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Sensitivity of buckling loads of anisotropic shells of revolution to geometric imperfections and design changes
Com,wers & Strucrures Vol. 31, No. 6, pp. 98>995, Printed in Great Britain.
1989 0
CnMs-7949/89 s3m+ 0.00 1989 Pergamon Press plc
SENSITIVITY OF BUCKLING LOADS OF ANISOTROPIC SHELLS OF REVOLUTION TO GEOMETRIC IMPERFECTIONS AND DESIGN CHANGES GERALD A. COHEN~ and RAPHAEL T. HAFTKA~ IDepartment
tStructures Research Associates, Laguna Beach, CA 92651, U.S.A. of Aerospace and Ocean Engineering, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061, U.S.A. (Received 2 March 1988)
Abstract-Buckling load sensitivity calculations in the shell-of-revolution program FASOR are discussed. This development is based on Koiter’s initial postbuckling theory, which has been generalized to include the effect of stiffness changes, as well as geometric imperfections. The implementation in FASOR is valid for anisotropic, as well as orthotropic, shells. Examples are presented for cylindrical panels under axial compression, complete cylindrical shells in torsion, and antisymmetric angle-ply cylindrical panels under
edge shear,
INTRODUCTION
In a recent paper [I] the principle of virtual work is used to derive for elastic structures sensitivity field equations, i.e., equations satisfied by the derivatives of stress, strain and displacement tensors with respect to a stiffness parameter. In particular, for nonlinear shell analysis, these equations display the fact that the sensitivity solution may be obtained by solving load-perturbation equations with fictitious thermal loads which depend on the derivative of the stiffness tensor. The sensitivity solution can, of course, be approximated by taking finite differences of two neighboring nonlinear response fields. However, this is considerably more work than a direct solution of the linear sensitivity equations, and introduces additional truncation and condition errors. In [2] the sensitivity of vibration frequency and bifurcation buckling load is discussed. The analysis is again based on the principle of virtual work, and it turns out that the second-order state in an asymptotic expansion of the postbuckling response [3] plays a central role in the calculation of the sensitivity of vibration frequency and buckling load. As discussed in [3], the second-order postbuckling state is also used to calculate the second postbuckling coefficient b, which in turn yields the buckling load knockdown due to geometric imperfections in accordance with the well-known Koiter theory. Therefore, it is convenient to implement both sensitivity calculations simultaneously in a structural analysis program. This has been done for buckling loads based on classical orthotropic and anisotropic shell theory in the shell-of-revolution program FASOR [4]. As in prebuckling sensitivity analysis, buckling load sensitivity can be calculated by taking finite differences of buckling loads for neighboring designs.
In this case the advantages of sensitivity analysis are even greater, since recalculation of buckling loads is a lengthy process involving, in general, both nonlinear prebuckling and eigenvalue reanalysis for each design change. In contrast, once the second-order postbuckling state is obtained for the nominal structure, a linear calculation needed anyway for imperfection-sensitivity, evaluation of the (first-order) effect of various design changes on buckling load is essentially instantaneous. FASOR results are presented and compared to previously published solutions for cylindrical panels under axial compression [S], complete cylindrical shells in torsion [6], and antisymmetric angle-ply cylindrical panels under edge shear [7]. ANALYTICALFORMULATION Buckling load sensitivity analysis consists essentially of two steps following the prerequisite prebuckling and buckling calculations [8]. These are:
(4 the solution of the linear boundary value problem (i.e., field equations and boundary conditions) for the second-order postbuckling state, and (b) the evaluation of certain functionals of the prebuckling state at the bifurcation load, buckling mode, and second-order postbuckling state. Each of these steps is discussed below for anisotropic shells of revolution, Field equations for second-order
postbuckling
state
Field equations for the second-order postbuckling state are derived for orthotropic shells of revolution in [9]. It was observed there that the homogeneous parts of these equations are identical to the corresponding buckling mode equations, the only difference being the addition of nonhomogeneous terms, 985
986
A. COHENand RAPHAELT. HAF~KA
GERALD
which are quadratically dependent on buckling mode variables, in the second-order state equations. In accordance with the general theory for arbitrary structures [3], the same is true for anisotropic shells, and the derivation in [9] is used as a template for the following development. With two modifications the derivation of the nonhomogeneous terms given in [9] is valid for anisotropic shells. First, that analysis is limited to axisymmetric torsionless prebuckling states, thereby allowing prebuckIing rotation components (about meridional and normal directions) associated with axisymmetric torsion to be neglected. Since anisotropy couples axisymmetric bending and torsion, all three buckling rotations are retained here. Secondly, whereas for orthotropic shells of revolu-0 tion the antisymmetric circumferential component of harmonic buckling modes is decoupled from the symmetric component and therefore, without loss of generality, is omitted, the full constitutive equations of anisotropic shells (as well as torsional Ioading of orthotropic shells) couple these components. In this context, cosine components of normal type variables (those associated with axisymmetric bending of orthotropic shells) and sine components of shear type variables (those associated with axisymmetric torsion of orthotropic shells) are symmetric components, and the reverse is true of antis~met~c components. Thus in the anisotropic case, both sine and cosine components of buckling mode variables must be retained. For example, buckling and postbuckling meridional strain equations (I 7) of [9] become, with the retention of the prebuckling rotation 0”) about the normal (Fig. I), for anisotropic shells cl” _ ,\I) _ x’o’X”’_ @‘#I = 0
(la)
E\2)_ 42) _ xwx(2) _ eoern = q
(lb)
where Eje)=(1/2)[p+e"q.
Here and in the following, superscripts (0), (1) and (2) refer to prebuckling variables at bifurcation, buckling mode variables, and second-order postbuckling variables, respectively, and the superscript (e) indicates a nonhomogeneous term. The buckling mode rotations x(i) and e(i) may be written in terms of their symmet~c and antisymmetric components as
x(‘)= Xc’)COSn,$
+ x(O)sinn,4
(34
@I) = P sin n,# f 8@)cos n,#
(3b)
where 4 is the circumferential angle and n, is the buckling mode harmonic number. Substituting eqns (3) into eqn (2) gives, with the help of the doubleangle sine and cosine trigonometric identities, clef= (l/4)fX’“‘Z+ x’“” + @2 + @@>2] + (1/4)1y’“‘2- x@)“‘2+ fP)z- P’] cos 2n,$ + (1/2)[x(s)x(a)+
we(a)] sin 2n c(jb’
n =
2n,.
In order to preserve the form of the field equations in passing from orthotropic shells to anisotropic shells, complex buckling and postbuckling variables are used, the real and imaginary parts of which represent symmetric and antisymmetric harmonic components [lo]. Specifically, denoting symmetric and antisymmetric harmonic amplitudes of a typical normal type variable by NC’)and NC”)and symmetric
f REFERENCE SURFACE MERIDIAN VARIABLES
(4)
The form of eqn (4), which is typical of all nonhomogeneous terms in the field equations, shows that response variables of the second-order postbuc~ing state are composed of two harmonics, n = 0 and
AXIS OF REVOLUTION
(a) PRIMARY
(2)
(b) STRESS RESULTANTS,
(c) STRESS COUPLES,
DISPLACEMENTS, AND SURFACE FORCES
Fig, I. Shell-of-revolution notation.
ROTATIONS, AND SURFACE MOMENTS
987
Sensitivity of buckling loads of anisotropic shells of revolution for n = 2n,:
and antisymmetric harmonic amplitudes of a typical shear type variable by S@)and S(“) we form complex normal and shear amplitudes as hollows, N = NC” + iN(“)
(sa)
S = SC”)_ is(“)
(5b)
&’ = -(l/4)($2+
eq
Eli’ = (1/2)Ph
(9b)
Xf’ = -(n,/r)(T,
+ T2)0
Xy)= -(r’/2r)(T,
In terms of complex amplitudes, the harmonic components of eqn (4) may be written as
+ T,)f?
xp=o
for n = 0:
EY)= (1/4)(x . x + e .e)
(64
for n = 2n,:
df’ = (1/4)(x2 - er)
(6b)
in which complex amplitudes are denoted by the same symbols as used previously for total quantities and, for simplicity in writing, the superscript (l), denoting buckling mode variables, has been dropped from x(r) and t?(r). Also, in eqn (6a) and the following, dot and cross products of two complex variables u and u are defined by u v = Re(ou*) = Re(u)Re(v) + Im(u)Im(v)
(7a)
u x u = Im(uu*) = Re(u)Im(o) - Re(u)Im(u)
(7b)
in which * denotes complex conjugate. Equations (6) are the counterpart for anisotropic shells of eqn (26a) of [9] for orthotropic shells. In a similar manner the remaining nonhomogeneous terms in the equilibrium and kinematic equations, analogous to eqns (25) and (26b, c) of [9] are for n = 0:
#=(1/4)($~$+0~e)
(8a)
.$I = (l/2)& x $
(8b)
XI’)=0
(8~)
Xf) = -i(r’/2r)(T,
+ T2) x 8
(8d)
XI’)=0
(8e)
Ly) = (1/2)i(T, x I&- T,, x x)
(8f1
JX’ = -(l/2)0-,
4,
aI? -i&
.x +
T,2.
(W
$)
iA23
1
- iA,,
1 33
I -iA,,
(1/WT,IcI
Li=‘=
-W)V,x
a14
a15
+
T,2x)
-
T,,$).
