Imperfection sensitivity and geometric effects in stiffened plates susceptible to cellular buckling

Structures 3 (2015) 172–186

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Imperfection sensitivity and geometric effects in stiffened plates susceptible to cellular buckling M. Ahmer Wadee ⁎, Maryam Farsi Department of Civil & Environmental Engineering, Imperial College London, South Kensington Campus, London SW7 2AZ, UK

a r t i c l e

i n f o

Article history: Received 21 November 2014 Received in revised form 24 April 2015 Accepted 25 April 2015 Available online 1 May 2015 Keywords: Stiffened plates Mode interaction Imperfection sensitivity Structural mechanics Structural stability Analytical modelling Nonlinear mechanics

a b s t r a c t An analytical model for axially loaded thin-walled stringer stiffened plates based on variational principles is exploited to study the sensitivity to initial geometric imperfections and the effects of altering geometric properties. Studies on different forms of global and local imperfections indicate that the post-buckling response governs the worst case imperfections. The investigation also focuses on the effect of changing the global and the local slendernesses on the post-buckling behaviour. The parametric space in which the stiffened plates are imperfection sensitive and susceptible to highly unstable cellular buckling is identified. © 2015 The Institution of Structural Engineers. Published by Elsevier Ltd. All rights reserved.

1. Introduction A stiffened plate is an exemplar of an optimized structural component because although it is well known to be highly efficient at carrying external loads, it is equally well known to be susceptible to complex instabilities under certain circumstances [1–4]. Stiffened plates are commonly found in long-span bridge decks [5], ships and offshore structures [6], and in aerospace structures [7]. Hence, understanding the behaviour of these components represents a structural problem of enormous practical significance [8–10]. Other significant structural components such as sandwich struts [11], built-up columns [12], corrugated plates [13] and other thin-walled components [14–18] are also similarly well-known to be vulnerable to such complex instabilities. In the current case, the interaction between global and local buckling modes is particularly pertinent. A recently developed nonlinear analytical model for an axially loaded thin-walled stiffened plate made from a linear elastic material [19,20] is exploited. The nonlinear mode interaction [1,3] between global Euler buckling and local buckling of the stiffener as well as the main plate was fully described, which was then validated through comparisons with a finite element (FE) model formulated in Abaqus [21] and with existing physical experiments [2]. The studies [18,19]

⁎ Corresponding author. E-mail addresses: [email protected] (M.A. Wadee), [email protected] (M. Farsi).

focused on the perfect elastic post-buckling response and highly unstable cellular buckling behaviour was highlighted [22]. In the current context, cellular buckling, also referred to as “snaking” in the applied mathematics literature [23–25], is a particular type of post-buckling response where a sequence of snap-backs is observed after an initial instability is triggered. The snap-backs tend to occur due to inherent destabilizing and stabilizing characteristics of the structure. In the present case, the primary source of destabilization is the nonlinear interaction between local and global buckling modes, whereas the primary source of restabilization is derived from the resulting plate buckling deformation. The physical signature of cellular buckling is a reduction in the load carrying capacity in conjunction with the gradual spreading of the local buckling mode, which begins as a localized mode and in the limit spreads throughout the structure and the post-buckling mode progressively changes (usually to a smaller) wavelength. The snap-backs, observed in the case where the main plate–stiffener joint was assumed to provide a rotationally flexible (or pinned) connection, have been found to diminish by increasing the joint rigidity, although the local buckling wavelength still reduces as the post-buckling deformation is increased [19]. The changing local buckling wavelength has been observed in physical experiments in closely related structures [26,17,18] but has been found to be difficult to detect using static finite element models [20,27]. The current work exploits the previously presented model [20] by studying the sensitivity to initial local and global geometric imperfections. This is followed by a parametric study to evaluate the most vulnerable geometric combinations of local and global slendernesses

http://dx.doi.org/10.1016/j.istruc.2015.04.004 2352-0124/© 2015 The Institution of Structural Engineers. Published by Elsevier Ltd. All rights reserved.

M.A. Wadee, M. Farsi / Structures 3 (2015) 172–186

Simply-supported stiffened panel:

173

b

L b

Focus on this portion Fig. 1. An axially compressed simply-supported stiffened plated panel of length L and evenly spaced stiffeners separated by a distance b.

main plate; a rigid end plate transfers the point load as a uniform compressive pressure through the cross-section before any instability occurs. As in previous works [19,20], the rigidity of the connection between the main plate and the stiffeners is modelled with a rotational spring of stiffness cp, as shown in Fig. 2(c). This spring reflects the relative rigidity of the actual connection, where a fully-penetrating buttweld connecting the stiffener onto the main plate may practically allow a fully-rigid connection to be assumed. However, a fillet-welded, spot-welded [17] or even a riveted connection between the stiffener and the main plate (the latter sometimes found in aircraft construction) may only allow a basic pinned connection to be assumed. The analytical model that governs the imperfect system has been fully developed previously [20], a summary of which follows.

