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Buckling of cylindrical shells with spiral stiffeners under uniform compression and torsion
compvrm &strnchllw vol.11, pp.m-595 Pergamon Press Ltd., MO.
Printed in Great Britain
BUCKLING OF CYLINDRICAL SHELLS WITH SPIRAL STIFFENERS UNDER UNIFORM COMPRESSION AND TORSIONS SHAO-WENYENS Douglas Aircraft Company. McDonnell Douglas Corporation, Long Beach, CA 90846, U.S.A. (Received 26 February 1979;received for publication 12 September 1979) Abstra&--This paper investigated the general instability of cylindrical shells in which the stiffeners formed spirals along the length and at an arbitrary angle with the axis. Two loading conditions were considered: unifo~ axial and lateral compressions and torsion. The stress-strain relations of the stiffeners were developed by rotation of the strain tensor. The buckling determinate was obtained by introducing into the equilibrium equations the admissible displacement functions consistent with the end constraints, thereby enforcing equilibrium by satisfying the characteristic equations. The buclking equations were programmed for a computer which rearched through a finite set of stress resultants for assigned values of spiral angle and modes and printed out the buckling load. The optimum structure weight of the stiffened shell was determined by iterating the design parameters at the required spiral angle so that the buckling load approached the applied load as a limit until the difference between these loads was within the design allowance.
NO~NCLA~
p detined by I$/(1 - v2) I defined by N@ rXy engineering shear strain eX,~iy engineering direct strains { defined by CAelDh 7) defined by f&/b B spiral angle of the stiffeners K,, Q, ~~~ change of curvatures ri, de&red by m&L Y Poisson’s ratio 4 defined by C&/Dir o;, u+ direct stresses T~+ shearing stress W defined by I127rR2
stiffener cross-sectional area defined by eqns (5lJ43) elements of matrices defined by 2hEld flexural rigidity of the shell stiffener spacing Young’s modulus defined by z dAIA I direct straintomponent of the shell direct strain components of the stiffeners with +@inclination direct strain components of the stiffeners with -0 inclination defined by eqns (80)-(83) stiffener depth coefficients of displacement functions moment of inertia of stiffeners about the middle surface of the shell length of the shell moments of forces number of half-waves in the axial direction stress resultants number of waves in the circumferential direction buckling load stiffener load in the W-axis stiffener Ioad in the 19”-axis coetIicients of displacement functions radius of the shelf length of the stiffener grid applied torsion buckling load stiffener width skin thickness of the shell components of displacement rectangular coordinates cylindrical coordinates tThis paper is part of the work performed at the Douglas Aircraft Company of the McDonnell Douglas Corporation under the sponsorship of its Independent Research and Development (IRAD) Program. SSenior Engineer/Scientist.
INTRODUCTION The buckling of stiffened cylindrical shells under specified geomet~cal characteristics has been investigated both theoretic~~y and experimentally by many authors[ l-201. Most of them placed the stiffeners on the interior of the shell for aerodynamic advantages, although the exterior-stiffened theoretical values for typical shell designs are about 30% higher than the interior theoretical values. Others, for geometrical simplicity, preferred the conventional ring-and-stringer shell design. It is doubtful whether this simple design can provide minimum shell structural weight. The necessity for weight-saving measures in the design of aeronautics and space structures makes it advisable that other feasible shell design configurations be examined for further improvement. This paper deals with the general instability of eccentrically stiffened cylindrical shells in which the spiral stiffeners with constant spacings are arranged symmetrically at an arbitrary angle with the axis of the shell. The buckling determinate is obtained by substitut~g into the ~~erential equatjons of equ~ib~um the admissible displacement functions consistent with the end conditions and enforcing equilibrium by satisfying the characteristic equation. The buckling equations are programmed for a computer which searches through a finite set of the stress
587
588
S.-W. YEN
resultants for assigned values of stiffener angles and modes and prints out the minimum stress resultant and buckling load. The optimum structural weight of the stiffened shell is obtained by iteration of design parameters at the required spiral angle in such a way that the buckling load approaches the applied load as a limit until the difference between these loads is within the design ahowance. ANALYSIS
In developing the analysis, the following assumptions are made: 1, ~formations are small and elastic. 2. Stiffeners have constant spacing, equal spiral angle, and the same elastic material properties. 3. In-plane shear stresses transverse to the stiffeners are resisted by the shell skin. 4. The Love-Kirchhof? hypothesis of normals remaining normal holds true. The cylindrical shell under consideration has a radius R, length Z_.and skin thickness, t,. The stiffeners have width tf, height h, spacing d, and spiral angles (? and -8 arranged symmetrically with the axis of the shell. The middle surface of the cylindrical shell skin is taken as the reference surface for the analysis (Fig. 1). ST-RAIN
Fig.
2. Rectangularcoordinatesrotation.
The distributed edge stresses of the stiffener grid can be written in terms of the discrete stiffener loads:
= Ct cos’ 6(e, co? @+ ev sin2 0).
RELATIONS
The stress-strain relations of the stiffeners are developed by rotation of the strain tensor. The transformation of strains from the xy-axis to the l’Z’-axis(Fig. 2) yields
Sim~arly,
x ( eYsin* Bt eXco? 8)
e;, = e,, cos2 8 + 2e,, sin @cos 8 i e,, sin2 8 e;, = e,, sin’ 8 - 2e,, sin 6 cos Bt e,,cos*8.1 (1) Similarly, the transformation of strains from the xy-axis to the I”2’‘-axisyields e:‘, = e,, cos* B - 2e,, sin @cos B+ e,, sin* B 1 e’;, = e,, sin’ 6 -t 2e,, sin 6 cos @t e,, cos’ B. i (2)
7Xr=!$$ei,
-eY,)= Cssin* f?cos2@yX,
e,(Z) = G -
ZKx
&)=Q-ZK+ ?‘x&) = %z+- 2ZK,&
(a) STIFFENED CYLIN~~AL SHELL AND ASSOCIATE0 STIFFENER LOADS P’, P”
%
V ;ENLAROED)
’
STIFFILNER GRID OF SIDES ‘2’ WHERE
A RltWBiC
S=d/SlN(28) ANDd IS STIFFENER SPACING
(5)
(6)
where the relations 2s sin @cos 8 = d, C = 2hE/d, and the engineering strain expressions of e,, = eX, e,, = cY,and eXY= l/2?,, have been used in eqns (4~(6). . , The Eve-Kirchhoff hvnothesis in cylind~cai coordinates (x, #, z) is given by’
Consider a rhombic stiffener grid ABCD of sides s, depth h, width Q, spacing d, and the associated stiffener loads P’ and P” (Fig. 1). The stiffener loads P’ and P” are related to the stiffener strains by the uniaxial Hooke’s law.
VIEW
(4)
‘7 SECTION A-A (ENLARGED)
(b)
STRESS COMPONENTS
Fig. 1. A cylindrical shell with eccentric spiral stiffeners.