(9f) (9g)
A24
A 25 -
iA,,
4, iA
1 36
(10)
I -114
-124
-
I
A*
145
iA
-&
-125
--iA,,
II
A,5
1 55
%6
iA
-A,,
’
-iA,,
- i&,
&6
(9d) (9e)
LI” =
--------___----J-------_-----___
U,,
(9c)
Comparison of eqns (6b) and (9) with eqns (25) and (26) of [9] shows that the unsymmetrical components of the complex nonhomogeneous terms are identical in form to their real counterparts for orthotropic shells. The nonhomogeneous terms given by eqns (6), (8), and (9) apply directly to the complex harmonic equilibrium and kinematic equations, which, as noted, have the same form as their real counterparts (in terms of symmetric amplitudes) for orthotropic shells. Additional nonhomogeneous terms arise from the “excess” variables in these equations, viz. T2, M2, and M,, in the equilibrium and c,, K~, and e,2 in the kinematic equations. (Excess variables are dependent variables which appear in the linearized equilibrium and kinematic equations, c$ eqns (11) and (13) of [9], but whose meridional derivatives are absent.) These variables are expressed in terms of the primary variables (i.e., variables whose meridional derivatives do appear, see Fig. 1) through partially inverted constitutive equations. Partially inverted constitutive equations in terms of complex harmonic amplitudes have been derived for anisotropic shells based on transverse shear deformation theory in [lo]. The same procedure with a slightly different ordering of stress and strain measures leads to the following complex constitutive equations, which are the analog of eqns (42a) of [lo], for anisotropic shells based on classical shell theory
a12 a22
iA
(9a)
1 66
GERALDA. COHENand RAPHAELT. HAFTKA
988
Equation (lo), being linear, applies as well to complex harmonic amplitudes of buckling or postbuckling variables. The modified twist K’\gand shear stress resultant s”(‘),complex amplitudes of which appear in eqn (lo), are given for orthotropic shells by eqns (21) of 191. For anisotropic shells, the corresponding equations are
bation forms of the equilibrium and kinematic equations, eqns (11) and (13) of [9], written in terms of harmonic amplitudes, to obtain the field equations for complex harmonic amplitudes of the secondorder postbuckling state. In standard matrix form suitable for solution by the field method [4], these are Y’ + 10 + c&M.Y + [b + B(&)lz =f z’+c?,+[~++(1,)]z=g
s”(2)-S’2’+(1/2)[T(P)+
2-$@]6’2’
+ (1/2)[7-~2’+ Z-@?‘“‘= s(P) (llb) where rZy;= -d#Rz
(124
3’ = (1/2)[T\” + l”ir’]e(‘).
(12b)
Equation (12a) is valid in terms of complex ha~onic amplitudes for n = 0 and n = 2n, where t”R is given by either eqn (8b) or eqn (9b). Complex harmonic amplitudes for S”(@), obtained from eqn (12b) in the same manner as was done previously for other nonhomogeneous terms, are for n = 0:
S”(e)= -(1/4)i(T,
+ Tz) x 0
(13a)
for n = 2n,:
@+= -(1/4)(T,
+ T,)B.
(13b)
(15a) (15b)
the homogeneous forms of which are buckling mode equations. (For the sake of simplicity, live pressure terms in the equilibrium equations (15a) are omitted.) Prime denotes differentiation with respect to meridional arc distance s, ir is the critical load factor, y and z are the 4-element complex force and displacement vectors r(P, Q, S, M,) and (& n, v, ~1 (Fig. 1), and a, b, c, d, K, /I, and 6 are 4 x 4 complex matrices (see Appendix) satisfying the Hermitian self-adjoint conditions (the superscript t signifies Hermitian transpose) a= -d’,
a= -6’
b = b’,
P=P
c=c’. The complex nonhomogeneous eqns (15) are
(161 vectors f and g of
-r(r/R2)X(le)--nr’Ly)-n2(r’/r)h4ff As in eqns (6) (8), and (9), the superscript (1) has been omitted from buckling mode variables of eqns (13). The nonhomogeneous terms tr), R’;1, and s”@‘) give rise, through eqn (lo), to additional nonhomogeneous terms for each of the excess variables given on the left-hand side of eqn (10). To distinguish the additional nonhomogeneous contributions to 6, and c,2 from cl’) and @J given by eqns (6), (8b), and (9b), those arising from eqn (10) are denoted by - Acf) and -Acl”,‘. Thus, from eqn (10) one obtains the following additional nonhomogeneous terms Ty) = L,,c~) + i&R# + iA,,P MS’) = A,,t?Ip’+ iA2g# + iA,,P 2Myd = -iA,,e!$ + A&~ + A&?@) -A.s~)=
-&t$-
U&@+
i&@)
KY) = --1,,6f) - iA&(;?:+ i&P -A#
= id,,c~) - ,I&‘$ + ;IS6gfe).
+ 2(n/r)Mfj f=
-rr’X(f) + n(r/R~)L~) + Tfze)
(17)
+ (n2/R2)M(2e) -r~~)~(rlR2)~~)+n~~)+(n/R2)~~) -rLf)
+ r’h@ - 2nMg + A@]
(18)
Wd (14b) (14c) (W (144 Wf)
The nonhomogeneous terms given by eqns (6) (8), (9). and (14) mav now be substituted into the pertur-
Analogous equations may be derived for discrete rings, providing boundary conditions for eqns (15) at ring locations. Evaluation of functionals General formulas for the second postbuckling coefficient 6, initial postbuckling stiffness K, and imperfection parameter a are given in f3]. (For axisymmetric structures with unique harmonic buckling modes, the first postbuckling coefficient is identically zero.) The formula for the first-order change A1.
989
Sensitivity of buckling loads of anisotropic shells of revolution in the buckling load due to a change AH in the stiffness tensor is given in [Z]. In the operator notation of those papers,
For shells of revolution, FJand t: may be interpreted as six-component vectors consisting of stress resultants and couples for 6 and reference surface strains and curvature changes for L, viz. ff = IT,,
K= K,f[l + (4,‘2H:)(a,
. q + 2q,. Q]
a= -Ia, . q -6, . L(wl~ dl/&m, Ai, = -PAH(c,) .c2 + AHk)
u,)
1t,l/‘F(u,, u,>
Wb) ww (194
6 =
T2,
T,2,
M,,
M2,
M,,)
@W
GW
i ~,,~2,~,2,iC,,K*,2K*t~.
Corresponding to eqn (24b), L*(U) and L,,(u,,u,) may be written in terms of shell element rotation components as
where F(u,, u,) is the inner product of the buckling mode with itself given by (with live pressure terms omitted)
L,(~)=(x~+B*,~*+~*,~x~L,O,O,O) (25a) L,,(u,, u2) = {X,X2+ W-&1+,+* +- V2, X,$2
f;(u,, u,) = 20, . L,,(ii,, %I +4.
L,(4).
+ $,x2, 0, O,O>.(25b)
(20)
In eqns (19) and (20) (T, c, and u represent stress, strain and displacement tensor fields, and L, is the homogeneous quadratic operator in the straindisplacement relation
Since only axisymmetric loading is considered, prebuckling variables are independent of the circumferential coordinate 4, whereas buckling mode variables have harmonic variation with, in general, both sine and cosine components, cJ: eqn (3). As discussed (21) previously, second-order ~stbuckling variables have t: = L,(u) + (1~2)~~(~) both axisymmetric and asymmetric harmonic components, which are denoted here by subscripts 0 and 2, in which L,(u) is the linear component. The bilinear respectively, written to the left of the corresponding operator L,,(u, v) is defined by symbol. For example, normal and shear type postbuckling variables such as T, and T,, may be written Lz(trfv)=L*(u)+2L1,(U,V)+Lz(u). (22) as Subscripts 0, I and 2 on 6, f, u and K indicate prebuckling, buckling and second-order postbuckling states, respectively, with the understanding that all prebuckling quantities are evaluated at the bifurcation load 1,. (Note that for individual components of 6, E, and U, superscripts (0), (1) and (2) are used to denote these states.) In eqn (20) bars signify derivatives of prebuckling fields with respect to the load factor I, evaluated at the bifurcation load. Also, the low-dot product [as opposed to the high-dot product of eqn (7a)] signifies the total work of the left-side field acting through the right-side field. Thus, for shell structures the low-dot products are double integrals over the reference surface of the shell. The imperfection parameter GIgiven by eqn (19c) is based on a small initial stress-free displacement field PU,, where U, is the buckling mode displacement and p is the imperfection amplitude. In FASOR the buckling mode is normalized such that its maximum normal deflection is the shell thickness, so that ~1 represents the imperfection amplitude-to-thickness ratio. (For variable thickness shells the thickness at the final point of the FASOR model is used.) If b is negative, Koiter theory gives the buckling load knockdo~ &/A,, c$ eqn (38) of [3], as A$/& = 1 + 3(a2$b/4)“3,
(23)
Tf’=,TI”)~zTI”‘cos2n,cp
+ zTf”‘sin2n,q5
T\? = ,,T@ + 2TB cos 2n,(t) + *pj sin 2n,&
(26a) (26b)
To evaluate rr, . L,(u,) of eqn (19a) one has from eqns (24a) and (25a)
+
T$2)[$,(,h
+
fj(,h]
+
2T\Z,'X"'$"')r d$ ds.
(27)
Substituting for the buckling and second-order postbuckling variables on the right-hand side of eqn (27) their harmonic component expressions, as in eqns (3) and (26), and performing the +-integration with the help of the following identities 2n
2n cos2n~
s 9-O 2x s
,=o
d# =
sit&$ d4 = n s .$=O
cos2nt$cos 2nd d#
(2ga)
990
GERALD
A. CXXIEN and kw%m.
sin2 n# cos 2n# d$ - n/2
=Z-
(28b)
cos2m# sin 2nrb d4
T. HAFTKA
a,. c2= 2x
[rl”) +o~l+ Go) *oe2- T# Im(&,) s + $01 1orci+ Mi”j aoicr
- 2M’# Im(,k,,)]r ds
(314
2n =
sin2 nd sin 24
d$ = 0
(28~)
s rp-0 + CT,, x x - T2 x
$>@”
reduces eqn (27) to - [(TI + T,) x @)@O))rds
(31d)
{oT(f)[X(++ p2 + @2 + fpq
a2. L&u,) = I[
ss
- 2[( T, + T2)
In terms of complex harmonic amplitudes, as in eqns (5), eqn (29) becomes
s
IoTdx-x>+oT2W+Icr)
a2.Wl)=n
x O]P(O )r
ds.
