc., since no particular solution_is needed for the unloaded region. Also, the continuity conditions at 0 = 0 are (37c) 0 I DII I II' I II DI II Examination of (30) reveals that condition (37a) is satisfied provided that the constant coefficients of the functions kerl (s) and keil (s) are set equal to zero at the outset. Hence, for region I, the solution is given by = ( )/)l+p(l-2t2)]berl (s)-Clbeils)J (38) =I (2t) {(l-t2 )[l+P (l-2t f' L' I1/2fCo beil s + Cl ber' 2a ( tan. E 2 (l+p2 cos2 0) The solution for region II, on the other hand, involves (30) only with all four functions retained. The six constants of integration are determined from the conditions (37bc) which result in the following six simultaneous equations: CO beri(s) - C1 beil(s) - Ao berl () +,A1 bei1(s) - Bo kerl(s) + B1 keil(s) = 0. Co berI(s) - C1 beil(s) - Ao ber{(F) + A1 beil(s) - Bo kerl(s) + B1 kei() = O Co beil(s) + C1 berl(s) - Ao beil(F) -A1 berl(s) - Bo keil(s) - B1 kerl(7) tan ~ (l+p2) ( (l+p2 cos2 0) 2 Co beil(s) + C1 berL(s) - Ao beiJl() - A1 berl(s) - Bo keil(s) - B1 kern(s) |= 0 (l+p2)5/4 1 2p2 sn2 tan 02 (2) A0 berl(s) - A1 beil(s) + B0 ker1(s) - B1 keil(s) = 0._ ~11

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN Ao[bei (s)-(X/2) + (212)1/2 beii(s)] + Al[berl(s)t(r/2) + (2k2)l/ ber( )]1 +Bo[kei'(ss);(X/2) + (2k2)1/2 kei.,(s)] + Bl[kerl(s)~(i/2) + (2 2)1/2kerl()] | - o ((l+p2 cos2 [2/ 2)] where = E () n' s = s s = s(i/2) and the primes in these equations denote differentiation with respect to'st. Taking a/h = 20, c/a = (p2 = 1), ~ = 10, and v = 0.3, then the constants of integration are determined as follows: Co = 0.065738 s2 C1 = -0.126213 Qo Bo = -0.016310 Go (40) B = 0.010468 ao Ao = -5.5574 x 10-8 ao Al = -2.5676 x 10-8 so Using (4), the stress distribution for the example treated is aa shown in Figure 2. It is noteworthy that, although the ratio of c/a in the present example is 4, the resulting bending stresses in the loaded region are in very good agreement with those of the corresponding example of shallow spherical shells 14]. 12

x Fig. 1 15

-0.25 1 1 -- 0 2 0 V 2 0 3 0 4 0 5 0 6 0 l o_0_ 9 0-8 b (a/b)pn -0.15 - ---- --- - c-b -0.10 ____ INDEGRE 0.05 (/h Pn 0.05 5~ 0. I5 0! O 20 30 40 50 60 I0 80 90 4 IN DEGREES Fig. 2

REFERENCES 1. "On the Deformation of Elastic Shells of Revolution". by P.M. Naghdi and C. Nevin De Silva to appear in Quart. Appl. Math.; see also O.O.R. Tech. Rept. No. 3, Contract DA-20-018-ORD-12099, Univ. of Mich.,(1953). 2. "On the Theory of Thin Elastic Shells"', by Eric Reissner, H. Reissner Anniv. Vol., 231-247 (1949). 3* "On the Asymptotic Solution of Ordinary Differential Equations, with Reference to the Stokes' Phenomenon about a Singular Point", by R.E. Langer, Trans. Am. Math. Soc. 37 397-416 (1935). 4. "Stresses and Small Displacements of Shallow Spherical Shells I -and II", by Eric Reissner, J. Math. Phys., 25, 80-85 and 279-300 -(1946). 5. "jUber Elastizitat und Festigkeit dunner Schalen", by E, Meissner, Vierteljahrsschrift der Naturforschende Gesellschaft in Zurich, 60, 2347 (1915)9 6. A Treatise on the Theory of Bessel Functions, by G.N. Watson, Cambridge Univ. Press (1952). 7. Tables of the Bessel Functions Jo(Z) and J1(Z) for Complex Arguments, Columbia Univ. Press (1943). 8. Tables of the Bessel Functions Yo(Z) and Y1(Z) for Complex Arguments, Columbia Univ. Press, (1950). 15