that trigger interactive buckling. The latter study provides a better practical understanding of where the interactive buckling behaviour of the stiffened plate is important in terms of the geometric properties, how it may be accounted and where it may be practically ignored. 2. Review of analytical model Consider a thin-walled simply-supported plated panel that has uniformly spaced stiffeners above and below the main plate, as shown in Fig. 1, with panel length L and the spacing between the stiffeners being b. It is made from a linear elastic, homogeneous and isotropic material with Young's modulus E, Poisson's ratio v and shear modulus G = E/[2(1 + ν)]. If the panel is significantly wider than long, i.e. L ≪ nsb, where ns is the number of stiffeners in the panel, the critical buckling behaviour of the panel would be strut-like with a half-sine wave eigenmode along the length. There would also be a half-sine wave eigenmode across the width of the panel, the curvature of which would be considerably smaller than the corresponding curvature along the length. This, in turn, would allow a portion of the panel that is representative of its entirety to be isolated as a strut, as depicted in Fig. 1, since the transverse bending curvature of the panel during initial post-buckling would be insignificant over the width b of the central portion of the stiffened plate. The coordinate system and the section properties for the strut are shown in Fig. 2. The axial load P is applied at the centroid of the whole cross-section denoted as the distance y from the centre line of the

2.1. Modal descriptions Two degrees of freedom, known as “sway” and “tilt” in the literature [28], are used to model the global buckling mode. The corresponding generalized coordinates are qs and qt respectively. The sway mode is represented by the lateral displacement W and the tilt mode is represented by the corresponding angle of inclination θ of the plane sections, as shown in Fig. 3(a). Based on linear theory, W(z) and θ(z) are given by the following expressions [19]:

W ðzÞ ¼ −qs L sin

πz ; L

θðzÞ ¼ qt π cos

πz : L

ð1Þ

y

Neutral axis of bending

P z

L

(a) h1

ts

h2

ts

y cp

y x b

(b)

tp

x

(c)

Fig. 2. (a) Elevation of the representative portion of the stiffened plate modelled as a pin-ended strut of length L that is compressed axially by a force P, the rigid end-plates shown transfer the point load into a uniform pressure through the cross-section depth but are also allowed to rotate. (b) Strut cross-section geometry. (c) Modelling the joint rigidity of the main plate– stiffener connection with a rotational spring of stiffness cp.

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Sway mode:

Tilt mode:

z

y

W(z)

y

θ (z)

z

(a) w sl(y,z) w s0(y,z) u(h 1-y,0)

u(h 1-y,L)

y

w pl(x,z)

y

w p0(x,z)

x

(b) L

ε

z

ε0

y

θ0

θ

z

W 0(z) z

W(z) Flexural rigidity EI p Stress relieved M

u t = -y (θ - θ 0) ∂u t ε z,overall = ∂z θ M dW χ

χ0

dθ

M

(c) Fig. 3. (a) Sway and tilt components of the global buckling mode. (b) Local in-plane deflection of the stiffener u(y, z) and local out-of-plane deflection of the stiffener wsl(y, z) with the initial imperfection ws0(y, z) and local out-of-plane deflection of the main plate wpl(x, z) with the initial imperfection wp0(x, z). (c) Introduction of the global imperfection functions W0 and θ0 by stress relieving the initially imperfect configuration; the initial tilt displacement ut is related to the global strain and the elemental bending moment M is related to the angle change dθ and the curvature χ showing that due to the stress relief M = 0 when χ = χ0.

To account for pre-buckling compressive displacements in the strut, an additional generalized coordinate Δ is introduced such that the prebuckling in-plane displacement is ΔL, which is assumed to be uniformly distributed throughout the cross-section and length. The pre-buckling compressive strain is therefore equal to Δ. For the local buckling mode, the in-plane displacement function u(y, z), the local out-of-plane displacement function of the stiffener wsl(y, z) and the sympathetic local out-of-plane displacement of the main plate wpl(x, z), are shown in Fig. 3(b). The general form of outof-plane displacement functions of the stiffener and the main plate are

wsl(y, z) = f(y)w(z) and wpl(x, z) = g(x)wp(z) respectively. The corresponding approximate shape functions f and g were derived in previous work [20] assuming a combination of polynomial and trigonometric terms in conjunction with appropriate boundary conditions: 
 π3 6 2Y−3Y 2 þ Y 3 − 3 sin ðπY Þ wðzÞ ¼ f ðyÞwðzÞ; Y− J s 6 π 
 1 wpl ðx; zÞ ¼ − sin ðπX Þ þ J p X þ ð−1Þi X 2 − sin ðπX Þ wp ðzÞ ¼ g ðxÞwp ðzÞ; 4 wsl ðy; zÞ ¼

ð2Þ

Fig. 4. Local imperfection profile w0. (a) Localized imperfections introduced by increasing α. (b) Periodic imperfections (α = 0) with different numbers of half sine waves by changing β. In both cases η = L/2.

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175

Fig. 5. Sketch of equilibrium diagrams for the global critical buckling case with a global imperfection: (left) initially weakly stable (flat) perfect post-buckling path with critical and secondary bifurcations C and S; (middle) imperfect path for load P versus qs and (right) imperfect path for load versus wmax both with ultimate load Pu with associated values of qs = qus and wmax = wumax, and a bifurcation point S where interactive buckling is triggered.

expressions, which are affine to the initial global eigenmode, are given by:

where: 
−1 Ds π2 π2 Js ¼ π þ −1 ; cp h1 3


 1 2Dp 1 −1 − − Jp ¼ ; 4 cp bπ π

ð3Þ

with Y ðyÞ ¼ ðy þ yÞ=h1 and X(x) = x/b. The shape function f(y) was selected such that the limiting cases yielded either a linear function or a quarter sine function for a pinned or a fixed joint at the main plate– stiffener connection respectively. Similarly, the shape function g(x) was selected such that the limiting case of a main plate–stiffener joint being pinned yielded a pure half sine wave [29]. The quantities Ds and Dp are the stiffener and the main plate flexural rigidities given by the expressions Et3s /[12(1 − ν2)] and Et3p/[12(1 − ν2)] respectively. The length y gives the location of the neutral-axis of bending measured from the centre line of the main plate and is expressed thus: h i 2 2 t s h1 −h2