(31e)
Note that prebuckling variables in eqns (31c-e) are real, as opposed to complex buckling and postbuckling harmonic amplitudes. Evaluation of the functionals in the numerator of eqn (19d) is similar to that of eqns (31b) and (31~) with the replacement of components of a0 = H(co) and CJ,= H(Q) by components of AH(%) and AH@,), respectively. Analogous contributions from discrete rings to the above functionals can be derived. EXAMPLES
I
tot+oT2)(e- 0) + 2 Im(oT,2)(xx $1
+
+ twx27A . x2- 2Tz.Jr”
- t27i f 2~2).e2i +
2Tl2 * (xJ/)b
(30)
d&
Similariy, the remaining functionals of eqns (19) and (20) reduce to
a,
.6]
=
x
s
(T, acl
i-
T2’
(i2 f
T12*cl2
s
+ Mr *~2 + 2M,z. K,& ds
-t
M, - KI
Numerical results are presented for three kinds of cylindrical shelf structures for which published results by other investigators are available. The first example, cylindrical panels under axial compression, verifies the applicability of large radius toroidal modeling of general cylindrical shells for initial postbuckling calculations. Since no published results of initial postbuckling of generally anisotropic shells are available, complete cylindrical shells in torsion, the second example, are used to verify this analysis. Although the cylinders treated are isotropic, shear loading produces coupling between circumferential symmetric and antisymmetric harmonic components of the buckling mode, just as anisotropy does. The third exampie, antisymmetric angle-ply panels under shear loading, illustrates buckling load sensitivity to changes in ply angle of a composite shell. The sensitivity calculations were verified by recakulation of the buckling load at different ply angles. However, in the special case of a cross-ply panel values of the postbuckling coefficient b obtained are in disagreement with those reported in 171. Cylindrical panels under axial compression
Koiter [5] analysed narrow cylindrical panels with idealized boundary conditions to simulate the
Sensitivity of buckling loads of anisotropic shells of revolution individual panels of stringer-stiffened cylindrical shells. His results are contained in the function G(8) = 1 - 2@ - 9S(O)
(32)
except in the neighborhood of G = - 1, which occurs at 0 = -0.77, where significance is lost due to the occurrence of the small difference of large numbers in the evaluation of eqns (l9b) and (34b). Cylindrical shells in torsion
where S(O), given by an infinite series, is tabulated in Table 1 of [5], and 0 is the curvature parameter defined by 0 = (1/2~}[12~1 - v*)]“~~/(R~)“*
991
(33)
in which B is the circumferential arc width of the panel, R is the panel radius, and h is the pane1 thickness. These results can be put in terms of the postbuckling coefficient b and the initial postbuckling stiffness ratio K/K0 through the formulas b/(1 - vz) = 3G/8(1 + 04)
(34a)
K/K, = G,‘(G + 1).
(34b)
A comparison of FASOR results with those of [S] are shown in Table 1. The FASOR results were obtained using a toroidal half-panel model with radius r/B = R,/B = 10,000 at the symmetry boundary and Poisson’s ratio v = 0.3. For 0 > 0, the m+dional radius of curvature R/h = 100, whereas for 0 = 0, corresponding to a flat panel, B/h = 10 was used. Model constraints at the “‘stringer edge” s = B/2 are: for buckling u = w = 0 (antisymmetric buckling mode about the stringer) and for postbuckling &/ad, = x = 0 (symmetric initial postbuckling state about the stringer) and ~~/~# = 0 (constraint due to out-of-surface stringer bending stiffness) (see Fig. 1). The postbuckling results of [5] are based on the classical buckling mode for a simply supported narrow curved panel. As shown in Table 1, FASOR buckling loads agree well with the classical values, the greatest relative difference being 1.4% at 0 = 1. It is noted that small discrepancies in the buckling mode tend to be magnified in the calculation of the postbuckling coefficient b. Thus the greatest relative difference for b in Table 1 is 4.3% at 0 = 1. The discrepancies shown for values of K/K0 are similar
Budiansky [6] analysed initial postbuckling of circular cylindrical shells under torsional edge loads for three sets of edge conditions: simply supported edges and two kinds of clamped edges with different insurface constraints. FASOR results are in good agreement with all three cases. However, for the sake of brevity, only the simple-support results are presented, the edge constraints for this case being &~/a# = w = 0 (see Fig. I). The analysis of [6] is based on the KarmanDonnell cylindrical shell equations and results are given for values of the curvature parameter 2 in the range 0 < Z < 10,000, where 2 = (1 - v2)‘12L2/rh,
L being the cylinder length. The critical shear stress z, and circumferential wave number n, are given by the dimensionless parameters AC= t,hL’/D
(W
1 = Lnn,jr
(W
where D is the flexural stiffness of the shell. A comparison of FASOR results with those of [6] is shown in Table 2. For the FASOR model, v = l/3, and n, = 10 (Z > IO) or 100 (0 < 2 < 10) were used. Thus for Z > 0, L/r and r/h were determined by eqns (36b) and (35) for various combinations of /I and Z. For Z = 0, corresponding to a narrow flat plate under edge shear, r/L = 10,000, L x h = 10, and n, = r/3/L = 25,100 were used. The agreement shown in Table 2 is good considering the approximate nature of the Ka~an-Donnell equations (see also discussion of Table 1). It may be noted that for a given value of Z, the complete shell theory of FASOR predicts values of b which are (weakly) dependent on r/h, whereas the Donnell theory results are independent of r/h. For example,
Table 1. Buckling and initial postbuckling of cylindrical panels under axial compression 0 0
fJ,ia: Classical? FASOR 1.000
I.000
b/(1 - v2) FASOR Ref. 151 0.375
0.315
0.355 1.006 0.355 0.231 i .os7 0.230 0.6 1.122 0.091 0.090 0.7 1.240 1.229 -0.109 -0.113 -0.364 0.8 1.410 1.394 -0.374 -0.662 0.9 1.656 1.636 -0.684 -0.997 1.0 2.000 1.972 - 1.040 * u0 = 4x2D/B2h is the critical stress for a flat plate. t The classical value of a& for a curved panel is (1 + @). 0.3 0.5
1.008 1.062 1.130
(35)
UK, Ref. I51 FASOR OS 0.488 0.395
0.5 0.488 0.394
0.213 -0.597 3.457 1.495 1.220
0.214 -0.550 3.902 1.536 1.238
GERALD
992
A. COHENand
RAPHAEL
T. HAFTKA
Table 2. Buckling and initial postbuckling of cylindrical shells in torsion B
Ref. [6]
FASOR
b/(1 -v*) Ref. [6] FASOR
Ref. [6]
UK, FASOR
2.51 2.51 3.97 9.29 19.75 37.4
52.61 53.18 14.29 271.6 1482 8389
52.61 53.18 74.53 271.5 1471 8295
0.2139 0.1508 -0.2364 -0.0756 -0.0122 -0.0015
0.671 0.582 3.42 -0.913 -0.175 -0.031
0.671 0.586 4.21 -0.725 -0.150 -0.052
4 Z 0
I 10 100 1000 10,000
for Z = 100 reducing equivalent to changing
n, from 10 to 5, which is r/h from 123 to 30.7, yields: A, = 259.6, b/(1 - v*) = -0.0759, and K/K, = -0.832. On the other hand, increasing n, to 100, equivalent to r/h = 12,300, yields A, = 271.5, b/(1 -v*) = -0.0665, and K/& = -0.718, which
are in better agreement with the results for n, = 10. Long antisymmetric angle-ply panels under shear Hui and Du [7] analysed initial postbuckling of long antisymmetric cross-ply cylindrical panels under edge shear for simply supported and clamped edges. Graphical results are presented in [7] for b versus the simplified curvature parameter 0, defined as [cJ eqn (33)l 0, = B/(m)“*.
(37)
A large radius toroidal model (Fig. 2) is used for FASOR analysis of the 2-layer clamped panel of [7]. Note that the circumferential (4) direction of the model corresponds to the longitudinal direction of the panel. The panel wall is a regular antisymmetric cross-ply (a = 0) laminate with the outer layer fibers running in the longitudinal direction. Although calculated critical shear stresses agree well with [7j, values of the postbuckling coefficient b do not. Figure 3 compares b-values calculated by FASOR with those reproduced from Fig. 2b of [7]. Thus, whereas the results of [7] predict imperfectionsensitive panels (i.e., b < 0) for 0, > -3.4, the
Boundary conditions at s = 0 and s = 6: out-of-surface, clamp (w = u/R-awlas = 0) in-surface, free (T,+ =Th = const., T, = 0)
I r>> 0
--
Lamina elastic moduli: EL/ET = 10 GLT/ET = ‘A VLT (major Poisson’s ratio) = 0.22
-+
0.2143 0.1545 -0.2209 -0.0661 -0.0107 -0.0015
present results predict imperfection-insensitive
panels
(b > 0) over the range of 0, considered.
