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Buckling analysis of discretely stringer-stiffened cylindrical shells
ARTICLE IN PRESS
International Journal of Mechanical Sciences 48 (2006) 1505–1515 www.elsevier.com/locate/ijmecsci
Buckling analysis of discretely stringer-stiffened cylindrical shells I.V. Andrianova, V.M. Verbonolb, J. Awrejcewiczc, a
Institut fu¨r Allgemeine Mechanik, RWTH Aachen, Templergraben 64, D-52056, Aachen, Germany Prydniprovskaya State Academy of Civil Engineering and Architecture, 24a Chernyshevskogo Street, Dnepropetrovsk UA-49005, Ukraine c Department of Automatics and Biomechanics, Technical University of Ło´dz´, 1/15 Stefanowskiego Street, Ło´dz´ PL-90-924, Poland
b
Received 27 November 2004; received in revised form 15 May 2006; accepted 28 June 2006 Available online 8 September 2006
Abstract Engineering approach for computation of stringer stiffened cylindrical shells is realized mainly using the structurally orthotropic theory with momentless pre-buckling state. On the other hand, experimental results suggest that in many cases the mentioned theory provides excessive values of buckling load. The influence of imperfections for stringer stiffened shells seems to be less important than in an isotropic case. Considering axially symmetric momentous components of pre-buckling state cannot essentially improve theoretical results. Specific experiments showed a significant influence of stringer discreteness on the buckling loads of reinforced shells. The mentioned influence can be divided into two parts: excitation of essentially non-axially symmetric pre-buckling and buckling states. Usually, only the latter phenomenon is taken into account. In this paper we show that the first factor dominates. We propose simple analytical expressions governing non-axially symmetric pre-buckling state components. We also propose an asymptotic simplification of the buckling boundary value problem. Results obtained are compared numerically with the known theoretical and experimental data. r 2006 Elsevier Ltd. All rights reserved. Keywords: Stringer-stiffened cylindrical shell; Buckling; Discreteness; Pre-buckling state; Homogenisation; Singular asymptotics
1. Introduction Our investigations follow an extended experimental study of buckling of axially compressed stringer stiffened cylindrical shells (SSCS) carried out at both Dnepropetrovsk State University and Prydniprovskaya State Academy of Civil Engineering and Architecture [1–5]. Unfortunately, the fundamental results of these precise experiments are published, as a rule, only in Russian (copies of these papers are available upon request at [email protected]). However, for readers’ convenience we cite here the main conclusions following from the experiments [1–5]. Testing of SSCS with central reinforcement (stringers posed without eccentricity of skin middle surface) has been focused on avoiding effects of non-axial-symmetry components of pre-buckling state caused by loading of skin only and stringer eccentricity, as well as on a careful study of both the influence of the number of stringers and boundary Corresponding author. Tel.: +48 42 631 23 78; fax: +48 42 631 22 25.
E-mail address: [email protected] (J. Awrejcewicz). 0020-7403/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijmecsci.2006.06.012
conditions on the SSCS buckling. During experimental investigations average stringer rigidity parameters NE 1 I=ð2pRDÞ and NE 1 F =ð2pRBÞ were constant while changing the number of stringers. It happened that in the pre-buckling state realized via central support, specific non-axisymmetric deformations were not found. General buckling was associated with the occurrence of large dents covering a few ribs and being extended on the whole shell length. For a small number of stringers general buckling precedes local buckling (i.e. skin buckling inside the stiffened region). Note that the structurally orthotropic theory (SOT) with momentless pre-buckling state gives good results for simply supported edges of shell and absence of local buckling. On the other hand, in the case of clamped edges experimental data are significantly fewer than those theoretically predicted. The main features of the pre-buckling deformation of SSCS with one-sided stringers have been also studied. Among others, a periodical bending of CS in circumferential direction has been found. Non-axisymmetric deformation plays the most essential role for simply supported
ARTICLE IN PRESS I.V. Andrianov et al. / International Journal of Mechanical Sciences 48 (2006) 1505–1515
E E1 e F h h1 h2 h3
Young modulus of skin material Young modulus of stringer material stringer eccentricity (Fig. 1) stringer cross-section area (Fig. 1) skin thickness (Fig. 1) depth of stringer cross section (Fig. 1) width of the web of stringer cross section (Fig. 1) width of the flange of stringer cross section (Fig. 1) first moment of stringer cross-section area shell length (Fig. 1)
I L l ¼ L/R N number of stringers N11, N22, N12 membrane stresses M11, M22, M12 bending and torsion moments Q1, Q2 transverse shearing forces
radius of shell (Fig. 1) first moment of stringer cross-section area (S40 for the internal stringers) T axial compressive load buckling load Tb x longitudinal coordinate (Fig. 1) y circumferential coordinate (Fig. 1) z normal to shell surface coordinate (directed to shell cross-section centre) (Fig. 1) u axial displacement (Fig. 1) v circumferential displacement (Fig. 1) w normal displacement (Fig. 1) w11, w22, w12 curvatures and twisting d(y) Dirac function e11, e22, e12 deformation of middle surface and shear Z ¼ y/R n Poisson coefficient of skin x ¼ x/R P(u, w, T) expression defined by Eq. (20)
(a maximal effect does not exceed 15%) improvement of the theoretical model. This conclusion is also valid for the pre-buckling state with axially symmetric bending.
A
A
x,u
y,v z,w
0 R
A-A 2πR N
h2
e
h z h3
Fig. 1. Stringer-stiffened cylindrical shell.
y
ends and loading transfer to skin only. In other words, a real stress–strain state of SSCS under axial loading significantly differs from axially symmetric one. The main conclusion from the experimental investigation is that modelling of pre-buckling state is correct only if a discrete lay-out of stringers is considered. Now, we briefly summarise state-of-the art of the considered problem reported in Refs. [6–19]. It has been shown that stringer reinforcements of CS essentially decrease the influence of initial imperfections and that more sensitive to initial imperfections are the shells with external reinforcement [6–13]. In theoretical considerations the most widely used is the concept of SOT with momentless pre-buckling state. The applicability of SOT was estimated on the basis of numerous experimental investigations. A comparison of experimental results obtained for integrally stiffened shells, when local buckling is absent, with theoretical results based on the SOT with momentless pre-buckling state shows good agreement [14,15]. Considerable discrepancy between experimental and theoretical results was observed for SSCS with stringers of large bending rigidity. In order to explain the described effect many researchers used the concept of axisymmetric pre-buckling deformation [14]. However, it appears that the influence of this factor in the case of SSCS is essentially lower in comparison with the isotropic case. The next step into achievement of a more accurate theoretical model of SSCS has been initiated by considering of the discrete character of stringers lay-out [16,17]. Computations show that neglecting the factor of discreteness does not lead to essential errors in the case of global buckling only. Furthermore, it was shown that considering of the discreteness of stringers in buckling equations with momentless pre-buckling state leads to insignificant
R S
L
Nomenclature
h1
1506
ARTICLE IN PRESS I.V. Andrianov et al. / International Journal of Mechanical Sciences 48 (2006) 1505–1515
Coupled instability caused by the buckling form of both skin and stringers interactions is important for high thinwalled stringers [8,9,13]. A theoretical evaluation of SSCS buckling is the subject of our paper. The main current open problem here is a sufficient difference between the experimental and theoretical results. Our aim was to clarify this effect and to obtain more accurate theoretical results suggested by the experimental investigations. We suspect that the cause of the discussed discrepancy is motivated rather by an assumption concerning pre-buckling deformation, which is usually treated as either momentless or axially symmetric momentous one. That is why investigation of the real pre-buckling state and the influence of stringer discreteness in buckling equations, is the key point of our paper. The pre-buckling deformation would be composed of an axially symmetric part (momentless and edge effect deformation), and non-axially symmetric component. The latter one is obtained using a homogenisation procedure [20–22]. Analysis of the buckling boundary value problem is based on both the homogenisation approach and the asymptotic simplification of the governing relation [20–22]. The remaining part of the paper is organised as follows. In Section 2, we present governing relations. In Section 3, we investigate pre-buckling state using the homogenisation approach. In Sections 4 and 5, we analyse the buckling problem using a singular asymptotic technique and homogenisation approach. Some of the numerical results and their analysis are proposed in Section 6. Finally, the obtained results are discussed in Section 7. 2. Governing relations
qN 11 qN 12 1 qM 12 1 q þ þ ðyðN 11 þ N 22 ÞÞ ¼ 0, 2R qZ 2 qZ qx qZ qN 12 qN 22 1 qM 12 þ Q2 þ ðyN 12 þ yN 22 Þ 2R qx qx qZ 1 q þ ðyðN 11 þ N 22 ÞÞ ¼ 0, 2 qx qQ1 qQ2 q þ N 22 þ ðy1 N 11 þ y2 N 12 Þ qx qx qZ q ðy1 N 12 þ y2 N 22 Þ ¼ 0, qZ qM 11 qM 12 qM 12 qM 22 þ RQ1 ¼ 0; RQ2 ¼ 0, qx qZ qx qZ where y1 ¼
1 qw ; 2 qx
y2 ¼
elastic continuum according to the Kirchhoff–Clebsch theory; the skin is considered as a 2D elastic continuum according to the theory of isotropic shells; the bond between skin and stringer is taken to be perfect and continuous; the stringer attachment is modelled as an ideal line-contact. The middle surface of the skin is chosen as a reference state [8]. Then, stress–strain relations are defined as follows [8]: N 11 ¼ B11 þ B12 22 þ E 1 ðF 11 þ Sw11 ÞR1 fðZÞ, N 22 ¼ B21 11 þ B22 ;
1 qw ; 2 qZ
y¼
N 12 ¼ N 21 ¼ B33 12 ,
M 11 ¼ Dw11 þ D12 w22 þ E1ðS11 þ Iw11 ÞR1 fðZÞ, M 22 ¼ D21 w11 þ Dw22 ;
M 12 ¼ M 21 ¼ D33 w12 ,
ð2Þ
where
fðZÞ ¼
N X 2pi d Z ; N i¼1
B12 ¼ B21 ¼ vB,
B33 ¼ ð1=2ÞEhð1 þ vÞ1 ; 2
B ¼ Eh=ð1 n Þ;
D12 ¼ D21 ¼ vD, D ¼ Eh3 = 12ð1 n2 Þ ,
D33 ¼ Dð1 vÞ. The following geometric relations hold [23]
11 12
We deal with discretely SSCS shown in Fig. 1. The equilibrium equations of SSCS are written in the form proposed by Sanders [23]
1507
w22
1 qu y21 þ y2 1 qu y2 þ y22 þ w þ 1 ; 22 ¼ ¼ R qx R qZ 2 2 1 qv qu 1 1 qy1 þ , ¼ þ y1 y2 ; w11 ¼ 2R qx qx 2 R qx 1 qy2 1 qy2 qy1 ; w12 ¼ þ y . ¼ R qZ 2 qx qZ
ð3Þ
Further we will use the following boundary conditions for x ¼ 0; l: w ¼ y1 ¼ 0;
N 11 ¼ T;
u ¼ 0,
(4)
or w ¼ M 1 ¼ 0;
N 11 T;
u ¼ 0.