: y¼  2 ðb−t s Þt p þ ðh1 þ h2 Þt s

ð4Þ

Similarly, in the current work, the sympathetic deflection of the main plate is assumed in terms of the relationship wp(z) = μ pw(z). From evaluating the total bending moment at the joint between the stiffener and the main plate using the expressions in Eq. (2), the relating parameter μ p is found to be given by the following expression: 9 !8
μp ¼

2b

1

p p

p

W 0 ðzÞ ¼ −qs0 L sin

πz ; L

θ0 ¼ qt0 π cos

πz ; L

ð6Þ

where qs0 and qt0 define the initial amplitudes of the global imperfection. The local out-of-plane imperfection for the stiffener ws0(y, z) and the sympathetic initial displacement of the main plate wp0(x, z) are also introduced. In an identical way to the local buckling displacements, these imperfection functions are decomposed such that ws0(x, z) = f(y)w0(z) and wp0 = g(x)μ pw0(z). The imperfection amplitude function w0(z) is assumed to be of the following form: w0 ðzÞ ¼ A0 sech



 α ðz−ηÞ βπðz−ηÞ cos ; L L

ð7Þ

where z = [0, L] and w0 is symmetric about z = η and the assumption that η = L/2 is made throughout due to the inherent component symmetry. The parameter A0 is the amplitude of the local imperfection with β (the periodicity parameter) and α (the localization parameter) being the parameters that control the number of waves along the length and the degree of localization of the imperfection profile respectively, the effects of which are shown in Fig. 4. Different imperfection shapes are studied since the post-buckling mode changes qualitatively as the deformation progresses [19], the worst case imperfection is therefore likely to vary for the same reason [30].

ð5Þ

p

the full derivation of which may be found in [20]. An initial out-of-straightness in the y-direction (W0) and an initial rotation of the plane section (θ0) are also introduced. The corresponding

2.2. Governing equilibrium equations The governing equations of equilibrium are derived from variational principles by minimizing the total potential energy V of the system. This comprises the contributions from global and local strain energies of

Fig. 6. Sketch of equilibrium diagrams for the local critical buckling case with a local imperfection. Features correspond to Fig. 5, apart from the fact that the initial post-buckling response is strongly stable.

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~ where dots now represent differentiation with respect to the rescaled w, variable ~z:

Table 1 Geometric properties of an example stiffened plate used in the imperfection sensitivity study, with h2 = tp/2, the stiffeners are assumed to be only one side of the plate. Plate breadth b Plate thickness tp Stiffener depth (top) h1 Stiffener depth (bottom) h2 Stiffener thickness ts Young's modulus E Poisson's ratio v

2 :::: e H e þ nL o w −w 0 2 2 f

120 mm 2.4 mm 38 mm tp/2 1.2 mm 210 kN/mm2 0.3

n 2 o 

 0 ~ðw− e e € w €0 þ k ~ w ~ 0Þ w− νf f f 00 gy −ð1−νÞ f y

y

n o 2 2

2Y f

 2 e þw e e e e e e e e e ~ n o 3w w− € ww € 0 −2w €w € 0 w0 w þ n o y u €u −D 2 2 f f "

n




f

4

o

y











y



y

n o 2

# 2 yf π π~z e π π~z e e € þ cos € n o y sin w w ðqt −qt0 Þ −2Δw−2 2 2 2 L f2 

bending, Ubo and Ubl respectively, the strain energy stored in the “membrane” of the stiffener Um arising from axial and shear stresses and strains, with the work done by the external load PE. The full formulation of the total potential energy V of the imperfect stiffened plate was established in previous work [19,20]. The governing equations are obtained by performing the calculus of variations on V, the integrand of which can be expressed as the Lagrangian (L) of the form: Z

L

V¼ 0

 € w; w; u; u; z dz; L w; 



ð8Þ

y

"



~ 2w ~ n 0 2 o e 2 1  0 e GL e 2 −w e e €0 þ ~ w− € w ~ 0w ff w þw f f yu − n o 0 2 y h1 2 f 





π2  0  π~z f f y sin þ ½ðqs −qs0 Þ−ðqt −qt0 Þ 2 L

þ

#

" # 3 2h  n o i

 μ2  2 e tp H −w H e þ L ν fgg00 g −ð1−ν Þ g0 2 e e € € n po w− w g x w 0 0 x 2 x ts 2 f y

where dots represent differentiation with respect to z. The first variation of V is given by: δV ¼

Z L


 ∂L ∂L ∂L ∂L ∂L €þ δw þ δu þ δw δw þ δu dz: € ∂w ∂w ∂u ∂w ∂u 



0





−

~ 2
2

   2 t p Dμ p e w e −2Δg2  w e w− e e e e € ww € 0 −2w € € 0w n o μ 2p g4 x 3w 0 x 2 ts f 





y

ð9Þ

Using the procedure detailed in previous work [11,19,20], the governing non-dimensional nonlinear differential equations are given by a fourth order equation for the rescaled local out-of-plane deflection



−

2 ~ 4 ~
n

 o 2 t p L Gμ p w 2 e þw e 2 −w e e €0 ~ w− € w ~ 0w n o ¼ 0; w ðgg0 Þ 0 2 x ts 2 f 



y

ð10Þ

Fig. 7. Numerical equilibrium paths for the pinned case (cp = 0). The graphs show a family of curves of the normalized force ratio p (=P/P Co ) versus (a) the global mode amplitude qs, (b) the normalized local mode amplitude wmax/ts and (c) the normalized end-shortening ε/L; (d) shows the local versus the global mode amplitude.