Reference [7] also gives results for simply supported isotropic panels under edge shear. For 0, + 0, this case becomes equivalent to cylinders of the preceding example for Z -+ 0, since both models approach a narrow isotropic simply supported plate under edge shear. (In the limit the panel width B plays the same role as the cylinder length L.) The critical shear stress for such a plate is well-known [ll], viz. A, = 5.35~~ = 52.8. The critical stress shown in Fig. la of [7] for 0, = 0 agrees with this value, as do the results in Table 2. However, Fig. lb of [7] gives b/(1 - v*) N 0.076 for 0, = 0, whereas [6] and FASOR (Table 2) give b(1 -v*) = 0.214. This disagreement is similar to that for the composite panel at 0, = 0 (Fig. 3) in that the present results exceed those of [7j by roughly a factor of 3. FASOR results for design sensitivity of panels of Fig. 2 are discussed below. The sensitivity parameter is the outer layer fiber angle CC,measured from the longitudinal direction, with positive a corresponding to rotation of the layer towards the direction of maximum tension under the pure shear load (see Fig. 2). As the outer layer rotates, it is assumed that the inner layer rotates by the same angle a, but in the opposite direction. Thus, in general, the stacking sequence is [90° - a, CC]. Table 3 gives dimensionless FASOR results for buckling, initial postbuckling and design sensitivity of cross-ply (a = 0) panels. In this table, 1 is the
b
0.4
0.2
Model dimensions: r/B = 10,000 lUh=lOOforOs~O B/h==lOforO,=O
Fig. 2. FASOR model of long 2-ply [90°-a, a] composite
cylindrical panel.
Fig. 3. Comparison of postbuckling coefficients for long cross-ply cylindrical panel.
993
Sensitivity of buckling loads of anisotropic shells of revolution Table 3. Buckling, initial postbuckling and design sensitivity of cross-ply panels
@s 1128 0.730 0.730 0.742 0.747 0.708 0.639 0.600
0 2 4 5 6 7 8
4 34.21 34.73 41.06 48.52 58.27 68.85 79.67
b
K/K,
0.874 0.820 0.419 0.178 0.091 0.148 0.185
0.936 0.933 0.875 0.742 0.554 0.611 0.644
130 -
aA,iattt AAS 4.942 4.880 4.895 4.960 5.234 5.821 6.380
5.315 5.227 5.149 5.143 5.329 5.856 6.414
* A, = r,hB2/D. t Semianalytical derivative based on eqn (19d) using AH
for Aa = 1”. $ Forward difference for Aa = 1”.
b - 0
70 0
10
-0.2 20
adegrees
Fig. 4. Critical shear stress and postbuckling coefficient of long angle-ply cylindrical panel.
longitudinal buckle wave length, A[ = r,hB’/D, where D = D,, = D,, is the flexural stiffness of the cross-ply panel, and AA, is the change in A, as a changes from 0 to 1 degree. As seen from Table 3, the difference between the sensitivity derivative aA,/& and AA, grows as 0, decreases. This is due to the fact that the variation of A, with 0: near a = 0 becomes more nonlinear as 0, decreases. In order to verify this, both aA,/& and AA, corresponding to Aa = 0.25” and 0.5” were calculated for 0, = 0. [Note that an,/& changes due to nonlinear variation of the stiffness tensor H with u, CJ eqn (19d).] Results of these calculations are shown in Table 4, which exhibits convergence of aA,/& and AA,jAu as Au approaches zero. It is interesting to note from the results in Table 3 that, initially, rotation of the fibers towards the direction of maximum tension (and hence away from the direction of maximum compression) increases panel stability. As a increases from zero, the panel is stabilized by the appearance of longitudinal tension arising from shear-extension coupling of the now generally anisotropic panel. Figure 4 shows critical stress A, and postbuckling coefficient I, for angle-ply panels (0, = 8) as a function of the ply angle a. Although the critical stress increases with a until tl = lo”, the panel becomes imperfection-sensitive (b c 0) at a =4’, so that further gain in critical stress may be illusory.
CONCLUDING
REMARKS
Buckling load sensitivity analysis implemented in the shell code FASOR has been presented. Sensitivity to shell wall design changes, as well as sensitivity to geometric imperfections, are treated. The analysis is based on classical orthotropic and Table 4. Effect of Aa on dA,/da and AA,/Aa for flat (S, = 0) cross-ply panels Aa (deg)
an,lact
dAdAa
Rel. diff. (%)
1.0
4.942
5.315 5.124 5.020
7.5 3.9 2.0
0.5
4.930
0.25
4.923
generally anisotropic shell theory. Illustrative examples are presented which verify the computerized analysis. FASOR also calculates buckling modes of anisotropic shells including transverse shear deformations. Since transverse shear deformations can be important for composite shells, the present work will be extended to include this effect. Acknowledgement-This work was partially supported by NASA Grant NAG- l-224.
REFERENCES
1. B. Barthelemy, R. T. Haftka and G. A. Cohen, Physically based sensitivity derivatives for structural analysis programs. Cotnput. Mech. (in press). 2. R. T. Haftka, G. A. Cohen and Z. Mroz, Derivatives of buckling loads and vibration frequencies with respect to stiffness and initial strain parameters. J. appl. Me& (submitted). 3. G. A. Cohen, Effect of a nonlinear prebuckling state on the postbuckling behavior and imperfection sensitivity of elastic structures. AIAA Jnl6, 16161619 (1968). [See also Reply by Author to J. R. Fitch and J. R. Hutchinson, AIAA Jnl7, 1407-1408 (1969).] 4. G. A. Cohen, FASOR-A program for stress, buckling, and vibration of shells of revolution. Adv. Engg Software 3, 155-162 (1981). 5. W. T. Koiter, Buckling and post-buckling behaviour of a cylindrical panel under axial compression. National Aeronautical Research Institute, Amsterdam, Report S. 476, May (1956). 6. B. Budiansky, Post-buckling behavior of cylinders in torsion. Harvard University Report SM-17, August (1967). 7. D. Hui and I. H. Y. Du, Imperfection-sensitivity of long antisymmetric cross-ply cylindrical panels under shear loads. J. appl. Mech. 54, 292-298 (1987). 8. G. A. Cohen, Buckling of laminated anisotropic shells including transverse shear deformation. Comput. Meth. appl. Mech. 26, 197-204 (1981). 9. G. A. Cohen, Computer analysis of imperfection sensitivity of ring-stiffened orthotropic shells of revolution. AIAA Jnl9, 1032-1039 (1971). second generation shell of 10. G. A. Cohen, FASOR-A revolution code. Compur. Struct. 10, 301-309 (1979). 11. S. Timoshenko and J. M. Gere, Theory of Elastic Stability, 2nd edn, pp. 382-383. McGraw-Hill, New York (1961).
where
O1= -
cz--
b = _!
~= 1 r
r
1
r
-(L
+ n*~*dR*)(rlR*)
I
a, = a* = a3 = a, = a, =
+ n*J.,,lR&‘+
I
- I,/R*)x(~’
.
45
-iA,
W
&r/R*
a3 i(l - ~36/R2)u(o~/R2
-n(r’/r)(l na,
46
i&r ’
i&r/R*
I,, + 2n*A,,/R, + n41,,/R:
-n[n(L,,
-in(r’/rR2)1,,#o’ ina, ia, -L,,u’“/R;
i(& + n*J*,/R#
I
+ n*~*,/R*)
-4, +1*5/R*) - I*,r’ - id,,
-(A,
n(nl*,r’ + i/l,,)/r
“‘(4,
44, + 23WR2 + &JR:)
+ (n* + l)A2/R2 +
n*MRil
-n21nM12 + A,lR*)r’ + i(A, + LIR*)lIr
nr’ - i(l,, + “*1,/R*)
+ inl*,r ‘/r )
OF FIELD EQUATIONS
r’ -WA, + MR*) nl, - U*,r ’
n (r /R* - I,&
MATRICES
-(A,, + n*l,/R&
+ i&)/r
COEFFICIENT
-+(A4 + J*,lR*)r’ -r*/R* - I,r’* - in&r
(nr’/rR2)[u@) + I,x@)] (I,/Ri),y(‘) - (A,, + r’*)d”‘/r2 [(r/R&(1 - I,/R*)f”) + &,ucO)/r]/r [I,,(r/R,)2X(o) - 2,,d0)]/r2 xc” - (~,/R$I’~’
r’aS
(r/R&%
‘*
-ir’a, inr’a, ir ‘a*
I,r
i,r’r/R,
-i(r/RJa, in (r /R*)% i(r/R,)a,
Hermitian
&&r/R*)*
-n(h + MR*)(rIR*) - l.,/R,)rr - i&r/R*
nr’(nl*,r’
APPENDIX.
-n[n(l,r’*
22
;[($d&/R2)r’
3,
- W,3
- in&
+
L/&)1
+ n*1,,/R*)
+ I.,,) - i(n* - l)l,,r’]/r
(A,,+ n*&,/R,)r
i
Hermitian
0
-iA,,(nZ-
1)/r*
+ iA,,)/rR:
- 24,r’fR;
-n[(l,,/r*
+ &/R;)
[n2(A13/rz + I,,/R:)
n&w
I
1
1
and normal
- i;c,,r’/r’
the meridional
+ 1,,/Ra
+ &,/RiJ - il,,nr’/rz
-nr’(l,,n + iA,,r’)/rR, n(A,,n + iA,,r’)/R: (I,,n + il,,r’)/R: 0
Note: In the above equations the prebuckling rotation components I)~“) and Oco)about prebuckling circumferential displacement u lo) by using JI(O)= v(“)/Rz and 0”) = v(‘b’/r.