(5)
3. Analysis of the pre-buckling state ð1Þ
1 qu qu . 2R qx qZ
Below, we apply Sanders’ shell theory because of its consistency and good accuracy. We use the following basic approximate assumptions: the stringer is treated as a 1D
Using SOT one can write equations of the pre-buckling state in the following form: dN 011 B11 d2 u0 1 dw0 d2 w0 B12 dw0 ¼ þ 2 2 R dx dx dx R dx R dx 3 K d w0 2 ¼ 0, R dx3
ð6Þ
ARTICLE IN PRESS I.V. Andrianov et al. / International Journal of Mechanical Sciences 48 (2006) 1505–1515
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( " #) 2 D11 d4 w0 K d3 u0 1 d2 w0 dw0 d3 w0 3 þ þ dx dx2 R dx4 R2 dx3 R dx2 " # B12 du0 1 dw0 2 þ R dx 2R dx ( " # B 1 d2 w0 B11 du0 1 dw0 2 w0 þ þ R R dx2 R dx 2R dx ) B12 K d2 w0 w0 2 ¼ 0, R R dx2
where e¼
ð7Þ
where components with subscript ‘‘0’’ belong to SOT. From Eq. (6) and taking into account boundary conditions (4) (or (5)) one obtains N 011
¼ T.
Eq. (7) may be rewritten as follows: D1
d4 w0 2 d2 w0 2 R t R B2 w0 ¼ v21 TR3 , dx4 dx2
(8)
where 2Kv12 vB ; B2 ¼ B12 ð1 vv21 Þ; v21 ¼ , B11 R K2 NE 1 F NE 1 I ; D11 ¼ D þ , ; B11 ¼ B þ D1 ¼ D11 2pR 2pR B11 NE 1 S . K¼ 2pR tT
A particular solution to Eq. (8) is wp ¼ TR v21 =B2 , whereas a longitudinal bending is governed by the following equation of the nonlinear edge effect 2 d4 wg 2 d wg 2 2 D1 þ R t R B wg ¼ 0. dx4 dx2
(9)
General solution of Eq. (9) is wg ¼ expða2 xÞ ðC 1 sinða1 xÞ þ C 2 cosða1 xÞ þ exp ða2 ðx lÞÞ ðC 3 sin ða1 ðx lÞÞ þ C 4 cos ða1 ðx lÞÞÞ,
ð10Þ
1=2 0:5½RD1 1 ð2D2
i
1=2
where ai ¼ þ ð1Þ tR , i ¼ 1, 2; C1–C4 are the constants. Two different cases of loading have to be considered [8]: (i) load transfer to the middle skin surface; (ii) load transfer to the centre of gravity of stringer-skin joint cross-section. In the case of simply supported ends (5) condition M1 ¼ 0 must be changed to the following one: d2 w0 TR2 e ð1 vv21 Þ ¼ 0 D1 dx2
for x ¼ 0; l; or
(11)
K . B11
If load is applied with eccentricity d (d40 for inside of the shell load action), additional moment (Td) should be incorporated into the relation governing the end moment behaviour. Note that taking into account the discreteness of stringers causes the occurrence of a non-symmetric component of the pre-buckling state. Displacements of the pre-buckling state can be represented as follows: w ¼ w0 þ w1 ;
(12)
v ¼ v1 .
Components with subscript ‘‘1’’ caused by the discreteness of stringers lay-out can be found in monograph [21, Chapter 6.1], and they are: 3 1 q u0 q4 w 0 w1 ¼ 2 3 g 3 a 4 F 4 ðjÞ, pa N qx qx 2 1 q u0 q3 w0 b 2 g 3 F 2 ðjÞ u¼ pNð1 vÞ qx qx 4 2v q u0 q5 w0 g 4 a 5 F 6 ðjÞ, ð13Þ qx pð1 vÞ2 a2 N 5 qx 3 1þv q u0 q4 w0 b 3 g 4 F 3 ðjÞ v1 ¼ pð1 vÞ2 N 2 qx qx 3 4 1 q u0 q w0 þ 2 5 þ g 3 a 4 F 5 ðjÞ pa N qx qx 5 vð1 þ vÞ q u0 q6 w0 g 5 a 6 F 7 ðjÞ, þ pð1 vÞa2 N 6 qx qx where 3 2 2 j p j pj 2 þ , F 2 ¼ 0:5pj þ 0:25j ; F 3 ¼ 6 4 12 3 p2 j2 pj j4 p4 j2 p2 j4 þ þ ; F6 ¼ þ , F4 ¼ 12 12 48 180 144 5 pj j6 þ ; F 5 ¼ F 6;j , 240 1440 functions with period 2p/N, j ¼ Fi are the periodic Z/(2pN),a2 ¼ h2 12ð1 nÞR2 , a ¼ E 1 I BR3 ; b ¼ E 1 F =BR; g ¼ E 1 S BR2 ; ð. . . Þ;j denotes derivative with respect to j. 4. Asymptotic simplification of buckling equations Buckling equations are obtained from relations (1)–(3) by using first a representation of displacements and components of stress–strain state as a sum of pre-buckling components w, u, v and their variations w; u; v w ¼ w þ w;
d2 w0 vv21 TR2 e 0 for x ¼ 0; l, D1 dx2
u ¼ u0 þ u1 ;
u ¼ u þ u;
v ¼ v þ v,
and then linearisation with respect to variations should be carried out.
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Let us consider the asymptotic simplification of buckling equations. We introduce the following non-dimensional parameters sffiffiffiffiffiffiffiffiffiffiffiffiffi D11 D 1 ¼ ; ; 2 ¼ 2 D11 B11 R B33 K 5 ¼ . ; 6 ¼ B11 R B11
3 ¼
D33 ; D11
4 ¼
B , B11
1509
equation has the form: 3 E1S q u 1 qw q3 w0 fðZÞ þ Pðu; w; T Þ ¼ R qx3 R qx qx3 E1I 1 q4 w D q4 w fðZÞ Dþ R R2 qx4 R2 qZ4 q2 w q2 w0 N þ N 11 11 qx2 qx2 q2 w qw qN 22 þ N 22 ¼ 0, þ qZ qZ qZ þ RN 22
For a SSCS typical for engineering practice, the following asymptotic estimations are valid:
ð20Þ
where
1=2 1 ;
2 3
1=2
;
4 5 o1;
6 1 .
(14)
N 11
Let us introduce indexes of variation [20–22] which characterise the rate of change of the stress–strain state both in longitudinal and circumferential directions q 1 ð. . .Þo 1 ð. . .Þ; qx
N 22
q 2 ð. . .Þo ð. . .Þ. 2 qZ
(15)
In order to estimate components of the pre-buckling state, we introduce parameters o3–o5 in the following way: 3 wo 1 R;
4 uo w; 1
5 vo 1 w.
(16)
For functions u0, w0 the following estimations hold: o1 o0:5;
o2 o0:5;
o3 ¼ 2;
o5 40:5,
o2 ¼ 1;
o3 ¼ 2:5;
o4 ¼ 1:5;
(17)
o5 ¼ 2.
