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A semi-analytical model for local post-buckling analysis of stringer- and frame-stiffened cylindrical panels
Thin-Walled Structures 44 (2006) 102–114 www.elsevier.com/locate/tws
A semi-analytical model for local post-buckling analysis of stringer- and frame-stiffened cylindrical panels Philipp Buermann a,*, Raimund Rolfes b, Jan Tessmer b, Martin Schagerl b a
DLR, Institute of Structural Mechanics, Structural Analysis Section, Lilienthalplatz 7, 38108 Braunschweig, Germany b Airbus Deutschland GmbH, ESAS, Kreetslag 10, 21129 Hamburg, Germany Received 3 February 2005; accepted 11 August 2005 Available online 14 November 2005
Abstract A fast semi-analytical model for the post-buckling analysis of stiffened cylindrical panels is presented. The panel is comprised of a skin (shell) and stiffeners in both longitudinal (stringers) and circumferential direction (frames). Local buckling modes are considered where the skin may buckle within a bay and may induce rotation of the stiffeners. Stringers and frames are considered as structural elements and are thus not ‘smeared’ onto the skin. Large out-of-plane deflections and thus non-linear strain–displacement relations of skin and stiffeners are taken into account. The displacements of skin and stiffeners are approximated by trigonometric functions (Fourier series). First, a linear buckling eigenvalue analysis is carried out and some combination of buckling eigenmodes is chosen as imperfection. Then the load history is started and the Fourier coefficients are determined by minimizing the stiffened panel’s energy at each load level. A curve-tracing algorithm, the Riks method, is used to solve the equations. The present model can be used to assess the post-buckling behavior of stiffened panels, for example, aircraft fuselage sections. q 2005 Elsevier Ltd. All rights reserved. Keywords: Panel buckling; Post-buckling; Semi-analytical; Stiffened cylindrical shell; Local buckling model; Large deflection theory
1. Motivation Conventional aluminium aircraft fuselage structures consist of stiffened shells. They are usually very thin-walled and therefore susceptible to buckling. Typical designs exhibit skin buckling first, followed by load re-distribution to the stiffeners. When the load is further increased, stiffener tripping and finally collapse occurs. In order to fully exploit structural reserves the post-buckling regime (at loads beyond skin buckling) needs to be considered in the design. 2. Previous work A large number of textbooks on plates and shells is available. Flu¨gge [1], Pflu¨ger [2] and Timoshenko [3] provide a comprehensive background on plate and shell theory. A short overview on publications on buckling loads is given. Bedair [7,8] presents buckling loads for plates that are stiffened in axial and transverse direction. However, the stiffeners are * Corresponding author. Tel.: C49 531 295 2344; fax: C49 531 295 2232. E-mail address: [email protected] (P. Buermann).
0263-8231/$ - see front matter q 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.tws.2005.08.010
‘smeared’ onto the plate and are thus only considered in terms of orthotropic material properties. Hughes [13] and Hu [14] focus on the investigation of stiffener tripping loads. In both papers, only the stiffener alone without the underlying skin is considered. Fujikubo [15] considers a stringer-stiffened plate, where the stringers are considered as structural elements and are thus not ‘smeared’ onto the skin. The interaction of skin and stringers is taken into account in terms of appropriate continuity conditions at their interface. In the following, some publications on the non-linear postbuckling behavior of stiffened and un-stiffened plates are presented. All of them employ non-linear strain–displacement relations that account for large out-of-plane plate deflections. Among the first publications on the non-linear large deflection post-buckling behavior of un-stiffened rectangular plates with linear elastic material behavior are the publications by Marguerre [4], Kromm and Marguerre [5] and Levy [6]. Paik [12] captures the non-linear large deflection response of un-stiffened plates by an incremental Galerkin method, an approach that had been outlined before by Ueda [9]. In two other publications [10,11], Paik considers stiffened plates. However, the stiffeners are not considered as structural elements, but are ‘smeared’ out on the plating. Thus, the stiffened plates are modelled as orthotropic un-stiffened plates.
P. Buermann et al. / Thin-Walled Structures 44 (2006) 102–114
Byklum’s publications [16–18] treat stringer-stiffened plates with large deflections. The stringers are considered as structural elements and are not ‘smeared’ onto the plating but considered as structural elements. This is done by choosing appropriate deflection functions in the form of Fourier series for both the skin and the stringers. The frames are considered through boundary conditions. The energy of the structure is formulated and the equations are solved by minimizing the system’s energy using an arc-length method [19,20].
103
frame ‘b’
stringer ‘b’ stringer ‘a’
frame‘a’
0 .5 a
z, w
frame doubler ‘b’ a x, u
3. Local buckling model
y, v 0 .5b
In this paper, the concept presented in [16,18] is extended. Whereas Byklum models only the stringers as structural elements, in the present paper, both stringers and frames and thus stiffeners in both longitudinal and circumferential direction are considered as structural elements. Furthermore, cylindrically curved stiffened panels are considered whereas Byklum’s model only deals with stiffened plates. When the stiffener tripping load is reached, the free upper edge of the stiffener deflects perpendicular to the direction of the load. Stiffener tripping typically occurs at loads beyond the skin buckling load. By introducing shape functions for the stiffeners, stiffener tripping is captured. In this paper, local buckling modes are considered, that is, skin buckling within a bay and skin-induced stiffener rotation are investigated. The stiffeners themselves are not allowed to deflect in out-of-plane direction. Global buckling, that is, buckling across several bays, is not considered here and is subject to future work. The model presented is limited to linear-elastic material behavior. In the present paper, imperfect shell structures are considered. In reality, all structures are imperfect. Furthermore, by including imperfections, the non-linear problem can be formulated and solved in a numerically efficient way. 3.1. A representative panel Since only local buckling modes are considered in the model, it is assumed that buckling of a large structure such as a fuselage may be described by a representative panel like the one depicted in Fig. 1. It is comprised of a skin of dimensions 2a!2b and thickness h, two stringers (in x-direction) at yZ0, b, two frames (in y-direction) at xZ0, a and skin doublers beneath both stringers and frames. Continuity in terms of in-plane displacements u(x, y) and v(x, y) and out-of-plane displacement w(x, y) must be ensured at all edges. In out-of-plane direction, C(1)-continuity is ensured, while in-plane continuity is achieved by requiring the structure’s edges to remain straight. Deflection functions that meet these requirements are given in Section 3.3.
frame doubler ‘a’ 0 .5 a
stringer b doubler ‘b’
0 .5b
stringer doubler ‘a’
Fig. 1. A typical stiffened structure. The bay is of dimensions a!b.
large deflections. They can be given as
F;xx C F;xx ðw C w0 Þ;yy K2F;xy ðw C w0 Þ;xy R i C F;yy ðw C w0 Þ;xx
KDDw Zh
h DDF ZE w2;xy Kw;xx w;yy C 2w;xy w0;xy ðw C w0 Þ;xx Kw;xx w0;yy Kw;yy w0;xx K R
(1)
(2)
where w(x, y) is the shell out-of-plane deflection and F(x, y) is a potential function that is related to the in-plane membrane stresses by sxx Z F;yy ;
syy Z F;xx ;
txy ZKF;xy :
(3)
When isotropic, linear-elastic material behavior is considered, the material law can be given as sxx Z
E ð3x C n3y Þ; 1Kn2
syy Z
E ð3y C n3x Þ; 1Kn2
(4)
txy Z Ggxy where E is the Young’s modulus, n is the Poisson’s ratio and GZE/(2(1Cn)) is the shear modulus. Since large deflections are considered, non-linear strain-displacement relations [1–3] are employed 1 3x Z u;x C w2;x C w;x w0;x ; 2 w 1 3y Z v;y K C w2;y C w;y w0;y ; R 2
(5)
3.2. Large deflection shell theory
gxy Z u;y C v;x C w;x w;y C w;x w0;y C w0;x w;y :
The large deflection shell theory presented in this section is valid for thin, slightly curved shells. The Donnell shell equations govern the behavior of such a shell undergoing
The Donnell Eqs. (1) and (2) can also be found in [2,16,18]. Since there is no general analytical solution, some approximation must be found. First, Eq. (2) is considered.
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P. Buermann et al. / Thin-Walled Structures 44 (2006) 102–114
A set of appropriate shape functions w(x, y) and w0(x, y) in the form of Fourier series wðx; yÞ Z
M X N X
Amn gm ðxÞgn ðyÞ
(6)
mZ1 nZ1
w0 ðx; yÞ Z
M X N X
Bmn gm ðxÞgn ðyÞ
(7)
mZ1 nZ1
is selected and is introduced in Eq. (2). A potential function F(x, y) satisfying Eq. (2) is found XX Fðx; yÞ Z fmn ðAmn ; Bmn Þhm ðxÞhn ðyÞ; (8) m
n
Fig. 2. Local buckling modes for weak (a) and heavy (b) stiffeners.
where the coefficients fmn depend on the Fourier coefficients in w(x, y) and w0(x, y), Amn and Bmn, respectively. The second Donnell Eq. (1) is not solved directly. Instead, the total energy of the system is stated and the principle of minimum potential energy is used to find a solution. This is equivalent to solving Eq. (1) directly [16,18], but much more convenient and numerically efficient. The total energy of the stiffened panel P can be given as P Z U KT
(9)
where U is the internal energy of the structure and T is the potential of the external loads, respectively. It is emphasized that both U and T and thus P depend on the fourier coefficients Amn and Bmn. However, since the Fourier coefficients of the imperfection are chosen more or less arbitrarily, they are not considered as unknowns. The system is at equilibrium when the first variation vanishes !
dP Z 0:
(10)
Since both U and T can be expressed by the unknown Fourier coefficients, the variational problem can be reformulated by using the Rayleigh–Ritz method vP ! dP Z dAmn Z 0: vAmn
(11)
functions for both simply supported and clamped boundary conditions. They can be given as wðx; yÞ Z wss ðx; yÞ C wcc ðx; yÞ where wss ðx; yÞ Z
(12)
It will be shown in the following sections that P is of 4th order in unknown Fourier coefficients Amn. Consequently, Eq. (12) is of 3rd order in unknown Fourier coefficients Amn and thus an appropriate numerical method must be found to solve the equations. An incremental arc length method, the Riks method, was used to solve Eq. (12). The Riks method is not described here due to lack of space. The reader is referred to [16,18] or [19,20].