(7) (8) (9)
Buckling
of cylindrical
shells with spiral stiffeners
where z is the inward normal coordinate and K~, K~ and Kx4 are changes of curvatures. The stress-strain relations of the cylindrical shell skin are expressed by Hooke’s law:
under uniform
cr,s = j&s
Z(K,
+
VK+)}
and torsion
589
(Is/21
I
M, =
-ct.q/z,
= D{f
{G - VQ -
compression
u,sz dz t
cos4
e&
t
IA
5 sin2
(10)
-
uxfz dz e COS’ eE,+ (U
t
5 Sin*
(1 t
8 CO?
5 coS4
o)K,
o)K.$}
(20)
(IS/Z)
M+ =
U~SZdz +
I -(1121
= D{[
Sin4
86
t
IA
[ Sin’
ufz dz e COS2 e&
-
(1 t
( Sin4
o)K&
-(Vt[Sin28COS2e)K,}
The stress-strain relations of the stiffener grid are from eqns (4H9) flX,= C tr cos2 0{e, cos2 0 t E+sin* 0 Z(K,
cos2
et
sin* O,}
K+
rx+sz dz -
D{- [ Sin2eCOS*
h
-
M,,=-M,,=-
(13)
eyx4 t (1 - u t
I
(21)
A r,+fz dz =
25 Sin28 COS2 o)K&
(22)
The following notations are used in eqns (20)_(22):
ffQr= CL sin* 9{e+ sin2 0 + eXcos’ 0 h
-
Z( K4
Sin2
Qf = C i
8 t
Sin2
Kx
COS2
8 COS’
o)}
&&
-
(14) 2ZK,r}.
(1%
and
STRESSRESULTANTS
The stress resultants of the stiffened shell are obtained by integration of the respective stress components through the shell skin thickness t,. Hence N, =
uxs dz t
IA
uxf dz
(16)
where tf dz = dA = elementary cross-sectional area of the stiffener. Substituting eqns (IO) and (13) into eqn (l6), one has N, = p{(l t n cos4 e)eXt (V t 77sin’ e COST
lo =
IA
z2 dA = moment of inertia of the stiffener.
The strains and change of curvatures are related to the components of displacement u, c, and w by the following expressions provided the effect of strain in the middle surface on curvature is neglected:
e)~+
Kx= WJ,, K+= -j$ (0.m+ W.&, -
Tjt? COS4 eKx -
TJ~? SiII*
e COS’ OK+,}
(23)
(17)
where P =
Et, 1_
y29
A= t,h,
e=
I i\
zdAIA,
where
and ( L =j$(
n=g
I,( Ax+=&(
ketc.
P’ Similarly,
Substituting eqn (23) into eqns (l7)-(22), the following expressions of stress resultants are obtained in terms of the components of the displacements u, v and w:
US/2)
N+ = =
u4.s dz t
I -us/z, p{(l +
q
sin’
IA
e)e+
- T,V Sin4 OK+.
-
aa dz
t (V t ge
Sin2
sin2 0 cost
7
0 COS’
OK,}
0)~
Nx=/?
(itqC0de)u., I
(18) ti
I
vtnsin2fIcos2e
- f (Vt q sin’ 4 cos’ &jJ! t n sin2 8
c0s2
e
yxl >
-
bj
Sit?
e COS’ OK,+
(19)
-z
1
qe sin2 e
cos2
(
e)w
ew,, I
-
1-i
>I
V.4
qe cos4 ew.,
(24)
590
S.-W. YEN
N.+= fl (Yt 7)sin* Bcos* e)u, [
The buckling equations of the spirally stiffened cyhndrical shell are obtained in terms of the displacements u, v and w by the substitution of eqns (24X29) into eqns (30)-(32):
+~[1tnsin48(1-~)}u,+ - $( 1t q sin4 e)w - ne sin* e cos* ew,,
(1 t n c0s4e)u.,+$
Rne sin4 ew,& I iVxb=N,,=p f +!+nsinZecosZe [( --I
t
{
~t~sin2ecos2e
(
- f qe sin* e co8 ew.,, M,=D
(25)
1
)
I-f
>I
u.+
-E
(26)
(27)
1
M, = D f sin* Bcos* Bu., - j$ 1 - (R[ - 5) sin4 r?}u,+ c I
-+(l+[sin’fI)w,]
(28)
r
(
v t 1 sin* Bco? 8 - o N:
P
w., - -Di cos4 ew.,
B
(33)
BR (~+2(tl-~)sin*Bcos*Bju,~,
D +RN: v-xxt-1 R ‘+@? I I P + v-z12Jtx sin48 u.drg I ( PR BR’> +~{l-3(R~-f)sin26(cos2B)Jw.XX~
tD{l+(~-R5)sin4e}w,~=0 PR3
(34)
(Y t 7]sin’ 8 cos2 e)u
- 5 sin’ e COS*eii,+ t 9 {
t E3 cos4eu +x P pxx
t 23 sin’ e ~0s~8~s&S BR
- (I?[ - 6)sin’ e cos* e u,,
I
f (1 - v t 25 sin’ Bcos* 8)w.,
>
-+[lt(n-$sin4e>w,,
-~~sin48w-(~t~sin2ecos28)w,,
MXb= - M& = p
1
t((+J)(Rt$J+(?R-~t$sin2Bcos2B
sin* 0 co2 ew - (1 t 5 cos4 e)w_,
-~(~t~sin28~osze)w.~
e c0s2e u.++ 1
3& -2 sin* e cos2 ew,, = 0
~~~~~e~,-~{~-(R~-~)sin~ecos~e)~,+ L - $
7jsir?
Ft
t~(~t2(~-~)sin2e~os2~-~N~}~,~~
u.x
I
(
I
.
-+
(1 t 3(5 - Rg sin’ e cos* e}u.,
(29
BUCKLINGEQUATIONS
Two loadings are considered: (1) uniform axial and lateral compressions, and (2) torsion. Uniformaxialand lateralcompressions The cylindrical shell is simply supported at the ends and is subject to uniform axial and lateral compressions. Assume in this case that only the resultant forces M: and N& are of great importance. All other resultant forces are considered smalt so that the products of these forces with the derivatives of the displacements u, tr and w can be neglected. The equations of equilibrium of the cylindrical shell under this assumption become [21]
t~{lt(n-$siu4e}u,+ -+{l-(RI--[)~in~B)v,++, tR
1 0 pN:-(l+nsin48) I
t
IN:
(
-y
w
I
sin* e cos* e
>
wTxx - $/$I + 5 ~0~4e)w,,
-+2 t
(
+ (l t 56) sin* Bcos* e}w,,,, ;N:-
$$sin4
e w,++ >
-+(lt[sin”e)w,++++=O
(35)
where o = N&/N:. SoMo~ of the buclking equationsfor uniform axial and lateralcompressions.It can be seen that the buckling eqns (33)_(35)are satisfied by assuming 0 = 0,
w/R= VS., = const.