We also apply the following estimation: 1=2
(19)
Using estimations (14), (17)–(19) one can conclude that the limiting (e1-0) systems are strongly dependant upon the values of the parameters oi, i ¼ 1, 2. Now we are going to scan all possible values of these parameters, for which limiting systems have mathematical and physical appropriate meanings. In order to establish main terms of the asymptotic expansions and to determine the parameters oi, it is convenient to apply the generalised Newton polygon algorithm [20,22]. As a result, one obtains the following indexes of variation: (1) o1o0.5, o2 ¼ 0.5. These indexes of variation characterise the buckling form for which variability in the circumferential direction is greater than that in the longitudinal direction. As experiments show, formation of patterns stretched longitudinally is typical for buckling of SSCS. A simplified buckling
for x ¼ 0; l,
whereas for boundary conditions (5) one obtains w ¼ M 11 ¼ 0
(18)
w1 1 w0 .
Since the obtained equation is of the fourth order with respect to x, one must asymptotically split input boundary conditions (see for details [20,22]). For boundary conditions (4) one gets w ¼ y1 ¼ 0
whereas for functions u1, v1, w1 one obtains o1 X0:5;
1 E1F qu 1 qw0 qw Bþ fðZÞ þ ¼ R R qx R qx qx E1S q2 w 3 fðZÞ 2 , R qx ZZ 2 ZZ 2 q N 11 qu qw ¼ ¼ dZ dZ; dZ dZ qx qx2 qx2 ZZ 2 ZZ 3 1 q w0 qw 1 q w1 qw dZ þ dZ dZ. 2 R R qx qZ qx2 qZ
for x ¼ 0; l.
Note that the discreteness of stringers is displayed in Eq. (20) in two ways. This equation contains nonaxisymmetric components of the pre-buckling state and sums of Dirac d-functions. 5. Solution of the buckling problem Now we use the homogenisation approach to Eq. (20). The role of a small parameter plays e ¼ 1/N. Owing to the multiple scale method [20–22] ‘‘fast’’ Z1 (Z1 ¼ e1Z) and ‘‘slow’’ Z0 (Z0 ¼ Z) independent variables are introduced, and the following relation is used: q q q ð. . .Þ ¼ ð. . .Þ þ 1 ð. . .Þ. qZ qZ0 qZ1 We present displacements as the sum of SOT components and rapidly changed (in the circumferential direction) correctors of the following form: w ¼ w0 ðx; ZÞ þ 1=2 w1 x; Z0 ; Z1 þ w2 x; Z0 ; Z1 þ . . . , u ¼ u0 ðx; ZÞ þ 1=2 u1 x; Z0 ; Z1 þ u2 x; Z0 ; Z1 þ . . . . Also the following expansion is applied T ¼ T 0 þ 1=2 T 1 þ T 2 þ . . . ,
(21)
ARTICLE IN PRESS I.V. Andrianov et al. / International Journal of Mechanical Sciences 48 (2006) 1505–1515
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where T0 is the SOT buckling load, Ti are the corrections determined by the influence of discreteness of stringers. One can subdivide coefficients in Eq. (20) by means of the sum of components corresponding to the SOT and additional terms yielded by the discreteness of stringers Pðu¯ ; w; ¯ T Þ þ P1 ðu¯ ; w; ¯ T Þ ¼ 0. ¯ T Þ ¼ P0 ðu¯ ; w; Expansion of the terms containing components of the pre-buckling state into e series gives h i ð0Þ ð1Þ 1=2 P ¼ Pð0Þ Pð1Þ þ . . . ¼ 0, (22) 0 þ P1 þ 0 þ P1 where PðjÞ i ¼
Ki R
3
3
q uj 1 qwj q w0 þ qx3 R qx qx3
ðjÞ
ð jÞ
N 22i
4
D11i q wj R qx4
qwj qN 22i q2 w0 ðjÞ q2 wj ¼ 0, N 11i þ 2 N 22i þ 2 qZ qZ qZ qx B11i quj 1 qw0 qwj K 11i q2 wj þ ¼ , R qx R qx qx R qx2 ZZ 2 ZZ 2 ð jÞ q wj q N 11i quð jÞ ¼ ¼ dZ dZ; dZ dZ qx qx2 qx2 Z ZZ 3 1 q2 w0 qwj 1 q w1 qwj dZ þ dZ dZ, 2 qZ R R qx qZqx3 qZ i ¼ 0; 1; j ¼ 0; 1.
ð23Þ
(24)
Homogenisation of Eq. (23) with respect to fast variable Z1 gives (25)
Eq. (24) with the homogenised boundary conditions: ð0Þ
w0 ¼ 0;
y1 ¼ 0 for x ¼ 0; l,
w0 ¼ 0;
M 11 ¼ 0 for x ¼ 0; l,
ð0Þ
An cos nZ.
n¼1
Since a buckling mode with one half-wave in the longitudinal direction is typical for SSCS, one can use function f(x) ¼ sin2 (px/l) for boundary conditions (26), and function f(x) ¼ sin (px/l) for boundary conditions (27). Using a standard procedure of the Bubnov–Galerkin method, one obtains a system of linear algebraic equations. Buckling load T0 is obtained numerically from the equation
0
For i ¼ 0 and for i ¼ 1 expressions (23) contain only homogenised and discrete part of rigidity parameters, respectively. It means that for i ¼ 0 and 1 we replace the function f(Z) by N/(2p) and by fðZÞ N=ð2pÞ , respectively. For j ¼ 0 and 1 expressions (23) contain only functions w0 , etc., and only functions w1 , etc., respectively. Now, when only terms of the order one remain in Eq. (22), the following relation is obtained:
Pð0Þ 0 ¼ 0.
1 X
where D is the determinant of our system. In order to define the correction term T1 in series (21), we multiply Eq. (24) by function w0, and then we integrate it on the shell surface. The mentioned procedure gives Z 1 Z 2p Pð0Þ (28) 1 w0 dx dZ ¼ 0.
q2 w j D22i q4 wj ðjÞ þ RN þ N 11i 22i R2 qZ4 qx2
ð0Þ Pð0Þ 0 þ P1 ¼ 0.
w0 ¼ f ðxÞ
D ¼ 0,
!
þ
N 11i
w0 is sought in the following form:
(26) (27)
allows to solve an eigenvalue problem and to define the eigenvalue T0. The Bubnov–Galerkin method is used to solve the obtained eigenvalue problem, and a normal displacement
0
From (28) the following estimation is found: p 4 T 1 2:5 1 T 0. l The obtained relation shows that the correction to the SOT buckling load T0 is essential for shells with lengths comparable to the length of the edge effect zone ðla1=4 Þ. For 1ba1=4 the equation of buckling can be written as follows: K q3 u0 1 qw0 q3 w0 D11 q4 w0 D22 q4 w0 þ 2 2 3 3 R qx R qx qx R qx4 R qZ4 2 2 2 q w0 q w0 q w0 þ RN 22 þ N 11 þ N 11 þ N 22 2 2 qZ2 qx qx qw0 qN 22 ¼ 0, þ qZ qZ where B11 qu0 1 qw0 qw0 K 11 q2 w0 N 11 ¼ þ , R qx3 R qx qx3 R2 qx2 ZZ 2 q N 22 N 22 ¼ dZ dZ, qx2 ZZ 2 ZZ 2 qu0 q w0 1 q w0 qw0 ¼ dZ dZ dZ 2 R qx qx qx2 qZ ZZ 3 1 q w1 qw0 dZ dZ. þ R qZqx2 qZ The above equations correspond to the following model: the pre-buckling deformation is considered taking into account a discrete character of the stringers arrangement, while the buckling state is analysed within the SOT. The results lead to the following estimation of the applicability of SOT for analysis of the SSCS buckling N 4 ba2 .
(29)
ARTICLE IN PRESS I.V. Andrianov et al. / International Journal of Mechanical Sciences 48 (2006) 1505–1515
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Table 1 Physical and geometric parameters of SSCS
-(T1/T)100%
Ref.
Shell parameters (m) (Fig. 1)
10 1
0
1 2 3 4 5
2
5
[3] [3] [3] [22] [23]
N
Stringer parameters (m) (Fig. 1)
R 103
h 103
L 103
h1 103
h2 103
h3 103
71.5 71.5 71.5 60 600
0.19 0.19 0.09 0.2 3.53
71.5 50 80 30 666
2.3 2.3 2.3 1.75 48
0.234 0.234 0.234 0.513 3.53
3 3 3 0.513 3.53
24 12 36 12 20
3 20
30
40 l1
Note that estimation (29) is in agreement with the known estimations obtained on the basis of numerical calculations [17]. Some numerical results are reported in Fig. 2. One may conclude that an inclusion of the stringers discreteness into the buckling equations with SOT prebuckling state does not influence significantly the values of buckling forces.
0.8 2 w1 w1max
Fig. 2. Correction to the SOT buckling loads. Pre-buckling state assumed to be momentless. Parameters of shell and stringer are B11/B ¼ 1.8, D11/ D ¼ 500, R/h ¼ 500. Curve I corresponds to stringer number N ¼ 24, II— N ¼ 36, III—N ¼ 48. Solid lines correspond to inner stringers, dashed—to 1=4 internal ones, and l 1 ¼ l a1=2 2 .
