Toroidal shells under nonsymmetric loading

ht. .I. Solids Structures Vol. 31, No. 19. pp. 2735-2750, 1994 Copyright 0 1994 Elsetier Science Ltd Printed in Great Britain. All riehts reserved 002&7683/9~37.00 + .OO

Pergamon

0020_7683(94)EOO3&Y

TOROIDAL SHELLS UNDER NONSYMMETRIC LOADING R. J. ZHANG Department of Engineering Mechanics, Tongji University, Shanghai 200092, People’s Republic of China (Received 19 August 1992; in revisedform 8 August 1993)

Abstract-The complete asymptotic expansions of four homogeneous solutions, and a particular solution of toroidal shells, based on Novozhilov’s thin shell equations, are given, which are valid for the stress and deformation of toroidal shells of circular cross section subjected to nonsymmetric loadings.

NOTATION wall thickness of shells (constant) radius of the meridian circle distance from top of toroid to axis of revolution rol& Fourier harmonic index elastic modulus Poisson ratio

tangential and circumferential angles of shell l+psinfI principal coordinates Lame’s coefficients principal radii of curvature tangential, circumferential and normal loading components stress resultants, shown in Fig. 1 stress couples, shown in Fig. 1

#f,,f‘k,)

change in curvature and change in twist N, - i&Ehtc,

N, - i&EhK, S+ieEhr w, +flz

tangential, circumferential and normal displacements, respectively.

Fig. 1.Notation of force resultants and moment resultants in shell coordinates. 2735 W

31:19-J

2736

R. J. Zhang 1. INTRODUCTION

The taroidal shell is characterized by the existence of the transition points at 6 = 0 and n. Therefore, the displacement at this point is discontinuous in the membrane theory and bending moments always exist. This feature makes it difficult to obtain the asymptotic solutions. The toroidal shell problems are generally divided into two cases to be investigated according to the loadings ; axisymmetric (m = 0) or nonsymmetric (m # 0). In the present paper, only the latter is discussed. The case of axisymmetric loadings has been investigated for many years. However, there are few papers on the case of nonsymmetric loadings. Steele (1959) solved this problem for the first time in his dissertation at Stanford University. He derived the nonhomogeneous integro-differential equation of the fourth order and obtained an asymptotic solution for large values of ~1= ~~r~/(RJz) and for m3/(,,k(r,,/h)) cc 1. The latter condition restricted his solution adaptable for only lower harmonics m = 0, 1, 2. . . . In the present investigation, it is identified that the comparison equation is, in fact, the certain generalization of Airy’s equation. Therefore, the generalized Airy functions, introduced by Drazin and Reid (1981) are used to obtain successfully the complete asymptotic expansions of all four homogeneous solutions and a particular solution. They are numerically satisfactory, uniformly valid and they satisfy the accuracy of the theory of thin shells. As for the case of axisymmetric loading, a novel solution has been found by the author. The paper will be dispatched separately. The goal of the author is to obtain a unified solution for the toroidal shells under arbitrary loadings. The solution of toroidal shells with nonsymmetric loading corresponds to a transition point problem for differential equations of fourth order. The transition point of higher order occurs in shell vibrations. Zhang and Zhang (1991) have obtained the complete uniformly valid solutions.

2. FUNDAMENTAL

EQUATIONS

Novozhilov (195 1) has given the thin shell equations in the complex form 1 aA&.?

aA&

__ + Al ah

at,

aA* - -N2+i+ at,

= -A,,4,q,

g I

I

(1)

For the case of the toroidal shells of circular cross section, as shown in Fig. 2, the parameters are now <,=9,

Furthermore,

r2=(p;

A,=ro,

with the introduction

o=

eqn (1) can be rewritten as

A,=R,cr;

R,=ro,

R,=R,a/sintI.