(36)
Buckling of cylindrical shells with spiral stiffeners under
uniformcompressionand torsion
We assume that U, u and w in the buclking equations represent very small displacements from the compressed cylindrical form of equilibrium. With the origin of the coordinates at one end of the shell and using the notations L and R for the length and radius of the shell, respectively, the general solution of eqns (33H3.5) can be obtained by means of the following displacement functions: u = P,, sin nf$ cos A,x R
(37)
v = Q,,,,,cos rt~$sin y
(38)
A& w = R,,,, sin n# sin R
(39)
+ q-z+$. (
a23 = +{
)
sin2Bcos2e
Am2
1- 3(R[ - 6) sin’ 0 cos2 B}A,2n
++{l-(R[-t)sin’f+“”
where m?rR L ’
A,=-
a3r=
WI (47)
al3
and a32=
m=1,2,3
,...,
n=O,i
These functions assume that during buckling, the generators of the shells subdivide into m half-waves and the circumference into 2n half-waves. For the cylindrical shell simply supported at the ends, the edge conditions are w=OandM,=Oatx=O,L. The solutions obtained in this case can be used for other end conditions because they have very small effect on the magnitude of the buckling load, provided the length of the cylindrical shell is not too small (say L/R > 2). Substitution of eqns (37)_(39) into- eqns (33H35) transforms the differential equations into a set of linear, homogeneous algebraic equations with coefficients P,,,,, Qmn and R,,,,. These algebraic equations are expressed in the following matrix form: a11
i
(48
a23
, 2I 3,***
a,z--wnA,r
a2] a22+ AiI’ a32 a31
+ t sin'ej-$#
a33 =(I
sin2 ecos2 A,*
D( 1 t.$cos4 B)Am4 +/?R2 +
+{2
+ (f + 55) sin2 e cos2 e}Am2n2
(49)
A nontrivial solution of eqn (40) can be obtained only if the determinate of the matrix of coefficients P,,,,, Qmn and R,,,, is equal to zero. This yields a quadratic equation in Am2r. A,,,. (A,2r)2 + B,, (Amar)+ C,,,”= 0
(50,
A,, = ada,% + a2tty3)+ a3ln2
(51)
in which in which r = IV://?. The elements aij of the matrix, which represent the stiffness of the shell, have the following expressions: all=(l
tqcos46)A$+
(
ptq
sin2Bcos26
)
n*
I
(41) a,2=(~+2(~-~)sin26cos20}A.,~
+ (KY
(42)
I
a21 a23 a31 a33
(52)
a11 012 a13 al3 = (v + q sin’ BCOS?&A, - 3
(531
cos4 eA:
a31
a32
a33
where -Z&X sin2 e
c0s2
eh m f12
BR
nl=lt(n2-1+, a21=
aI2
m
a2=$,
.,=o” nl
A,’
592
S.-W. YEN
resultant shearing force NXm becomes extremely important. It can be expressed
The two roots of eqn (50) are 1 nln2r, = -2A,” ( - Em, + x4B”,n - 4A,,C,,))
(54)
and AmZTZ = $-
mn
(- B,, - tie&I
- 4A,,X,,,,)).
(55)
Therefore,
N,,=N,,=‘P+&.
(59)
where V = T/27rR2is the re_sultant shearing force due to the applied torsion T, and NXmis the small change in this force due to buckling. Other resultant forces are considered small so that the products of these forces with the derivatives of the displacements H, v and w can be neglected. The equations of equilibrium for the cylindrical shell subject to torsion are as follows[21~: NY, f $ K&d, -I-VI f li.,* - v., = 0 ( )
and
(60)
f N,, f &m.x f ; h&,.x - $ M,,
NL2= PI-2= &
(- B,, - \/(Bz,,, - 4A,,C’,,)). “8” I?? (57)
A computer program has been developed for eqns (56) and (57) that searches through a finite set of values of N:,(m, n. 8) and ~V,&m,n, 8) for m = 1,2,3,. . . r: n = 0,1,2,... s: and O> 6, > n/2 and prints out both minimum N:,(m, n),;,, and minimum Nk(m, n)minfor each 0. The critical axial stress resultant N:,., (or the critica lateral stress resultant N & which is defined by wN:,,) at the required spiral angle 0 is the value of either N\r or NL2 (wN:, or tiNi*) in eqn (56) or (57), whichever is smaller. In the case of a cylindrical shell with open ends under the action of an external uniform axial compression (Fig. 3), w in eqns (51)-(53) is equal to zero, and the criticat axial compression per unit circumferential length is NL,. On the other hand, when a cylindrical shell with closed ends is expanded by an internal pressure of intensity q, w in eqns (51)-(53) is equal to two. In this case, the intensity of the critical pressure is given by q,, = N&.,/R = ZN:,,IR.
(58)
Torsion at the ends The cylindrical shells are usually designed for torsion applied at the ends. Under the action of this loading, the
t ; vl(“,, - w.,) = 0
(61)
ML - + k&,x+ -t j$ MA& + f Nm + f Y’(o, f w.,,) = 0.
62)
By the substitution of eqns (24)-(29) into eqns (60)_(62), the buckling equations expressed in terms of the displacements u, v and w have the following forms: 1 1-u + 7j sit? 8 cos2 6 u.& (1 t 7jcos4B)u.x,+ i?” 2 1 ( +LQu BR
J*
+.!. Ifu R{ 2 t2
q-z
(
)
sin’ Bcos’ B t’s+ 1
- f VI., - $ (V+ n sin2 8 ~0s’ 6) we, - F cos4 Bw,,,, 3Dl sin . z 0 cos* Bw,~~ = 0 - jgj
(63)
[~+2(~-~)sidBc*s’B]rr,~+~u,~~ Nx
+ ~~~)(Rt~) 2& Dt sin2Bcos2B v.~~ i-R v-p~ip~z ( ) 1 +~{I+~+(~-2~+~)sin4~)~,~-~tW.,
+g{l-3(R[-~)sin2B(cos2B}w,,
-i[l+(n-$)sin’f?}w.+
Fig. 3. Cylinder under uniform axial compression.
+~{l-(Rg-l)sin’Blw,~=O
593
Buckling of cylindrical shells with spiral stiffeners under uniform compression and torsion
(V
t
q
sin’ 8 cos’ e)u,, + 7
magnitude of the critical stress. The substitution of eqns (66~(68) into eqns (63)-&S) reduces the differential equations to a set of linear, homogeneous algebraic equations in coefficients I,,,“, A,,, and K,,. These equations are expressed in matrix form:
COs”~u,xxx
+ 305 sin’ e cos2 tYu.,++ -t 2, tr
P
m
*x
-~(l-3(R5_5)sin’6coszs)v.,,,
(69) The elements bij in Matrix (69) have the following expressions: -f(l
t n sin4 t+v
-~sin2t?coszflw P
.xX+:Yw .xrb P - ?(I
- ;{2
+ ([ + 55)
+ 6 cos4 f?)w,
sin’ Bcos2 +v,,ti
-Zj$$sin’ew,~
- +(
1 t 6 sin4 f?)w,++++= 0.