0.6 1 0.4 0.2 0
π/6
Fig. 3. Mode of skin normal displacement between stringers for SSCS type no. 4 from Table 1; curve 1 depicts results of calculations with net method [24], curve 2 depicts results of calculations with formula (12).
For numerical investigations geometrical parameters of specimens tested both experimentally and theoretically [3,24,25] are used. They are collected in Table 1. Some numerical results are shown in Figs. 3–10. In order to verify formula (12) we have compared our computational results with those obtained by the finite difference method (Fig. 3) as well as experimental data (Fig. 4). It is obvious that formula (12) yields good results. Figs. 5 and 6 make it possible to estimate contributions to the pre-buckling state of axially symmetric and nonaxially symmetric components. It is obvious that the contribution of non-axially symmetric state is important. A comparison of numerical results for internal and external reinforcement shows that the influence of pre-buckling state is more sufficient for external reinforcement. Formula (12) enables a description of the basic features of deformation, observed in the experiment [3–5]: bending outward the normal of stringers for external reinforcement, and bending inward the normal for internal reinforcement (see Fig. 4b). An example of loading with significant load eccentricity for SSCS with strong internal reinforcement is shown in Fig. 6. Results of calculations reported in Ref. [3] reveal a significant influence of axially symmetric pre-buckling bending. However, an account of discreteness of reinforcement quantitatively changes pre-buckling factors (Fig. 7b)
.
6. Numerical and experimental results
Fig. 4. Distribution of circumferential deformations in axial direction for SSCS type no. 5 from Table 1 for different load intensity. Experimental data from [25] are marked by signs (J), (+) and (n), solid curves depict results of calculations with formula (12). Curve 1 and sign (J) correspond to load 8.10 g kN (g is free fall acceleration); curve 2 and sign (+) 1.6 102 g kN; curve 3 and sign (D) 2.4 105 g N; curve 4 and sign (n) 3.6 102 g kN; and j1 ¼ j=2p.
ARTICLE IN PRESS I.V. Andrianov et al. / International Journal of Mechanical Sciences 48 (2006) 1505–1515
.
1512
(a)
.
(a)
(b) Fig. 6. Circumferential stresses under axisymmetric (a) and non-axisymmetric (b) pre-buckling deformations (solid lines correspond to inner stringers, dashed—to internal ones; j1 ¼ j=2p, N ¼ 102 N 22 g1 N=m.
(b) Fig. 5. Pre-buckling displacement under axisymmetric (a) and nonaxisymmetric (b) deformations for SSCS type No. 1 from Table 1 for T ¼ g kN (solid lines correspond to inner stringers, dashed—to internal ones; and j1 ¼ j=2p).
in comparison with axially symmetric idealisations (Fig. 7a). Fig. 8 shows variations of non-axially symmetric components of normal pre-buckling displacements with both the number of variation stringers and the rate of average stresses. As it is illustrated in Fig. 9, taking into account only the axially symmetric pre-buckling bending does not give a possibility to get reasonable coincidence with the experimental data in the case of strong momentous type loading.
For analysis of buckling loads let us introduce new non1=4 dimensional parameters l 1 ¼ l a1=2 2 , N 1 ¼ N a1=2 . They allow us to reflect general dependencies of the pre-buckling state, typical for specimens of different sizes. Fig. 10b illustrates the influence of discrete reinforcement on buckling loads. For N141.8 it is less than 5%. The greater the length the smaller is this influence, because the corresponding length of edge effect zone decreases (namely it is essentially influenced by a discrete character of stringers lay-out). At N1o1 usually a local buckling of skin between stringers takes place so that in this case SOT cannot be applied. In such a way, the range, in which it is necessary to take into account the influence of discreteness, is relatively narrow 1oN1o2. However, one can be convinced that this range corresponds to numerous series of real structures (see Table 1). For internal reinforcement, the range of applicability of the presented method is the widest.
ARTICLE IN PRESS I.V. Andrianov et al. / International Journal of Mechanical Sciences 48 (2006) 1505–1515
1513
.
(a)
Fig. 8. Dependence of non-axisymmetric component of normal displacement versus number of stringers for different levels of average stresses for SSCS type no. 2 from Table 1 w ¼ 103 ðwjj¼0 wjf¼0:5 Þ; m .
(b) Fig. 7. Pre-buckling normal displacements and circumferential stresses under eccentric loading for SSCS type no. 3 from Table 1 (curves 1 and 2 correspond to values d/h ¼ 0 and 5).
Let us consider now the influence of axially symmetric pre-buckling state components (Fig. 10a) depending on the shell length and eccentricity signs. At l1o10 (the shell length is comparable with the length of edge effect zone) for internal reinforcement a significant decrease of buckling loads is observed. It is associated with the fact that circumferential loads coming from the boundary moment are influential only at the skin. Under external reinforcement the following factors are predominant: for short
Fig. 9. Dependence of buckling load on eccentricity of loading d for SSCS type no. 3 from Table 1. Curve II corresponds to pre-buckling state considering axisymmetric deformation only (calculations with finite differences method [1]), curve II corresponds to pre-buckling state considering axisymmetric deformation only calculated using our results; curve III obtained with consideration of pre-buckling axisymmetric and non-axisymmetric deformation; sign o marks experimental results [1]).
shells (l1p8) barrel-shaped configuration plays a key role, whereas for long shells (8pl1p16) circumferential stresses are the most important. The influence of axially symmetric pre-buckling state components for l1415–18 is minor in general and leads to an insignificant increase of buckling loads. Thus, numerical
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(a)
4. Asymptotic simplification of this model leads to the eigenvalue problem of the fourth order regarding the longitudinal variable. 5. A discrete character of reinforcement in the prebuckling state can be taken into account in an analytical way using the homogenisation approach. 6. Our present work concerns only linear buckling behaviour of SSCS. But it is obvious that an analysis of post-buckling behaviour must be supplemented by consideration of discrete lay-out of stringers, and hence the methodology proposed by us can be useful for this analysis. 7. For SSCS stiffened with thin stringers the effect of coupled instability can be sufficient. However, it seems that a further combination of the results obtained in this paper with the theory of coupled instability is important too.
Acknowledgements The authors are grateful to the anonymous referee, whose valuable suggestions and comments helped to improve the paper. References (b) Fig. 10. Buckling loads for different models of pre-buckling deformations for SSCS shells with parameters B11/B ¼ 1.8, D11/D ¼ 500. Here TI is buckling load for momentless prebuckling state; TII is buckling load for pre-buckling state with nonlinear edge effect considered; TIII is buckling load for pre-buckling state with nonlinear edge effect and non-axisymmetric components considered.
calculations 1 allow us to improve the earlier asymptotic estimations: the influence of non-axially symmetric prebuckling bending on the buckling is of minor importance if N4(1.5–2) a1/2, and the influence of longitudinal prebuckling bending is of minor importance if l4(15–18) 1=4 a1=2 2 . Out of these limits the influence of pre-buckling bending factors plays a substantial role. 7. Concluding remarks Finally, as a result of our theoretical considerations supported by the experimental data the following conclusions are formulated: 1. Considering the discrete lay-out of stringers in the buckling problem without taking it in the pre-buckling state seems to be not appropriate. 2. Account for only axially symmetric edge effect components in the pre-buckling state leads to wrong conclusions. 3. Reliable results can be obtained using the following model: SOT equations of buckling with pre-buckling state considering the discrete character of reinforcement.
[1] Kostyrko VV, Krasovsky VL. Investigation of influence of load eccentricity to the stability of stringer shells. Dnepropetrovsk: DGU, Gidroaeromehanika i Teoriya Uprugosti; 1988. p. 75–81 [in Russian]. [2] Kostyrko VV, Krasovsky VL. Pre-buckling deformation of axially compressed longitudinally reinforced shells. Theoretical Foundations of Civil Engineering 1995: 87–96 [in Russian]. [3] Krasovsky VL. Stability of isotropic and reinforced cylinders under axial compression. Theoretical Foundations of Civil Engineering 1997: 123–32 [in Russian]. [4] Krasovsky VL. Pre-buckling deformation and buckling of isotropic and reinforced thin-walled cylinders under axial compression. Proceedings of the PGASA 1998(6):19–28 Dnepropetrovsk [in Russian]. [5] Krasovsky VL, Kostyrko VV. The investigation of effect of axial compressive load eccentricity on the stability of stringer shells. In: Proceedings of the VIII symposium on stability of structures, Zakopane (Poland) 2000. p. 137–42. [6] Singer J, Arbocz J, Weller T. Buckling experiments, shells, built-up structures, composites and additional topics. New York: Wiley; 2002. [7] Amiro IYa, Zarutskii VA. Experimental study of the stability of stiffened shells (Review). International Applied Mechanics 1996;32(9):659–68. [8] Manevitch AI. Stability and optimal design of reinforced shells. KievDonetzk: Visha Shkola; 1972 [in Russian]. [9] Manevitch AI. Coupled instability of cylindrical shells stiffened with thin ribs (theoretical models and experimental results). In: Zaras J, Kowal-Michalska K, Rhodes J, editors. Thin-walled structures. Advances and developments. Proceedings of the third international conference thin-walled structures. London: Elsevier; 2001. p. 683–91. [10] Walker AC, Andronnikov A, Stridharan S. Experimental investigation of the buckling of stiffened shells using small scale models. Buckling of the Shells in off-shore Structures. Granada: London et al.; 1982. p. 45–71. [11] Ellinas CP, Croll JGA. Experimental and theoretical correlations of elastic buckling of axially compressed stringer stiffened cylinders. Journal of Strain Analysis for Engineering Design 1983;18(1):41–67.