(2)

of a new complex variable

R,asin8fl,-hR$cosOg,

(3)

2131

Toroidal shells under nonsymmetric loading

Fig. 2. Toroidal shells of circular cross section.

a20

1

--

1

-+ Roa sin 8 a$

1 -ic rOo sin’

81

where

(qncOse-ql

sin@

z]+

$r,Ric’}.

(5)

Eliminating 0 from eqn (4), we obtain

o’g+gpricose$+

i$cr3sin8ca2(1+14p2-4psinB-19p2sin20)

1

[ + i~~2cos8(3+7psin8)+~~cos8(2+4p2-6~sin0-12p2sin28)

g

af

1

ae

aV aT aT +p4- +2/M---+4P3ac0seae2aq2 aeaq2 aq4 2”

+ i+02+~3(p-2sintI-2~sin2f7) [

1

fi aq2

(6) where j@,cp) = -i%

0 f”
go

-?-

a (qna2 sin 0) + -!L E (qna2 sin 8) . 1 a sin 8 aV2 I

80 [ sin’ 8 ae

(7) The loadings and all the variables in eqn (6) can be expanded in Fourier series along the circular direction of toroidal shells. We let their mth order harmonic components be

2738

R. J. Zhang

(~:41,4”,NI,N*,M,,M*,u,w)(e,cp)

= ~~~m,4,m,4nm,~,,,~2m,~lm,~2m.~m,~m~(~~~~~~cp

(42, S, K 4UA qo) = km,

&, K,

GJ(@

sinw.

(8)

Inserting eqn (8) in eqns (6) and (7) we obtain the equation d4F d3T A--$ + 8/Lo3cose---1+ de3 + i~-a2c0s8(3+7~sinB)+~~c0s8(2+4~2-4~2~2-6~sine-12~2sin28) [

+

I$$

i$~[7~-~m2+(9pZ--p2m2-2)sin~-12~sin2e-12~2sin3~]

1

+ b4m4 - p4m2 + 2p3m2 sin 8+ 2p4m2sin* e] Tm = Pm(e)

(9)

where

In terms of r,,,, the mth order harmonic components of stress resultants are expressed as

+ ~(1+2~2-~2mz-3~2sinz6))Im

% (

-1

+cosB(Z+3/rsinB)Re(~,,J-~cosOIm(~,,,)-3q~,,,~o~cosO 0 2 +

y1-v)cosBRe -T(l--v)Re(i;,)--s(o--vpsinO)Im(P,) 0

M2,,,=

-z(l-v)oRe

-$(l-v)cosORe + F(l-v)Re(pm)+s(Psin8--vo)Im(pm) 0

2739

Toroidal shells under nonsymmetric loading

mZi,=

E2

-4;(l-v)scoseRe

-$I-v)o’Re

-&(l-v)(1-t-zpi-p’m’-3pisin’R)Re

2

+&(I--v)osineIm

(

)

(

$$ *!

+ ~~l-v)coseReiT,)fEjl-v~cose(2+3irsin8)Im(i;,,).

(11)

0

The mth order harmonic components of displacements u, and w,,,are the respective solutions of the equation

and the equation

$f =$p,-vN,,)-w,.

(13)

The displacement ~7,is

+(l+v)zcos0Im

mo,=~~~(l+v)Im~)

-(l+v)$

de -t-[l+(l+v)psinf3]Re(F,,,) (dFm)

Im(Fm)-q,,(l+v)r,o

0

-(u,cosBfw,sin@.

(14)

I

For the asymptotic solutions of eqn (9) to be valid uniformly in the entire toroid, we define the transformation by

(15)

and

(16) Thus, eqn (9) is transformed to

d$fL,~+&J,

=;12G,,+G,

(17)

in which Iz is a large complex-valued parameter, L,, L2, L3, L4, Lg, G,, and G, are the realvalued functions of z and are analytic at the transition point z = 0. The nonhomogeneous term G, vanishes if the loadings have only normal components. The expressions for these coefficients are in Appendix B. However, it is necessary to indicate that

2740

R. J. Zhang L,(O)

=

3.