(65)
Solutionsof the bucklingequationsfor torsion.Unlike the previous case of uniform axial compression in which the generators of the cylindrical shell remain straight during buckling and form a system of straight nodal lines for a buckled surface, nodal lines of spiral form along the buckled surface are expected in the present case of torsion. To fulfill this condition, the following expressions for the displacements u, u and w are taken[21]:
u=z~.cos(+p)
b,l= -a11
(70)
b,z = UIZ
(71)
6,s = -a13
(72)
bzt = biz
(73)
bz2= - a22
(74)
bz3= 023
(75)
bf, = b,,
(76)
bx = bz3
(77)
b33= -
(78)
a33
where ali, a12, o13, az3 and up3 are given by eqns (41), (42f,(43), (45),(46)and (49),res~c~iv~ly. The condition for the existence of a nontrivial solution of eqn (69) is that the determinate of the matrix of coefficients I_ I,,, and K,,,, vanishes. This yields a cubic equation in sl@A,,,:
(66) where
&&os(+t))
(67)
~=K~*sin(~-~~)
(68)
RI = 4n(nZ- 1)
(80)
& = 2(2(n*-- I)bt, -n(A,bz, -nbn- &h-
bu)
nb3,- n*b&)
(81) nb,3+ b,z nbss+ bx I
where, as before, bz3 bs3
A, = mrRiL.
We assume, as in the previous case, that the cylindrical shell under consideration is long enough so that the constraints at the ends have negligible effect on the
H4=
b,, ba b,3 bz, bzz bz3 . I bs, bst b33
I
(82)
183)
594
S.-W. YEN
The three roots of eqn (79) are $A,=2J(-;)cos($)
(84
(85)
(861
where
x=;(3q-p’),
y=;(2p3-9pq+27r),
(87)
(88)
Therefore,
mined from the previous analysis can be obtained by iteration of the design parameters, and a computer program for this purpose has been developed. The primary design parameters of a stiffened shell are: shellmaterial, length, radius, and skin thickness; stiffenersmaterial, width, depth, and spacing. Among them, the material cannot be iterated once it has been selected. The shell length and radius are determined mainly by other design requirements and also cannot be iterated. There remain only the shell skin thickness, the stiffener width, depth, and spacing as candidates for iteration which is conducted as foliows: l Take one candidate at a time as the variable for iteration and keep the other candidates constant at their initial values. 0 Assign a proper increment AX for this variable. 0 Use a permissible minimum initial value of this variable to start the iteration. 0 The iteration continues until the buckling load appraoches the predetermined applied load as a limit and the difference between the two loads is within the design allowance. 0 Calculate and print out the shell structure weight based on the final value of iteration. Label this weight Wf,. l Use the other candidates in turn and repeat the process. Label the calculated structure weights ET,, M;, etc. l Compare Wt,, Wfz, Wt3, etc. The least value of these weights is the optimum structure weight of the designed shell.
(89) CONCLUSION
A computer program has also been developed for eqns (89)-(91), searching through a finite set of values of \U,(m, n, 131, Wm, n, 19) and Y&n, II, 0) for m = ,... s. 0 < 0 < 7r/2 and printing out 1,2,3,... r,nt=2,3,4 Yrtm, fl)min+, q,?(4 n)min+r and Y3(m, n)min+ for each 0. The buckling shearing stress resultant Y,, at the optimum spiral angle 19,~is the value of qyI, or Qz, or YJ mentioned above, whichever is the smallest. Under the action of the external torsion T appleid at the ends, the buckling shearing stress resultant ‘PC, is equal to 7727rR’. Therefore, the buckling load in this case is T,, = 27rR”@k,, where
R
(92)
is the radius of the cylindrical shell.
SHELL STRUCTURE WRIGHT OPTIMIZATION
The optimum structural weight of the spirally stiffened cylindrical shell at the required spiral angle &, deter-
tn cannot be 0 or 1; otherwise HI becomes zero and p, q and I becomes infinity.
The theory advanced here deals with the general instability of stiffened shells of isotropic materials simply supported at the ends. Other end conditions can be applied as well, provided the Iength-to-radius ratio of the shell is greater than 2. The theory can be readily extended to stiffened shells of composite materials which have pseudo-isotropic layup skin and unidirectiona1 fiberreinforced stiffeners, when the flexural rigidity of the shell (expressed by D) and the sectional rigidity of the stiffeners (expressed by Ctr) are modified by the appropriate material properties.
REFERENCES
1. J. Singer, M. Baruch and 0. Harira, On a stability of eccen-
trically stiffened cylindrical shells under axial compression. hf. J. Sol& Structures 3.445470 (1967). 2. Michael F. Card, P~limiaa~ Resaits of Compression Test on Cyijnders with Eccentric ~on~itudiaa~ Sti~eaers, NASA TM x-1004 (1964). S. B. Batdorf et al., Critical Stress of Thin-bawled Cylinders in Axiaf Compression, NACA Report 887 (1947). J. M. Hedgepeth and D. 8. Hall, Stability of stiffened cylinders. AIAA J. 3(12)(1%5). Michael F. Card and R. M. Jones, Experimental and Theoretical Cylinders.
Results for
Buckling
of Eccentrically
Stiffened
NASA TN D-3639(1966). 6. A. Van Der Neut, General Instability of Orthogonally Stiffened Cylindrical Shells. NASA TN D-1510(1%2). 7. D. L. Block et al., Buckling of Eccentrically Stiflened Orthotrovic Cylinders. NASA TN D-2%0 (1%5). 8. Hshe-Shkn Tsien, A theory for the dock&g of thin shells. J. Aero. Sci. 9.373-384 (1942).
595
Buckling of cylindrical shells with spiral stiffeners under uniform compression and torsion 9. R. R. Meyer, Buckling of 45” eccentric-stiffened waffle cylinders, J. Roy. Aero. Sot. 71, 516-520(1%7). IO. T. H. Bon Karman and H. S. Tsien, The buckling of thin cylindrical shells under axial compression. J. Aero. Sci. E(8). 303-312(1941). II. T. C. Soong, Buckling of cylindrical shells with eccentric spiral-type stiffeners. AIAA J. 7(l), 65-72 (1%9). 12. M. Baruch et al., Effect of Eccentricity of Stiffeners on the General Instability of Stiffened Cylindtical Shells under Torsion. AF EOAR 63-58. SR-3 (Aug. 1%5). 13. J. Singer et al., Inversion of the eccentricity effect in
stiffened cylindrical shells buckling under external pressure. .I Mech. Engng Sci. 8(4), 363-373 (1%6). 14. T. H. Von Karman, The Strength of Thin P/ales in Compression. Trans. ASME, 53-57 (1932). 15. M. Baruch and J. Singer, Effect of eccentricity of stiffeners
on the general instability of stiffened cylindrical shells under hydrostatic pressure. J. Mech. Engng Sci. 5(l) (1953). 16. M. Stein and J. Mayers, A small-deflection theory for curved sandwich plates. NACA Rep. IO08(1951). 17. C. Libore and S. B. Batdorf, A general small-deflection theory for flat sandwich plates. NACA Rep. 899 (1948). 18. J. P. Peterson and J. K. Anderson, Bending tests for largediameter ring-stiffened corrugated cylinders. NASA TN D3336(1%6). 19. D. L. Block, Buckling of eccentrically stiffened orthotropic cvlinders under oure bending. NASA TN D-3351 (1966). 20. L: H. Donnell, Stability of Thin-walled tubes under torsion, NACA Rep. No. 479 (1934). 21. S. Timoshenko, Theory of
E/astic
Stability,
McGraw-Hill, New York. London (1936).