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177

Fig. 8. Imperfection sensitivity graph for (a) a pinned joint and (b) a rigid joint between the stiffener and the main plate with only a global imperfection. The normalized ultimate load ratio pu (=Pu/PCo) versus the initial out-of-straightness coordinate qs0 is plotted in both graphs.

and a second order equation for the rescaled local in-plane deflection ũ:   ~


  e 3G π~z e e €− ~ 0w ~ w−w ~ þ f f0 y w u −2π ½ðqs −qs0 Þ−ðqt −qt0 Þ cos ψ ψ u 0 ~ 4D 2 ( )

2

 ~ 3Y f 1 3y πz e w ew 3 e e € þw cos ¼ 0; − ψ− w 0 € 0 þ ðqt −qt0 Þπ 2 2L 2 h1 







y

ð11Þ

where the rescaled variables and constants are: 2z 2u 2w 2w0 h ~ Et s L2 ~ Gt s L2 ~¼ ~ ¼ ~0 ¼ ;ψ¼ ;D ;G¼ ; ; u ; w ; w ¼ L 8Ds 8Ds L L L L
n 4 o n o h i 2  L 3 2 00 2 0 ~¼ n o k f þ μ 2p t p =t s g 00 þ cp f ð−yÞ−μ p g0 ð0Þ =Ds : 2 y x 16 f y

ð12Þ

It is worth noting that primes (′) denote differentiation with respect to the respective subscript outside the closing brace, with the terms within the braces being definite integrals, thus:

f F ðyÞgy ¼

h1 −y −y

Z F ðyÞ dy;

fH ðxÞgx ¼

b=2 −b=2

~Γ1 ¼

Lh1 ð2h1 −3yÞ

; 3 3 ðh1h−yÞ þ ðh2 þ yÞ i 2 2 6L ðh1 −yÞ −ðh2 þ yÞ ~Γ3 ¼ h i; 3 3 π ðh1 −yÞ þ ðh2 þ yÞ ~s ¼

Gt s ðh1 þ h2 ÞL2 ; EI p

H ðxÞ dx;

ð13Þ

Ip ¼

∂V PL ¼ π2 ðqs −qs0 Þ þ ~s½ðqs −qs0 Þ−ðqt −qt0 Þ− q EI p s ∂qs Z i ~ 2  0 e ~sϕ π~z h e ~þ ff y w ~ w−w ~ 0w u − cos d ~z ¼ 0; 0 2π 0 2 Z 2n ∂V 1 π~z h~ e ¼ π2 ðqt −qt0 Þ þ ~ Γ3 Δ−~t ½ðqs −qs0 Þ−ðqt −qt0 Þ− Γ1 u sin 2 2 ∂qt 0 2

h

io ~ 2 i ~   ~ π z t ϕ e −w e e e −w ~ 0w ~w ~ þ f f0 y w þ cos d~z ¼ 0; þ~ Γ2 w u 0 0 L π
h i ∂V h2 t p ðb−t s Þ P π 2 2 − þ ðqt −qt0 Þ ðh1 −yÞ −ðh2 þ yÞ ¼Δ 1þ þ t s h1 h1 Et s h1 Lh1 ∂Δ

 2 Z 2


 Z
 2 e 2 tp μ p 1 2 e 1 n 2o e 2 2 2 uþ w −w0 d~z− − f g x w −w0 d~z y 4 0 h1 t s 4h1 0 













3

;

L ; h1 þ h2

ð15Þ

ðb−t s Þt 3p 12

þ ðb−t s Þt p y2

ð16Þ

being the second moment of area of the plate about the global x-axis. Since the stiffened plate is an integral member, the first two integral equations provide a relationship linking qs and qt before any interactive buckling occurs, i.e. when w = u = 0. This relationship is assumed to hold also between qs0 and qt0, which has the beneficial effect of reducing the number of imperfection amplitude parameters associated with the global mode to one; it is given by the following expression:  qs0 ¼ 1 þ π2 =~t qt0 :

ð17Þ

~ and ũ and their derivatives are for The boundary conditions for w pinned conditions at ~z ¼ 0 and for symmetry at ~z ¼ 1: e ð1Þ ¼ w r e ð1Þ ¼ u e ~ ð1Þ ¼ 0; ~ ð0Þ ¼ w € ð0Þ ¼ w w







¼ 0;

ð14Þ

ð18Þ

with a further condition that arises from minimizing V, which relates the in-plane strains to each other at the boundary:   
2

1e 1 Y 2 h2 þ y e ð0Þ−w2 ð0Þ − 1 Δ þ P f ¼ 0: ð19Þ w uð0Þ þ 0 3 2 h1 2 Et s h1 h1 þ h2 y 

2



~¼ ϕ

y

3

ðh1 −yÞ þ ðh2 þ yÞ

2 ~t ¼ h 3GL ðh1 þ h2 Þ i; 3 3 E ðh1 −yÞ þ ðh2 þ yÞ



where F(y) and H(x) are example functions representing the actual expressions within the braces. Similarly, three integral equations are obtained by minimizing V with respect to the generalized coordinates qs, qt and Δ; in non-dimensional form these are as follows:



~Γ2 ¼

n o 2 3L yf

with:

~z ¼

Z

where the rescaled quantities are given by the following expressions:





It is worth remembering that these boundary conditions do not affect the main plate–stiffener joint characteristic, which applies in a completely different plane. The global critical load, PCo , determined by linear eigenvalue analysis of the perfect system, remains the same as before [19,20] and is given by the expression below:


 ~s π2 EI p P Co ¼ 2 1 þ : ð20Þ π2 þ ~t L This expression accounts for Timoshenko beam theory and reduces to the classical Euler load expression for a pin-ended strut [31] if the

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Fig. 9. Numerical equilibrium paths for the rigid joint case (cp → ∞). Graphs (a–d) are as described in the Fig. 7 caption.

Fig. 10. Imperfection sensitivity for the pinned case (cp = 0). Normalized imperfection size E 0 =L against: (a) the ultimate load ratio pu (=Pu/P Co) and (b–d) the normalized local deflection amplitude A0/ts, the periodicity β and the localization α parameters. Cross (×) symbols correspond to the imperfection form of the plate linear eigenvalue solution (α = 0, β = 1); asterisk (∗) symbols correspond to the periodic imperfection ( α= 0, β N 1); circle (∘) symbols correspond to the modulated imperfection (α, β N 1).

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179

Fig. 11. Interactive buckling regions while varying: (a–b) the stiffened plate lengths L and (c–d) the stiffener height h1. The length L = Lc and height hc are defined when P Co = P Cl , whereas D stiffened plates with L N Lo (or h1 b ho) and L b Ll (or h1 N hl) are assumed to exhibit pure global buckling and pure local buckling respectively. The values qD s and wmax define the limiting global and local mode amplitudes at the ultimate load Pu in the post-buckling range for pure global and local buckling respectively.

zero shear strain assumption (G → ∞) is imposed. However, the introduction of shear strains from Timoshenko beam theory is essential for the modelling of local–global mode interaction [11,17–19]. 3. Imperfection sensitivity study

Fig. 12. The idealized strength curve in terms of normalized slenderness and stress. The subscript “x” may be changed to “o” or “l” to denote global or local buckling respectively.

The sensitivity to initial geometric imperfections is first studied by varying an initial global buckling out-of-straightness amplitude to study the effect of an initial global imperfection. Subsequently, the effect of a purely local imperfection is studied by introducing the amplitude and then altering the qualitative profile by varying the periodicity and the localization parameters. The numerical solution is performed in the powerful numerical continuation and bifurcation software Auto [32], which can not only solve nonlinear differential equations subject to boundary and integral conditions but also can track the evolution of solutions with varying system parameters. Moreover, it has the key

Fig. 13. Pinned case (cp = 0), varying length L. Graphs show (a) the normalized lateral displacement qus and (b) the normalized maximum local out-of-plane displacement wumax/ts, both at the normalized ultimate load ratio pu, versus the strut length L, for the cases where global and local buckling are critical, respectively. The vertical dashed line with label Lc represents the D length where PCo = PCl . The horizontal dot–dashed lines represent the amount of displacement, above which interactive buckling is deemed to be insignificant (qD s , wmax). The vertical dashed line Ls represents the strut length where the first snap back occurs while the mechanical stiffness is still positive. The interactive buckling region is therefore defined as L = [Lo, Ll].

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Fig. 14. Pinned case (cp = 0), varying length L. The central graph shows the ratio of the ultimate load scaled with the local critical buckling load ratio Pu/PCl versus the normalized global critical load P Co /P Cl . The solid line represents the actual numerical solutions whereas the dashed lines representing Ll, Lc, Ls and Lo correspond directly to Fig. 13. The surrounding graphs show examples of the equilibrium paths of systems with a small imperfection corresponding to the different parts of the central graph, separated by the dashed lines. Anti-clockwise from top-left: L = [Lc, Lo]; L N Lo; L = Lc; L = [Ll, Lc]; L = Ll and L b Ll.

capability for nonlinear structural instability problems that it can pinpoint different kinds of bifurcation points and has the facility to swap between different solution branches, which in the present case are the equilibrium paths. The solution strategies for the aforementioned cases are the same as those in previous work [19,20]. Figs. 5 and 6 show sketches of examples of perfect and imperfect paths for critical modes that trigger global buckling or local buckling first. The critical and secondary bifurcations C and S respectively, the ultimate load of the imperfect systems Pu and the corresponding values of qus and wumax that define the load carrying capacity of an imperfect stiffened plate are highlighted. 3.1. Effects of initial imperfections The initial set of numerical results consider the post-buckling behaviour of the stiffened plate by varying the global imperfection amplitude only (qs0 ≠ 0, w0 = 0) for both pinned and rigid cases. The subsequent

set of numerical results focus on the local imperfections only (w0 ≠ 0, qs0 = 0) for the case where the main plate–stiffener joint is pinned (cp = 0). The local imperfection amplitude A0, the periodicity and the localization parameters β and α respectively, are varied to determine the worst case combination that gives the lowest ultimate load. The material and section properties that are used in the following examples are given in Table 1. The strut length L is chosen to be 5000 mm and so global buckling is critical for this set of imperfection sensitivity studies. 3.1.1. Global imperfection only (qs0 ≠ 0, w0 = 0) In this section, only the global initial out-of-straightness W0 is introduced. A set of values for the normalized initial out-of-straightness amplitude qs0 is assumed between zero (perfect case) and 1/500. Initially, the results are presented for the pinned case (cp = 0). Fig. 7 shows a series of graphs of the normalized axial load p (=P/PCo ) versus (a) the global mode amplitude qs, (b) the local mode amplitude wmax/ts and (c) the normalized end-shortening E=L, with (d) showing