(18)

Equation (17) is the fundamental equation. The present derivation is a modified version of the account given by Xia and Zhang (1986).

3. THE FORM OFTHE

ASYMPTOTIC

EXPANSION

We find first the “local solutions” which are valid only in the vicinity of the transition point z = 0. It is evident that they are reduced by letting z + 0 in the solutions which are valid everywhere in the entire toroid. Therefore, they can help us infer the extensive forms of the uniformly valid solutions. In order to find the local solutions we derive the “local equation” which is valid also only in the vicinity of the transition point by letting ( =

-J_2’3z

(19)

and by using the constants 12 =

~52(0),

13 =

L,(O),

instead of the correspondent

1,

=

b(O),

1s =

W8,

gn

=

G(O),

gr

=

‘Z(O)

(20)

coefficients L,(z), L*(z) . . . G,(z), in the form

where use has been made of eqn (18). Primes indicate differentiation with respect to c. The terms of order G(J.-“l’) are omitted, since, they are beyond the accuracy of the theory of thin shells. 3.1. Expansions of the homogeneous equations Substituting

(22) into the homogeneous obtain

part of eqn (21), equating coefficients of like powers of 1-2’3, we

lrnhj,

a&b,=

(23)

= 0

-1211/,_I-13~~_2+14~~_3(n= 1,2...)

(24)

where all coefficients with negative subscripts are defined to be zero. The differential operator % is of the form -I- = D34lD-3,

d

D =-@.

The fourth order comparison eqn (23) has four homogeneous seen in Appendix A, they are

H’ = Arc+,(i, wherek = 1, 2.

l),

1(1(03) = B,([, - l),

solutions. As can be

$6”’ = &(C, 1) = 1

(26)

2741

Toroidai shells under nonsymmetric loading

The higher approximation can be found by substituting eqn (26) into eqn (24). For example, substituting I,@ into eqn (24) of n = 1 and comparing with eqn (A6) ofp = - 1, we obtain

$F’ = /*A,+ 1 (L 0).

(27)

Then, substituting $6”’and I@ into eqn (24) of n = 2, noting eqn (A5) and comparing with eqn (A6) ofp = 0 orp = -3, respectively, we obtain

$1”’= $:A,+ (k 1)- b&c+ (c, -230) I

(28)

L

where Ak+ , (5, -2,0) has been used according to eqn (A24), instead of Ak+ ,(f,‘, -2). Moreover, substituting $bk’,tip) and +(‘) into eqn (24) of n = 3, noting eqn (A5) and comparing with eqn (A6) of p = 1 and eqn (A22) of p = - 2, q = 1, respectively, we obtain

ti’:’ = ~l:A,+,(r,2)+1,A,+,(i,-1,l). Thus, we can obtain $ik), $ik) . . . by repeating in the same way. Substituting all of them into eqn (22) and using eqns (A12) and (A23), we conclude that the formal expansions are of the form

+‘k’(kA) = a&+,(<, - l)fIZ-2’3BAk+l(r,0)+~-4’3yAk+,(5, 1) +~-2{~~~+~(~,-l,l)+~-2’3b~~+~(~,O,l)+~-4’3C~k+,(~,l,l)}+...

(30)

where the coefficients a, /3 . . . are the known constants composed of the coefficients in eqn (21) and possess expansions for A-*, for example

(31)

In a similar way, the formal expansion of the third independent solution to eqn (21) whose first approximation is 11/L’) shown in eqn (26) is of the form

r1/‘3’(LIz) = aB,(L - 1)+1-2’3BB,(r,0)+~--4’3yB,(r, +L-*{aB,(r,

1, 1)+AV36

-1,2)+1-2’36B,(5,0,2)+~-4’3CB*(5,1,2)}+

...