& = -v(O)
n(l,,
rF,O’
inr’T\p,’ -in(r/R,)T$’ -i(r/RJT(y
-n&,/R, + iA,,nr’)/rR, i&(n2 - 1)/R: 0
n(r’/RdTp n (r ‘* T’,” + Tj”)/r (r ‘l T$) + T$@)/r
- il,,nr’)/r’
+ U,,r’)/rR,
0
-t&n/R, 0
n2(r’/R,)T\o’ n ‘(r ‘2T\o)+ Tj’))/r
n*(A,, - iE.,,nr’)/r’
[ Hermitian
0
+ Tp)]/r
%,(nlr~2r’lR,
i Hermitian
& = _ $0’
B, = -
n*[(r,‘R$Ff)
B = i-4+ 82+ B,
where
directions
have been replaced
in terms of the
1989 0
CnMs-7949/89 s3m+ 0.00 1989 Pergamon Press plc
SENSITIVITY OF BUCKLING LOADS OF ANISOTROPIC SHELLS OF REVOLUTION TO GEOMETRIC IMPERFECTIONS AND DESIGN CHANGES GERALD A. COHEN~ and RAPHAEL T. HAFTKA~ IDepartment
tStructures Research Associates, Laguna Beach, CA 92651, U.S.A. of Aerospace and Ocean Engineering, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061, U.S.A. (Received 2 March 1988)
Abstract-Buckling load sensitivity calculations in the shell-of-revolution program FASOR are discussed. This development is based on Koiter’s initial postbuckling theory, which has been generalized to include the effect of stiffness changes, as well as geometric imperfections. The implementation in FASOR is valid for anisotropic, as well as orthotropic, shells. Examples are presented for cylindrical panels under axial compression, complete cylindrical shells in torsion, and antisymmetric angle-ply cylindrical panels under
edge shear,
INTRODUCTION
In a recent paper [I] the principle of virtual work is used to derive for elastic structures sensitivity field equations, i.e., equations satisfied by the derivatives of stress, strain and displacement tensors with respect to a stiffness parameter. In particular, for nonlinear shell analysis, these equations display the fact that the sensitivity solution may be obtained by solving load-perturbation equations with fictitious thermal loads which depend on the derivative of the stiffness tensor. The sensitivity solution can, of course, be approximated by taking finite differences of two neighboring nonlinear response fields. However, this is considerably more work than a direct solution of the linear sensitivity equations, and introduces additional truncation and condition errors. In [2] the sensitivity of vibration frequency and bifurcation buckling load is discussed. The analysis is again based on the principle of virtual work, and it turns out that the second-order state in an asymptotic expansion of the postbuckling response [3] plays a central role in the calculation of the sensitivity of vibration frequency and buckling load. As discussed in [3], the second-order postbuckling state is also used to calculate the second postbuckling coefficient b, which in turn yields the buckling load knockdown due to geometric imperfections in accordance with the well-known Koiter theory. Therefore, it is convenient to implement both sensitivity calculations simultaneously in a structural analysis program. This has been done for buckling loads based on classical orthotropic and anisotropic shell theory in the shell-of-revolution program FASOR [4]. As in prebuckling sensitivity analysis, buckling load sensitivity can be calculated by taking finite differences of buckling loads for neighboring designs.
In this case the advantages of sensitivity analysis are even greater, since recalculation of buckling loads is a lengthy process involving, in general, both nonlinear prebuckling and eigenvalue reanalysis for each design change. In contrast, once the second-order postbuckling state is obtained for the nominal structure, a linear calculation needed anyway for imperfection-sensitivity, evaluation of the (first-order) effect of various design changes on buckling load is essentially instantaneous. FASOR results are presented and compared to previously published solutions for cylindrical panels under axial compression [S], complete cylindrical shells in torsion [6], and antisymmetric angle-ply cylindrical panels under edge shear [7]. ANALYTICALFORMULATION Buckling load sensitivity analysis consists essentially of two steps following the prerequisite prebuckling and buckling calculations [8]. These are:
(4 the solution of the linear boundary value problem (i.e., field equations and boundary conditions) for the second-order postbuckling state, and (b) the evaluation of certain functionals of the prebuckling state at the bifurcation load, buckling mode, and second-order postbuckling state. Each of these steps is discussed below for anisotropic shells of revolution, Field equations for second-order
postbuckling
state
Field equations for the second-order postbuckling state are derived for orthotropic shells of revolution in [9]. It was observed there that the homogeneous parts of these equations are identical to the corresponding buckling mode equations, the only difference being the addition of nonhomogeneous terms, 985
986
A. COHENand RAPHAELT. HAF~KA
GERALD
which are quadratically dependent on buckling mode variables, in the second-order state equations. In accordance with the general theory for arbitrary structures [3], the same is true for anisotropic shells, and the derivation in [9] is used as a template for the following development. With two modifications the derivation of the nonhomogeneous terms given in [9] is valid for anisotropic shells. First, that analysis is limited to axisymmetric torsionless prebuckling states, thereby allowing prebuckIing rotation components (about meridional and normal directions) associated with axisymmetric torsion to be neglected. Since anisotropy couples axisymmetric bending and torsion, all three buckling rotations are retained here. Secondly, whereas for orthotropic shells of revolu-0 tion the antisymmetric circumferential component of harmonic buckling modes is decoupled from the symmetric component and therefore, without loss of generality, is omitted, the full constitutive equations of anisotropic shells (as well as torsional Ioading of orthotropic shells) couple these components. In this context, cosine components of normal type variables (those associated with axisymmetric bending of orthotropic shells) and sine components of shear type variables (those associated with axisymmetric torsion of orthotropic shells) are symmetric components, and the reverse is true of antis~met~c components. Thus in the anisotropic case, both sine and cosine components of buckling mode variables must be retained. For example, buckling and postbuckling meridional strain equations (I 7) of [9] become, with the retention of the prebuckling rotation 0”) about the normal (Fig. I), for anisotropic shells cl” _ ,\I) _ x’o’X”’_ @‘#I = 0
(la)
E\2)_ 42) _ xwx(2) _ eoern = q
(lb)
where Eje)=(1/2)[p+e"q.
Here and in the following, superscripts (0), (1) and (2) refer to prebuckling variables at bifurcation, buckling mode variables, and second-order postbuckling variables, respectively, and the superscript (e) indicates a nonhomogeneous term. The buckling mode rotations x(i) and e(i) may be written in terms of their symmet~c and antisymmetric components as
x(‘)= Xc’)COSn,$
+ x(O)sinn,4
(34
@I) = P sin n,# f 8@)cos n,#
(3b)
where 4 is the circumferential angle and n, is the buckling mode harmonic number. Substituting eqns (3) into eqn (2) gives, with the help of the doubleangle sine and cosine trigonometric identities, clef= (l/4)fX’“‘Z+ x’“” + @2 + @@>2] + (1/4)1y’“‘2- x@)“‘2+ fP)z- P’] cos 2n,$ + (1/2)[x(s)x(a)+
we(a)] sin 2n c(jb’
n =
2n,.
In order to preserve the form of the field equations in passing from orthotropic shells to anisotropic shells, complex buckling and postbuckling variables are used, the real and imaginary parts of which represent symmetric and antisymmetric harmonic components [lo]. Specifically, denoting symmetric and antisymmetric harmonic amplitudes of a typical normal type variable by NC’)and NC”)and symmetric
f REFERENCE SURFACE MERIDIAN VARIABLES
(4)
The form of eqn (4), which is typical of all nonhomogeneous terms in the field equations, shows that response variables of the second-order postbuc~ing state are composed of two harmonics, n = 0 and
AXIS OF REVOLUTION
(a) PRIMARY
(2)
(b) STRESS RESULTANTS,
(c) STRESS COUPLES,
DISPLACEMENTS, AND SURFACE FORCES
Fig, I. Shell-of-revolution notation.
ROTATIONS, AND SURFACE MOMENTS
987
Sensitivity of buckling loads of anisotropic shells of revolution for n = 2n,:
and antisymmetric harmonic amplitudes of a typical shear type variable by S@)and S(“) we form complex normal and shear amplitudes as hollows, N = NC” + iN(“)
(sa)
S = SC”)_ is(“)
(5b)
&’ = -(l/4)($2+
eq
Eli’ = (1/2)Ph
(9b)
Xf’ = -(n,/r)(T,
+ T2)0
Xy)= -(r’/2r)(T,
In terms of complex amplitudes, the harmonic components of eqn (4) may be written as
+ T,)f?
xp=o
for n = 0:
EY)= (1/4)(x . x + e .e)
(64
for n = 2n,:
df’ = (1/4)(x2 - er)
(6b)
in which complex amplitudes are denoted by the same symbols as used previously for total quantities and, for simplicity in writing, the superscript (l), denoting buckling mode variables, has been dropped from x(r) and t?(r). Also, in eqn (6a) and the following, dot and cross products of two complex variables u and u are defined by u v = Re(ou*) = Re(u)Re(v) + Im(u)Im(v)
(7a)
u x u = Im(uu*) = Re(u)Im(o) - Re(u)Im(u)
(7b)
in which * denotes complex conjugate. Equations (6) are the counterpart for anisotropic shells of eqn (26a) of [9] for orthotropic shells. In a similar manner the remaining nonhomogeneous terms in the equilibrium and kinematic equations, analogous to eqns (25) and (26b, c) of [9] are for n = 0:
#=(1/4)($~$+0~e)
(8a)
.$I = (l/2)& x $
(8b)
XI’)=0
(8~)
Xf) = -i(r’/2r)(T,
+ T2) x 8
(8d)
XI’)=0
(8e)
Ly) = (1/2)i(T, x I&- T,, x x)
(8f1
JX’ = -(l/2)0-,
4,
aI? -i&
.x +
T,2.
(W
$)
iA23
1
- iA,,
1 33
I -iA,,
(1/WT,IcI
Li=‘=
-W)V,x
a14
a15
+
T,2x)
-
T,,$).