ARTICLE IN PRESS I.V. Andrianov et al. / International Journal of Mechanical Sciences 48 (2006) 1505–1515 [12] Steinman I, Simitses GJ. Buckling analysis of geometrically imperfect stiffened cylinders under axial compression. AIAA Journal 1977;15(3):374–82. [13] Byskov E, Hansen JC. Postbuckling and imperfection sensitivity analysis of axially stiffened cylindrical shells with mode interaction. Journal of Structural Mechanics 1980;8(2):205–24. [14] Stein M. Some recent advances in the investigation of shell buckling. AIAA Journal 1968;6(12):2339–45. [15] Singer J. Stiffened cylindrical shells. In: Teng JG, Rotter JM, editors. Buckling of thin metal structures. London: Spon; 2004. p. 286–343. [16] Wang JTS, Lin YJ. Stability of discretely stringer-stiffened cylindrical shells. AIAA Journal 1973;11(6):810–4. [17] Zarutskii VA. Features of the buckling of reinforced shells. International Applied Mechanics 2000;36(5):559–85. [18] Stuhlman C, Luzio A, Almroth BD. Influence of stiffeners eccentricity and end moment on stability of cylinders under axial compression. AIAA Journal 1966;4(5):872–7. [19] Weller T, Singer J, Batterman SC. Influence of eccentricity of loading on buckling of stringer-stiffened cylindrical shells. In: Fung YC,
[20]
[21]
[22]
[23] [24] [25]
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Sechler EE, editors. Thin-shell structures. theory, experiment and design. Englewood Cliffs, NJ: Prentice-Hall; 1974. p. 305–24. Awrejcewicz J, Andrianov IV, Manevitch LI. Asymptotic approaches in nonlinear dynamics: new trends and applications. Heidelberg, Berlin: Springer; 1998. Andrianov IV, Awrejcewicz J, Manevitch LI. Asymptotical mechanics of thin-walled structures: a handbook. Heidelberg, Berlin: Springer; 2004. Manevitch LI, Andrianov IV, Oshmyan VO. Mechanics of periodically heterogeneous structures. Berlin, Heidelberg: Springer; 2002. Sanders JL. Nonlinear theories of thin shells. Quartly Applied Mathematics 1963;21(1):21–36. Gavrilenko GD. Stability of stringer shells under non-uniform stress–strain state. Soviet Applied Mechanics 1980;16(11):41–6. Dowling PJ, Harding TE, Agelidis N, Fary W. Buckling of orthogonally stiffened cylindrical shells used in offshore engineering. Buckling of shells. Proceedings of the state-of-the-art colloquium. Berlin: Springer; 1982. p. 239–73.
International Journal of Mechanical Sciences 48 (2006) 1505–1515 www.elsevier.com/locate/ijmecsci
Buckling analysis of discretely stringer-stiffened cylindrical shells I.V. Andrianova, V.M. Verbonolb, J. Awrejcewiczc, a
Institut fu¨r Allgemeine Mechanik, RWTH Aachen, Templergraben 64, D-52056, Aachen, Germany Prydniprovskaya State Academy of Civil Engineering and Architecture, 24a Chernyshevskogo Street, Dnepropetrovsk UA-49005, Ukraine c Department of Automatics and Biomechanics, Technical University of Ło´dz´, 1/15 Stefanowskiego Street, Ło´dz´ PL-90-924, Poland
b
Received 27 November 2004; received in revised form 15 May 2006; accepted 28 June 2006 Available online 8 September 2006
Abstract Engineering approach for computation of stringer stiffened cylindrical shells is realized mainly using the structurally orthotropic theory with momentless pre-buckling state. On the other hand, experimental results suggest that in many cases the mentioned theory provides excessive values of buckling load. The influence of imperfections for stringer stiffened shells seems to be less important than in an isotropic case. Considering axially symmetric momentous components of pre-buckling state cannot essentially improve theoretical results. Specific experiments showed a significant influence of stringer discreteness on the buckling loads of reinforced shells. The mentioned influence can be divided into two parts: excitation of essentially non-axially symmetric pre-buckling and buckling states. Usually, only the latter phenomenon is taken into account. In this paper we show that the first factor dominates. We propose simple analytical expressions governing non-axially symmetric pre-buckling state components. We also propose an asymptotic simplification of the buckling boundary value problem. Results obtained are compared numerically with the known theoretical and experimental data. r 2006 Elsevier Ltd. All rights reserved. Keywords: Stringer-stiffened cylindrical shell; Buckling; Discreteness; Pre-buckling state; Homogenisation; Singular asymptotics
1. Introduction Our investigations follow an extended experimental study of buckling of axially compressed stringer stiffened cylindrical shells (SSCS) carried out at both Dnepropetrovsk State University and Prydniprovskaya State Academy of Civil Engineering and Architecture [1–5]. Unfortunately, the fundamental results of these precise experiments are published, as a rule, only in Russian (copies of these papers are available upon request at [email protected]). However, for readers’ convenience we cite here the main conclusions following from the experiments [1–5]. Testing of SSCS with central reinforcement (stringers posed without eccentricity of skin middle surface) has been focused on avoiding effects of non-axial-symmetry components of pre-buckling state caused by loading of skin only and stringer eccentricity, as well as on a careful study of both the influence of the number of stringers and boundary Corresponding author. Tel.: +48 42 631 23 78; fax: +48 42 631 22 25.
E-mail address: [email protected] (J. Awrejcewicz). 0020-7403/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijmecsci.2006.06.012
conditions on the SSCS buckling. During experimental investigations average stringer rigidity parameters NE 1 I=ð2pRDÞ and NE 1 F =ð2pRBÞ were constant while changing the number of stringers. It happened that in the pre-buckling state realized via central support, specific non-axisymmetric deformations were not found. General buckling was associated with the occurrence of large dents covering a few ribs and being extended on the whole shell length. For a small number of stringers general buckling precedes local buckling (i.e. skin buckling inside the stiffened region). Note that the structurally orthotropic theory (SOT) with momentless pre-buckling state gives good results for simply supported edges of shell and absence of local buckling. On the other hand, in the case of clamped edges experimental data are significantly fewer than those theoretically predicted. The main features of the pre-buckling deformation of SSCS with one-sided stringers have been also studied. Among others, a periodical bending of CS in circumferential direction has been found. Non-axisymmetric deformation plays the most essential role for simply supported
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E E1 e F h h1 h2 h3
Young modulus of skin material Young modulus of stringer material stringer eccentricity (Fig. 1) stringer cross-section area (Fig. 1) skin thickness (Fig. 1) depth of stringer cross section (Fig. 1) width of the web of stringer cross section (Fig. 1) width of the flange of stringer cross section (Fig. 1) first moment of stringer cross-section area shell length (Fig. 1)
I L l ¼ L/R N number of stringers N11, N22, N12 membrane stresses M11, M22, M12 bending and torsion moments Q1, Q2 transverse shearing forces
radius of shell (Fig. 1) first moment of stringer cross-section area (S40 for the internal stringers) T axial compressive load buckling load Tb x longitudinal coordinate (Fig. 1) y circumferential coordinate (Fig. 1) z normal to shell surface coordinate (directed to shell cross-section centre) (Fig. 1) u axial displacement (Fig. 1) v circumferential displacement (Fig. 1) w normal displacement (Fig. 1) w11, w22, w12 curvatures and twisting d(y) Dirac function e11, e22, e12 deformation of middle surface and shear Z ¼ y/R n Poisson coefficient of skin x ¼ x/R P(u, w, T) expression defined by Eq. (20)
(a maximal effect does not exceed 15%) improvement of the theoretical model. This conclusion is also valid for the pre-buckling state with axially symmetric bending.