Fig. 15. Pinned case (cp = 0), varying stiffener height h1. Graphs show (a) the normalized lateral displacement qus and (b) the normalized maximum local out-of-plane displacement wumax/ ts at the ultimate load ratio pu, versus the stiffener height h1, for the cases where global and local buckling are critical, respectively. The vertical dashed line with label hc represents the critical stiffened plate length where P Co = P Cl . The horizontal dot–dashed line represents the amount of displacement, above which interactive buckling is assumed to be insignificant D (qD s , wmax); the interactive buckling region is therefore defined as h1 = [ho, hl].

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Fig. 16. Pinned case (cp = 0), varying stiffener height h1. The central graph shows the normalized ultimate load ratio Pu/P Cl versus the normalized global critical load P Co/P Cl . The solid line represents the actual numerical solutions whereas the dashed lines representing hl, hc and ho correspond directly to Fig. 15. The surrounding graphs show examples of the equilibrium paths of systems with a small imperfection corresponding to the different parts of the central graph, separated by the dashed lines. Anti-clockwise from top-left: h1 = [ho, hc]; h1 b ho; h1 = hc; h1 = [hc, hl] and h1 N hl.

the normalized local versus global mode amplitudes. Note that the total end-shortening E is given by the following expression [19]: E¼



 Z L
1 πz h2 þ y Δ þ q2s π2 cos 2 − u dz: 2 L h1 þ h2 0 

ð21Þ

Fig. 8(a) shows a scatter plot of the ultimate load ratio pu (=P/PCo ) against the initial global imperfection amplitudes. It is clearly observed that pu decreases as the size of the imperfection increases beyond qs0 = 4 × 10−4 for the example presented, which shows that the stiffened plate is sensitive to global imperfections of sufficient size. The flatter region where qs0 b 4 × 10−4 shows that the stiffened plate can carry the global critical buckling and is not particularly sensitive to imperfections of that size. This is because of the significant interval between the critical and secondary bifurcations C and S (sketched in Fig. 5) for this particular geometry. Global imperfection sensitivity is also studied for the case where the main plate–stiffener joint is assumed to be rigid (cp → ∞). The same set of material and geometric properties as the pinned case is considered

and, again, the global imperfection amplitude qs0 is varied to study the post-buckling behaviour. Of course, given that a rigid joint effectively increases the local buckling critical stress, in the following examples global buckling is still critical. Fig. 9 shows a series of graphs in an identical format to Fig. 7 with Fig. 8(b) showing a scatter plot of the ultimate load ratio pu against the initial global imperfection amplitudes. It is clearly observed that pu decreases once again as the size of the imperfection increases, demonstrating that the stiffened plate is sensitive to global imperfections for sufficiently high values of qs0. It also seems that for very small values of qs0 the rigid case is more sensitive than the pinned case. However, closer scrutiny of this, in conjunction with the equilibrium paths for both the pinned (Fig. 7) and rigid cases (Fig. 9), reveals that although the ultimate loads for the pinned case are higher for very small qs0 values, the subsequent snap-backs destabilize the component much more strongly. For higher values of qs0 the behaviours of the pinned and rigid case converge somewhat — the pinned case still exhibits snap-backs but with progressively reduced severity as shown with the post-buckling slope generally being flatter as qs0 is increased.

Fig. 17. Rigid case (cp → ∞), varying length L. Features corresponding to Fig. 13.

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Fig. 18. Rigid case (cp → ∞), varying length L. Features corresponding to Fig. 14. Surrounding equilibrium diagrams, anti-clockwise from top-left: L = [Lc, Lo]; L N Lo; L = Lc; L = Ll.

3.1.2. Local imperfection only (w0 ≠ 0, qs0 = 0, cp = 0) In this section, three types of local imperfection w0 are studied. Initially, the periodicity and the localization parameters β and α respectively, are kept at constant values of 1 and 0 respectively. Hence, the initial study also replicates the introduction of imperfection affine to the first eigenmode of the flange, this time considering a local buckling mode — a common methodology used to investigate instabilities in structures particularly when FE methods are employed [33]. The initial end-shortening E 0 due to the local imperfection, which is evaluated from the first-order approximation: E0 ¼

1 2

Z

L 0



2

w0 dz;

ð22Þ

is used as the measure for the size of the initial imperfection because it has been shown to provide a fairer comparison between different qualitative shapes of imperfection [30]. Note that this measure of imperfection size can be established in practice since the evaluation of E 0 is relatively straightforward through numerical integration of the measured initially imperfect profile. The local imperfection amplitude A0 is increased and the ultimate loads are recorded for each value of E 0.