Mss X Nss X
Ass mn sin
mZ1 nZ1
wcc ðx;yÞ Z
h mpx i h npy i sin a b
(14)
2mpx 2npy Acc 1Kcos 1Kcos 4 mn a b mZ1 nZ1 Mcc X Ncc X 1
(15) It can be concluded that C(1)-continuity is satisfied along all edges of the structure when a stiffened panel of dimensions K(a/2)%x%(3a/2) and K(b/2)%y%(3b/2) such as the one depicted in Fig. 1 is considered. The shell imperfections were chosen accordingly cc w0 ðx; yÞ Z wss 0 ðx; yÞ C w0 ðx; yÞ
(16)
where
and thus vP ! Z 0: vAmn
(13)
wss 0 ðx; yÞ Z
Mss X Nss X
Bss mn sin
mZ1 nZ1
wcc 0 ðx;yÞZ
h mpx i h npy i sin a b
(17)
2mpx 2npy Bcc 1Kcos 1Kcos : 4 mn a b mZ1 nZ1 Mcc X Ncc X 1
(18)
3.3. Skin shape functions
The shape functions w(x, y) and w0(x, y) from Eqs. (13) and (16) are introduced into Eq. (1). After re-arranging and comparing coefficients a closed form solution for the potential function F(x, y) can be given as
In order to allow for a wide range of deflection patterns (weak/heavy stiffeners, Fig. 2) it was decided to include shape
Fðx;yÞZF ss ðx;yÞCF cc ðx;yÞCF sscc ðx;yÞ
(19)
P. Buermann et al. / Thin-Walled Structures 44 (2006) 102–114 2 2 bcc pqrsZCpqrsKp s ; 8 ððpCrZmÞhðqCsZnÞÞ > > < gððjpKrjZmÞhðjqKsjZnÞhðpsmÞhðssnÞÞ > > : gððjpKrjZmÞhðjqKsjZnÞhðpZmÞhðsZnÞÞ
where 2Mss X 2Nss X
F ss ðx;yÞZ
ss fmn cos
mZ0 nZ0 Mss X Nss X
C
h mpx i h npy i cos a b
gss mn sin
h mpx i a
mZ1 nZ1
cc
F ðx;yÞZ
2Mcc X 2Ncc X
cc fmn cos
mZ0 nZ0 Mcc X Ncc X
C
gcc mn cos
MssX C2Mcc NssX C2Ncc mZ0
h npy i b
(20)
nZ0
2mpx 2npy cos a b
h mpx i h npy i sscc fmn sin sin a b
(21)
(22)
2
2
X
2
Rp ðb m Ca n Þ
r;s
E
X
ss ss css rs ðArs CBrs Þ;
(
cc CAcc rs Bpq Þ;
2 2 bcc pqrs ZK2p s ;
(25) X
Ea2 b4 2
2
2 2 2
2
Rp ð4b m C4a n Þ
cc cc ccc rs ðArs CBrs Þ;
(26)
cc CAss rs Bpq Þ
gððjpKrjZmÞhðjqKsjsnÞhðpZmÞhðsZnÞÞ
2 2 bcc pqrs ZK4p s ; ðjpKrjsmÞhðjqKsjsnÞhðpZmÞhðsZnÞ
bcc pqrs Z0;
else
2 2 2 2 bsscc pqrs ZKq r C2pqrsKp s ; 8 ðð2pKr Z mÞhð2qKs Z nÞhðpsrÞhðqssÞÞ > > < gððrK2p Z mÞhðsK2q ZnÞÞ > > : gðð2pCr Z mÞhð2qCs Z nÞÞ
(27)
ss cc sscc where fð0Þð0Þ Zfð0Þð0Þ Zfð0Þð0Þ Z0 and ( ððjpKrjZmÞhðjqKsjZnÞÞ 2 2 bss pqrs ZpqrsKr q ; gððpCrZmÞhðqCsZnÞÞ
(
bss pqrs Z0; else
ððjpKrjsmÞhðjqKsjZnÞhðpZmÞhðsZnÞÞ
r;s
X E sscc cc ss cc ss fmn Z bsscc 2 pqrs ðApq Ars CApq Brs 2b 2a p;q;r;s 4 m Cn a b
2 2 bss pqrs ZpqrsCr q ;
> > gððpCrZmÞhðjqKsjsnÞhðsZnÞÞ > > > > : gððjpKrjZmÞhðjqKsjsnÞhðpsmÞhðsZnÞÞ
(24)
cc cc cc cc cc fmn Z bcc 2 pqrs ðApq Ars CApq Brs 2b 2a p;q;r;s 4 4m C4n a b
gcc mn Z
2 2 bcc pqrsZCpqrsCp s ; ( ððjpKrjZmÞhðjqKsjZnÞhðpZmÞhðssnÞÞ
2 2 bcc pqrsZC2p s ; 8 ððjpKrjsmÞhðqCsZnÞhðpZmÞÞÞ > > > > > >
(23)
2 2 2
gððpCrZmÞhðjqKsjZnÞhðsZnÞÞ
gððjpKrjZmÞhðjqKsjZnÞhðpsmÞhðsZnÞÞ
ss CAss rs Bpq Þ;
Ea2 b4
gððpCrZmÞhðjqKsjZnÞhðssnÞÞ
( ððjpKrjZmÞhðqCsZnÞhðpZmÞÞ
2 2 bcc pqrsZKpqrsCp s ;
ss cc cc sscc , gss The operators fmn mn , fmn , gmn and fmn are X E ss ss ss ss ss Z bss fmn 2 pqrs ðApq Ars CApq Brs b a p;q;r;s 4 m2 Cn2 a b
gss mn Z
( ððjpKrjZmÞhðqCsZnÞhðpsmÞÞ
2 2 bcc pqrsZKpqrsKp s ;
2mpx 2npy cos a b
mZ1 nZ0
F sscc ðx;yÞZ
sin
105
ððjpKrjZmÞhðqCsZnÞÞ gððpCrZmÞhðjqKsjZnÞÞ
2 2 2 2 bsscc pqrs ZCq r K2pqrs Cp s ; ( ðð2pKr ZmÞhðsK2q Z nÞhðpsrÞÞ
gððrK2p Z mÞhð2qKs Z nÞhðqssÞÞ 2 2 2 2 bsscc pqrs ZCq r C2pqrs Cp s ; ( ðð2pCr Z mÞhð2qKs ZnÞhðqssÞÞ
gðð2pKr Z mÞhð2qCs Z nÞhðpsrÞÞ ( 2 2 2 2 bsscc pqrs ZKq r K2pqrsKp s ;
ðð2pCr Z mÞhðsK2q Z nÞÞ gððrK2p Z mÞhð2qCs Z nÞÞ
106
P. Buermann et al. / Thin-Walled Structures 44 (2006) 102–114
2 2 2 2 bsscc pqrs ZK3q r C2pqrsKp s ;
Txy
ð2pKr ZmÞhð2qKs Z nÞhðp ZrÞhðqssÞ Tx 2 2 2 2 bsscc pqrs ZKq r C2pqrsK3p s ;
ð2pKr ZmÞhð2qKs Z nÞhðpsrÞhðq Z sÞ 2 2 2 2 bsscc pqrs ZK3q r C2pqrsK3p s ;
Ty
z
ð2pKr ZmÞhð2qKs Z nÞhðp ZrÞhðq ZsÞ x
Ty
y 2 2 2 2 bsscc pqrs ZC3q r K2pqrsCp s ;
ð2pKr ZmÞhðsK2qZnÞhðpZrÞ Tx
2 2 2 2 bsscc pqrs ZCq r K2pqrsC3p s ;
Txy
ðrK2pZmÞhð2qKsZnÞhðqZsÞ Fig. 3. Panel under external loads (defined positive in compression).
2 2 2 2 bsscc pqrs ZCq r C2pqrsC3p s ;
ð2pCrZmÞhð2qKsZnÞhðqZsÞ
2 2 2 2 bsscc pqrs ZC3q r C2pqrsCp s ; ð2pKrZmÞhð2qCsZnÞhðpZrÞ
2 2 bsscc pqrs ZC2q r ;
ððr ZmÞhðsK2qZnÞhðpsrÞÞ
2 2 bsscc pqrs ZK2p s ; ðsZnÞhð2pKrZmÞhðqssÞ
2 2 bsscc pqrs ZC2p s ;
(30)
For both stringer and both frames independent shape functions are selected. The selection of the shape functions is explained here for stringer ‘a’. As for the skin, it is assumed that the deflection of the stringer web is a combination of mode shapes
gððrZmÞhð2qCsZnÞhðpsrÞÞ
(
D m D m D sm xx ZKTx CF;yy ; syy ZKTy CF;xx ; txy ZKTxy KF;xy :
3.4. Stiffener shape functions
2 2 bsscc pqrs ZK2q r ; ðrZmÞhð2qKsZnÞhðpsrÞ
(
which satisfies the compatibility condition (2). According to Eq. (3), the modified membrane stresses can then be given as
va ðx; zÞ Z va1 ðx; zÞ C va2 ðx; zÞ C/
(31)
where va1 ðxZ const:; zÞ and va2 ðxZ const:; zÞ are depicted in Fig. 4. va1 represents pure rotation (torsion) of the stringer while va2 is associated with stringer web bending. In x-direction a sinusoidal function is chosen. The shape function for stringer web ‘a’ can be given as h Mss X z a pz mpx i a a v ðx; zÞ Z V C 1Kcos : V sin 1m 2m dwx 2dwx a mZ1
ððsZnÞhðrK2pZmÞhðqssÞÞ gððsZnÞhð2pCrZmÞhðqssÞÞ
bsscc pqrs Z0; else 2 css rs Zr ; ðrZmÞhðsZnÞ
(32) This shape function can also be found in the literature [15, 16,18] and is justified by non-linear FE-analysis. It was found that the present choice of va ðx; zÞ captures the two most important effects of stiffener deflection: stringer torsion and stringer bending. Like the skin, the stringers are assumed to be
css rs Z0; else 2 ccc rs ZCp ; ðpZmÞhðqZnÞ
2 ccc rs ZKp ; ðpZmÞhðnZ0Þ
ccc rs Z0; else In Fig. 3, the external loads that act on the panel are depicted. No external loads have been considered so far. They will be considered on a macroscopic level. Following Levy’s and Byklum’s approach [6,16,18], a modified potential function Fðx;yÞZF F ðx;yÞCF D ðx;yÞ is introduced where F F ðx;yÞZK
Tx 2 Ty 2 y K x CTxy xy 2 2
F D ðx;yÞZF ss ðx;yÞCF cc ðx;yÞCF sscc ðx;yÞ
(28) (29)
Fig. 4. Stringer deflection shapes va1 ðx; zÞ and va2 ðx; zÞ (left). Stringer geometry and frame geometry (right).
P. Buermann et al. / Thin-Walled Structures 44 (2006) 102–114
107
Mb Mt σb y0 Mt z
Mb
y,v(x,z)
x
Fig. 5. Un-deformed stringer (left) and deformed stringer (right).
initially imperfect. The stringer imperfection can be given as h Mss X z a pz mpx i a va0 ðx; zÞ Z V C 1Kcos : V 01m 02m sin x x dw 2dw a mZ1 (33) Continuity in terms of a right angle between skin and stiffeners must be enforced at their connection, that is, the relations
dw
dva
dw0
dva0
(34) K Z and K Z
dy yZ0 dz zZ0 dy yZ0 dz z¼0 a a must be satisfied. It follows that the coefficients V1m and V01m are not independent and thus do not enter the system as unknowns. For stringer ‘b’ at yZb, the same shape functions as for stringer ‘a’ are chosen (Eq. (32)), but with independent coefficients. For the frames ‘a’ and ‘b’, according shape functions are selected, that is, a torsional mode and a bending mode are superposed. As for the stringers, a right angle at the interface of frame and skin must be ensured. For stringer doublers ‘a’ and ‘b’ and frame doublers ‘a’ and ‘b’, no independent shape functions are chosen. Instead, they are modelled as Saint-Venant torsion bars.
3.5. Stress and strain assumptions In this section, the stresses and strains for all structural elements of the panel are given. The superscripts ‘sk’ stand for ‘skin’, ‘st’ for ‘stringer’, ‘fr’ for ‘frame’, ‘dst’ for ‘skin doubler beneath the stringer’ and ‘dfr’ for ‘skin doubler beneath the frame’. It is assumed that all stresses (and consequently, all strains) may be divided into a contribution from membrane stretching (superscript ‘m’) and a contribution from bending (superscript ‘b’). Thus stresses and strains can be re-written as ð,Þ;m ð,Þ;b sð,Þ ð,Þ Z sð,Þ C sð,Þ
4
where w(x, y) is defined in Eq. (13). Using Hooke’s law (4) the skin bending stresses can be found. For the stringer and frame webs (superscript ‘w’), the bending stresses are assumed to behave accordingly—a twodimensional bending stress state about the neutral plane of the stringer web is assumed. Thus, for the stringers, the bending strains are given as ZKyva;xx ; 3st;w;b x
3st;w;b ZKyva;zz ; z
gst;w;b ZK2yva;xz xz
(36)
for aZa, b and the bending stresses are once again found using Hooke’s law. For the frame webs, the strains are found accordingly. No independent shape functions are selected for the stringer flanges and the frame flanges. Instead, the flange deflections are governed by the respective web deflections. In Fig. 5, a stringer and the assumed bending stress distribution in the flange is depicted. When the stringer web is deflected by v(x, z), a bending moment Mb (about z in the xyplane) and a torsional moment Mt (about x in the yz-plane) is acting on the stringer flange. Due to the un-symmetric ‘L’-shape of the stringer, the bending moment Mb causes skew bending, that is, bending about the axes of principal moments of inertia h and z, as depicted in Fig. 6. Point ‘S’ is the center of gravity of the stringer, located at yZy0 from the neutral plane of the stringer web. In the model, the effects of skew bending are neglected and instead, plane bending about an axis parallel to the z-axis y0
S
η
S
ð,Þ;m ð,Þ;b 3ð,Þ ð,Þ Z 3ð,Þ C 3ð,Þ
Mb ς
3.5.1. Bending stresses and strains The skin is taken as a two-dimensional structural element and thus strain components in x- and y-direction and in-plane shear must be accounted for. The bending strains are 3sk;b x
ZKzw;xx ;
3sk;b y
ZKzw;yy ;
gsk;b xy
ZK2zw;xy
(35)
z
z
y
y
Fig. 6. Skew bending of the stringer (left); assumed plane bending (right).