(9f) (9g)
A24
A 25 -
iA,,
4, iA
1 36
(10)
I -114
-124
-
I
A*
145
iA
-&
-125
--iA,,
II
A,5
1 55
%6
iA
-A,,
’
-iA,,
- i&,
&6
(9d) (9e)
LI” =
--------___----J-------_-----___
U,,
(9c)
Comparison of eqns (6b) and (9) with eqns (25) and (26) of [9] shows that the unsymmetrical components of the complex nonhomogeneous terms are identical in form to their real counterparts for orthotropic shells. The nonhomogeneous terms given by eqns (6), (8), and (9) apply directly to the complex harmonic equilibrium and kinematic equations, which, as noted, have the same form as their real counterparts (in terms of symmetric amplitudes) for orthotropic shells. Additional nonhomogeneous terms arise from the “excess” variables in these equations, viz. T2, M2, and M,, in the equilibrium and c,, K~, and e,2 in the kinematic equations. (Excess variables are dependent variables which appear in the linearized equilibrium and kinematic equations, c$ eqns (11) and (13) of [9], but whose meridional derivatives are absent.) These variables are expressed in terms of the primary variables (i.e., variables whose meridional derivatives do appear, see Fig. 1) through partially inverted constitutive equations. Partially inverted constitutive equations in terms of complex harmonic amplitudes have been derived for anisotropic shells based on transverse shear deformation theory in [lo]. The same procedure with a slightly different ordering of stress and strain measures leads to the following complex constitutive equations, which are the analog of eqns (42a) of [lo], for anisotropic shells based on classical shell theory
a12 a22
iA
(9a)
1 66
GERALDA. COHENand RAPHAELT. HAFTKA
988
Equation (lo), being linear, applies as well to complex harmonic amplitudes of buckling or postbuckling variables. The modified twist K’\gand shear stress resultant s”(‘),complex amplitudes of which appear in eqn (lo), are given for orthotropic shells by eqns (21) of 191. For anisotropic shells, the corresponding equations are
bation forms of the equilibrium and kinematic equations, eqns (11) and (13) of [9], written in terms of harmonic amplitudes, to obtain the field equations for complex harmonic amplitudes of the secondorder postbuckling state. In standard matrix form suitable for solution by the field method [4], these are Y’ + 10 + c&M.Y + [b + B(&)lz =f z’+c?,+[~++(1,)]z=g
s”(2)-S’2’+(1/2)[T(P)+
2-$@]6’2’
+ (1/2)[7-~2’+ Z-@?‘“‘= s(P) (llb) where rZy;= -d#Rz
(124
3’ = (1/2)[T\” + l”ir’]e(‘).
(12b)
Equation (12a) is valid in terms of complex ha~onic amplitudes for n = 0 and n = 2n, where t”R is given by either eqn (8b) or eqn (9b). Complex harmonic amplitudes for S”(@), obtained from eqn (12b) in the same manner as was done previously for other nonhomogeneous terms, are for n = 0:
S”(e)= -(1/4)i(T,
+ Tz) x 0
(13a)
for n = 2n,:
@+= -(1/4)(T,
+ T,)B.
(13b)
(15a) (15b)
the homogeneous forms of which are buckling mode equations. (For the sake of simplicity, live pressure terms in the equilibrium equations (15a) are omitted.) Prime denotes differentiation with respect to meridional arc distance s, ir is the critical load factor, y and z are the 4-element complex force and displacement vectors r(P, Q, S, M,) and (& n, v, ~1 (Fig. 1), and a, b, c, d, K, /I, and 6 are 4 x 4 complex matrices (see Appendix) satisfying the Hermitian self-adjoint conditions (the superscript t signifies Hermitian transpose) a= -d’,
a= -6’
b = b’,
P=P
c=c’. The complex nonhomogeneous eqns (15) are
(161 vectors f and g of
-r(r/R2)X(le)--nr’Ly)-n2(r’/r)h4ff As in eqns (6) (8), and (9), the superscript (1) has been omitted from buckling mode variables of eqns (13). The nonhomogeneous terms tr), R’;1, and s”@‘) give rise, through eqn (lo), to additional nonhomogeneous terms for each of the excess variables given on the left-hand side of eqn (10). To distinguish the additional nonhomogeneous contributions to 6, and c,2 from cl’) and @J given by eqns (6), (8b), and (9b), those arising from eqn (10) are denoted by - Acf) and -Acl”,‘. Thus, from eqn (10) one obtains the following additional nonhomogeneous terms Ty) = L,,c~) + i&R# + iA,,P MS’) = A,,t?Ip’+ iA2g# + iA,,P 2Myd = -iA,,e!$ + A&~ + A&?@) -A.s~)=
-&t$-
U&@+
i&@)
KY) = --1,,6f) - iA&(;?:+ i&P -A#
= id,,c~) - ,I&‘$ + ;IS6gfe).
+ 2(n/r)Mfj f=
-rr’X(f) + n(r/R~)L~) + Tfze)
(17)
+ (n2/R2)M(2e) -r~~)~(rlR2)~~)+n~~)+(n/R2)~~) -rLf)
+ r’h@ - 2nMg + A@]
(18)
Wd (14b) (14c) (W (144 Wf)
The nonhomogeneous terms given by eqns (6) (8), (9). and (14) mav now be substituted into the pertur-
Analogous equations may be derived for discrete rings, providing boundary conditions for eqns (15) at ring locations. Evaluation of functionals General formulas for the second postbuckling coefficient 6, initial postbuckling stiffness K, and imperfection parameter a are given in f3]. (For axisymmetric structures with unique harmonic buckling modes, the first postbuckling coefficient is identically zero.) The formula for the first-order change A1.
989
Sensitivity of buckling loads of anisotropic shells of revolution in the buckling load due to a change AH in the stiffness tensor is given in [Z]. In the operator notation of those papers,
For shells of revolution, FJand t: may be interpreted as six-component vectors consisting of stress resultants and couples for 6 and reference surface strains and curvature changes for L, viz. ff = IT,,
K= K,f[l + (4,‘2H:)(a,
. q + 2q,. Q]
a= -Ia, . q -6, . L(wl~ dl/&m, Ai, = -PAH(c,) .c2 + AHk)
u,)
1t,l/‘F(u,, u,>
Wb) ww (194
6 =
T2,
T,2,
M,,
M2,
M,,)
@W
GW
i ~,,~2,~,2,iC,,K*,2K*t~.
Corresponding to eqn (24b), L*(U) and L,,(u,,u,) may be written in terms of shell element rotation components as
where F(u,, u,) is the inner product of the buckling mode with itself given by (with live pressure terms omitted)
L,(~)=(x~+B*,~*+~*,~x~L,O,O,O) (25a) L,,(u,, u2) = {X,X2+ W-&1+,+* +- V2, X,$2
f;(u,, u,) = 20, . L,,(ii,, %I +4.
L,(4).
+ $,x2, 0, O,O>.(25b)
(20)
In eqns (19) and (20) (T, c, and u represent stress, strain and displacement tensor fields, and L, is the homogeneous quadratic operator in the straindisplacement relation
Since only axisymmetric loading is considered, prebuckling variables are independent of the circumferential coordinate 4, whereas buckling mode variables have harmonic variation with, in general, both sine and cosine components, cJ: eqn (3). As discussed (21) previously, second-order ~stbuckling variables have t: = L,(u) + (1~2)~~(~) both axisymmetric and asymmetric harmonic components, which are denoted here by subscripts 0 and 2, in which L,(u) is the linear component. The bilinear respectively, written to the left of the corresponding operator L,,(u, v) is defined by symbol. For example, normal and shear type postbuckling variables such as T, and T,, may be written Lz(trfv)=L*(u)+2L1,(U,V)+Lz(u). (22) as Subscripts 0, I and 2 on 6, f, u and K indicate prebuckling, buckling and second-order postbuckling states, respectively, with the understanding that all prebuckling quantities are evaluated at the bifurcation load 1,. (Note that for individual components of 6, E, and U, superscripts (0), (1) and (2) are used to denote these states.) In eqn (20) bars signify derivatives of prebuckling fields with respect to the load factor I, evaluated at the bifurcation load. Also, the low-dot product [as opposed to the high-dot product of eqn (7a)] signifies the total work of the left-side field acting through the right-side field. Thus, for shell structures the low-dot products are double integrals over the reference surface of the shell. The imperfection parameter GIgiven by eqn (19c) is based on a small initial stress-free displacement field PU,, where U, is the buckling mode displacement and p is the imperfection amplitude. In FASOR the buckling mode is normalized such that its maximum normal deflection is the shell thickness, so that ~1 represents the imperfection amplitude-to-thickness ratio. (For variable thickness shells the thickness at the final point of the FASOR model is used.) If b is negative, Koiter theory gives the buckling load knockdo~ &/A,, c$ eqn (38) of [3], as A$/& = 1 + 3(a2$b/4)“3,
(23)
Tf’=,TI”)~zTI”‘cos2n,cp
+ zTf”‘sin2n,q5
T\? = ,,T@ + 2TB cos 2n,(t) + *pj sin 2n,&
(26a) (26b)
To evaluate rr, . L,(u,) of eqn (19a) one has from eqns (24a) and (25a)
+
T$2)[$,(,h
+
fj(,h]
+
2T\Z,'X"'$"')r d$ ds.
(27)
Substituting for the buckling and second-order postbuckling variables on the right-hand side of eqn (27) their harmonic component expressions, as in eqns (3) and (26), and performing the +-integration with the help of the following identities 2n
2n cos2n~
s 9-O 2x s
,=o
d# =
sit&$ d4 = n s .$=O
cos2nt$cos 2nd d#
(2ga)
990
GERALD
A. CXXIEN and kw%m.
sin2 n# cos 2n# d$ - n/2
=Z-
(28b)
cos2m# sin 2nrb d4
T. HAFTKA
a,. c2= 2x
[rl”) +o~l+ Go) *oe2- T# Im(&,) s + $01 1orci+ Mi”j aoicr
- 2M’# Im(,k,,)]r ds
(314
2n =
sin2 nd sin 24
d$ = 0
(28~)
s rp-0 + CT,, x x - T2 x
$>@”
reduces eqn (27) to - [(TI + T,) x @)@O))rds
(31d)
{oT(f)[X(++ p2 + @2 + fpq
a2. L&u,) = I[
ss
- 2[( T, + T2)
In terms of complex harmonic amplitudes, as in eqns (5), eqn (29) becomes
s
IoTdx-x>+oT2W+Icr)
a2.Wl)=n
x O]P(O )r
ds.