Fig. 10(a) shows the normalized initial end-shortening E0/L versus the ultimate load ratio pu = Pu/PCo. By increasing the E0 value, the corresponding ultimate load ratios pu are plotted with the cross (×) symbols when β = 1 and α = 0. A very small reduction is found in pu, which indicates that the stiffened plate is not particularly sensitive to the imperfection affine to the first local eigenmode. In the next set of results, periodic flange imperfections of shorter wavelengths (α = 0, β N 1) are introduced. Hence, the procedure involves choosing a value for the initial end-shortening E 0, fixing a value of β and determining A0; the graphs in Fig. 10(b) and (c) show the respective distributions of A0 and β versus E 0. It is worth noting that β only takes odd integer values to satisfy the symmetry conditions at mid-span. The combination of β and A0 that gives the lowest ultimate load is shown with the (∗) symbol in Fig. 10 graphs. It is observed that, by increasing the β value, the ultimate load is reduced significantly asE0 increases. This is to be expected since the post-buckling response in the perfect case exhibits a progressively changing wavelength, which is a remnant of the cellular buckling behaviour found in the pinned joint case [19], with the mode spreading as the deformation progresses. In the final set of results, localized imperfections are considered by increasing the localization parameter α, while E 0 is varied and β is

Fig. 19. Rigid case (cp → ∞), varying stiffener height h1. Features corresponding to Fig. 15.

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Fig. 20. Rigid case (cp → ∞), varying stiffener height h1. Features corresponding to Fig. 16. Surrounding equilibrium diagrams, anti-clockwise from top-left: h1 = [ho, hc]; h1 b ho; h1 = hc; h1 = hl.

kept at the corresponding value shown in Fig. 10(c). Therefore, as α is increased it is necessary to increase A0, as shown in Fig. 10(b) with the circle (∘) symbol. The combination of α and A0 that gives the lowest ultimate load is determined. A significant further reduction in the ultimate loads is observed when the imperfection is more localized (modulated) by varying α, for all values of E 0 as shown in Fig. 10(a). The corresponding α values for each value of normalized endshortening E 0/L are shown in Fig. 10(d). For the largest E 0 value, the amplitude A0 for the worst case periodic imperfection is slightly larger than ts/10, whereas for the worst modulated imperfection, A0 increases to approximately ts/4. Note that for higher values of E0, the most severe case α begins to reduce; this would be expected since, as explained above, the perfect case shows cellular buckling and hence the postbuckling mode amplitude envelope begins to spread outwards from the mid-span. Moreover, the mode shape begins to appear to be more periodic for higher deformations. This is in contrast to the more localized post-buckling mode that initially emerges after the secondary bifurcation. 4. Geometric effects on post-buckling behaviour Hitherto, the stiffened plate with the geometric properties given in Table 1 has been considered and global buckling was critical. In this section, however, a parametric study on the aspect ratio of the stiffened plate is conducted by varying the length L as well as the stiffener height h1. These variations affect the normalized global and local slenderness ratios, the effects of which are then discussed. 4.1. Parametric definitions Initially, the parametric study is conducted such that the global slenderness ratio L/r is varied by changing the strut length L. Note that pffiffiffiffiffiffiffi r is the section radius of gyration, which is equal to I=A, where I is the cross-section total second moment of area about the neutral axis of bending parallel to the main plate, and A is the total cross-sectional area of the representative portion of the stiffened plate (see Figs. 1–2). The other geometric properties of the representative portion of the

stiffened plate are kept constant, as given in Table 1. The expression for the global buckling critical load PCo is given by Eq. (20), whereas the local buckling critical stress of the stiffener σCl can be evaluated using the well-known formula: σ Cl ¼ kp

Ds π2 2

h1 t s

;

ð23Þ

where the coefficient kp depends on the plate boundary conditions. By increasing cp, the relative rigidity of the joint connecting the main plate and the stiffener varies from being completely pinned (cp = 0) to rigid (cp → ∞). Therefore, limiting values for kp may be identified as 0.426 or 1.247 for a long stiffener connected to the main plate with one edge free and the edge defining the junction between the stiffener and the main plate being taken to be pinned or fixed respectively [29]. The local buckling critical load is therefore equal to the local buckling critical stress of the stiffener multiplied by the entire cross-sectional area, P Cl = AσCl . However, the value of the global critical buckling load P Co remains the same since it is independent of cp. Hitherto, the full set of equilibrium equations was solved when global buckling was critical (i.e. P Co b P Cl ). It is clear that varying the Table 2 Summary of findings for the pinned case with the first and last cases being ultimately limited by plasticity. The expressions for σ state the ultimate strength of an imperfect stiffened plate to estimate the practical strength. Parameter range

Critical mode

Post-buckling behaviour

L N Lo h1 b ho L = [Lc, Lo] h1 = [ho, hc] L = Lc h1 = hc L = [Ls, Lc] h1 = [hc, hs] L = [Ll, Ls] h1 = [hs, hl] L b Ll h1 N hl

Global

Weakly stable (limited by material failure) σ ≈σ o Weakly stable followed by cellular buckling σ b σo Strongly unstable cellular buckling σ ≪ fσ o ; σ l g Weakly unstable followed by cellular buckling σ b σ o but σ ≈ σ l Strongly stable followed by cellular buckling σ b σ o but σ N σ l Strongly stable (limited by material failure) σ N σl

Global Simultaneous Local Local Local

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Table 3 Summary of values for λl , λc , and λo , for struts with the properties given in the previous section while varying the stiffened plate length L (h1 = hc) or the stiffener height h1 (L = Lc) for both the pinned and the rigid main plate–stiffener joint cases. Parametric variation