108
P. Buermann et al. / Thin-Walled Structures 44 (2006) 102–114
and located at yZy0 is considered. This assumption is depicted in Fig. 6. Thus, the strains due to Euler–Bernoulli bending and SaintVenant torsion can be given as ;b 3st;f ZKðyKy0 Þva;xx jzZdwx ; x
3yst;f ;b Z 0; (37)
;b gst;f ZK2zva;xz jzZdwx yz
where aZa, b. An uni-axial stress distribution is assumed and thus the stresses are given as st; f ;b f ;b sxx Z E3st; ; x
st; f ;b syy Z 0;
f ;b f ;b tst; Z Ggst; yz yz
(38)
For the frame flanges, analogous stresses and strains are taken. As for the stringer and frame flanges, there are no independent shape functions for the stringer and frame doublers. It is assumed that their deflection is governed by the skin shape functions, that is, the doublers exhibit the same deflections as the skin. Consequently, the doublers are modelled as Saint-Venant torsion bars. Since only local buckling modes are considered there is no bending in the doublers. However, there exists an angle of twist w,xy causing a torsional strain in the doublers. For the stringer doublers, the strains are given as 3xdst; f ; b Z 3ydst; f ; b Z 0;
f;b gdst; ZK2zw;xy jyZ0;b xy
(39)
and thus the stress components are f ;b sdst; x
Z sydst; f ;b
Z 0;
dst; f ;b txy
f ;b Z Ggdst; xy
(40)
3.5.2. Membrane stresses and strains It is assumed that all membrane stresses (and consequently, all membrane strains) may be divided into a part due to deflection, indicated by the superscript ‘D’, and a part due to external force, indicated by the superscript ‘F’ [16,18]. As will be seen, the ‘D’-parts depend on the deformed state of the structure in terms of the Fourier coefficients, while the ‘F’-parts depend on the load state. They can be given as 4
ð,Þ;D ð,Þ;F 3ð,Þ;m ð,Þ Z 3ð,Þ C 3ð,Þ
Since the ‘F’-parts only depend on the load state, they are taken as constant. First, the skin membrane stresses and strains are discussed. Using Hooke’s law (4) and Eq. (30), the skin stress state can be given as ssk;m xx
(41)
E D ð3sk;F C n3sk;F y x Þ C F;xx ; 1Kn2
(42)
D tsk;m xy ZKTxy KF;xy
(44)
3st;m Z gst;m z xz Z 0 This is justified for two reasons: firstly, the stringers are typically very slender and thus an uni-axial stress state may be assumed. Secondly and more important, the stringers are neither constrained nor loaded in z-direction and thus the membrane stresses and strains developing in z-direction are negligible. Following Byklum [18] the assumption 3st;F Z ust;x ; x
1 3st;D Z ðva;x Þ2 C va;x va0;x x 2
(45)
is made. For the stringer membrane stresses and strains, no distinction between web and flange needs to be made since v(x, z) is defined on the entire stringer. The stringer stresses are thus (46)
The frames are treated analogously, and thus no explicit results are given here. For the stringer doublers and the frame doublers, a uni-axial membrane stress distribution is assumed, too. When the stresses are expressed through the strains (Eq. (4)), the stringer doubler membrane stress is dst;D sdst;m Z sdst;F Z Eð3dst;F C 3xdst;D Þ xx xx C sxx x
(47)
where 3dst;D Z x
1 D D ðF KnF;xx ÞjyZ0;b E ;yy
(48)
The stresses and strains for the frame doublers are found analogously. st;F In this section, the yet unknown strains 3sk;F and 3dst;F x , 3x x sk;F fr;F dfr;F and accordingly 3y , 3y and 3y have been introduced. They can be calculated by using additional relations, as outlined in the next section. 3.6. Connecting the stiffeners to the skin
E D Z ð3sk;F C n3sk;F x y Þ C F;yy ; 1Kn2
ssk;m yy Z
1 st;D Z 3st;F Z ust;x C ðva;x Þ2 C va;x va0;x ; 3st;m x x C 3x 2
st;F st;D st;F st;D sst;m xx Z sxx C sxx Z Eð3x C 3x Þ
Similarly, the results for the frame doublers are found.
ð,Þ;D ð,Þ;F sð,Þ;m ð,Þ Z sð,Þ C sð,Þ
Unlike the skin (which is supported along all edges), the stiffener has a free edge and thus there exists no potential function for the stringer and frame webs [18]. Instead, it was decided to express the membrane stresses in stringers and frames through the corresponding membrane strains which are related to the respective out-of-plane displacements. For reasons of simplification, it is assumed that all stringer membrane strains other than the one in longitudinal direction vanish [18], that is,
(43)
As mentioned earlier and depicted in Fig. 3, the skin may be loaded in-plane by the normal stresses Tx and Ty and a shear stress Txy. When the skin buckling load is exceeded, the load is internally redistributed across the structure, that is, the stiffeners attract the load. This must be accounted for by two requirements:
P. Buermann et al. / Thin-Walled Structures 44 (2006) 102–114
† the global force equilibrium must be satisfied in x- and y-direction, that is, the external loads on the skin and the internal reaction forces of skin and stiffeners must be at equilibrium † both longitudinal and transverse continuity of skin and stiffeners must be ensured, that is, skin and stiffeners are stretched/ compressed at the same rate The force equilibrium can be given as ð3=2Þb ð h=2 ð
C
sdst;a xx dAdst
C
C
ð
ssk yy dzdx C
sfr;a yy dAfr C
sst;b xx dAst
ZK2bhTx
cx
C
Adfr
K2ð1KnÞðw;xx w;yy Kw2;xy Þdydx where Ksk is the shell bending stiffness. The membrane term is
ð
(54)
(49)
For the stringers, the internal energy can be given as U st Z Ubst;a C Ubst;b C Umst , where the bending energy is sfr;b yy dAfr
K Ubst;a Z st
Afr
ð
x ð3=2Þa ð dðw
2
½ðva;xx Cva;zz Þ2 K2ð1KnÞðva;xx va;zz Kðva;xz Þ2 Þdzdx
Ka=2 0
sdfr;b yy dAdfr
ZK2ahTy
c y;
(50)
EIfx C 2
Adfr
Integration is carried out over the respective cross sections, where Ast Z dwx twx C dfx tfx , Afr Z dwy twy C dfy tfy , Adst Z ddst tdst and Adfr Z ddfr tdst . Here, stringer doublers and frame doublers are of equal thickness. The second requirement states that in-plane displacements of skin and stiffeners in axial and circumferential direction must be equal, that is, Dusk Z Dust;a Z Dust;b Z Dudst;a Z Dudst;b Z Du Z
ð3=2Þa ð
uð,Þ ;x dx
Ka=2
(51)
ð3=2Þa ð
GJfx ½va;xx jzZdwx 2 dxC
Dv Z Dv
fr;a
Z Dv
fr;b
Z Dv
dfr;a
Z Dv
dfr;b
ð3=2Þb ð
Z Dv Z
ð3=2Þa ð
2
Ka=2
½va;xz jzZdwx 2 dx
Ka=2
(55) Ifx
for aZa, b. Kst is the web bending stiffness, is the flange bending stiffness and Jfx is the flange torsional stiffness, respectively. The flange stress assumptions from Eqs. (37) and (38) have been used. The stringer membrane energy is 1 sq 2 st;F Umst Z 2aEAst ð3st;F x Þ C E3x Ist C EIst 2
(56)
where ð ð 1 a 2 1 b 2 ðv;x Þ C va;x va0;x dVst C ðv;x Þ C vb;x vb0;x dVst Ist Z 2 2 Vst
sk
(53)
Ka=2 Kb=2
Ka=2 Kb=2
sdst;b xx dAdst
Afr
sdfr;a yy dAdfr
½ðw;xx C w;yy Þ2
2abhE sk;F 2 1Kn sk;F 2 2 sk;F sk;F ½ð3x Þ Cð3sk;F ðgxy Þ y Þ C2n3x 3y C 2 1Kn2 ð3=2Þa ð ð ð3=2Þb h D D 2 D D D 2 ½ðF;xx CF;yy Þ K2ð1CnÞðF;xx F;yy KðFxy Þ Þdydx C 2E
Adst
Ka=2 Kh=2
ð
ð
ð3=2Þa ð ð ð3=2Þb
K Z sk 2
Umsk Z
Ast
ð
Adst
ð3=2Þa ð h=2 ð
sst;a xx dAst C
Ast
Kb=2 Kh=2
ð
ð
ssk xx dzdy C
Ubsk
109
Vst
(57) vð,Þ ;y dy Istsq Z
Kb=2
(52) When the strains from the previous section are inserted in Eqs. (51) and (52), additional relations for 3xst;a;F and 3dst;F , and x accordingly for 3yfr;a;F and 3dfr;F are found. The two remaining y unknown strains 3sk;F and 3sk;F are determined from solving x y Eqs. (49) and (50). Then, all stresses and strains can be expressed through the unknown Fourier coefficients and the energy of the structure can be provided.
ð
1 a 2 ðv Þ Cva;x va0;x 2 ;x
2
ð dVst C
Vst
1 b 2 ðv Þ Cvb;x vb0;x 2 ;x
2 dVst
Vst
(58) fr
and VstZ2aAst. The frame energy U is obtained analogously. The stringer doubler energy can be given as U dst ZUbdst;a CUbdst;b CUmdst where GJ Ubdst;a Z dst
ð3=2Þa ð
2
½w;xy jyZ0;b 2 dx;
(59)
Ka=2
3.7. The internal energy
is the bending term for aZa, b. Jdst is the doubler torsional stiffness. The membrane term is
The skin energy can be given as U sk Z Ubsk C Umsk . The bending energy is
1 sq 2 Umdst Z 2aEAdst ð3dst;F Þ CE3dst;F Idst C EIdst x x 2
(60)
110
P. Buermann et al. / Thin-Walled Structures 44 (2006) 102–114
with ð Idst Z
D D
F;yy KnF;xx
E
Vdst
ð C Vdst
sq Z Idst
ð
(a/2)%x%(3a/2) and K(b/2)%y%3(b/2) to 0%x%a and 0%y%b since the integral in Eq. (67) vanishes for the former limits of integration. More details can be found in [18]. The contribution of the lateral pressure, Tp can be given as
dVdst yZ0
D D
F;yy KnF;xx
dVdst
E yZb
(61)
Tp Z pl
w dydx
(68)
Ka=2 Kb=2
2
32
D D 6 F;yy KnF;xx
4
ð
2
E
32
D D 6 F;yy KnF;xx
7
yZ0 5 dVdst C 4
Vdst
Vdst
E
7
yZb 5 dVdst
(62) dfr
and VdstZ2aAdst. The energy of the frame doublers, U , is obtained analogously and the total internal energy can be given as UZU CU CU CU CU : sk
ð3=2Þa ð ð3=2Þb ð
st
fr
dst
dfr
(63)
3.8. The potential of external loads The potential of external loads T can be written as the sum of three components. T Z Tc C Tt C Tp
(64)
Tc is the contribution from axial and transversal compression, Tt is the contribution from shear loading and Tp is the energy due to a lateral pressure pl (not depicted in Fig. 3), respectively. The lateral pressure is assumed to act on the panel’s skin and is defined positive in positive z-direction. The potential of external loads due to compression can be given as ð3=2Þa ð ð ð3=2Þb
1 3sk x K
Tc ZKTx h
2
w2;x Kw;x w0;x
dydx
where only the shape functions for clamped boundary conditions wcc(x, y) give a contribution due to their quilt-like deflection pattern. 3.9. Refined flange model The stringer flanges and the frame flanges are both modelled as Euler–Bernoulli beams and Saint-Venant torsion bars. However, non-linear elastic FE-analyses show that the free edge of the flange is sensitive to buckling, that is, flange tripping may occur. In order to predict the onset of flange tripping, a refined flange model is presented here. Its derivation is outlined here for the stringer flange. In the refined flange model, the flange is modelled as a plate that is loaded by a linearly varying compression stress " Tx ðyÞ Z lfl
# y Tx1 C x ðTx2 KTx1 Þ df
where lfl is a load parameter. The flange is simply supported along the edges xZ0, a and yZ0 and free along the edge yZ dfx . It is depicted in Fig. 7. For the refined flange, a shape function similar to that for the web is chosen:
Ka=2 Kb=2 ð3=2Þa ð ð ð3=2Þb
KTy h
1 2 w C w0 3sk dydx y K w;y Kw;y w0;y C 2 R
(69)
wðx; yÞ Z
X m
"
" #! # h mpx i y py V C 1Kcos : V sin 4m dfx 3m 2dfx a
Ka=2 Kb=2
(70) (65)
The contribution of the shear stress, Tt, requires some discussion. Since the stringer doublers and the frame doublers are located in the same plane as the skin, they absorb part of the shear energy. This is accounted for by ‘smearing’ the doublers onto the skin. An equivalent skin thickness h0 Z
4abh C 2addst tdst C 2bddfr tdst h 4abh
In analogy to the stringer web bending energy, the flange bending energy can be given as x
Ub Z
Kfl 2
ð3=2Þa ð dðf
ðw;xx C w;yy Þ2 K2ð1KnÞðw;xx w;yy Kw2;xy Þdydx
Ka=2 0
(66)
(71)
is calculated and is used instead of the skin thickness h. It was shown by Byklum [18] that the potential of external loads due to shear can be given as
where Kfl Z Eðtfx Þ3 =ð12ð1Kn2 ÞÞ is the flange bending stiffness. Timoshenko [3] gives the potential of external loads as
Tt Z 4Txy h 0
ða ðb
ðw;x w;y C w;x w0;y C w0;x w;y Þdydx;
(67)
0 0
where only the sine-type shape functions wss(x, y) give a contribution. The limits of integration have been changed from
Fig. 7. The stringer flange seen from above.