(32)

in which the coefficients, except 6, are denoted in terms of the same notations as in eqn (30) because they are equal to each other. To derive the formal expansion of the fourth independent solution to eqn (21) whose first approximation is tji4) = 1 shown in eqn (26) we begin with the nonhomogeneous equation of the form

uw,(4) =

-1,

which is obtained by substituting I,@ = 1 into eqn (24) of n = 1. It is easily seen, by lettingp = 1 in eqn (A8) and noting eqn (AlO), that -fB,([, a particular solution of the nonhomogeneous equation TDu = 1. Thus, $‘14’= $,B,(i, 2).

(33)

2) is

(34)

In a similar way, we can find I+$$~‘, $i4) . . . which are expressed in terms of B,(c, p) (p=l,2,3.. .). In addition, we note from eqn (AlO) that B,(~,p) are polynomials in c.

2742

R. J. Zhang

Then, by noting eqns (19) and (22) we know that the complete expansion of the fourth solution must be of the form lp4) = lp4’(2, /I) = f

(n-‘)“lg4’(z)

(35)

which is a slowly varying function. 3.2. Expansion of a particular solution We assume a particular solution to eqn (21) having the form

.

(36)

Substituting eqn (36) into eqn (21) yields a series of equations in the form

uw,* = gn

(37)

We find from eqns (24), (33) and (35) that the particular solution $* has the same expansion as i,G4)of the form

ij* = lj*(z,

4 = f. W2Pfw)

(39)

which is also a slowly varying function. This is totally different than the situation in the case of axisymmetric loadings where the particular solution was described by the Lommel function which is a rapidly varying function [see Clark (1958)]. As mentioned before, all of these solutions are valid only in the vicinity of the transition point. However, we may reasonably think that they are also uniformly valid in the entire toroid if the coefficients in them are the unknown functions of z instead of the known constants. In other words, we will find that such uniformly valid solutions of eqn (17) have the same expressions as eqns (30), (32), (35) and (39), in which, however, the coefficients are the unknown functions of z, but the known constants have also the same expansions of eqn (31).

4. UNIFORMLY

VALID EXPANSIONS

IN THE ENTIRE TOROID

We only need to determine the coefficients in eqns (30), (32), (35) and (39). 4.1. TheJirst and second homogeneous solutions Substituting eqn (30) into eqn (17), using eqns (A5), (A21), (A12), (A23) and (A24) then equating the coefficients of Ak+ I (c,p, q) (p = - 1, 0, 1; q = 0, 1, 2 . . .) we obtain the ordinary differential equations in terms of the unknown coefficients in eqn (30). All of these unknown coefficients can be written in the form of the asymptotic series as eqn (31). Substituting them into the equations, we then obtain 2z2ab + 3zao - zL,cro = 0 2zp’,-(L,

-2)/I.

= Szai+8ab-L,a&-L,u,+zL,a,

zyb+ L,yb + Lzyo = 0

(40) (41)

(42)

2143

Toroidal shells under nonsymmetric loading

2z2ab + 3za, -zL,aO

= 0.

(43)

The solution of these equations yields the complete asymptotic accuracy of the theory of thin shells in the form

V%,J-) = ~o&+~(L-1)+~-2'380Ak+1(~,0)+~-4'3yOAk+l(~,

expansions within the

1)+1-2aoAk+l(L

-1, l), (44)

where

so(z) = so(z) = ~-~‘~exp

is

1 =LWdrl

2

-

0

PO(z)=$z-‘exp

M(r) = ~“~[27-

{i [ Fdr)

r

J

liV(r)dr

18L,(r)+3L:(r)]+2r-3’2[5L;(r)-2L&)]+4L3(r).

(45)

(46)

(47)

It is argued that so(z) and PO(z) are analytic at the transition point z = 0. In addition, eqn (42) is the homogeneous counterpart of the membrane equation zqY+L,@+L&

= G,,

(48)

which is obtained by putting 1 + co in the fundamental eqn (17). It can be shown that only one homogeneous solution of eqn (48) specified by c#J{‘)(z),is analytic at the transition point z = 0 and the other independent homogeneous solution, specified by 4$‘)(z), is singular at z = 0. Moreover, there exists such a particular solution of eqn (48), specified by &,?(z), which is analytic at z = 0. We let

YOM= 4!“‘(z).