1st Edn.
Printed in Great Britain
BUCKLING OF CYLINDRICAL SHELLS WITH SPIRAL STIFFENERS UNDER UNIFORM COMPRESSION AND TORSIONS SHAO-WENYENS Douglas Aircraft Company. McDonnell Douglas Corporation, Long Beach, CA 90846, U.S.A. (Received 26 February 1979;received for publication 12 September 1979) Abstra&--This paper investigated the general instability of cylindrical shells in which the stiffeners formed spirals along the length and at an arbitrary angle with the axis. Two loading conditions were considered: unifo~ axial and lateral compressions and torsion. The stress-strain relations of the stiffeners were developed by rotation of the strain tensor. The buckling determinate was obtained by introducing into the equilibrium equations the admissible displacement functions consistent with the end constraints, thereby enforcing equilibrium by satisfying the characteristic equations. The buclking equations were programmed for a computer which rearched through a finite set of stress resultants for assigned values of spiral angle and modes and printed out the buckling load. The optimum structure weight of the stiffened shell was determined by iterating the design parameters at the required spiral angle so that the buckling load approached the applied load as a limit until the difference between these loads was within the design allowance.
NO~NCLA~
p detined by I$/(1 - v2) I defined by N@ rXy engineering shear strain eX,~iy engineering direct strains { defined by CAelDh 7) defined by f&/b B spiral angle of the stiffeners K,, Q, ~~~ change of curvatures ri, de&red by m&L Y Poisson’s ratio 4 defined by C&/Dir o;, u+ direct stresses T~+ shearing stress W defined by I127rR2
stiffener cross-sectional area defined by eqns (5lJ43) elements of matrices defined by 2hEld flexural rigidity of the shell stiffener spacing Young’s modulus defined by z dAIA I direct straintomponent of the shell direct strain components of the stiffeners with +@inclination direct strain components of the stiffeners with -0 inclination defined by eqns (80)-(83) stiffener depth coefficients of displacement functions moment of inertia of stiffeners about the middle surface of the shell length of the shell moments of forces number of half-waves in the axial direction stress resultants number of waves in the circumferential direction buckling load stiffener load in the W-axis stiffener Ioad in the 19”-axis coetIicients of displacement functions radius of the shelf length of the stiffener grid applied torsion buckling load stiffener width skin thickness of the shell components of displacement rectangular coordinates cylindrical coordinates tThis paper is part of the work performed at the Douglas Aircraft Company of the McDonnell Douglas Corporation under the sponsorship of its Independent Research and Development (IRAD) Program. SSenior Engineer/Scientist.
INTRODUCTION The buckling of stiffened cylindrical shells under specified geomet~cal characteristics has been investigated both theoretic~~y and experimentally by many authors[ l-201. Most of them placed the stiffeners on the interior of the shell for aerodynamic advantages, although the exterior-stiffened theoretical values for typical shell designs are about 30% higher than the interior theoretical values. Others, for geometrical simplicity, preferred the conventional ring-and-stringer shell design. It is doubtful whether this simple design can provide minimum shell structural weight. The necessity for weight-saving measures in the design of aeronautics and space structures makes it advisable that other feasible shell design configurations be examined for further improvement. This paper deals with the general instability of eccentrically stiffened cylindrical shells in which the spiral stiffeners with constant spacings are arranged symmetrically at an arbitrary angle with the axis of the shell. The buckling determinate is obtained by substitut~g into the ~~erential equatjons of equ~ib~um the admissible displacement functions consistent with the end conditions and enforcing equilibrium by satisfying the characteristic equation. The buckling equations are programmed for a computer which searches through a finite set of the stress
587
588
S.-W. YEN
resultants for assigned values of stiffener angles and modes and prints out the minimum stress resultant and buckling load. The optimum structural weight of the stiffened shell is obtained by iteration of design parameters at the required spiral angle in such a way that the buckling load approaches the applied load as a limit until the difference between these loads is within the design ahowance. ANALYSIS
In developing the analysis, the following assumptions are made: 1, ~formations are small and elastic. 2. Stiffeners have constant spacing, equal spiral angle, and the same elastic material properties. 3. In-plane shear stresses transverse to the stiffeners are resisted by the shell skin. 4. The Love-Kirchhof? hypothesis of normals remaining normal holds true. The cylindrical shell under consideration has a radius R, length Z_.and skin thickness, t,. The stiffeners have width tf, height h, spacing d, and spiral angles (? and -8 arranged symmetrically with the axis of the shell. The middle surface of the cylindrical shell skin is taken as the reference surface for the analysis (Fig. 1). ST-RAIN
Fig.
2. Rectangularcoordinatesrotation.
The distributed edge stresses of the stiffener grid can be written in terms of the discrete stiffener loads:
= Ct cos’ 6(e, co? @+ ev sin2 0).
RELATIONS
The stress-strain relations of the stiffeners are developed by rotation of the strain tensor. The transformation of strains from the xy-axis to the l’Z’-axis(Fig. 2) yields
Sim~arly,
x ( eYsin* Bt eXco? 8)
e;, = e,, cos2 8 + 2e,, sin @cos 8 i e,, sin2 8 e;, = e,, sin’ 8 - 2e,, sin 6 cos Bt e,,cos*8.1 (1) Similarly, the transformation of strains from the xy-axis to the I”2’‘-axisyields e:‘, = e,, cos* B - 2e,, sin @cos B+ e,, sin* B 1 e’;, = e,, sin’ 6 -t 2e,, sin 6 cos @t e,, cos’ B. i (2)
7Xr=!$$ei,
-eY,)= Cssin* f?cos2@yX,
e,(Z) = G -
ZKx
&)=Q-ZK+ ?‘x&) = %z+- 2ZK,&
(a) STIFFENED CYLIN~~AL SHELL AND ASSOCIATE0 STIFFENER LOADS P’, P”
%
V ;ENLAROED)
’
STIFFILNER GRID OF SIDES ‘2’ WHERE
A RltWBiC
S=d/SlN(28) ANDd IS STIFFENER SPACING
(5)
(6)
where the relations 2s sin @cos 8 = d, C = 2hE/d, and the engineering strain expressions of e,, = eX, e,, = cY,and eXY= l/2?,, have been used in eqns (4~(6). . , The Eve-Kirchhoff hvnothesis in cylind~cai coordinates (x, #, z) is given by’
Consider a rhombic stiffener grid ABCD of sides s, depth h, width Q, spacing d, and the associated stiffener loads P’ and P” (Fig. 1). The stiffener loads P’ and P” are related to the stiffener strains by the uniaxial Hooke’s law.