P. Buermann et al. / Thin-Walled Structures 44 (2006) 102–114 Table 1 Bay dimensions in mm
x
tfx TZ 2
ð3=2Þa ð dðf
Tx ðyÞw2;x dydx
111
(72) Panel /
1a
1b
2a
2b
3a
3b
a b
400 200
400 400
600 200
600 400
800 200
800 400
Ka=2 0
where Tx(y) is defined in Eq. (69). The stresses Tx1 and Tx2 are the actual internal stresses in the flange at the current load step. They are calculated by superposing the membrane and bending stresses in the flange according to the stress and strain assumptions from Section 3.5. Then the energy of the refined flange can be stated and a buckling eigenvalue analysis is carried out. A buckling eigenvalue lfl,cr is calculated and compared to the actual value of l at the current load step. Since Tx1 and Tx2 are different for each l, so is lfl,cr. When l exceeds lfl,cr, flange tripping is assumed to occur and thus the flange is deflecting is a way that cannot be described by the present model.
4. Implementation and results The present model was implemented in the computer code ‘IBUCK’ developed at the Institute of Structural Mechanics at DLR Braunschweig. The code was written in the C programming language. Standard clapack routines from the LAPACK (Linear Algebra PACKage) package [21] were used to carry out the eigenvalue analysis and for solving the equations at each load step. Computations were carried out for structures of six different geometries. For all computations, an aluminium structure (EZ64935 MPa, nZ0.34) was considered. Linear elastic material with a yield stress of sfZ435 MPa was assumed. The shell radius was set to RZ2000 mm. A total of six different stiffened panels was investigated. They are depicted in Fig. 8 and their dimensions can be taken from Table 1. All other geometric quantities are specified in Table 2. All panels were loaded axially by a compressive stress of T x Z 2000 MPa, where the load parameter l was increased from 0 to 1. As depicted in Fig. 3, the load was applied to the skin only. Consequently, the mean load acting on the entire cross section of the panel is less than the load on the skin. The mean load can be given as Tx Z bh=ðbhC Ast C Adst ÞT x . In the plots, it is additionally scaled by the yield stress sf. The FE analysis was carried out with ABAQUS 6.4.1. The structure was modelled using S4 shell elements with 10 mm
Table 2 Panel geometry h
dwx
dfx
twx
tfx
dwy
dfy
2.7
60
20
3.5
3.5
160 35
twy
tfy
ddst
ddfr
tdst
tdfr
2.5
2.5
40
40
2.3
2.3
All dimensions are in mm.
edge length. In ABAQUS, the equations were solved using the damped Newton method (*static,stabilize) with a damping parameter of 5!10K5. The lowest five buckling eigenmodes with a magnitude of 0.1 h each were chosen as imperfection. The material was modelled as linear-elastic. 4.1. Panels with aspect ratio a/bO2 In this section, the results for panels with aspect ratio a/bO2 (panels 1a, 2a and 3a) are presented. It will be seen that IBUCK is able to produce fast and reliable results at low computational costs. The load–displacement curve for panel 2a is given in Fig. 9. Point A indicates the skin buckling load. When the load is further increased, the stringer tripping load is reached (point B), and the free edge of the stringer undergoes large deflections. For the stringer geometry considered here, the stringer tripping mode is associated with a larger wavelength than the skin buckling mode. In the deep post-buckling range (point C), the skin has adopted a 6!1 buckling pattern while the stringer is tripping in a 2-wave pattern, cf. Fig. 10. The horizontal dashed line and the horizontal solid line in Fig. 9 indicate the flange tripping load as calculated by IBUCK and ABAQUS, respectively. It can be seen that 2 ABAQUS IBUCK
1.75 C 1.5 1.25
ABAQUS flange tripping IBUCK flange tripping
1
B 0.75 0.5 0.25
A
–30
Fig. 8. Panels investigated.
–25
–20
–15
–10
Fig. 9. Load–deflection curve for panel 2a.
–5
0
112
P. Buermann et al. / Thin-Walled Structures 44 (2006) 102–114 Table 3 Computation summary for panels with aspect ratio a/bO2 Panel
ABAQUS
1a 2a 3a
Fig. 10. Panel 2a, deep post-buckling range (point C in Fig. 9): IBUCK result and ABAQUS result.
the load–displacement behavior of panel 2a is captured very well by IBUCK. Although IBUCK over-estimates the stiffness of the structure at loads beyond the flange tripping load, good correspondence with non-linear FE can be achieved by including a large enough number of degrees of freedom in the model. For panels 1a and 3a, the correspondence between IBUCK and ABAQUS is also very good. The corresponding loaddisplacement curves are given in Fig. 11. Although no plots of 2 1.75
ABAQUS IBUCK
C
1.5
ABAQUS flange tripping
1.25 IBUCK flange tripping
1
B
0.75 0.5 0.25
A –20
–15
–10
–5
IBUCK
Advantage
DOF
Time (s)
DOF
Time (s)
40,470 55,110 69,750
7052 8267 16,620
50 56 68
193 352 742
97.26% 95.74% 95.49%
the deformed state are presented here, the analyses showed that the deflection patterns of IBUCK and ABAQUS corresponded very well. For panels 1a, 2a and 3a, the computational costs for ABAQUS and IBUCK are compared in Table 3. 4.2. Panels with aspect ratio a/b!2 Here, the results for panels with aspect ratio a/b!2 (panels 1b, 2b and 3b) are presented. It will be seen that for these panels, IBUCK cannot be fast and reliable at the same time but produces too stiff results when the number of degrees of freedom is selected too small. The load–displacement curve for panel 2b is given in Fig. 12. It can be seen that two different IBUCK results are depicted, one for a small model with 44 degrees of freedom (solid line), and one for a large model with 161 degrees of freedom (dashed line). For the small IBUCK model (solid line), the correspondence between IBUCK and ABAQUS is not satisfying in the deep post-buckling range (point C): the predicted results are too stiff. However, it was found that by increasing the number of Fourier terms in IBUCK, better results can be achieved at the cost of much higher computation times: the dashed line in Fig. 12 corresponds well with the ABAQUS solution. In the deep post-buckling range, broad bays tend to deflect in a short-wave pattern that requires many Fourier terms to describe it. The deflection pattern is at short wavelengths and it
0 1.6
2 1.75
C
ABAQUS IBUCK (DOF=44) - - IBUCK (DOF=161)
1.4
ABAQUS IBUCK
C
1.2
1.5 1
1.25
IBUCK flange tripping
ABAQUS flange tripping
0.8
IBUCK flange tripping
1
ABAQUS flange tripping B
B
0.6
0.75
0.4
0.5
A
0.2
0.25
A –40
–30
–20
–10
Fig. 11. Load–deflection curve for panel 1a and 3a.
0
–40
–30
–20
–10
Fig. 12. Load–deflection curve for panel 2b.
0
P. Buermann et al. / Thin-Walled Structures 44 (2006) 102–114
113
5. Concluding remarks
1.6 1.4
ABAQUS IBUCK (DOF=44) - - IBUCK (DOF=136)
C
1.2 1
IBUCK flange tripping ABAQUS flange tripping
0.8
B 0.6 0.4 0.2
A –30
–25
–20
–15
–10
–5
0
1.6 ABAQUS IBUCK (DOF=44) - - IBUCK (DOF=161)
1.4
C 1.2 1
IBUCK flange tripping
0.8
ABAQUS flange tripping
B
0.6 0.4 0.2
A
–50
–40
–30
–20
–10
0
Fig. 13. Load–deflection curve for panel 1b and 3b.
is thus very costly to approximate by semi-analytical methods based on Fourier deflection functions. The load–displacement curves for panels 1b and 3b are given in Fig. 13. As for panel 2b, the correspondence between IBUCK and ABAQUS is not satisfying for the small IBUCK models but very good for the large models. For panels 1b, 2b and 3b, the computational costs for ABAQUS and IBUCK are compared in Table 4. It can be seen that the computational costs for IBUCK increase dramatically when larger models with more than 100 degrees of freedom are considered.