(49)

4.2. The third homogeneous solution Similarly, substituting eqn (32) into eqn (17) we obtain the complete expansion of the third homogeneous solution within the accuracy of the theory of thin shells in the form

IC/“‘K,4= aoK(i, -1)+~-2’3BoB,(r,0)+~-4’3yOBl(r, 1,1)+1-%,(z).

(50)

It is remarkable that the last term in eqn (50) is a slowly varying function. clo, PO and y. remain to be determined by eqn (45), (46) and (49), respectively. The slowly varying function 6, is a solution of the nonhomogeneous equation

where x(z) = -6cr~-L3a,-z~~+4P’o-L~B’o-L2/30+2zy;,+~,yo-~o.

Thus, SAS 31:19-K

(52)

2744

R. J. Zhang

(53)

4.3. The fourth homogeneous solution and the particular solution It is sufficient to take the leading terms in the expansions (35) and (39) in powers of A-*. Substituting eqns (35) and (39) into eqn (17), we find that the leading terms l(/J’and $ b4’satisfy the membrane eqn (48) and its homogeneous counterpart, respectively. Therefore, we let (54) and

Ic/*z

+o= @yz).

(55)

All of them are slowly varying functions.

5. THE FINAL RESULTS

A general solution of the fundamental eqn (17) is of the form tyG((,1) = c,lp+

c,$‘*‘+ c,lp3’+ c41p4)+lj*

(56)

where the homogeneous solutions $(I), ICI(*), $(3) and e(4) and the particular solution JI* are shown in eqns (44), (5,0), (54) and (55), respectively. Four complex constants Ci will be determined by eight boundary conditions. With the knowledge of the general solution, the complex variable F,,, will follow from eqn (16). Then the stress resultants and displacements can be determined using eqns (1 l)(14).

6. COMPARISON

Let m = 1 in eqn (8), the loading becomes 41 =

4~~(Qcoscp,q2 =

q21(@sincp,

qn = q.1(8)coscp,

(57)

which is referred to as “wind-type” loading. Novozhilov (1951) in his monograph has given the asymptotic expressions of the force and moment resultants and the strains for the general shell of revolution, subjected to the wind-type loading based on the limit that the special circumferences 8 = 0 and 8 = rr are not. on the shell. .Obviously, his expressions are still valid for a segment of toroidal shell with positive curvature between and away from 8 = 0 and 8 = n. In the other respects, the results in the present paper are adaptable for the toroidal shell with any nonsymmetric loadings. Their special case of m = 1 and 8 # 0 and 0 # ?I of course must coincide with Novozhilov’s results. Novozhilov’s results are only first approximation. All the small quantities higher than order O(G) N O(G) in them are omitted. With the same accuracy formula (56) becomes

ll/(c,

n)

=

aO(z)[ClA2(~,

-

1) +

C2A3(C?

-

I)]

+

c44(P’(z)

+

in which, according to eqns (A14)-(A16), A, and A3 are expressed as

dd?(z)

(58)

Toroidal shells under nonsymmetric loading A&,

-

2145

1) xf(8,s)[ie(1-9”+e-(1-‘~Y

A,([, - 1) =f”((B,&)[--ie(‘-9”+e-(‘-9w]

(W

where

(61) and 8, # 0 is the upper boundary circumference.. Substituting eqns (59)-(61) into eqns (58) and (16) yields 1=

T

1 sin0 -‘I4 - Re, I()a2 az

&)~o(z)[c”, e-“-‘l”+ C2 e(l-fl”] 1 -314 K4d’pYz) +

&?@)I (62)

where

c,= c,+c*,

C2 = i(C, -C,).