VIEW
(4)
‘7 SECTION A-A (ENLARGED)
(b)
STRESS COMPONENTS
Fig. 1. A cylindrical shell with eccentric spiral stiffeners.
(7) (8) (9)
Buckling
of cylindrical
shells with spiral stiffeners
where z is the inward normal coordinate and K~, K~ and Kx4 are changes of curvatures. The stress-strain relations of the cylindrical shell skin are expressed by Hooke’s law:
under uniform
cr,s = j&s
Z(K,
+
VK+)}
and torsion
589
(Is/21
I
M, =
-ct.q/z,
= D{f
{G - VQ -
compression
u,sz dz t
cos4
e&
t
IA
5 sin2
(10)
-
uxfz dz e COS’ eE,+ (U
t
5 Sin*
(1 t
8 CO?
5 coS4
o)K,
o)K.$}
(20)
(IS/Z)
M+ =
U~SZdz +
I -(1121
= D{[
Sin4
86
t
IA
[ Sin’
ufz dz e COS2 e&
-
(1 t
( Sin4
o)K&
-(Vt[Sin28COS2e)K,}
The stress-strain relations of the stiffener grid are from eqns (4H9) flX,= C tr cos2 0{e, cos2 0 t E+sin* 0 Z(K,
cos2
et
sin* O,}
K+
rx+sz dz -
D{- [ Sin2eCOS*
h
-
M,,=-M,,=-
(13)
eyx4 t (1 - u t
I
(21)
A r,+fz dz =
25 Sin28 COS2 o)K&
(22)
The following notations are used in eqns (20)_(22):
ffQr= CL sin* 9{e+ sin2 0 + eXcos’ 0 h
-
Z( K4
Sin2
Qf = C i
8 t
Sin2
Kx
COS2
8 COS’
o)}
&&
-
(14) 2ZK,r}.
(1%
and
STRESSRESULTANTS
The stress resultants of the stiffened shell are obtained by integration of the respective stress components through the shell skin thickness t,. Hence N, =
uxs dz t
IA
uxf dz
(16)
where tf dz = dA = elementary cross-sectional area of the stiffener. Substituting eqns (IO) and (13) into eqn (l6), one has N, = p{(l t n cos4 e)eXt (V t 77sin’ e COST
lo =
IA
z2 dA = moment of inertia of the stiffener.
The strains and change of curvatures are related to the components of displacement u, c, and w by the following expressions provided the effect of strain in the middle surface on curvature is neglected:
e)~+
Kx= WJ,, K+= -j$ (0.m+ W.&, -
Tjt? COS4 eKx -
TJ~? SiII*
e COS’ OK+,}
(23)
(17)
where P =
Et, 1_
y29
A= t,h,
e=
I i\
zdAIA,
where
and ( L =j$(
n=g
I,( Ax+=&(
ketc.
P’ Similarly,
Substituting eqn (23) into eqns (l7)-(22), the following expressions of stress resultants are obtained in terms of the components of the displacements u, v and w:
US/2)
N+ = =
u4.s dz t
I -us/z, p{(l +
q
sin’
IA
e)e+
- T,V Sin4 OK+.
-
aa dz
t (V t ge
Sin2
sin2 0 cost
7
0 COS’
OK,}
0)~
Nx=/?
(itqC0de)u., I
(18) ti
I
vtnsin2fIcos2e
- f (Vt q sin’ 4 cos’ &jJ! t n sin2 8
c0s2
e
yxl >
-
bj
Sit?
e COS’ OK,+
(19)
-z
1
qe sin2 e
cos2
(
e)w
ew,, I
-
1-i
>I
V.4
qe cos4 ew.,
(24)
590
S.-W. YEN
N.+= fl (Yt 7)sin* Bcos* e)u, [
The buckling equations of the spirally stiffened cyhndrical shell are obtained in terms of the displacements u, v and w by the substitution of eqns (24X29) into eqns (30)-(32):
+~[1tnsin48(1-~)}u,+ - $( 1t q sin4 e)w - ne sin* e cos* ew,,
(1 t n c0s4e)u.,+$
Rne sin4 ew,& I iVxb=N,,=p f +!+nsinZecosZe [( --I
t
{
~t~sin2ecos2e
(
- f qe sin* e co8 ew.,, M,=D
(25)
1
)
I-f
>I
u.+
-E
(26)
(27)
1
M, = D f sin* Bcos* Bu., - j$ 1 - (R[ - 5) sin4 r?}u,+ c I
-+(l+[sin’fI)w,]
(28)
r
(
v t 1 sin* Bco? 8 - o N:
P
w., - -Di cos4 ew.,
B
(33)
BR (~+2(tl-~)sin*Bcos*Bju,~,
D +RN: v-xxt-1 R ‘+@? I I P + v-z12Jtx sin48 u.drg I ( PR BR’> +~{l-3(R~-f)sin26(cos2B)Jw.XX~
tD{l+(~-R5)sin4e}w,~=0 PR3
(34)
(Y t 7]sin’ 8 cos2 e)u
- 5 sin’ e COS*eii,+ t 9 {
t E3 cos4eu +x P pxx
t 23 sin’ e ~0s~8~s&S BR
- (I?[ - 6)sin’ e cos* e u,,
I
f (1 - v t 25 sin’ Bcos* 8)w.,
>
-+[lt(n-$sin4e>w,,
-~~sin48w-(~t~sin2ecos28)w,,
MXb= - M& = p
1
t((+J)(Rt$J+(?R-~t$sin2Bcos2B
sin* 0 co2 ew - (1 t 5 cos4 e)w_,
-~(~t~sin28~osze)w.~
e c0s2e u.++ 1
3& -2 sin* e cos2 ew,, = 0
~~~~~e~,-~{~-(R~-~)sin~ecos~e)~,+ L - $
7jsir?
Ft
t~(~t2(~-~)sin2e~os2~-~N~}~,~~
u.x
I
(
I
.