In the present publication, IBUCK, a fast semi-analytical post-buckling tool developed at DLR, is presented. IBUCK was developed for the design process of stiffened shells. The effects of local skin buckling and stiffener tripping are captured. In the stiffeners, a uni-axial membrane stress state is assumed while plane stress state is assumed for the skin. Linear-elastic material is considered. Shape functions are selected for the skin and the stiffener webs, that is, for stringers and frames. The stiffener flanges are considered as Euler–Bernoulli beams and Saint-Venant torsion bars while the skin doublers beneath the stiffeners are modelled as Saint-Venant torsion bars. Since the flange’s bending beam/torsion bar formulation does not provide information about flange tripping, the critical flange tripping load is determined in an external procedure. For validation, six different panels were loaded axially. FEresults (ABAQUS 6.4.1) were compared to the results obtained with IBUCK. It was found that IBUCK performed very well for panels with aspect ratios a/bO2 (panels 1a, 2a and 3a). These panels exhibited a regular, long-wave post-buckling behavior that could be represented by a relatively small number of Fourier terms. Thus, IBUCK finished orders of magnitude faster than ABAQUS. For stiffened panels with aspect ratio a/b!2 (panels 1b, 2b and 3b), the post-buckling deflection pattern proved to be rather irregular and of short wavelengths, thus requiring a large number (more than 100) Fourier terms and resulting in long IBUCK computation times. However, correspondence between IBUCK and ABAQUS was very good when a large enough number of Fourier terms was considered. It is very likely that the high computational costs for models with more than 100 Fourier terms could be reduced by implementing a more efficient memory management. However, at this time, this is not the focus of this work. Acknowledgements The present model was developed as part of a sponsorship contract between Airbus Germany and DLR. All support is gratefully acknowledged. References
Table 4 Computation summary for panels with aspect ratio a/b!2 Panel
1b 2b 3b
ABAQUS
IBUCK
Advantage
DOF
Time (s)
DOF
Time (s)
70,470 94,710 118,950
8928 38,592 19,333
44 (136) 56 (161) 68 (161)
207 (7622) 247 (16,547) 568 (14,832)
97.68 (14.63) 99.36 (57.12) 97.06 (23.28)
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A semi-analytical model for local post-buckling analysis of stringer- and frame-stiffened cylindrical panels Philipp Buermann a,*, Raimund Rolfes b, Jan Tessmer b, Martin Schagerl b a
DLR, Institute of Structural Mechanics, Structural Analysis Section, Lilienthalplatz 7, 38108 Braunschweig, Germany b Airbus Deutschland GmbH, ESAS, Kreetslag 10, 21129 Hamburg, Germany Received 3 February 2005; accepted 11 August 2005 Available online 14 November 2005
Abstract A fast semi-analytical model for the post-buckling analysis of stiffened cylindrical panels is presented. The panel is comprised of a skin (shell) and stiffeners in both longitudinal (stringers) and circumferential direction (frames). Local buckling modes are considered where the skin may buckle within a bay and may induce rotation of the stiffeners. Stringers and frames are considered as structural elements and are thus not ‘smeared’ onto the skin. Large out-of-plane deflections and thus non-linear strain–displacement relations of skin and stiffeners are taken into account. The displacements of skin and stiffeners are approximated by trigonometric functions (Fourier series). First, a linear buckling eigenvalue analysis is carried out and some combination of buckling eigenmodes is chosen as imperfection. Then the load history is started and the Fourier coefficients are determined by minimizing the stiffened panel’s energy at each load level. A curve-tracing algorithm, the Riks method, is used to solve the equations. The present model can be used to assess the post-buckling behavior of stiffened panels, for example, aircraft fuselage sections. q 2005 Elsevier Ltd. All rights reserved. Keywords: Panel buckling; Post-buckling; Semi-analytical; Stiffened cylindrical shell; Local buckling model; Large deflection theory
1. Motivation Conventional aluminium aircraft fuselage structures consist of stiffened shells. They are usually very thin-walled and therefore susceptible to buckling. Typical designs exhibit skin buckling first, followed by load re-distribution to the stiffeners. When the load is further increased, stiffener tripping and finally collapse occurs. In order to fully exploit structural reserves the post-buckling regime (at loads beyond skin buckling) needs to be considered in the design. 2. Previous work A large number of textbooks on plates and shells is available. Flu¨gge [1], Pflu¨ger [2] and Timoshenko [3] provide a comprehensive background on plate and shell theory. A short overview on publications on buckling loads is given. Bedair [7,8] presents buckling loads for plates that are stiffened in axial and transverse direction. However, the stiffeners are * Corresponding author. Tel.: C49 531 295 2344; fax: C49 531 295 2232. E-mail address: [email protected] (P. Buermann).
0263-8231/$ - see front matter q 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.tws.2005.08.010
‘smeared’ onto the plate and are thus only considered in terms of orthotropic material properties. Hughes [13] and Hu [14] focus on the investigation of stiffener tripping loads. In both papers, only the stiffener alone without the underlying skin is considered. Fujikubo [15] considers a stringer-stiffened plate, where the stringers are considered as structural elements and are thus not ‘smeared’ onto the skin. The interaction of skin and stringers is taken into account in terms of appropriate continuity conditions at their interface. In the following, some publications on the non-linear postbuckling behavior of stiffened and un-stiffened plates are presented. All of them employ non-linear strain–displacement relations that account for large out-of-plane plate deflections. Among the first publications on the non-linear large deflection post-buckling behavior of un-stiffened rectangular plates with linear elastic material behavior are the publications by Marguerre [4], Kromm and Marguerre [5] and Levy [6]. Paik [12] captures the non-linear large deflection response of un-stiffened plates by an incremental Galerkin method, an approach that had been outlined before by Ueda [9]. In two other publications [10,11], Paik considers stiffened plates. However, the stiffeners are not considered as structural elements, but are ‘smeared’ out on the plating. Thus, the stiffened plates are modelled as orthotropic un-stiffened plates.
P. Buermann et al. / Thin-Walled Structures 44 (2006) 102–114
Byklum’s publications [16–18] treat stringer-stiffened plates with large deflections. The stringers are considered as structural elements and are not ‘smeared’ onto the plating but considered as structural elements. This is done by choosing appropriate deflection functions in the form of Fourier series for both the skin and the stringers. The frames are considered through boundary conditions. The energy of the structure is formulated and the equations are solved by minimizing the system’s energy using an arc-length method [19,20].
103
frame ‘b’
stringer ‘b’ stringer ‘a’
frame‘a’
0 .5 a
z, w
frame doubler ‘b’ a x, u
3. Local buckling model
y, v 0 .5b
In this paper, the concept presented in [16,18] is extended. Whereas Byklum models only the stringers as structural elements, in the present paper, both stringers and frames and thus stiffeners in both longitudinal and circumferential direction are considered as structural elements. Furthermore, cylindrically curved stiffened panels are considered whereas Byklum’s model only deals with stiffened plates. When the stiffener tripping load is reached, the free upper edge of the stiffener deflects perpendicular to the direction of the load. Stiffener tripping typically occurs at loads beyond the skin buckling load. By introducing shape functions for the stiffeners, stiffener tripping is captured. In this paper, local buckling modes are considered, that is, skin buckling within a bay and skin-induced stiffener rotation are investigated. The stiffeners themselves are not allowed to deflect in out-of-plane direction. Global buckling, that is, buckling across several bays, is not considered here and is subject to future work. The model presented is limited to linear-elastic material behavior. In the present paper, imperfect shell structures are considered. In reality, all structures are imperfect. Furthermore, by including imperfections, the non-linear problem can be formulated and solved in a numerically efficient way. 3.1. A representative panel Since only local buckling modes are considered in the model, it is assumed that buckling of a large structure such as a fuselage may be described by a representative panel like the one depicted in Fig. 1. It is comprised of a skin of dimensions 2a!2b and thickness h, two stringers (in x-direction) at yZ0, b, two frames (in y-direction) at xZ0, a and skin doublers beneath both stringers and frames. Continuity in terms of in-plane displacements u(x, y) and v(x, y) and out-of-plane displacement w(x, y) must be ensured at all edges. In out-of-plane direction, C(1)-continuity is ensured, while in-plane continuity is achieved by requiring the structure’s edges to remain straight. Deflection functions that meet these requirements are given in Section 3.3.
frame doubler ‘a’ 0 .5 a
stringer b doubler ‘b’
0 .5b
stringer doubler ‘a’
Fig. 1. A typical stiffened structure. The bay is of dimensions a!b.
large deflections. They can be given as
F;xx C F;xx ðw C w0 Þ;yy K2F;xy ðw C w0 Þ;xy R i C F;yy ðw C w0 Þ;xx
KDDw Zh
h DDF ZE w2;xy Kw;xx w;yy C 2w;xy w0;xy ðw C w0 Þ;xx Kw;xx w0;yy Kw;yy w0;xx K R
(1)
(2)
where w(x, y) is the shell out-of-plane deflection and F(x, y) is a potential function that is related to the in-plane membrane stresses by sxx Z F;yy ;
syy Z F;xx ;
txy ZKF;xy :
(3)
When isotropic, linear-elastic material behavior is considered, the material law can be given as sxx Z
E ð3x C n3y Þ; 1Kn2
syy Z
E ð3y C n3x Þ; 1Kn2
(4)
txy Z Ggxy where E is the Young’s modulus, n is the Poisson’s ratio and GZE/(2(1Cn)) is the shear modulus. Since large deflections are considered, non-linear strain-displacement relations [1–3] are employed 1 3x Z u;x C w2;x C w;x w0;x ; 2 w 1 3y Z v;y K C w2;y C w;y w0;y ; R 2
(5)
3.2. Large deflection shell theory
gxy Z u;y C v;x C w;x w;y C w;x w0;y C w0;x w;y :
The large deflection shell theory presented in this section is valid for thin, slightly curved shells. The Donnell shell equations govern the behavior of such a shell undergoing
The Donnell Eqs. (1) and (2) can also be found in [2,16,18]. Since there is no general analytical solution, some approximation must be found. First, Eq. (2) is considered.
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P. Buermann et al. / Thin-Walled Structures 44 (2006) 102–114
A set of appropriate shape functions w(x, y) and w0(x, y) in the form of Fourier series wðx; yÞ Z
M X N X
Amn gm ðxÞgn ðyÞ
(6)
mZ1 nZ1
w0 ðx; yÞ Z
M X N X
Bmn gm ðxÞgn ðyÞ
(7)
mZ1 nZ1
is selected and is introduced in Eq. (2). A potential function F(x, y) satisfying Eq. (2) is found XX Fðx; yÞ Z fmn ðAmn ; Bmn Þhm ðxÞhn ðyÞ; (8) m
n
Fig. 2. Local buckling modes for weak (a) and heavy (b) stiffeners.
where the coefficients fmn depend on the Fourier coefficients in w(x, y) and w0(x, y), Amn and Bmn, respectively. The second Donnell Eq. (1) is not solved directly. Instead, the total energy of the system is stated and the principle of minimum potential energy is used to find a solution. This is equivalent to solving Eq. (1) directly [16,18], but much more convenient and numerically efficient. The total energy of the stiffened panel P can be given as P Z U KT
(9)
where U is the internal energy of the structure and T is the potential of the external loads, respectively. It is emphasized that both U and T and thus P depend on the fourier coefficients Amn and Bmn. However, since the Fourier coefficients of the imperfection are chosen more or less arbitrarily, they are not considered as unknowns. The system is at equilibrium when the first variation vanishes !
dP Z 0:
(10)
Since both U and T can be expressed by the unknown Fourier coefficients, the variational problem can be reformulated by using the Rayleigh–Ritz method vP ! dP Z dAmn Z 0: vAmn
(11)
functions for both simply supported and clamped boundary conditions. They can be given as wðx; yÞ Z wss ðx; yÞ C wcc ðx; yÞ where wss ðx; yÞ Z
(12)
It will be shown in the following sections that P is of 4th order in unknown Fourier coefficients Amn. Consequently, Eq. (12) is of 3rd order in unknown Fourier coefficients Amn and thus an appropriate numerical method must be found to solve the equations. An incremental arc length method, the Riks method, was used to solve Eq. (12). The Riks method is not described here due to lack of space. The reader is referred to [16,18] or [19,20].