(63)

In the first braces of eqn (62), the coefficient l/a2(sin 8/az)-3’43((8, s)cro(z) has to be seen as a constant, since only the primary term is retained in the derivative of r, with respect to 8. It can be absorbed by the arbitrary complex constants c, and c2. Thus, eqn (62) is rewritten as

in which z” =

c,

e-‘l-9”+C

,e

(I-90 (65)

and T(o) = L

a*

sin 8 H az

-3/4 [c4~1°‘(z)+&%~)1.

As it can be seen that eqn (65) is the same as Novozhilov’s formula (4.13.24) and that eqn (66) must satisfy the membrane eqn (9) or the membrane eqn (1) replaced with eqns (2) and (8) and m = 1. The reason of the latter is because C,#J{“’+ $5”’ satisfies the membrane eqn (48) and because of eqn (16). All the stress resultants and stress couples can be obtained by substituting eqn (64) into eqn (11). As an example we consider N,, =

-!f

r.

coseIm (67)

where

2146

R. J. Zhang

_iiEcoseIm(!L& -l_J-g

cos e {[(C;

- C’;) cos w - (C; + C;‘) sin 01 e-“’

r.

-[(Ci-C;)co~to+(CS+C~)sinw]e~j.

(68)

It can be seen that eqn (68) is identical to Novozhilov’s formula (3.14.9) if cos 0 is replaced with cot 8. However, both cos 8 and cot 0 can be absorbed in the arbitrary constants C;, C’,‘,C; and C; as mentioned above. Thus, we can see eqn (68) as the same as Novozhilov’s resultant. In addition, the second summand of eqn (67) is equal to the membrane stress resultant N’iq’.The real part of the complex constant C, in M,‘j [see eqn (66)] can be determined by giving the boundary membrane stress resultant

-~sin8Re(~-~sin8Re(T(“))+q,,roo 8=0,

(69)

As for the determination of the imaginary part of Cd, we have to give the boundary shear stress resultant S*. This is the same as Novozhilov‘s approach. Finally, we have

N,,

=

-1

Jz

Esin e

J

-cose([(c;-C;‘)cosw-(C;+C;‘)sinw]e-”’

Rag

-[(C;-C;‘)cosw+(C;+C~)sino]e”}+Niq’

(70)

which is identical to Novozhilov’s formulae (3.14.9) or (3.14.22). Note that there are only four constants C;, C;, C; and C’g to be determined in the present case instead of eight constants in the other nonsymmetric cases. This is a special feature of the wind-type loading on which Novozhilov has made a detailed explanation.

7. CONCLUSIONS

(1) The fact that the particular solution $*(z) satisfies the membrane eqn (48) shows that nonaxisymmetric loadings are equilibrated totally by membrane stress resultants. This differs completely from the axisymmetric loading situation in which the axisymmetric surface loadings are equilibrated by both the membrane stress resultants and the bending moment. (2) The existence of the third homogeneous solution I//~) shows that bending moments exist everywhere in the entire shell and their existence is independent from the loadings but dependent on the boundary conditions. This is completely different from the axisymmetric loading situation in which the bending moments existing in the entire toroid are induced by the loadings. Acknowledgemenrs-This work was supported by the National Natural Science Foundation Wei Zhang was in charge of this work. The author gratefully acknowledges him.

of China.

Professor

REFERENCES Clark, R. A. (1958). Asymptotic solutions of toroidal shell problems. Quart. Appl. Math. 16,47-60. Drazin. P. D. and Reid, W. H. (1981). Hydrodynamic Stability. Cambridge University Press, Cambridge. Novozhilov, V. V. (1951). Theory of Elastic Shell. State Press of Naval Architecture (in Russian). Steele, C. R. (1959). Toroidal shells with nonsymmetric loading. Ph.D. Dissertation, Stanford University. Xia, Z. H. and Zhang, W. (1986). The general solution for thin-walled curved tubes with arbitrary loadings and various boundary conditions. Int. J. Press. I/es. Piping 26, 129-144. Zhang, R. J. and Zhang, W. (1991). Turning point solution for thin shell vibrations. Int. J. Solids Structures 27(10), 1311-1326.