-+
(1 t 3(5 - Rg sin’ e cos* e}u.,
(29
BUCKLINGEQUATIONS
Two loadings are considered: (1) uniform axial and lateral compressions, and (2) torsion. Uniformaxialand lateralcompressions The cylindrical shell is simply supported at the ends and is subject to uniform axial and lateral compressions. Assume in this case that only the resultant forces M: and N& are of great importance. All other resultant forces are considered smalt so that the products of these forces with the derivatives of the displacements u, tr and w can be neglected. The equations of equilibrium of the cylindrical shell under this assumption become [21]
t~{lt(n-$siu4e}u,+ -+{l-(RI--[)~in~B)v,++, tR
1 0 pN:-(l+nsin48) I
t
IN:
(
-y
w
I
sin* e cos* e
>
wTxx - $/$I + 5 ~0~4e)w,,
-+2 t
(
+ (l t 56) sin* Bcos* e}w,,,, ;N:-
$$sin4
e w,++ >
-+(lt[sin”e)w,++++=O
(35)
where o = N&/N:. SoMo~ of the buclking equationsfor uniform axial and lateralcompressions.It can be seen that the buckling eqns (33)_(35)are satisfied by assuming 0 = 0,
w/R= VS., = const.
(36)
Buckling of cylindrical shells with spiral stiffeners under
uniformcompressionand torsion
We assume that U, u and w in the buclking equations represent very small displacements from the compressed cylindrical form of equilibrium. With the origin of the coordinates at one end of the shell and using the notations L and R for the length and radius of the shell, respectively, the general solution of eqns (33H3.5) can be obtained by means of the following displacement functions: u = P,, sin nf$ cos A,x R
(37)
v = Q,,,,,cos rt~$sin y
(38)
A& w = R,,,, sin n# sin R
(39)
+ q-z+$. (
a23 = +{
)
sin2Bcos2e
Am2
1- 3(R[ - 6) sin’ 0 cos2 B}A,2n
++{l-(R[-t)sin’f+“”
where m?rR L ’
A,=-
a3r=
WI (47)
al3
and a32=
m=1,2,3
,...,
n=O,i
These functions assume that during buckling, the generators of the shells subdivide into m half-waves and the circumference into 2n half-waves. For the cylindrical shell simply supported at the ends, the edge conditions are w=OandM,=Oatx=O,L. The solutions obtained in this case can be used for other end conditions because they have very small effect on the magnitude of the buckling load, provided the length of the cylindrical shell is not too small (say L/R > 2). Substitution of eqns (37)_(39) into- eqns (33H35) transforms the differential equations into a set of linear, homogeneous algebraic equations with coefficients P,,,,, Qmn and R,,,,. These algebraic equations are expressed in the following matrix form: a11
i
(48
a23
, 2I 3,***
a,z--wnA,r
a2] a22+ AiI’ a32 a31
+ t sin'ej-$#
a33 =(I
sin2 ecos2 A,*
D( 1 t.$cos4 B)Am4 +/?R2 +
+{2
+ (f + 55) sin2 e cos2 e}Am2n2
(49)
A nontrivial solution of eqn (40) can be obtained only if the determinate of the matrix of coefficients P,,,,, Qmn and R,,,, is equal to zero. This yields a quadratic equation in Am2r. A,,,. (A,2r)2 + B,, (Amar)+ C,,,”= 0
(50,
A,, = ada,% + a2tty3)+ a3ln2
(51)
in which in which r = IV://?. The elements aij of the matrix, which represent the stiffness of the shell, have the following expressions: all=(l
tqcos46)A$+
(
ptq
sin2Bcos26
)
n*
I
(41) a,2=(~+2(~-~)sin26cos20}A.,~
+ (KY
(42)
I
a21 a23 a31 a33
(52)
a11 012 a13 al3 = (v + q sin’ BCOS?&A, - 3
(531
cos4 eA:
a31
a32
a33
where -Z&X sin2 e
c0s2
eh m f12
BR
nl=lt(n2-1+, a21=
aI2
m
a2=$,
.,=o” nl
A,’
592
S.-W. YEN
resultant shearing force NXm becomes extremely important. It can be expressed
The two roots of eqn (50) are 1 nln2r, = -2A,” ( - Em, + x4B”,n - 4A,,C,,))
(54)
and AmZTZ = $-
mn
(- B,, - tie&I
- 4A,,X,,,,)).
(55)
Therefore,
N,,=N,,=‘P+&.
(59)
where V = T/27rR2is the re_sultant shearing force due to the applied torsion T, and NXmis the small change in this force due to buckling. Other resultant forces are considered small so that the products of these forces with the derivatives of the displacements H, v and w can be neglected. The equations of equilibrium for the cylindrical shell subject to torsion are as follows[21~: NY, f $ K&d, -I-VI f li.,* - v., = 0 ( )
and
(60)
f N,, f &m.x f ; h&,.x - $ M,,
NL2= PI-2= &
(- B,, - \/(Bz,,, - 4A,,C’,,)). “8” I?? (57)
A computer program has been developed for eqns (56) and (57) that searches through a finite set of values of N:,(m, n. 8) and ~V,&m,n, 8) for m = 1,2,3,. . . r: n = 0,1,2,... s: and O> 6, > n/2 and prints out both minimum N:,(m, n),;,, and minimum Nk(m, n)minfor each 0. The critical axial stress resultant N:,., (or the critica lateral stress resultant N & which is defined by wN:,,) at the required spiral angle 0 is the value of either N\r or NL2 (wN:, or tiNi*) in eqn (56) or (57), whichever is smaller. In the case of a cylindrical shell with open ends under the action of an external uniform axial compression (Fig. 3), w in eqns (51)-(53) is equal to zero, and the criticat axial compression per unit circumferential length is NL,. On the other hand, when a cylindrical shell with closed ends is expanded by an internal pressure of intensity q, w in eqns (51)-(53) is equal to two. In this case, the intensity of the critical pressure is given by q,, = N&.,/R = ZN:,,IR.
(58)
Torsion at the ends The cylindrical shells are usually designed for torsion applied at the ends. Under the action of this loading, the
t ; vl(“,, - w.,) = 0
(61)
ML - + k&,x+ -t j$ MA& + f Nm + f Y’(o, f w.,,) = 0.
62)
By the substitution of eqns (24)-(29) into eqns (60)_(62), the buckling equations expressed in terms of the displacements u, v and w have the following forms: 1 1-u + 7j sit? 8 cos2 6 u.& (1 t 7jcos4B)u.x,+ i?” 2 1 ( +LQu BR
J*
+.!. Ifu R{ 2 t2
q-z
(
)
sin’ Bcos’ B t’s+ 1
- f VI., - $ (V+ n sin2 8 ~0s’ 6) we, - F cos4 Bw,,,, 3Dl sin . z 0 cos* Bw,~~ = 0 - jgj
(63)
[~+2(~-~)sidBc*s’B]rr,~+~u,~~ Nx
+ ~~~)(Rt~) 2& Dt sin2Bcos2B v.~~ i-R v-p~ip~z ( ) 1 +~{I+~+(~-2~+~)sin4~)~,~-~tW.,
+g{l-3(R[-~)sin2B(cos2B}w,,
-i[l+(n-$)sin’f?}w.+
Fig. 3. Cylinder under uniform axial compression.