Mss X Nss X
Ass mn sin
mZ1 nZ1
wcc ðx;yÞ Z
h mpx i h npy i sin a b
(14)
2mpx 2npy Acc 1Kcos 1Kcos 4 mn a b mZ1 nZ1 Mcc X Ncc X 1
(15) It can be concluded that C(1)-continuity is satisfied along all edges of the structure when a stiffened panel of dimensions K(a/2)%x%(3a/2) and K(b/2)%y%(3b/2) such as the one depicted in Fig. 1 is considered. The shell imperfections were chosen accordingly cc w0 ðx; yÞ Z wss 0 ðx; yÞ C w0 ðx; yÞ
(16)
where
and thus vP ! Z 0: vAmn
(13)
wss 0 ðx; yÞ Z
Mss X Nss X
Bss mn sin
mZ1 nZ1
wcc 0 ðx;yÞZ
h mpx i h npy i sin a b
(17)
2mpx 2npy Bcc 1Kcos 1Kcos : 4 mn a b mZ1 nZ1 Mcc X Ncc X 1
(18)
3.3. Skin shape functions
The shape functions w(x, y) and w0(x, y) from Eqs. (13) and (16) are introduced into Eq. (1). After re-arranging and comparing coefficients a closed form solution for the potential function F(x, y) can be given as
In order to allow for a wide range of deflection patterns (weak/heavy stiffeners, Fig. 2) it was decided to include shape
Fðx;yÞZF ss ðx;yÞCF cc ðx;yÞCF sscc ðx;yÞ
(19)
P. Buermann et al. / Thin-Walled Structures 44 (2006) 102–114 2 2 bcc pqrsZCpqrsKp s ; 8 ððpCrZmÞhðqCsZnÞÞ > > < gððjpKrjZmÞhðjqKsjZnÞhðpsmÞhðssnÞÞ > > : gððjpKrjZmÞhðjqKsjZnÞhðpZmÞhðsZnÞÞ
where 2Mss X 2Nss X
F ss ðx;yÞZ
ss fmn cos
mZ0 nZ0 Mss X Nss X
C
h mpx i h npy i cos a b
gss mn sin
h mpx i a
mZ1 nZ1
cc
F ðx;yÞZ
2Mcc X 2Ncc X
cc fmn cos
mZ0 nZ0 Mcc X Ncc X
C
gcc mn cos
MssX C2Mcc NssX C2Ncc mZ0
h npy i b
(20)
nZ0
2mpx 2npy cos a b
h mpx i h npy i sscc fmn sin sin a b
(21)
(22)
2
2
X
2
Rp ðb m Ca n Þ
r;s
E
X
ss ss css rs ðArs CBrs Þ;
(
cc CAcc rs Bpq Þ;
2 2 bcc pqrs ZK2p s ;
(25) X
Ea2 b4 2
2
2 2 2
2
Rp ð4b m C4a n Þ
cc cc ccc rs ðArs CBrs Þ;
(26)
cc CAss rs Bpq Þ
gððjpKrjZmÞhðjqKsjsnÞhðpZmÞhðsZnÞÞ
2 2 bcc pqrs ZK4p s ; ðjpKrjsmÞhðjqKsjsnÞhðpZmÞhðsZnÞ
bcc pqrs Z0;
else
2 2 2 2 bsscc pqrs ZKq r C2pqrsKp s ; 8 ðð2pKr Z mÞhð2qKs Z nÞhðpsrÞhðqssÞÞ > > < gððrK2p Z mÞhðsK2q ZnÞÞ > > : gðð2pCr Z mÞhð2qCs Z nÞÞ
(27)
ss cc sscc where fð0Þð0Þ Zfð0Þð0Þ Zfð0Þð0Þ Z0 and ( ððjpKrjZmÞhðjqKsjZnÞÞ 2 2 bss pqrs ZpqrsKr q ; gððpCrZmÞhðqCsZnÞÞ
(
bss pqrs Z0; else
ððjpKrjsmÞhðjqKsjZnÞhðpZmÞhðsZnÞÞ
r;s
X E sscc cc ss cc ss fmn Z bsscc 2 pqrs ðApq Ars CApq Brs 2b 2a p;q;r;s 4 m Cn a b
2 2 bss pqrs ZpqrsCr q ;
> > gððpCrZmÞhðjqKsjsnÞhðsZnÞÞ > > > > : gððjpKrjZmÞhðjqKsjsnÞhðpsmÞhðsZnÞÞ
(24)
cc cc cc cc cc fmn Z bcc 2 pqrs ðApq Ars CApq Brs 2b 2a p;q;r;s 4 4m C4n a b
gcc mn Z
2 2 bcc pqrsZCpqrsCp s ; ( ððjpKrjZmÞhðjqKsjZnÞhðpZmÞhðssnÞÞ
2 2 bcc pqrsZC2p s ; 8 ððjpKrjsmÞhðqCsZnÞhðpZmÞÞÞ > > > > > >
(23)
2 2 2
gððpCrZmÞhðjqKsjZnÞhðsZnÞÞ
gððjpKrjZmÞhðjqKsjZnÞhðpsmÞhðsZnÞÞ
ss CAss rs Bpq Þ;
Ea2 b4
gððpCrZmÞhðjqKsjZnÞhðssnÞÞ
( ððjpKrjZmÞhðqCsZnÞhðpZmÞÞ
2 2 bcc pqrsZKpqrsCp s ;
ss cc cc sscc , gss The operators fmn mn , fmn , gmn and fmn are X E ss ss ss ss ss Z bss fmn 2 pqrs ðApq Ars CApq Brs b a p;q;r;s 4 m2 Cn2 a b
gss mn Z
( ððjpKrjZmÞhðqCsZnÞhðpsmÞÞ
2 2 bcc pqrsZKpqrsKp s ;
2mpx 2npy cos a b
mZ1 nZ0
F sscc ðx;yÞZ
sin
105
ððjpKrjZmÞhðqCsZnÞÞ gððpCrZmÞhðjqKsjZnÞÞ
2 2 2 2 bsscc pqrs ZCq r K2pqrs Cp s ; ( ðð2pKr ZmÞhðsK2q Z nÞhðpsrÞÞ
gððrK2p Z mÞhð2qKs Z nÞhðqssÞÞ 2 2 2 2 bsscc pqrs ZCq r C2pqrs Cp s ; ( ðð2pCr Z mÞhð2qKs ZnÞhðqssÞÞ
gðð2pKr Z mÞhð2qCs Z nÞhðpsrÞÞ ( 2 2 2 2 bsscc pqrs ZKq r K2pqrsKp s ;
ðð2pCr Z mÞhðsK2q Z nÞÞ gððrK2p Z mÞhð2qCs Z nÞÞ
106
P. Buermann et al. / Thin-Walled Structures 44 (2006) 102–114
2 2 2 2 bsscc pqrs ZK3q r C2pqrsKp s ;
Txy
ð2pKr ZmÞhð2qKs Z nÞhðp ZrÞhðqssÞ Tx 2 2 2 2 bsscc pqrs ZKq r C2pqrsK3p s ;
ð2pKr ZmÞhð2qKs Z nÞhðpsrÞhðq Z sÞ 2 2 2 2 bsscc pqrs ZK3q r C2pqrsK3p s ;
Ty
z
ð2pKr ZmÞhð2qKs Z nÞhðp ZrÞhðq ZsÞ x
Ty
y 2 2 2 2 bsscc pqrs ZC3q r K2pqrsCp s ;
ð2pKr ZmÞhðsK2qZnÞhðpZrÞ Tx
2 2 2 2 bsscc pqrs ZCq r K2pqrsC3p s ;
Txy
ðrK2pZmÞhð2qKsZnÞhðqZsÞ Fig. 3. Panel under external loads (defined positive in compression).
2 2 2 2 bsscc pqrs ZCq r C2pqrsC3p s ;
ð2pCrZmÞhð2qKsZnÞhðqZsÞ
2 2 2 2 bsscc pqrs ZC3q r C2pqrsCp s ; ð2pKrZmÞhð2qCsZnÞhðpZrÞ
2 2 bsscc pqrs ZC2q r ;
ððr ZmÞhðsK2qZnÞhðpsrÞÞ
2 2 bsscc pqrs ZK2p s ; ðsZnÞhð2pKrZmÞhðqssÞ
2 2 bsscc pqrs ZC2p s ;
(30)
For both stringer and both frames independent shape functions are selected. The selection of the shape functions is explained here for stringer ‘a’. As for the skin, it is assumed that the deflection of the stringer web is a combination of mode shapes
gððrZmÞhð2qCsZnÞhðpsrÞÞ
(
D m D m D sm xx ZKTx CF;yy ; syy ZKTy CF;xx ; txy ZKTxy KF;xy :
3.4. Stiffener shape functions
2 2 bsscc pqrs ZK2q r ; ðrZmÞhð2qKsZnÞhðpsrÞ
(
which satisfies the compatibility condition (2). According to Eq. (3), the modified membrane stresses can then be given as
va ðx; zÞ Z va1 ðx; zÞ C va2 ðx; zÞ C/
(31)
where va1 ðxZ const:; zÞ and va2 ðxZ const:; zÞ are depicted in Fig. 4. va1 represents pure rotation (torsion) of the stringer while va2 is associated with stringer web bending. In x-direction a sinusoidal function is chosen. The shape function for stringer web ‘a’ can be given as h Mss X z a pz mpx i a a v ðx; zÞ Z V C 1Kcos : V sin 1m 2m dwx 2dwx a mZ1
ððsZnÞhðrK2pZmÞhðqssÞÞ gððsZnÞhð2pCrZmÞhðqssÞÞ
bsscc pqrs Z0; else 2 css rs Zr ; ðrZmÞhðsZnÞ
(32) This shape function can also be found in the literature [15, 16,18] and is justified by non-linear FE-analysis. It was found that the present choice of va ðx; zÞ captures the two most important effects of stiffener deflection: stringer torsion and stringer bending. Like the skin, the stringers are assumed to be
css rs Z0; else 2 ccc rs ZCp ; ðpZmÞhðqZnÞ
2 ccc rs ZKp ; ðpZmÞhðnZ0Þ
ccc rs Z0; else In Fig. 3, the external loads that act on the panel are depicted. No external loads have been considered so far. They will be considered on a macroscopic level. Following Levy’s and Byklum’s approach [6,16,18], a modified potential function Fðx;yÞZF F ðx;yÞCF D ðx;yÞ is introduced where F F ðx;yÞZK
Tx 2 Ty 2 y K x CTxy xy 2 2
F D ðx;yÞZF ss ðx;yÞCF cc ðx;yÞCF sscc ðx;yÞ
(28) (29)
Fig. 4. Stringer deflection shapes va1 ðx; zÞ and va2 ðx; zÞ (left). Stringer geometry and frame geometry (right).
P. Buermann et al. / Thin-Walled Structures 44 (2006) 102–114
107
Mb Mt σb y0 Mt z
Mb
y,v(x,z)
x
Fig. 5. Un-deformed stringer (left) and deformed stringer (right).
initially imperfect. The stringer imperfection can be given as h Mss X z a pz mpx i a va0 ðx; zÞ Z V C 1Kcos : V 01m 02m sin x x dw 2dw a mZ1 (33) Continuity in terms of a right angle between skin and stiffeners must be enforced at their connection, that is, the relations
dw
dva
dw0
dva0
(34) K Z and K Z
dy yZ0 dz zZ0 dy yZ0 dz z¼0 a a must be satisfied. It follows that the coefficients V1m and V01m are not independent and thus do not enter the system as unknowns. For stringer ‘b’ at yZb, the same shape functions as for stringer ‘a’ are chosen (Eq. (32)), but with independent coefficients. For the frames ‘a’ and ‘b’, according shape functions are selected, that is, a torsional mode and a bending mode are superposed. As for the stringers, a right angle at the interface of frame and skin must be ensured. For stringer doublers ‘a’ and ‘b’ and frame doublers ‘a’ and ‘b’, no independent shape functions are chosen. Instead, they are modelled as Saint-Venant torsion bars.
3.5. Stress and strain assumptions In this section, the stresses and strains for all structural elements of the panel are given. The superscripts ‘sk’ stand for ‘skin’, ‘st’ for ‘stringer’, ‘fr’ for ‘frame’, ‘dst’ for ‘skin doubler beneath the stringer’ and ‘dfr’ for ‘skin doubler beneath the frame’. It is assumed that all stresses (and consequently, all strains) may be divided into a contribution from membrane stretching (superscript ‘m’) and a contribution from bending (superscript ‘b’). Thus stresses and strains can be re-written as ð,Þ;m ð,Þ;b sð,Þ ð,Þ Z sð,Þ C sð,Þ
4
where w(x, y) is defined in Eq. (13). Using Hooke’s law (4) the skin bending stresses can be found. For the stringer and frame webs (superscript ‘w’), the bending stresses are assumed to behave accordingly—a twodimensional bending stress state about the neutral plane of the stringer web is assumed. Thus, for the stringers, the bending strains are given as ZKyva;xx ; 3st;w;b x
3st;w;b ZKyva;zz ; z
gst;w;b ZK2yva;xz xz
(36)
for aZa, b and the bending stresses are once again found using Hooke’s law. For the frame webs, the strains are found accordingly. No independent shape functions are selected for the stringer flanges and the frame flanges. Instead, the flange deflections are governed by the respective web deflections. In Fig. 5, a stringer and the assumed bending stress distribution in the flange is depicted. When the stringer web is deflected by v(x, z), a bending moment Mb (about z in the xyplane) and a torsional moment Mt (about x in the yz-plane) is acting on the stringer flange. Due to the un-symmetric ‘L’-shape of the stringer, the bending moment Mb causes skew bending, that is, bending about the axes of principal moments of inertia h and z, as depicted in Fig. 6. Point ‘S’ is the center of gravity of the stringer, located at yZy0 from the neutral plane of the stringer web. In the model, the effects of skew bending are neglected and instead, plane bending about an axis parallel to the z-axis y0
S
η
S
ð,Þ;m ð,Þ;b 3ð,Þ ð,Þ Z 3ð,Þ C 3ð,Þ
Mb ς
3.5.1. Bending stresses and strains The skin is taken as a two-dimensional structural element and thus strain components in x- and y-direction and in-plane shear must be accounted for. The bending strains are 3sk;b x
ZKzw;xx ;
3sk;b y
ZKzw;yy ;
gsk;b xy
ZK2zw;xy
(35)
z
z
y
y
Fig. 6. Skew bending of the stringer (left); assumed plane bending (right).