A
A
x,u
y,v z,w
0 R
A-A 2πR N
h2
e
h z h3
Fig. 1. Stringer-stiffened cylindrical shell.
y
ends and loading transfer to skin only. In other words, a real stress–strain state of SSCS under axial loading significantly differs from axially symmetric one. The main conclusion from the experimental investigation is that modelling of pre-buckling state is correct only if a discrete lay-out of stringers is considered. Now, we briefly summarise state-of-the art of the considered problem reported in Refs. [6–19]. It has been shown that stringer reinforcements of CS essentially decrease the influence of initial imperfections and that more sensitive to initial imperfections are the shells with external reinforcement [6–13]. In theoretical considerations the most widely used is the concept of SOT with momentless pre-buckling state. The applicability of SOT was estimated on the basis of numerous experimental investigations. A comparison of experimental results obtained for integrally stiffened shells, when local buckling is absent, with theoretical results based on the SOT with momentless pre-buckling state shows good agreement [14,15]. Considerable discrepancy between experimental and theoretical results was observed for SSCS with stringers of large bending rigidity. In order to explain the described effect many researchers used the concept of axisymmetric pre-buckling deformation [14]. However, it appears that the influence of this factor in the case of SSCS is essentially lower in comparison with the isotropic case. The next step into achievement of a more accurate theoretical model of SSCS has been initiated by considering of the discrete character of stringers lay-out [16,17]. Computations show that neglecting the factor of discreteness does not lead to essential errors in the case of global buckling only. Furthermore, it was shown that considering of the discreteness of stringers in buckling equations with momentless pre-buckling state leads to insignificant
R S
L
Nomenclature
h1
1506
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Coupled instability caused by the buckling form of both skin and stringers interactions is important for high thinwalled stringers [8,9,13]. A theoretical evaluation of SSCS buckling is the subject of our paper. The main current open problem here is a sufficient difference between the experimental and theoretical results. Our aim was to clarify this effect and to obtain more accurate theoretical results suggested by the experimental investigations. We suspect that the cause of the discussed discrepancy is motivated rather by an assumption concerning pre-buckling deformation, which is usually treated as either momentless or axially symmetric momentous one. That is why investigation of the real pre-buckling state and the influence of stringer discreteness in buckling equations, is the key point of our paper. The pre-buckling deformation would be composed of an axially symmetric part (momentless and edge effect deformation), and non-axially symmetric component. The latter one is obtained using a homogenisation procedure [20–22]. Analysis of the buckling boundary value problem is based on both the homogenisation approach and the asymptotic simplification of the governing relation [20–22]. The remaining part of the paper is organised as follows. In Section 2, we present governing relations. In Section 3, we investigate pre-buckling state using the homogenisation approach. In Sections 4 and 5, we analyse the buckling problem using a singular asymptotic technique and homogenisation approach. Some of the numerical results and their analysis are proposed in Section 6. Finally, the obtained results are discussed in Section 7. 2. Governing relations
qN 11 qN 12 1 qM 12 1 q þ þ ðyðN 11 þ N 22 ÞÞ ¼ 0, 2R qZ 2 qZ qx qZ qN 12 qN 22 1 qM 12 þ Q2 þ ðyN 12 þ yN 22 Þ 2R qx qx qZ 1 q þ ðyðN 11 þ N 22 ÞÞ ¼ 0, 2 qx qQ1 qQ2 q þ N 22 þ ðy1 N 11 þ y2 N 12 Þ qx qx qZ q ðy1 N 12 þ y2 N 22 Þ ¼ 0, qZ qM 11 qM 12 qM 12 qM 22 þ RQ1 ¼ 0; RQ2 ¼ 0, qx qZ qx qZ where y1 ¼
1 qw ; 2 qx
y2 ¼
elastic continuum according to the Kirchhoff–Clebsch theory; the skin is considered as a 2D elastic continuum according to the theory of isotropic shells; the bond between skin and stringer is taken to be perfect and continuous; the stringer attachment is modelled as an ideal line-contact. The middle surface of the skin is chosen as a reference state [8]. Then, stress–strain relations are defined as follows [8]: N 11 ¼ B11 þ B12 22 þ E 1 ðF 11 þ Sw11 ÞR1 fðZÞ, N 22 ¼ B21 11 þ B22 ;
1 qw ; 2 qZ
y¼
N 12 ¼ N 21 ¼ B33 12 ,
M 11 ¼ Dw11 þ D12 w22 þ E1ðS11 þ Iw11 ÞR1 fðZÞ, M 22 ¼ D21 w11 þ Dw22 ;
M 12 ¼ M 21 ¼ D33 w12 ,
ð2Þ
where
fðZÞ ¼
N X 2pi d Z ; N i¼1
B12 ¼ B21 ¼ vB,
B33 ¼ ð1=2ÞEhð1 þ vÞ1 ; 2
B ¼ Eh=ð1 n Þ;
D12 ¼ D21 ¼ vD, D ¼ Eh3 = 12ð1 n2 Þ ,
D33 ¼ Dð1 vÞ. The following geometric relations hold [23]
11 12
We deal with discretely SSCS shown in Fig. 1. The equilibrium equations of SSCS are written in the form proposed by Sanders [23]
1507
w22
1 qu y21 þ y2 1 qu y2 þ y22 þ w þ 1 ; 22 ¼ ¼ R qx R qZ 2 2 1 qv qu 1 1 qy1 þ , ¼ þ y1 y2 ; w11 ¼ 2R qx qx 2 R qx 1 qy2 1 qy2 qy1 ; w12 ¼ þ y . ¼ R qZ 2 qx qZ
ð3Þ
Further we will use the following boundary conditions for x ¼ 0; l: w ¼ y1 ¼ 0;
N 11 ¼ T;
u ¼ 0,
(4)
or w ¼ M 1 ¼ 0;
N 11 T;
u ¼ 0.
(5)
3. Analysis of the pre-buckling state ð1Þ
1 qu qu . 2R qx qZ
Below, we apply Sanders’ shell theory because of its consistency and good accuracy. We use the following basic approximate assumptions: the stringer is treated as a 1D
Using SOT one can write equations of the pre-buckling state in the following form: dN 011 B11 d2 u0 1 dw0 d2 w0 B12 dw0 ¼ þ 2 2 R dx dx dx R dx R dx 3 K d w0 2 ¼ 0, R dx3
ð6Þ
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1508
( " #) 2 D11 d4 w0 K d3 u0 1 d2 w0 dw0 d3 w0 3 þ þ dx dx2 R dx4 R2 dx3 R dx2 " # B12 du0 1 dw0 2 þ R dx 2R dx ( " # B 1 d2 w0 B11 du0 1 dw0 2 w0 þ þ R R dx2 R dx 2R dx ) B12 K d2 w0 w0 2 ¼ 0, R R dx2
where e¼
ð7Þ
where components with subscript ‘‘0’’ belong to SOT. From Eq. (6) and taking into account boundary conditions (4) (or (5)) one obtains N 011
¼ T.
Eq. (7) may be rewritten as follows: D1
d4 w0 2 d2 w0 2 R t R B2 w0 ¼ v21 TR3 , dx4 dx2
(8)
where 2Kv12 vB ; B2 ¼ B12 ð1 vv21 Þ; v21 ¼ , B11 R K2 NE 1 F NE 1 I ; D11 ¼ D þ , ; B11 ¼ B þ D1 ¼ D11 2pR 2pR B11 NE 1 S . K¼ 2pR tT
A particular solution to Eq. (8) is wp ¼ TR v21 =B2 , whereas a longitudinal bending is governed by the following equation of the nonlinear edge effect 2 d4 wg 2 d wg 2 2 D1 þ R t R B wg ¼ 0. dx4 dx2
(9)
General solution of Eq. (9) is wg ¼ expða2 xÞ ðC 1 sinða1 xÞ þ C 2 cosða1 xÞ þ exp ða2 ðx lÞÞ ðC 3 sin ða1 ðx lÞÞ þ C 4 cos ða1 ðx lÞÞÞ,
ð10Þ
1=2 0:5½RD1 1 ð2D2
i
1=2
where ai ¼ þ ð1Þ tR , i ¼ 1, 2; C1–C4 are the constants. Two different cases of loading have to be considered [8]: (i) load transfer to the middle skin surface; (ii) load transfer to the centre of gravity of stringer-skin joint cross-section. In the case of simply supported ends (5) condition M1 ¼ 0 must be changed to the following one: d2 w0 TR2 e ð1 vv21 Þ ¼ 0 D1 dx2
for x ¼ 0; l; or
(11)
K . B11
If load is applied with eccentricity d (d40 for inside of the shell load action), additional moment (Td) should be incorporated into the relation governing the end moment behaviour. Note that taking into account the discreteness of stringers causes the occurrence of a non-symmetric component of the pre-buckling state. Displacements of the pre-buckling state can be represented as follows: w ¼ w0 þ w1 ;
(12)
v ¼ v1 .
Components with subscript ‘‘1’’ caused by the discreteness of stringers lay-out can be found in monograph [21, Chapter 6.1], and they are: 3 1 q u0 q4 w 0 w1 ¼ 2 3 g 3 a 4 F 4 ðjÞ, pa N qx qx 2 1 q u0 q3 w0 b 2 g 3 F 2 ðjÞ u¼ pNð1 vÞ qx qx 4 2v q u0 q5 w0 g 4 a 5 F 6 ðjÞ, ð13Þ qx pð1 vÞ2 a2 N 5 qx 3 1þv q u0 q4 w0 b 3 g 4 F 3 ðjÞ v1 ¼ pð1 vÞ2 N 2 qx qx 3 4 1 q u0 q w0 þ 2 5 þ g 3 a 4 F 5 ðjÞ pa N qx qx 5 vð1 þ vÞ q u0 q6 w0 g 5 a 6 F 7 ðjÞ, þ pð1 vÞa2 N 6 qx qx where 3 2 2 j p j pj 2 þ , F 2 ¼ 0:5pj þ 0:25j ; F 3 ¼ 6 4 12 3 p2 j2 pj j4 p4 j2 p2 j4 þ þ ; F6 ¼ þ , F4 ¼ 12 12 48 180 144 5 pj j6 þ ; F 5 ¼ F 6;j , 240 1440 functions with period 2p/N, j ¼ Fi are the periodic Z/(2pN),a2 ¼ h2 12ð1 nÞR2 , a ¼ E 1 I BR3 ; b ¼ E 1 F =BR; g ¼ E 1 S BR2 ; ð. . . Þ;j denotes derivative with respect to j. 4. Asymptotic simplification of buckling equations Buckling equations are obtained from relations (1)–(3) by using first a representation of displacements and components of stress–strain state as a sum of pre-buckling components w, u, v and their variations w; u; v w ¼ w þ w;
d2 w0 vv21 TR2 e 0 for x ¼ 0; l, D1 dx2
u ¼ u0 þ u1 ;
u ¼ u þ u;
v ¼ v þ v,
and then linearisation with respect to variations should be carried out.