+~{l-(Rg-l)sin’Blw,~=O
593
Buckling of cylindrical shells with spiral stiffeners under uniform compression and torsion
(V
t
q
sin’ 8 cos’ e)u,, + 7
magnitude of the critical stress. The substitution of eqns (66~(68) into eqns (63)-&S) reduces the differential equations to a set of linear, homogeneous algebraic equations in coefficients I,,,“, A,,, and K,,. These equations are expressed in matrix form:
COs”~u,xxx
+ 305 sin’ e cos2 tYu.,++ -t 2, tr
P
m
*x
-~(l-3(R5_5)sin’6coszs)v.,,,
(69) The elements bij in Matrix (69) have the following expressions: -f(l
t n sin4 t+v
-~sin2t?coszflw P
.xX+:Yw .xrb P - ?(I
- ;{2
+ ([ + 55)
+ 6 cos4 f?)w,
sin’ Bcos2 +v,,ti
-Zj$$sin’ew,~
- +(
1 t 6 sin4 f?)w,++++= 0.
(65)
Solutionsof the bucklingequationsfor torsion.Unlike the previous case of uniform axial compression in which the generators of the cylindrical shell remain straight during buckling and form a system of straight nodal lines for a buckled surface, nodal lines of spiral form along the buckled surface are expected in the present case of torsion. To fulfill this condition, the following expressions for the displacements u, u and w are taken[21]:
u=z~.cos(+p)
b,l= -a11
(70)
b,z = UIZ
(71)
6,s = -a13
(72)
bzt = biz
(73)
bz2= - a22
(74)
bz3= 023
(75)
bf, = b,,
(76)
bx = bz3
(77)
b33= -
(78)
a33
where ali, a12, o13, az3 and up3 are given by eqns (41), (42f,(43), (45),(46)and (49),res~c~iv~ly. The condition for the existence of a nontrivial solution of eqn (69) is that the determinate of the matrix of coefficients I_ I,,, and K,,,, vanishes. This yields a cubic equation in sl@A,,,:
(66) where
&&os(+t))
(67)
~=K~*sin(~-~~)
(68)
RI = 4n(nZ- 1)
(80)
& = 2(2(n*-- I)bt, -n(A,bz, -nbn- &h-
bu)
nb3,- n*b&)
(81) nb,3+ b,z nbss+ bx I
where, as before, bz3 bs3
A, = mrRiL.
We assume, as in the previous case, that the cylindrical shell under consideration is long enough so that the constraints at the ends have negligible effect on the
H4=
b,, ba b,3 bz, bzz bz3 . I bs, bst b33
I
(82)
183)
594
S.-W. YEN
The three roots of eqn (79) are $A,=2J(-;)cos($)
(84
(85)
(861
where
x=;(3q-p’),
y=;(2p3-9pq+27r),
(87)
(88)
Therefore,
mined from the previous analysis can be obtained by iteration of the design parameters, and a computer program for this purpose has been developed. The primary design parameters of a stiffened shell are: shellmaterial, length, radius, and skin thickness; stiffenersmaterial, width, depth, and spacing. Among them, the material cannot be iterated once it has been selected. The shell length and radius are determined mainly by other design requirements and also cannot be iterated. There remain only the shell skin thickness, the stiffener width, depth, and spacing as candidates for iteration which is conducted as foliows: l Take one candidate at a time as the variable for iteration and keep the other candidates constant at their initial values. 0 Assign a proper increment AX for this variable. 0 Use a permissible minimum initial value of this variable to start the iteration. 0 The iteration continues until the buckling load appraoches the predetermined applied load as a limit and the difference between the two loads is within the design allowance. 0 Calculate and print out the shell structure weight based on the final value of iteration. Label this weight Wf,. l Use the other candidates in turn and repeat the process. Label the calculated structure weights ET,, M;, etc. l Compare Wt,, Wfz, Wt3, etc. The least value of these weights is the optimum structure weight of the designed shell.
(89) CONCLUSION
A computer program has also been developed for eqns (89)-(91), searching through a finite set of values of \U,(m, n, 131, Wm, n, 19) and Y&n, II, 0) for m = ,... s. 0 < 0 < 7r/2 and printing out 1,2,3,... r,nt=2,3,4 Yrtm, fl)min+, q,?(4 n)min+r and Y3(m, n)min+ for each 0. The buckling shearing stress resultant Y,, at the optimum spiral angle 19,~is the value of qyI, or Qz, or YJ mentioned above, whichever is the smallest. Under the action of the external torsion T appleid at the ends, the buckling shearing stress resultant ‘PC, is equal to 7727rR’. Therefore, the buckling load in this case is T,, = 27rR”@k,, where
R
(92)
is the radius of the cylindrical shell.
SHELL STRUCTURE WRIGHT OPTIMIZATION
The optimum structural weight of the spirally stiffened cylindrical shell at the required spiral angle &, deter-
tn cannot be 0 or 1; otherwise HI becomes zero and p, q and I becomes infinity.
The theory advanced here deals with the general instability of stiffened shells of isotropic materials simply supported at the ends. Other end conditions can be applied as well, provided the Iength-to-radius ratio of the shell is greater than 2. The theory can be readily extended to stiffened shells of composite materials which have pseudo-isotropic layup skin and unidirectiona1 fiberreinforced stiffeners, when the flexural rigidity of the shell (expressed by D) and the sectional rigidity of the stiffeners (expressed by Ctr) are modified by the appropriate material properties.
REFERENCES
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Buckling of cylindrical shells with spiral stiffeners under uniform compression and torsion 9. R. R. Meyer, Buckling of 45” eccentric-stiffened waffle cylinders, J. Roy. Aero. Sot. 71, 516-520(1%7). IO. T. H. Bon Karman and H. S. Tsien, The buckling of thin cylindrical shells under axial compression. J. Aero. Sci. E(8). 303-312(1941). II. T. C. Soong, Buckling of cylindrical shells with eccentric spiral-type stiffeners. AIAA J. 7(l), 65-72 (1%9). 12. M. Baruch et al., Effect of Eccentricity of Stiffeners on the General Instability of Stiffened Cylindtical Shells under Torsion. AF EOAR 63-58. SR-3 (Aug. 1%5). 13. J. Singer et al., Inversion of the eccentricity effect in
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on the general instability of stiffened cylindrical shells under hydrostatic pressure. J. Mech. Engng Sci. 5(l) (1953). 16. M. Stein and J. Mayers, A small-deflection theory for curved sandwich plates. NACA Rep. IO08(1951). 17. C. Libore and S. B. Batdorf, A general small-deflection theory for flat sandwich plates. NACA Rep. 899 (1948). 18. J. P. Peterson and J. K. Anderson, Bending tests for largediameter ring-stiffened corrugated cylinders. NASA TN D3336(1%6). 19. D. L. Block, Buckling of eccentrically stiffened orthotropic cvlinders under oure bending. NASA TN D-3351 (1966). 20. L: H. Donnell, Stability of Thin-walled tubes under torsion, NACA Rep. No. 479 (1934). 21. S. Timoshenko, Theory of
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1st Edn.