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P. Buermann et al. / Thin-Walled Structures 44 (2006) 102–114
and located at yZy0 is considered. This assumption is depicted in Fig. 6. Thus, the strains due to Euler–Bernoulli bending and SaintVenant torsion can be given as ;b 3st;f ZKðyKy0 Þva;xx jzZdwx ; x
3yst;f ;b Z 0; (37)
;b gst;f ZK2zva;xz jzZdwx yz
where aZa, b. An uni-axial stress distribution is assumed and thus the stresses are given as st; f ;b f ;b sxx Z E3st; ; x
st; f ;b syy Z 0;
f ;b f ;b tst; Z Ggst; yz yz
(38)
For the frame flanges, analogous stresses and strains are taken. As for the stringer and frame flanges, there are no independent shape functions for the stringer and frame doublers. It is assumed that their deflection is governed by the skin shape functions, that is, the doublers exhibit the same deflections as the skin. Consequently, the doublers are modelled as Saint-Venant torsion bars. Since only local buckling modes are considered there is no bending in the doublers. However, there exists an angle of twist w,xy causing a torsional strain in the doublers. For the stringer doublers, the strains are given as 3xdst; f ; b Z 3ydst; f ; b Z 0;
f;b gdst; ZK2zw;xy jyZ0;b xy
(39)
and thus the stress components are f ;b sdst; x
Z sydst; f ;b
Z 0;
dst; f ;b txy
f ;b Z Ggdst; xy
(40)
3.5.2. Membrane stresses and strains It is assumed that all membrane stresses (and consequently, all membrane strains) may be divided into a part due to deflection, indicated by the superscript ‘D’, and a part due to external force, indicated by the superscript ‘F’ [16,18]. As will be seen, the ‘D’-parts depend on the deformed state of the structure in terms of the Fourier coefficients, while the ‘F’-parts depend on the load state. They can be given as 4
ð,Þ;D ð,Þ;F 3ð,Þ;m ð,Þ Z 3ð,Þ C 3ð,Þ
Since the ‘F’-parts only depend on the load state, they are taken as constant. First, the skin membrane stresses and strains are discussed. Using Hooke’s law (4) and Eq. (30), the skin stress state can be given as ssk;m xx
(41)
E D ð3sk;F C n3sk;F y x Þ C F;xx ; 1Kn2
(42)
D tsk;m xy ZKTxy KF;xy
(44)
3st;m Z gst;m z xz Z 0 This is justified for two reasons: firstly, the stringers are typically very slender and thus an uni-axial stress state may be assumed. Secondly and more important, the stringers are neither constrained nor loaded in z-direction and thus the membrane stresses and strains developing in z-direction are negligible. Following Byklum [18] the assumption 3st;F Z ust;x ; x
1 3st;D Z ðva;x Þ2 C va;x va0;x x 2
(45)
is made. For the stringer membrane stresses and strains, no distinction between web and flange needs to be made since v(x, z) is defined on the entire stringer. The stringer stresses are thus (46)
The frames are treated analogously, and thus no explicit results are given here. For the stringer doublers and the frame doublers, a uni-axial membrane stress distribution is assumed, too. When the stresses are expressed through the strains (Eq. (4)), the stringer doubler membrane stress is dst;D sdst;m Z sdst;F Z Eð3dst;F C 3xdst;D Þ xx xx C sxx x
(47)
where 3dst;D Z x
1 D D ðF KnF;xx ÞjyZ0;b E ;yy
(48)
The stresses and strains for the frame doublers are found analogously. st;F In this section, the yet unknown strains 3sk;F and 3dst;F x , 3x x sk;F fr;F dfr;F and accordingly 3y , 3y and 3y have been introduced. They can be calculated by using additional relations, as outlined in the next section. 3.6. Connecting the stiffeners to the skin
E D Z ð3sk;F C n3sk;F x y Þ C F;yy ; 1Kn2
ssk;m yy Z
1 st;D Z 3st;F Z ust;x C ðva;x Þ2 C va;x va0;x ; 3st;m x x C 3x 2
st;F st;D st;F st;D sst;m xx Z sxx C sxx Z Eð3x C 3x Þ
Similarly, the results for the frame doublers are found.
ð,Þ;D ð,Þ;F sð,Þ;m ð,Þ Z sð,Þ C sð,Þ
Unlike the skin (which is supported along all edges), the stiffener has a free edge and thus there exists no potential function for the stringer and frame webs [18]. Instead, it was decided to express the membrane stresses in stringers and frames through the corresponding membrane strains which are related to the respective out-of-plane displacements. For reasons of simplification, it is assumed that all stringer membrane strains other than the one in longitudinal direction vanish [18], that is,
(43)
As mentioned earlier and depicted in Fig. 3, the skin may be loaded in-plane by the normal stresses Tx and Ty and a shear stress Txy. When the skin buckling load is exceeded, the load is internally redistributed across the structure, that is, the stiffeners attract the load. This must be accounted for by two requirements:
P. Buermann et al. / Thin-Walled Structures 44 (2006) 102–114
† the global force equilibrium must be satisfied in x- and y-direction, that is, the external loads on the skin and the internal reaction forces of skin and stiffeners must be at equilibrium † both longitudinal and transverse continuity of skin and stiffeners must be ensured, that is, skin and stiffeners are stretched/ compressed at the same rate The force equilibrium can be given as ð3=2Þb ð h=2 ð
C
sdst;a xx dAdst
C
C
ð
ssk yy dzdx C
sfr;a yy dAfr C
sst;b xx dAst
ZK2bhTx
cx
C
Adfr
K2ð1KnÞðw;xx w;yy Kw2;xy Þdydx where Ksk is the shell bending stiffness. The membrane term is
ð
(54)
(49)
For the stringers, the internal energy can be given as U st Z Ubst;a C Ubst;b C Umst , where the bending energy is sfr;b yy dAfr
K Ubst;a Z st
Afr
ð
x ð3=2Þa ð dðw
2
½ðva;xx Cva;zz Þ2 K2ð1KnÞðva;xx va;zz Kðva;xz Þ2 Þdzdx
Ka=2 0
sdfr;b yy dAdfr
ZK2ahTy
c y;
(50)
EIfx C 2
Adfr
Integration is carried out over the respective cross sections, where Ast Z dwx twx C dfx tfx , Afr Z dwy twy C dfy tfy , Adst Z ddst tdst and Adfr Z ddfr tdst . Here, stringer doublers and frame doublers are of equal thickness. The second requirement states that in-plane displacements of skin and stiffeners in axial and circumferential direction must be equal, that is, Dusk Z Dust;a Z Dust;b Z Dudst;a Z Dudst;b Z Du Z
ð3=2Þa ð
uð,Þ ;x dx
Ka=2
(51)
ð3=2Þa ð
GJfx ½va;xx jzZdwx 2 dxC
Dv Z Dv
fr;a
Z Dv
fr;b
Z Dv
dfr;a
Z Dv
dfr;b
ð3=2Þb ð
Z Dv Z
ð3=2Þa ð
2
Ka=2
½va;xz jzZdwx 2 dx
Ka=2
(55) Ifx
for aZa, b. Kst is the web bending stiffness, is the flange bending stiffness and Jfx is the flange torsional stiffness, respectively. The flange stress assumptions from Eqs. (37) and (38) have been used. The stringer membrane energy is 1 sq 2 st;F Umst Z 2aEAst ð3st;F x Þ C E3x Ist C EIst 2
(56)
where ð ð 1 a 2 1 b 2 ðv;x Þ C va;x va0;x dVst C ðv;x Þ C vb;x vb0;x dVst Ist Z 2 2 Vst
sk
(53)
Ka=2 Kb=2
Ka=2 Kb=2
sdst;b xx dAdst
Afr
sdfr;a yy dAdfr
½ðw;xx C w;yy Þ2
2abhE sk;F 2 1Kn sk;F 2 2 sk;F sk;F ½ð3x Þ Cð3sk;F ðgxy Þ y Þ C2n3x 3y C 2 1Kn2 ð3=2Þa ð ð ð3=2Þb h D D 2 D D D 2 ½ðF;xx CF;yy Þ K2ð1CnÞðF;xx F;yy KðFxy Þ Þdydx C 2E
Adst
Ka=2 Kh=2
ð
ð
ð3=2Þa ð ð ð3=2Þb
K Z sk 2
Umsk Z
Ast
ð
Adst
ð3=2Þa ð h=2 ð
sst;a xx dAst C
Ast
Kb=2 Kh=2
ð
ð
ssk xx dzdy C
Ubsk
109
Vst
(57) vð,Þ ;y dy Istsq Z
Kb=2
(52) When the strains from the previous section are inserted in Eqs. (51) and (52), additional relations for 3xst;a;F and 3dst;F , and x accordingly for 3yfr;a;F and 3dfr;F are found. The two remaining y unknown strains 3sk;F and 3sk;F are determined from solving x y Eqs. (49) and (50). Then, all stresses and strains can be expressed through the unknown Fourier coefficients and the energy of the structure can be provided.
ð
1 a 2 ðv Þ Cva;x va0;x 2 ;x
2
ð dVst C
Vst
1 b 2 ðv Þ Cvb;x vb0;x 2 ;x
2 dVst
Vst
(58) fr
and VstZ2aAst. The frame energy U is obtained analogously. The stringer doubler energy can be given as U dst ZUbdst;a CUbdst;b CUmdst where GJ Ubdst;a Z dst
ð3=2Þa ð
2
½w;xy jyZ0;b 2 dx;
(59)
Ka=2
3.7. The internal energy
is the bending term for aZa, b. Jdst is the doubler torsional stiffness. The membrane term is
The skin energy can be given as U sk Z Ubsk C Umsk . The bending energy is
1 sq 2 Umdst Z 2aEAdst ð3dst;F Þ CE3dst;F Idst C EIdst x x 2
(60)
110
P. Buermann et al. / Thin-Walled Structures 44 (2006) 102–114
with ð Idst Z
D D
F;yy KnF;xx
E
Vdst
ð C Vdst
sq Z Idst
ð
(a/2)%x%(3a/2) and K(b/2)%y%3(b/2) to 0%x%a and 0%y%b since the integral in Eq. (67) vanishes for the former limits of integration. More details can be found in [18]. The contribution of the lateral pressure, Tp can be given as
dVdst yZ0
D D
F;yy KnF;xx
dVdst
E yZb
(61)
Tp Z pl
w dydx
(68)
Ka=2 Kb=2
2
32
D D 6 F;yy KnF;xx
4
ð
2
E
32
D D 6 F;yy KnF;xx
7
yZ0 5 dVdst C 4
Vdst
Vdst
E
7
yZb 5 dVdst
(62) dfr
and VdstZ2aAdst. The energy of the frame doublers, U , is obtained analogously and the total internal energy can be given as UZU CU CU CU CU : sk
ð3=2Þa ð ð3=2Þb ð
st
fr
dst
dfr
(63)
3.8. The potential of external loads The potential of external loads T can be written as the sum of three components. T Z Tc C Tt C Tp
(64)
Tc is the contribution from axial and transversal compression, Tt is the contribution from shear loading and Tp is the energy due to a lateral pressure pl (not depicted in Fig. 3), respectively. The lateral pressure is assumed to act on the panel’s skin and is defined positive in positive z-direction. The potential of external loads due to compression can be given as ð3=2Þa ð ð ð3=2Þb
1 3sk x K
Tc ZKTx h
2
w2;x Kw;x w0;x
dydx
where only the shape functions for clamped boundary conditions wcc(x, y) give a contribution due to their quilt-like deflection pattern. 3.9. Refined flange model The stringer flanges and the frame flanges are both modelled as Euler–Bernoulli beams and Saint-Venant torsion bars. However, non-linear elastic FE-analyses show that the free edge of the flange is sensitive to buckling, that is, flange tripping may occur. In order to predict the onset of flange tripping, a refined flange model is presented here. Its derivation is outlined here for the stringer flange. In the refined flange model, the flange is modelled as a plate that is loaded by a linearly varying compression stress " Tx ðyÞ Z lfl
# y Tx1 C x ðTx2 KTx1 Þ df
where lfl is a load parameter. The flange is simply supported along the edges xZ0, a and yZ0 and free along the edge yZ dfx . It is depicted in Fig. 7. For the refined flange, a shape function similar to that for the web is chosen:
Ka=2 Kb=2 ð3=2Þa ð ð ð3=2Þb
KTy h
1 2 w C w0 3sk dydx y K w;y Kw;y w0;y C 2 R
(69)
wðx; yÞ Z
X m
"
" #! # h mpx i y py V C 1Kcos : V sin 4m dfx 3m 2dfx a
Ka=2 Kb=2
(70) (65)
The contribution of the shear stress, Tt, requires some discussion. Since the stringer doublers and the frame doublers are located in the same plane as the skin, they absorb part of the shear energy. This is accounted for by ‘smearing’ the doublers onto the skin. An equivalent skin thickness h0 Z
4abh C 2addst tdst C 2bddfr tdst h 4abh
In analogy to the stringer web bending energy, the flange bending energy can be given as x
Ub Z
Kfl 2
ð3=2Þa ð dðf
ðw;xx C w;yy Þ2 K2ð1KnÞðw;xx w;yy Kw2;xy Þdydx
Ka=2 0
(66)
(71)
is calculated and is used instead of the skin thickness h. It was shown by Byklum [18] that the potential of external loads due to shear can be given as
where Kfl Z Eðtfx Þ3 =ð12ð1Kn2 ÞÞ is the flange bending stiffness. Timoshenko [3] gives the potential of external loads as
Tt Z 4Txy h 0
ða ðb
ðw;x w;y C w;x w0;y C w0;x w;y Þdydx;
(67)
0 0
where only the sine-type shape functions wss(x, y) give a contribution. The limits of integration have been changed from
Fig. 7. The stringer flange seen from above.