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Let us consider the asymptotic simplification of buckling equations. We introduce the following non-dimensional parameters sffiffiffiffiffiffiffiffiffiffiffiffiffi D11 D 1 ¼ ; ; 2 ¼ 2 D11 B11 R B33 K 5 ¼ . ; 6 ¼ B11 R B11
3 ¼
D33 ; D11
4 ¼
B , B11
1509
equation has the form: 3 E1S q u 1 qw q3 w0 fðZÞ þ Pðu; w; T Þ ¼ R qx3 R qx qx3 E1I 1 q4 w D q4 w fðZÞ Dþ R R2 qx4 R2 qZ4 q2 w q2 w0 N þ N 11 11 qx2 qx2 q2 w qw qN 22 þ N 22 ¼ 0, þ qZ qZ qZ þ RN 22
For a SSCS typical for engineering practice, the following asymptotic estimations are valid:
ð20Þ
where
1=2 1 ;
2 3
1=2
;
4 5 o1;
6 1 .
(14)
N 11
Let us introduce indexes of variation [20–22] which characterise the rate of change of the stress–strain state both in longitudinal and circumferential directions q 1 ð. . .Þo 1 ð. . .Þ; qx
N 22
q 2 ð. . .Þo ð. . .Þ. 2 qZ
(15)
In order to estimate components of the pre-buckling state, we introduce parameters o3–o5 in the following way: 3 wo 1 R;
4 uo w; 1
5 vo 1 w.
(16)
For functions u0, w0 the following estimations hold: o1 o0:5;
o2 o0:5;
o3 ¼ 2;
o5 40:5,
o2 ¼ 1;
o3 ¼ 2:5;
o4 ¼ 1:5;
(17)
o5 ¼ 2.
We also apply the following estimation: 1=2
(19)
Using estimations (14), (17)–(19) one can conclude that the limiting (e1-0) systems are strongly dependant upon the values of the parameters oi, i ¼ 1, 2. Now we are going to scan all possible values of these parameters, for which limiting systems have mathematical and physical appropriate meanings. In order to establish main terms of the asymptotic expansions and to determine the parameters oi, it is convenient to apply the generalised Newton polygon algorithm [20,22]. As a result, one obtains the following indexes of variation: (1) o1o0.5, o2 ¼ 0.5. These indexes of variation characterise the buckling form for which variability in the circumferential direction is greater than that in the longitudinal direction. As experiments show, formation of patterns stretched longitudinally is typical for buckling of SSCS. A simplified buckling
for x ¼ 0; l,
whereas for boundary conditions (5) one obtains w ¼ M 11 ¼ 0
(18)
w1 1 w0 .
Since the obtained equation is of the fourth order with respect to x, one must asymptotically split input boundary conditions (see for details [20,22]). For boundary conditions (4) one gets w ¼ y1 ¼ 0
whereas for functions u1, v1, w1 one obtains o1 X0:5;
1 E1F qu 1 qw0 qw Bþ fðZÞ þ ¼ R R qx R qx qx E1S q2 w 3 fðZÞ 2 , R qx ZZ 2 ZZ 2 q N 11 qu qw ¼ ¼ dZ dZ; dZ dZ qx qx2 qx2 ZZ 2 ZZ 3 1 q w0 qw 1 q w1 qw dZ þ dZ dZ. 2 R R qx qZ qx2 qZ
for x ¼ 0; l.
Note that the discreteness of stringers is displayed in Eq. (20) in two ways. This equation contains nonaxisymmetric components of the pre-buckling state and sums of Dirac d-functions. 5. Solution of the buckling problem Now we use the homogenisation approach to Eq. (20). The role of a small parameter plays e ¼ 1/N. Owing to the multiple scale method [20–22] ‘‘fast’’ Z1 (Z1 ¼ e1Z) and ‘‘slow’’ Z0 (Z0 ¼ Z) independent variables are introduced, and the following relation is used: q q q ð. . .Þ ¼ ð. . .Þ þ 1 ð. . .Þ. qZ qZ0 qZ1 We present displacements as the sum of SOT components and rapidly changed (in the circumferential direction) correctors of the following form: w ¼ w0 ðx; ZÞ þ 1=2 w1 x; Z0 ; Z1 þ w2 x; Z0 ; Z1 þ . . . , u ¼ u0 ðx; ZÞ þ 1=2 u1 x; Z0 ; Z1 þ u2 x; Z0 ; Z1 þ . . . . Also the following expansion is applied T ¼ T 0 þ 1=2 T 1 þ T 2 þ . . . ,
(21)
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where T0 is the SOT buckling load, Ti are the corrections determined by the influence of discreteness of stringers. One can subdivide coefficients in Eq. (20) by means of the sum of components corresponding to the SOT and additional terms yielded by the discreteness of stringers Pðu¯ ; w; ¯ T Þ þ P1 ðu¯ ; w; ¯ T Þ ¼ 0. ¯ T Þ ¼ P0 ðu¯ ; w; Expansion of the terms containing components of the pre-buckling state into e series gives h i ð0Þ ð1Þ 1=2 P ¼ Pð0Þ Pð1Þ þ . . . ¼ 0, (22) 0 þ P1 þ 0 þ P1 where PðjÞ i ¼
Ki R
3
3
q uj 1 qwj q w0 þ qx3 R qx qx3
ðjÞ
ð jÞ
N 22i
4
D11i q wj R qx4
qwj qN 22i q2 w0 ðjÞ q2 wj ¼ 0, N 11i þ 2 N 22i þ 2 qZ qZ qZ qx B11i quj 1 qw0 qwj K 11i q2 wj þ ¼ , R qx R qx qx R qx2 ZZ 2 ZZ 2 ð jÞ q wj q N 11i quð jÞ ¼ ¼ dZ dZ; dZ dZ qx qx2 qx2 Z ZZ 3 1 q2 w0 qwj 1 q w1 qwj dZ þ dZ dZ, 2 qZ R R qx qZqx3 qZ i ¼ 0; 1; j ¼ 0; 1.
ð23Þ
(24)
Homogenisation of Eq. (23) with respect to fast variable Z1 gives (25)
Eq. (24) with the homogenised boundary conditions: ð0Þ
w0 ¼ 0;
y1 ¼ 0 for x ¼ 0; l,
w0 ¼ 0;
M 11 ¼ 0 for x ¼ 0; l,
ð0Þ
An cos nZ.
n¼1
Since a buckling mode with one half-wave in the longitudinal direction is typical for SSCS, one can use function f(x) ¼ sin2 (px/l) for boundary conditions (26), and function f(x) ¼ sin (px/l) for boundary conditions (27). Using a standard procedure of the Bubnov–Galerkin method, one obtains a system of linear algebraic equations. Buckling load T0 is obtained numerically from the equation
0
For i ¼ 0 and for i ¼ 1 expressions (23) contain only homogenised and discrete part of rigidity parameters, respectively. It means that for i ¼ 0 and 1 we replace the function f(Z) by N/(2p) and by fðZÞ N=ð2pÞ , respectively. For j ¼ 0 and 1 expressions (23) contain only functions w0 , etc., and only functions w1 , etc., respectively. Now, when only terms of the order one remain in Eq. (22), the following relation is obtained:
Pð0Þ 0 ¼ 0.
1 X
where D is the determinant of our system. In order to define the correction term T1 in series (21), we multiply Eq. (24) by function w0, and then we integrate it on the shell surface. The mentioned procedure gives Z 1 Z 2p Pð0Þ (28) 1 w0 dx dZ ¼ 0.
q2 w j D22i q4 wj ðjÞ þ RN þ N 11i 22i R2 qZ4 qx2
ð0Þ Pð0Þ 0 þ P1 ¼ 0.
w0 ¼ f ðxÞ
D ¼ 0,
!
þ
N 11i
w0 is sought in the following form:
(26) (27)
allows to solve an eigenvalue problem and to define the eigenvalue T0. The Bubnov–Galerkin method is used to solve the obtained eigenvalue problem, and a normal displacement
0
From (28) the following estimation is found: p 4 T 1 2:5 1 T 0. l The obtained relation shows that the correction to the SOT buckling load T0 is essential for shells with lengths comparable to the length of the edge effect zone ðla1=4 Þ. For 1ba1=4 the equation of buckling can be written as follows: K q3 u0 1 qw0 q3 w0 D11 q4 w0 D22 q4 w0 þ 2 2 3 3 R qx R qx qx R qx4 R qZ4 2 2 2 q w0 q w0 q w0 þ RN 22 þ N 11 þ N 11 þ N 22 2 2 qZ2 qx qx qw0 qN 22 ¼ 0, þ qZ qZ where B11 qu0 1 qw0 qw0 K 11 q2 w0 N 11 ¼ þ , R qx3 R qx qx3 R2 qx2 ZZ 2 q N 22 N 22 ¼ dZ dZ, qx2 ZZ 2 ZZ 2 qu0 q w0 1 q w0 qw0 ¼ dZ dZ dZ 2 R qx qx qx2 qZ ZZ 3 1 q w1 qw0 dZ dZ. þ R qZqx2 qZ The above equations correspond to the following model: the pre-buckling deformation is considered taking into account a discrete character of the stringers arrangement, while the buckling state is analysed within the SOT. The results lead to the following estimation of the applicability of SOT for analysis of the SSCS buckling N 4 ba2 .