(31e)
Note that prebuckling variables in eqns (31c-e) are real, as opposed to complex buckling and postbuckling harmonic amplitudes. Evaluation of the functionals in the numerator of eqn (19d) is similar to that of eqns (31b) and (31~) with the replacement of components of a0 = H(co) and CJ,= H(Q) by components of AH(%) and AH@,), respectively. Analogous contributions from discrete rings to the above functionals can be derived. EXAMPLES
I
tot+oT2)(e- 0) + 2 Im(oT,2)(xx $1
+
+ twx27A . x2- 2Tz.Jr”
- t27i f 2~2).e2i +
2Tl2 * (xJ/)b
(30)
d&
Similariy, the remaining functionals of eqns (19) and (20) reduce to
a,
.6]
=
x
s
(T, acl
i-
T2’
(i2 f
T12*cl2
s
+ Mr *~2 + 2M,z. K,& ds
-t
M, - KI
Numerical results are presented for three kinds of cylindrical shelf structures for which published results by other investigators are available. The first example, cylindrical panels under axial compression, verifies the applicability of large radius toroidal modeling of general cylindrical shells for initial postbuckling calculations. Since no published results of initial postbuckling of generally anisotropic shells are available, complete cylindrical shells in torsion, the second example, are used to verify this analysis. Although the cylinders treated are isotropic, shear loading produces coupling between circumferential symmetric and antisymmetric harmonic components of the buckling mode, just as anisotropy does. The third exampie, antisymmetric angle-ply panels under shear loading, illustrates buckling load sensitivity to changes in ply angle of a composite shell. The sensitivity calculations were verified by recakulation of the buckling load at different ply angles. However, in the special case of a cross-ply panel values of the postbuckling coefficient b obtained are in disagreement with those reported in 171. Cylindrical panels under axial compression
Koiter [5] analysed narrow cylindrical panels with idealized boundary conditions to simulate the
Sensitivity of buckling loads of anisotropic shells of revolution individual panels of stringer-stiffened cylindrical shells. His results are contained in the function G(8) = 1 - 2@ - 9S(O)
(32)
except in the neighborhood of G = - 1, which occurs at 0 = -0.77, where significance is lost due to the occurrence of the small difference of large numbers in the evaluation of eqns (l9b) and (34b). Cylindrical shells in torsion
where S(O), given by an infinite series, is tabulated in Table 1 of [5], and 0 is the curvature parameter defined by 0 = (1/2~}[12~1 - v*)]“~~/(R~)“*
991
(33)
in which B is the circumferential arc width of the panel, R is the panel radius, and h is the pane1 thickness. These results can be put in terms of the postbuckling coefficient b and the initial postbuckling stiffness ratio K/K0 through the formulas b/(1 - vz) = 3G/8(1 + 04)
(34a)
K/K, = G,‘(G + 1).
(34b)
A comparison of FASOR results with those of [S] are shown in Table 1. The FASOR results were obtained using a toroidal half-panel model with radius r/B = R,/B = 10,000 at the symmetry boundary and Poisson’s ratio v = 0.3. For 0 > 0, the m+dional radius of curvature R/h = 100, whereas for 0 = 0, corresponding to a flat panel, B/h = 10 was used. Model constraints at the “‘stringer edge” s = B/2 are: for buckling u = w = 0 (antisymmetric buckling mode about the stringer) and for postbuckling &/ad, = x = 0 (symmetric initial postbuckling state about the stringer) and ~~/~# = 0 (constraint due to out-of-surface stringer bending stiffness) (see Fig. 1). The postbuckling results of [5] are based on the classical buckling mode for a simply supported narrow curved panel. As shown in Table 1, FASOR buckling loads agree well with the classical values, the greatest relative difference being 1.4% at 0 = 1. It is noted that small discrepancies in the buckling mode tend to be magnified in the calculation of the postbuckling coefficient b. Thus the greatest relative difference for b in Table 1 is 4.3% at 0 = 1. The discrepancies shown for values of K/K0 are similar
Budiansky [6] analysed initial postbuckling of circular cylindrical shells under torsional edge loads for three sets of edge conditions: simply supported edges and two kinds of clamped edges with different insurface constraints. FASOR results are in good agreement with all three cases. However, for the sake of brevity, only the simple-support results are presented, the edge constraints for this case being &~/a# = w = 0 (see Fig. I). The analysis of [6] is based on the KarmanDonnell cylindrical shell equations and results are given for values of the curvature parameter 2 in the range 0 < Z < 10,000, where 2 = (1 - v2)‘12L2/rh,
L being the cylinder length. The critical shear stress z, and circumferential wave number n, are given by the dimensionless parameters AC= t,hL’/D
(W
1 = Lnn,jr
(W
where D is the flexural stiffness of the shell. A comparison of FASOR results with those of [6] is shown in Table 2. For the FASOR model, v = l/3, and n, = 10 (Z > IO) or 100 (0 < 2 < 10) were used. Thus for Z > 0, L/r and r/h were determined by eqns (36b) and (35) for various combinations of /I and Z. For Z = 0, corresponding to a narrow flat plate under edge shear, r/L = 10,000, L x h = 10, and n, = r/3/L = 25,100 were used. The agreement shown in Table 2 is good considering the approximate nature of the Ka~an-Donnell equations (see also discussion of Table 1). It may be noted that for a given value of Z, the complete shell theory of FASOR predicts values of b which are (weakly) dependent on r/h, whereas the Donnell theory results are independent of r/h. For example,
Table 1. Buckling and initial postbuckling of cylindrical panels under axial compression 0 0
fJ,ia: Classical? FASOR 1.000
I.000
b/(1 - v2) FASOR Ref. 151 0.375
0.315
0.355 1.006 0.355 0.231 i .os7 0.230 0.6 1.122 0.091 0.090 0.7 1.240 1.229 -0.109 -0.113 -0.364 0.8 1.410 1.394 -0.374 -0.662 0.9 1.656 1.636 -0.684 -0.997 1.0 2.000 1.972 - 1.040 * u0 = 4x2D/B2h is the critical stress for a flat plate. t The classical value of a& for a curved panel is (1 + @). 0.3 0.5
1.008 1.062 1.130
(35)
UK, Ref. I51 FASOR OS 0.488 0.395
0.5 0.488 0.394
0.213 -0.597 3.457 1.495 1.220
0.214 -0.550 3.902 1.536 1.238
GERALD
992
A. COHENand
RAPHAEL
T. HAFTKA
Table 2. Buckling and initial postbuckling of cylindrical shells in torsion B
Ref. [6]
FASOR
b/(1 -v*) Ref. [6] FASOR
Ref. [6]
UK, FASOR
2.51 2.51 3.97 9.29 19.75 37.4
52.61 53.18 14.29 271.6 1482 8389
52.61 53.18 74.53 271.5 1471 8295
0.2139 0.1508 -0.2364 -0.0756 -0.0122 -0.0015
0.671 0.582 3.42 -0.913 -0.175 -0.031
0.671 0.586 4.21 -0.725 -0.150 -0.052
4 Z 0
I 10 100 1000 10,000
for Z = 100 reducing equivalent to changing
n, from 10 to 5, which is r/h from 123 to 30.7, yields: A, = 259.6, b/(1 - v*) = -0.0759, and K/K, = -0.832. On the other hand, increasing n, to 100, equivalent to r/h = 12,300, yields A, = 271.5, b/(1 -v*) = -0.0665, and K/& = -0.718, which
are in better agreement with the results for n, = 10. Long antisymmetric angle-ply panels under shear Hui and Du [7] analysed initial postbuckling of long antisymmetric cross-ply cylindrical panels under edge shear for simply supported and clamped edges. Graphical results are presented in [7] for b versus the simplified curvature parameter 0, defined as [cJ eqn (33)l 0, = B/(m)“*.
(37)
A large radius toroidal model (Fig. 2) is used for FASOR analysis of the 2-layer clamped panel of [7]. Note that the circumferential (4) direction of the model corresponds to the longitudinal direction of the panel. The panel wall is a regular antisymmetric cross-ply (a = 0) laminate with the outer layer fibers running in the longitudinal direction. Although calculated critical shear stresses agree well with [7j, values of the postbuckling coefficient b do not. Figure 3 compares b-values calculated by FASOR with those reproduced from Fig. 2b of [7]. Thus, whereas the results of [7] predict imperfectionsensitive panels (i.e., b < 0) for 0, > -3.4, the
Boundary conditions at s = 0 and s = 6: out-of-surface, clamp (w = u/R-awlas = 0) in-surface, free (T,+ =Th = const., T, = 0)
I r>> 0
--
Lamina elastic moduli: EL/ET = 10 GLT/ET = ‘A VLT (major Poisson’s ratio) = 0.22
-+
0.2143 0.1545 -0.2209 -0.0661 -0.0107 -0.0015
present results predict imperfection-insensitive
panels
(b > 0) over the range of 0, considered.