P. Buermann et al. / Thin-Walled Structures 44 (2006) 102–114 Table 1 Bay dimensions in mm
x
tfx TZ 2
ð3=2Þa ð dðf
Tx ðyÞw2;x dydx
111
(72) Panel /
1a
1b
2a
2b
3a
3b
a b
400 200
400 400
600 200
600 400
800 200
800 400
Ka=2 0
where Tx(y) is defined in Eq. (69). The stresses Tx1 and Tx2 are the actual internal stresses in the flange at the current load step. They are calculated by superposing the membrane and bending stresses in the flange according to the stress and strain assumptions from Section 3.5. Then the energy of the refined flange can be stated and a buckling eigenvalue analysis is carried out. A buckling eigenvalue lfl,cr is calculated and compared to the actual value of l at the current load step. Since Tx1 and Tx2 are different for each l, so is lfl,cr. When l exceeds lfl,cr, flange tripping is assumed to occur and thus the flange is deflecting is a way that cannot be described by the present model.
4. Implementation and results The present model was implemented in the computer code ‘IBUCK’ developed at the Institute of Structural Mechanics at DLR Braunschweig. The code was written in the C programming language. Standard clapack routines from the LAPACK (Linear Algebra PACKage) package [21] were used to carry out the eigenvalue analysis and for solving the equations at each load step. Computations were carried out for structures of six different geometries. For all computations, an aluminium structure (EZ64935 MPa, nZ0.34) was considered. Linear elastic material with a yield stress of sfZ435 MPa was assumed. The shell radius was set to RZ2000 mm. A total of six different stiffened panels was investigated. They are depicted in Fig. 8 and their dimensions can be taken from Table 1. All other geometric quantities are specified in Table 2. All panels were loaded axially by a compressive stress of T x Z 2000 MPa, where the load parameter l was increased from 0 to 1. As depicted in Fig. 3, the load was applied to the skin only. Consequently, the mean load acting on the entire cross section of the panel is less than the load on the skin. The mean load can be given as Tx Z bh=ðbhC Ast C Adst ÞT x . In the plots, it is additionally scaled by the yield stress sf. The FE analysis was carried out with ABAQUS 6.4.1. The structure was modelled using S4 shell elements with 10 mm
Table 2 Panel geometry h
dwx
dfx
twx
tfx
dwy
dfy
2.7
60
20
3.5
3.5
160 35
twy
tfy
ddst
ddfr
tdst
tdfr
2.5
2.5
40
40
2.3
2.3
All dimensions are in mm.
edge length. In ABAQUS, the equations were solved using the damped Newton method (*static,stabilize) with a damping parameter of 5!10K5. The lowest five buckling eigenmodes with a magnitude of 0.1 h each were chosen as imperfection. The material was modelled as linear-elastic. 4.1. Panels with aspect ratio a/bO2 In this section, the results for panels with aspect ratio a/bO2 (panels 1a, 2a and 3a) are presented. It will be seen that IBUCK is able to produce fast and reliable results at low computational costs. The load–displacement curve for panel 2a is given in Fig. 9. Point A indicates the skin buckling load. When the load is further increased, the stringer tripping load is reached (point B), and the free edge of the stringer undergoes large deflections. For the stringer geometry considered here, the stringer tripping mode is associated with a larger wavelength than the skin buckling mode. In the deep post-buckling range (point C), the skin has adopted a 6!1 buckling pattern while the stringer is tripping in a 2-wave pattern, cf. Fig. 10. The horizontal dashed line and the horizontal solid line in Fig. 9 indicate the flange tripping load as calculated by IBUCK and ABAQUS, respectively. It can be seen that 2 ABAQUS IBUCK
1.75 C 1.5 1.25
ABAQUS flange tripping IBUCK flange tripping
1
B 0.75 0.5 0.25
A
–30
Fig. 8. Panels investigated.
–25
–20
–15
–10
Fig. 9. Load–deflection curve for panel 2a.
–5
0
112
P. Buermann et al. / Thin-Walled Structures 44 (2006) 102–114 Table 3 Computation summary for panels with aspect ratio a/bO2 Panel
ABAQUS
1a 2a 3a
Fig. 10. Panel 2a, deep post-buckling range (point C in Fig. 9): IBUCK result and ABAQUS result.
the load–displacement behavior of panel 2a is captured very well by IBUCK. Although IBUCK over-estimates the stiffness of the structure at loads beyond the flange tripping load, good correspondence with non-linear FE can be achieved by including a large enough number of degrees of freedom in the model. For panels 1a and 3a, the correspondence between IBUCK and ABAQUS is also very good. The corresponding loaddisplacement curves are given in Fig. 11. Although no plots of 2 1.75
ABAQUS IBUCK
C
1.5
ABAQUS flange tripping
1.25 IBUCK flange tripping
1
B
0.75 0.5 0.25
A –20
–15
–10
–5
IBUCK
Advantage
DOF
Time (s)
DOF
Time (s)
40,470 55,110 69,750
7052 8267 16,620
50 56 68
193 352 742
97.26% 95.74% 95.49%
the deformed state are presented here, the analyses showed that the deflection patterns of IBUCK and ABAQUS corresponded very well. For panels 1a, 2a and 3a, the computational costs for ABAQUS and IBUCK are compared in Table 3. 4.2. Panels with aspect ratio a/b!2 Here, the results for panels with aspect ratio a/b!2 (panels 1b, 2b and 3b) are presented. It will be seen that for these panels, IBUCK cannot be fast and reliable at the same time but produces too stiff results when the number of degrees of freedom is selected too small. The load–displacement curve for panel 2b is given in Fig. 12. It can be seen that two different IBUCK results are depicted, one for a small model with 44 degrees of freedom (solid line), and one for a large model with 161 degrees of freedom (dashed line). For the small IBUCK model (solid line), the correspondence between IBUCK and ABAQUS is not satisfying in the deep post-buckling range (point C): the predicted results are too stiff. However, it was found that by increasing the number of Fourier terms in IBUCK, better results can be achieved at the cost of much higher computation times: the dashed line in Fig. 12 corresponds well with the ABAQUS solution. In the deep post-buckling range, broad bays tend to deflect in a short-wave pattern that requires many Fourier terms to describe it. The deflection pattern is at short wavelengths and it
0 1.6
2 1.75
C
ABAQUS IBUCK (DOF=44) - - IBUCK (DOF=161)
1.4
ABAQUS IBUCK
C
1.2
1.5 1
1.25
IBUCK flange tripping
ABAQUS flange tripping
0.8
IBUCK flange tripping
1
ABAQUS flange tripping B
B
0.6
0.75
0.4
0.5
A
0.2
0.25
A –40
–30
–20
–10
Fig. 11. Load–deflection curve for panel 1a and 3a.
0
–40
–30
–20
–10
Fig. 12. Load–deflection curve for panel 2b.
0
P. Buermann et al. / Thin-Walled Structures 44 (2006) 102–114
113
5. Concluding remarks
1.6 1.4
ABAQUS IBUCK (DOF=44) - - IBUCK (DOF=136)
C
1.2 1
IBUCK flange tripping ABAQUS flange tripping
0.8
B 0.6 0.4 0.2
A –30
–25
–20
–15
–10
–5
0
1.6 ABAQUS IBUCK (DOF=44) - - IBUCK (DOF=161)
1.4
C 1.2 1
IBUCK flange tripping
0.8
ABAQUS flange tripping
B
0.6 0.4 0.2
A
–50
–40
–30
–20
–10
0
Fig. 13. Load–deflection curve for panel 1b and 3b.
is thus very costly to approximate by semi-analytical methods based on Fourier deflection functions. The load–displacement curves for panels 1b and 3b are given in Fig. 13. As for panel 2b, the correspondence between IBUCK and ABAQUS is not satisfying for the small IBUCK models but very good for the large models. For panels 1b, 2b and 3b, the computational costs for ABAQUS and IBUCK are compared in Table 4. It can be seen that the computational costs for IBUCK increase dramatically when larger models with more than 100 degrees of freedom are considered.
In the present publication, IBUCK, a fast semi-analytical post-buckling tool developed at DLR, is presented. IBUCK was developed for the design process of stiffened shells. The effects of local skin buckling and stiffener tripping are captured. In the stiffeners, a uni-axial membrane stress state is assumed while plane stress state is assumed for the skin. Linear-elastic material is considered. Shape functions are selected for the skin and the stiffener webs, that is, for stringers and frames. The stiffener flanges are considered as Euler–Bernoulli beams and Saint-Venant torsion bars while the skin doublers beneath the stiffeners are modelled as Saint-Venant torsion bars. Since the flange’s bending beam/torsion bar formulation does not provide information about flange tripping, the critical flange tripping load is determined in an external procedure. For validation, six different panels were loaded axially. FEresults (ABAQUS 6.4.1) were compared to the results obtained with IBUCK. It was found that IBUCK performed very well for panels with aspect ratios a/bO2 (panels 1a, 2a and 3a). These panels exhibited a regular, long-wave post-buckling behavior that could be represented by a relatively small number of Fourier terms. Thus, IBUCK finished orders of magnitude faster than ABAQUS. For stiffened panels with aspect ratio a/b!2 (panels 1b, 2b and 3b), the post-buckling deflection pattern proved to be rather irregular and of short wavelengths, thus requiring a large number (more than 100) Fourier terms and resulting in long IBUCK computation times. However, correspondence between IBUCK and ABAQUS was very good when a large enough number of Fourier terms was considered. It is very likely that the high computational costs for models with more than 100 Fourier terms could be reduced by implementing a more efficient memory management. However, at this time, this is not the focus of this work. Acknowledgements The present model was developed as part of a sponsorship contract between Airbus Germany and DLR. All support is gratefully acknowledged. References
Table 4 Computation summary for panels with aspect ratio a/b!2 Panel
1b 2b 3b
ABAQUS
IBUCK
Advantage
DOF
Time (s)
DOF
Time (s)
70,470 94,710 118,950
8928 38,592 19,333
44 (136) 56 (161) 68 (161)
207 (7622) 247 (16,547) 568 (14,832)
97.68 (14.63) 99.36 (57.12) 97.06 (23.28)
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