(29)
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Table 1 Physical and geometric parameters of SSCS
-(T1/T)100%
Ref.
Shell parameters (m) (Fig. 1)
10 1
0
1 2 3 4 5
2
5
[3] [3] [3] [22] [23]
N
Stringer parameters (m) (Fig. 1)
R 103
h 103
L 103
h1 103
h2 103
h3 103
71.5 71.5 71.5 60 600
0.19 0.19 0.09 0.2 3.53
71.5 50 80 30 666
2.3 2.3 2.3 1.75 48
0.234 0.234 0.234 0.513 3.53
3 3 3 0.513 3.53
24 12 36 12 20
3 20
30
40 l1
Note that estimation (29) is in agreement with the known estimations obtained on the basis of numerical calculations [17]. Some numerical results are reported in Fig. 2. One may conclude that an inclusion of the stringers discreteness into the buckling equations with SOT prebuckling state does not influence significantly the values of buckling forces.
0.8 2 w1 w1max
Fig. 2. Correction to the SOT buckling loads. Pre-buckling state assumed to be momentless. Parameters of shell and stringer are B11/B ¼ 1.8, D11/ D ¼ 500, R/h ¼ 500. Curve I corresponds to stringer number N ¼ 24, II— N ¼ 36, III—N ¼ 48. Solid lines correspond to inner stringers, dashed—to 1=4 internal ones, and l 1 ¼ l a1=2 2 .
0.6 1 0.4 0.2 0
π/6
Fig. 3. Mode of skin normal displacement between stringers for SSCS type no. 4 from Table 1; curve 1 depicts results of calculations with net method [24], curve 2 depicts results of calculations with formula (12).
For numerical investigations geometrical parameters of specimens tested both experimentally and theoretically [3,24,25] are used. They are collected in Table 1. Some numerical results are shown in Figs. 3–10. In order to verify formula (12) we have compared our computational results with those obtained by the finite difference method (Fig. 3) as well as experimental data (Fig. 4). It is obvious that formula (12) yields good results. Figs. 5 and 6 make it possible to estimate contributions to the pre-buckling state of axially symmetric and nonaxially symmetric components. It is obvious that the contribution of non-axially symmetric state is important. A comparison of numerical results for internal and external reinforcement shows that the influence of pre-buckling state is more sufficient for external reinforcement. Formula (12) enables a description of the basic features of deformation, observed in the experiment [3–5]: bending outward the normal of stringers for external reinforcement, and bending inward the normal for internal reinforcement (see Fig. 4b). An example of loading with significant load eccentricity for SSCS with strong internal reinforcement is shown in Fig. 6. Results of calculations reported in Ref. [3] reveal a significant influence of axially symmetric pre-buckling bending. However, an account of discreteness of reinforcement quantitatively changes pre-buckling factors (Fig. 7b)
.
6. Numerical and experimental results
Fig. 4. Distribution of circumferential deformations in axial direction for SSCS type no. 5 from Table 1 for different load intensity. Experimental data from [25] are marked by signs (J), (+) and (n), solid curves depict results of calculations with formula (12). Curve 1 and sign (J) correspond to load 8.10 g kN (g is free fall acceleration); curve 2 and sign (+) 1.6 102 g kN; curve 3 and sign (D) 2.4 105 g N; curve 4 and sign (n) 3.6 102 g kN; and j1 ¼ j=2p.
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(a)
.
(a)
(b) Fig. 6. Circumferential stresses under axisymmetric (a) and non-axisymmetric (b) pre-buckling deformations (solid lines correspond to inner stringers, dashed—to internal ones; j1 ¼ j=2p, N ¼ 102 N 22 g1 N=m.
(b) Fig. 5. Pre-buckling displacement under axisymmetric (a) and nonaxisymmetric (b) deformations for SSCS type No. 1 from Table 1 for T ¼ g kN (solid lines correspond to inner stringers, dashed—to internal ones; and j1 ¼ j=2p).
in comparison with axially symmetric idealisations (Fig. 7a). Fig. 8 shows variations of non-axially symmetric components of normal pre-buckling displacements with both the number of variation stringers and the rate of average stresses. As it is illustrated in Fig. 9, taking into account only the axially symmetric pre-buckling bending does not give a possibility to get reasonable coincidence with the experimental data in the case of strong momentous type loading.
For analysis of buckling loads let us introduce new non1=4 dimensional parameters l 1 ¼ l a1=2 2 , N 1 ¼ N a1=2 . They allow us to reflect general dependencies of the pre-buckling state, typical for specimens of different sizes. Fig. 10b illustrates the influence of discrete reinforcement on buckling loads. For N141.8 it is less than 5%. The greater the length the smaller is this influence, because the corresponding length of edge effect zone decreases (namely it is essentially influenced by a discrete character of stringers lay-out). At N1o1 usually a local buckling of skin between stringers takes place so that in this case SOT cannot be applied. In such a way, the range, in which it is necessary to take into account the influence of discreteness, is relatively narrow 1oN1o2. However, one can be convinced that this range corresponds to numerous series of real structures (see Table 1). For internal reinforcement, the range of applicability of the presented method is the widest.
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.
(a)
Fig. 8. Dependence of non-axisymmetric component of normal displacement versus number of stringers for different levels of average stresses for SSCS type no. 2 from Table 1 w ¼ 103 ðwjj¼0 wjf¼0:5 Þ; m .
(b) Fig. 7. Pre-buckling normal displacements and circumferential stresses under eccentric loading for SSCS type no. 3 from Table 1 (curves 1 and 2 correspond to values d/h ¼ 0 and 5).
Let us consider now the influence of axially symmetric pre-buckling state components (Fig. 10a) depending on the shell length and eccentricity signs. At l1o10 (the shell length is comparable with the length of edge effect zone) for internal reinforcement a significant decrease of buckling loads is observed. It is associated with the fact that circumferential loads coming from the boundary moment are influential only at the skin. Under external reinforcement the following factors are predominant: for short
Fig. 9. Dependence of buckling load on eccentricity of loading d for SSCS type no. 3 from Table 1. Curve II corresponds to pre-buckling state considering axisymmetric deformation only (calculations with finite differences method [1]), curve II corresponds to pre-buckling state considering axisymmetric deformation only calculated using our results; curve III obtained with consideration of pre-buckling axisymmetric and non-axisymmetric deformation; sign o marks experimental results [1]).
shells (l1p8) barrel-shaped configuration plays a key role, whereas for long shells (8pl1p16) circumferential stresses are the most important. The influence of axially symmetric pre-buckling state components for l1415–18 is minor in general and leads to an insignificant increase of buckling loads. Thus, numerical
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4. Asymptotic simplification of this model leads to the eigenvalue problem of the fourth order regarding the longitudinal variable. 5. A discrete character of reinforcement in the prebuckling state can be taken into account in an analytical way using the homogenisation approach. 6. Our present work concerns only linear buckling behaviour of SSCS. But it is obvious that an analysis of post-buckling behaviour must be supplemented by consideration of discrete lay-out of stringers, and hence the methodology proposed by us can be useful for this analysis. 7. For SSCS stiffened with thin stringers the effect of coupled instability can be sufficient. However, it seems that a further combination of the results obtained in this paper with the theory of coupled instability is important too.
Acknowledgements The authors are grateful to the anonymous referee, whose valuable suggestions and comments helped to improve the paper. References (b) Fig. 10. Buckling loads for different models of pre-buckling deformations for SSCS shells with parameters B11/B ¼ 1.8, D11/D ¼ 500. Here TI is buckling load for momentless prebuckling state; TII is buckling load for pre-buckling state with nonlinear edge effect considered; TIII is buckling load for pre-buckling state with nonlinear edge effect and non-axisymmetric components considered.
calculations 1 allow us to improve the earlier asymptotic estimations: the influence of non-axially symmetric prebuckling bending on the buckling is of minor importance if N4(1.5–2) a1/2, and the influence of longitudinal prebuckling bending is of minor importance if l4(15–18) 1=4 a1=2 2 . Out of these limits the influence of pre-buckling bending factors plays a substantial role. 7. Concluding remarks Finally, as a result of our theoretical considerations supported by the experimental data the following conclusions are formulated: 1. Considering the discrete lay-out of stringers in the buckling problem without taking it in the pre-buckling state seems to be not appropriate. 2. Account for only axially symmetric edge effect components in the pre-buckling state leads to wrong conclusions. 3. Reliable results can be obtained using the following model: SOT equations of buckling with pre-buckling state considering the discrete character of reinforcement.
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