Reference [7] also gives results for simply supported isotropic panels under edge shear. For 0, + 0, this case becomes equivalent to cylinders of the preceding example for Z -+ 0, since both models approach a narrow isotropic simply supported plate under edge shear. (In the limit the panel width B plays the same role as the cylinder length L.) The critical shear stress for such a plate is well-known [ll], viz. A, = 5.35~~ = 52.8. The critical stress shown in Fig. la of [7] for 0, = 0 agrees with this value, as do the results in Table 2. However, Fig. lb of [7] gives b/(1 - v*) N 0.076 for 0, = 0, whereas [6] and FASOR (Table 2) give b(1 -v*) = 0.214. This disagreement is similar to that for the composite panel at 0, = 0 (Fig. 3) in that the present results exceed those of [7j by roughly a factor of 3. FASOR results for design sensitivity of panels of Fig. 2 are discussed below. The sensitivity parameter is the outer layer fiber angle CC,measured from the longitudinal direction, with positive a corresponding to rotation of the layer towards the direction of maximum tension under the pure shear load (see Fig. 2). As the outer layer rotates, it is assumed that the inner layer rotates by the same angle a, but in the opposite direction. Thus, in general, the stacking sequence is [90° - a, CC]. Table 3 gives dimensionless FASOR results for buckling, initial postbuckling and design sensitivity of cross-ply (a = 0) panels. In this table, 1 is the
b
0.4
0.2
Model dimensions: r/B = 10,000 lUh=lOOforOs~O B/h==lOforO,=O
Fig. 2. FASOR model of long 2-ply [90°-a, a] composite
cylindrical panel.
Fig. 3. Comparison of postbuckling coefficients for long cross-ply cylindrical panel.
993
Sensitivity of buckling loads of anisotropic shells of revolution Table 3. Buckling, initial postbuckling and design sensitivity of cross-ply panels
@s 1128 0.730 0.730 0.742 0.747 0.708 0.639 0.600
0 2 4 5 6 7 8
4 34.21 34.73 41.06 48.52 58.27 68.85 79.67
b
K/K,
0.874 0.820 0.419 0.178 0.091 0.148 0.185
0.936 0.933 0.875 0.742 0.554 0.611 0.644
130 -
aA,iattt AAS 4.942 4.880 4.895 4.960 5.234 5.821 6.380
5.315 5.227 5.149 5.143 5.329 5.856 6.414
* A, = r,hB2/D. t Semianalytical derivative based on eqn (19d) using AH
for Aa = 1”. $ Forward difference for Aa = 1”.
b - 0
70 0
10
-0.2 20
adegrees
Fig. 4. Critical shear stress and postbuckling coefficient of long angle-ply cylindrical panel.
longitudinal buckle wave length, A[ = r,hB’/D, where D = D,, = D,, is the flexural stiffness of the cross-ply panel, and AA, is the change in A, as a changes from 0 to 1 degree. As seen from Table 3, the difference between the sensitivity derivative aA,/& and AA, grows as 0, decreases. This is due to the fact that the variation of A, with 0: near a = 0 becomes more nonlinear as 0, decreases. In order to verify this, both aA,/& and AA, corresponding to Aa = 0.25” and 0.5” were calculated for 0, = 0. [Note that an,/& changes due to nonlinear variation of the stiffness tensor H with u, CJ eqn (19d).] Results of these calculations are shown in Table 4, which exhibits convergence of aA,/& and AA,jAu as Au approaches zero. It is interesting to note from the results in Table 3 that, initially, rotation of the fibers towards the direction of maximum tension (and hence away from the direction of maximum compression) increases panel stability. As a increases from zero, the panel is stabilized by the appearance of longitudinal tension arising from shear-extension coupling of the now generally anisotropic panel. Figure 4 shows critical stress A, and postbuckling coefficient I, for angle-ply panels (0, = 8) as a function of the ply angle a. Although the critical stress increases with a until tl = lo”, the panel becomes imperfection-sensitive (b c 0) at a =4’, so that further gain in critical stress may be illusory.
CONCLUDING
REMARKS
Buckling load sensitivity analysis implemented in the shell code FASOR has been presented. Sensitivity to shell wall design changes, as well as sensitivity to geometric imperfections, are treated. The analysis is based on classical orthotropic and Table 4. Effect of Aa on dA,/da and AA,/Aa for flat (S, = 0) cross-ply panels Aa (deg)
an,lact
dAdAa
Rel. diff. (%)
1.0
4.942
5.315 5.124 5.020
7.5 3.9 2.0
0.5
4.930
0.25
4.923
generally anisotropic shell theory. Illustrative examples are presented which verify the computerized analysis. FASOR also calculates buckling modes of anisotropic shells including transverse shear deformations. Since transverse shear deformations can be important for composite shells, the present work will be extended to include this effect. Acknowledgement-This work was partially supported by NASA Grant NAG- l-224.
REFERENCES
1. B. Barthelemy, R. T. Haftka and G. A. Cohen, Physically based sensitivity derivatives for structural analysis programs. Cotnput. Mech. (in press). 2. R. T. Haftka, G. A. Cohen and Z. Mroz, Derivatives of buckling loads and vibration frequencies with respect to stiffness and initial strain parameters. J. appl. Me& (submitted). 3. G. A. Cohen, Effect of a nonlinear prebuckling state on the postbuckling behavior and imperfection sensitivity of elastic structures. AIAA Jnl6, 16161619 (1968). [See also Reply by Author to J. R. Fitch and J. R. Hutchinson, AIAA Jnl7, 1407-1408 (1969).] 4. G. A. Cohen, FASOR-A program for stress, buckling, and vibration of shells of revolution. Adv. Engg Software 3, 155-162 (1981). 5. W. T. Koiter, Buckling and post-buckling behaviour of a cylindrical panel under axial compression. National Aeronautical Research Institute, Amsterdam, Report S. 476, May (1956). 6. B. Budiansky, Post-buckling behavior of cylinders in torsion. Harvard University Report SM-17, August (1967). 7. D. Hui and I. H. Y. Du, Imperfection-sensitivity of long antisymmetric cross-ply cylindrical panels under shear loads. J. appl. Mech. 54, 292-298 (1987). 8. G. A. Cohen, Buckling of laminated anisotropic shells including transverse shear deformation. Comput. Meth. appl. Mech. 26, 197-204 (1981). 9. G. A. Cohen, Computer analysis of imperfection sensitivity of ring-stiffened orthotropic shells of revolution. AIAA Jnl9, 1032-1039 (1971). second generation shell of 10. G. A. Cohen, FASOR-A revolution code. Compur. Struct. 10, 301-309 (1979). 11. S. Timoshenko and J. M. Gere, Theory of Elastic Stability, 2nd edn, pp. 382-383. McGraw-Hill, New York (1961).
where
O1= -
cz--
b = _!
~= 1 r
r
1
r
-(L
+ n*~*dR*)(rlR*)
I
a, = a* = a3 = a, = a, =
+ n*J.,,lR&‘+
I
- I,/R*)x(~’
.
45
-iA,
W
&r/R*
a3 i(l - ~36/R2)u(o~/R2
-n(r’/r)(l na,
46
i&r ’
i&r/R*
I,, + 2n*A,,/R, + n41,,/R:
-n[n(L,,
-in(r’/rR2)1,,#o’ ina, ia, -L,,u’“/R;
i(& + n*J*,/R#
I
+ n*~*,/R*)
-4, +1*5/R*) - I*,r’ - id,,
-(A,
n(nl*,r’ + i/l,,)/r
“‘(4,
44, + 23WR2 + &JR:)
+ (n* + l)A2/R2 +
n*MRil
-n21nM12 + A,lR*)r’ + i(A, + LIR*)lIr
nr’ - i(l,, + “*1,/R*)
+ inl*,r ‘/r )
OF FIELD EQUATIONS
r’ -WA, + MR*) nl, - U*,r ’
n (r /R* - I,&
MATRICES
-(A,, + n*l,/R&
+ i&)/r
COEFFICIENT
-+(A4 + J*,lR*)r’ -r*/R* - I,r’* - in&r
(nr’/rR2)[u@) + I,x@)] (I,/Ri),y(‘) - (A,, + r’*)d”‘/r2 [(r/R&(1 - I,/R*)f”) + &,ucO)/r]/r [I,,(r/R,)2X(o) - 2,,d0)]/r2 xc” - (~,/R$I’~’
r’aS
(r/R&%
‘*
-ir’a, inr’a, ir ‘a*
I,r
i,r’r/R,
-i(r/RJa, in (r /R*)% i(r/R,)a,
Hermitian
&&r/R*)*
-n(h + MR*)(rIR*) - l.,/R,)rr - i&r/R*
nr’(nl*,r’
APPENDIX.
-n[n(l,r’*
22
;[($d&/R2)r’
3,
- W,3
- in&
+
L/&)1
+ n*1,,/R*)
+ I.,,) - i(n* - l)l,,r’]/r
(A,,+ n*&,/R,)r
i
Hermitian
0
-iA,,(nZ-
1)/r*
+ iA,,)/rR:
- 24,r’fR;
-n[(l,,/r*
+ &/R;)
[n2(A13/rz + I,,/R:)
n&w
I
1
1
and normal
- i;c,,r’/r’
the meridional
+ 1,,/Ra
+ &,/RiJ - il,,nr’/rz
-nr’(l,,n + iA,,r’)/rR, n(A,,n + iA,,r’)/R: (I,,n + il,,r’)/R: 0
Note: In the above equations the prebuckling rotation components I)~“) and Oco)about prebuckling circumferential displacement u lo) by using JI(O)= v(“)/Rz and 0”) = v(‘b’/r.
& = -v(O)
n(l,,
rF,O’
inr’T\p,’ -in(r/R,)T$’ -i(r/RJT(y
-n&,/R, + iA,,nr’)/rR, i&(n2 - 1)/R: 0
n(r’/RdTp n (r ‘* T’,” + Tj”)/r (r ‘l T$) + T$@)/r
- il,,nr’)/r’
+ U,,r’)/rR,
0
-t&n/R, 0
n2(r’/R,)T\o’ n ‘(r ‘2T\o)+ Tj’))/r
n*(A,, - iE.,,nr’)/r’
[ Hermitian
0
+ Tp)]/r
%,(nlr~2r’lR,
i Hermitian
& = _ $0’
B, = -
n*[(r,‘R$Ff)
B = i-4+ 82+ B,
where
directions
have been replaced
in terms of the