2141

Toroidal shells under nonsymmetric loading APPENDIX

A

Drazin and Reid (198 1) have shown that the generalized Airy functions a&p)=&

&&P)

t-Pexp(jf-ft3)dt s ‘k =

(AlI

(k=1,2,3:p=O,~l.f2...)

t-Pexp(&ft3)dt

642)

(k=1.2,3;p=O,-I,-2,...)

sc

and t-Pexp(jt-ftj)dt s ‘”

@=0,*1.+2...)

(A3)

are solutions of the differential equation (D’-jD+p-

l)u = 0

(A4)

where the contours are shown in Fig. Al. The derivatives of these solutions satisfy the relation D”‘%(l#) = ‘%(i,P-n),

D”&(i.P) = &(i.P-n),

D”&(l,P)

= Bdi,P-n).

W)

It is not difficult from eqn (A4) to arrive at the following relations of identical form T[LDAk(l.Pf 1) = -@+2)‘4,(LP)

(‘46)

aw,(i>p+

1) = -@+2)B,(i.P)

(‘47)

TD&(Lp+

1) = -(pfW,(i,P)

w3)

and

where T = D’-[IDi-3.

(A9)

Note from eqn (A3) that E,([, p) = 0, if p < 0; otherwise, it is a polynomial in < of degree p- 1 which, by the residue theorem, is simply the coefficient of fP-’ in the expansion of exp ([t -f?). The first few terms of these polynomials are B,(L 1) = 1

1 WY.4) = $-

&(i,2)

B&5)

= i

B,(L3)=+

1 =$-5’

1 5 1

B,(i,6)=$‘-;i2-

(AlO)

It is seen by putting p = - 2 in eqns (A6) and (A7) and p = 0 in eqn (A8), that AP([, - 1). Ek([, - 1) and B,,({, 1) (k = 1, 2, 3) are non-trivial solutions of the homogeneous equation UDu = 0. Moreover, we indicate that

Fig. A 1. The paths of integration in the f-plane.

2748

R. J. Zhang

A,((, -l), A,((, -l), B,(j, - 1) and It,&, 1) = 1 are a linearly independent Wronskian is a constant and is given by W(A*,Ar,B,,

set of solutions, because their

l)(C, -1) = - ;.

(All)

By using cqn (A6) we can immediately obtain the particular integrals of the inhomogeneous TDu = A&p) for all values ofp except p = -2, and have the recursion formula A&p-3)-&4&P--

I)+@-- J)&(i,P)

= 0.

equation

(A12)

Thus, for other values of p, A,([,p) can be expressed as a linear combination of A,((, 0) and A,({, f 1) with polynomial coefficients. The asymptotic behaviour of A&p) is given by A,(~,P) - A-(LP)

(in Tru Tr)

A&P)

- iA+

(ie T3 u T,)

A,W)

-

-A-(LP) -iA+

(i~T,l (CETZ)

(A13)

and -i

-1

-1

A+(Lp)

-i

-1

-1

A-(LP)

B&P)

(A14)

where Tk are sectors shown in Fig. A2 ;

6415) and

a,@) = -&12pz+24p+5)

a&)

The functions B&p)

=-&(144p*+1344p3+3864pz+3504p+385)+

also satisfy eqn (A12) with A&p) B&P-3)-iBdi,p-

I)+@--

(A161

replaced by B&p), ~)B&,P)

= 0

i.e.

(P Q 0).

(A17)

The other important recursion formula is &(i> - 2) -i&(i, The asymptotic behaviour of B&p)

0) - 1 = 0.

in T, is given by

Fig. A2. The sectors for the Airy functions in the c-plane.

(Af8)

2749

Toroidal shells under nonsymmetric loading B&p)

_ (-l)‘_‘(-p)![P_‘{1-f(p-1)@-2)[-3+

(A19)

. ..}.

TO obtain the particular integrals of the nonhomogeneous equation TDu = Ak(<, -2), we consider the third generalized Airy functions defined by Drazin and Reid (198 1) as follows :

A,(Lp, q) =

s

y$ ‘kP

(In t)“exp ([t -fr’) df

(A2O)

and a branch cut has been placed along the positive real axis where/c= 1,2,3;P=O, kl,2...;q=O,l,2...; in the t-plane so that 0 Q pht -z 2n. Similarly, the derivatives satisfy the relation D”Ml,p, The function A&p,

(A211

4) = A&p--n,q).

q) are solutions of the nonhomogeneous

equation

6422)

= -C~+2)AdL~,q)+qAdLp>q-l)

TDM,p+l,q) from which we have the recursion formula

A,(r,p-3,q)-rA,(r,p-l,q)+Go-l)A,(r,~,q)

(~23)

= qA,(Lp>q-1).

It is easily seen that A&P,

The asymptotic expansion of A,([,p, A,(LP,

1)

(~24)

1) is

_ L(_~)p~-‘zp+lwe-<

m ~~~(-I)S[(~lnr+ni)ns@)-a~@)l~-s

2J;;

The corresponding expansions for A&p, A&p,

0) = A,(i,p).

1) and A&p,

KET~uT~).

(A25)

1) then follow from the recursion formula

1) = e-2y-‘)nii3[A,(je2n’13,p, 1) +$A,

([e’““‘,p)]

1)-$iA,(~e-2”i~3,p)], A3(Lp. 1) = e 2(p-‘)n’i3[A,(ie-2n’i3,p,

(A26)

To deal with the third homogeneous solution we need to use the functions defined by Drazin and Reid (1981) as follows :

&(LP, 4) = &

0,) ,)2x,3, s m.cip,*(k-

wherek=l,2,3;p=O,fl,f2...;q=0,1,2

t-’ (In t)4exp (it-ft*)

dr

(~27)

. . . . Wemaynotethat B&,P,O)

= WLP)

@ = 0, k 1, k2..

.)

6428)

and Bk(c,P,l)=~k(LP)

@=O,-1,-Z...)

for all values of k. The functions E&p, q) also satisfy eqns (A21)-(A23) with A&p, The asymptotic expansions of B, (c, p, q) in the sector T, are Bi(Ll,l)

_ -lni-y+

m (3n-1Yr_,” C “=I 3”n!

(~29) q) replaced by Ek (5, p?q).

(A3O)

where y is the Euler constant.

APPENDIX

B

The coefficients in eqn (17) are as follows : L,

C-J- -

1

m46 qL/

[(Q’ +2&z’) sin B+ 34,~’ cos O]

031)

(B2)

2750

R. J. Zhang

Lg =$~(l+2p’-pimi+8psin~+5~2sin’B)+-

’ (44 &z” + 3&,~“~+ 12r#+‘z”+ 6&P) 4 02‘,I

(B3)

(c#++“+4~~z”‘+6~~z”+4~~z’)

L, = + &Z

[

1

+~(~oz”+2&,z’)(l+2~‘-2~2m2+8psinB+.5p2sin20)

(41 4: Ls =- 40 + -(1+2p2-2~2m2+8~sinO+5p2sin20)+

f#%Z’* -

.21#h)Z’4

$[(6-4p2-p2

+ %(6-4$+4p’m2+2~sinO)

(B4)

dbltc0se ~(6-4/12+4~2m2+2~sinO) rJ3f$oZ’4

m 4+5~2m2)+2~2sine-(l2-2~2+2~2m2)sin*e-12~sin3B-4~2sin4e]

(B5)

and

where -3’4

and

The expression of G, is omitted

(B7)