Joint type

λl

λc

λo

L varied, h1 = hc

Pinned Rigid Pinned Rigid

2.09 1.22 2.52 1.25

2.09 1.23 2.09 1.23

5.62 3.39 3.19 2.06

h1 varied, L = Lc

strut length L only affects the global critical load P Co, whereas varying the stiffener height h1 simultaneously affects both the global and the local critical loads, P Co and P Cl , respectively. Since it is perfectly reasonable for local buckling of the stiffener to be critical, in the following sections for those geometries, an imperfect stiffened plate is considered with an initial global imperfection in conjunction with a local imperfection. This has been done to avoid the necessity of formulating a model that also captured perfect local buckling behaviour [18], since it would complicate the current model considerably as the function wp(z) could not then be assumed to be purely dependent on w(z). The global imperfection amplitude qs0 is equal to 1/2000 and the initial local amplitude A0/t is equal to 1/100 with β = 1 and α = 0. These imperfections are chosen to be of sufficiently small magnitude to provide results in close proximity to the geometrically perfect case (W0, w0 = 0). 4.2. Variation in length While varying the strut length L, several bounds become apparent. These are illustrated schematically in Fig. 11(a–b). The length Ll is the largest length for which P Cl b P Co and interactive buckling does not occur, i.e. local buckling does not trigger interactive buckling until the deflections are very large. Hence, a stiffened plate with L b Ll exhibits effectively pure local buckling. Conversely, the length Lo is the smallest length for which P Co b P Cl and exhibits effectively pure global buckling with no interactive buckling occurring (the critical and secondary bifurcations being sufficiently far apart). A critical length Lc is also determined for the condition where P Co = P Cl and nonlinear interactive buckling is triggered simultaneously with the primary instability. For the sake of simplicity, a limiting mid-span deflection to length ratio qD s (taken presently as 1/125) is taken as a “sufficiently large” limiting value for the global lateral deflection for both pinned and rigid cases; hence cases where global buckling is critical and qus N qD s are considered to have effectively negligible mode interaction; the length where qus = qD s defines L = Lo. Similarly, a limiting local stiffener deflection to thickness ratio, where wumax/ts = wD max/ts defines the length where the pure local buckling occurs L = Ll; for the length variation, the limit of wD max/ ts ≈ 2.0 is used and defined where the graph corresponding to

Fig. 11(b) shows a distinct change in slope. Note that qus and wumax were defined in Figs. 5 and 6 respectively. Therefore interactive buckling, where local buckling is critical, occurs in the range of lengths: L = [Ll, Lc]. However, where global buckling is critical, interactive buckling occurs when the length lies in the range: L = [Lc, Lo]. For the range L = [Lc, Lo], the load carrying capacity ratio pu for the perfect case is determined by the global critical load PCo since the initial post-buckling characteristic of global buckling is weakly stable. However, for the range where local buckling is critical, the load carrying capacity ratio pu is determined as the peak load (normalized by the global critical buckling load) exhibited in the equilibrium path; this may be above the local buckling critical load due to the inherently stable post-buckling characteristic of elastic local buckling. However, subsequent interactive buckling will ultimately limit the load carrying capacity. The study is conducted for both cases where the main plate– stiffener joint is pinned and rigid respectively. 4.3. Variation in stiffener height In this part, the stiffener height h1 is varied whereas the thickness ts and the panel length L are kept constant, as given in Table 1. Similarly, the critical stiffener height hc is determined when P Co = P Cl . By increasing h1, the local critical load P Cl decreases and the global critical load PCo increases. The latter is due to the increase in the second moment of area I about the bending axis parallel to the main plate. It is considered that when h1 N hl effectively pure local buckling occurs; the interactive region where local buckling is critical is in the range where h1 = [hc, hl]. However, decreasing h1 implies that P Cl increases and P Co decreases and therefore global buckling can become critical. Continuing the reduction in h1, the condition is reached when h1 b ho, this is where effectively pure global buckling is exhibited. Therefore the region where mode interaction is important when global buckling is critical is defined by: h1 = [ho, hc]. The diagrammatic graphs presented in Fig. 11(c–d) show these features. It is worth noting that the limiting value for the stiffener local out-of-plane deflection is identified where a kink is observed in the equilibrium path, which signifies the beginning of interactive buckling, for both the pinned or rigid joint cases. The same limit is used for qD s as for the length variation study described above. The limit wD max/ts is as described above for the pinned main plate–stiffener joint case, but for the rigid joint case it needed to be set much lower (to approximately 0.35) purely due to the increased intrinsic stiffness of the stiffener constraining the local buckling deflection. 4.4. Buckling strength curve It is well-known that the failure mechanism is primarily due to structural instability in slender members and plasticity in stocky members.

Fig. 21. The idealized strength curves with the symbols representing the global and the local normalized slendernesses from varying the length L (left) and from varying the stiffener height h1 (right) respectively. Symbols (∗) and (×) represent the cases where the main plate–stiffener joint is assumed to be rigid and pinned respectively. Note the correspondingly higher slenderness values for the pinned cases.

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The normalized global and local slendernesses λo and λl respectively can be obtained from [34]: sffiffiffiffiffiffi fy ; λo ¼ σ Co

λl ¼

sffiffiffiffiffiffi fy σ Cl

ð24Þ

where fy and σCo (=P Co /A) are the yield and the average global buckling stresses, respectively. Currently, fy is taken as 355 N/mm2. In addition, σCl is the critical stress of the stiffener which was given in Eq. (23) and therefore the normalized local slenderness is given by:  sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 12 1−ν 2 f y h1 : λl ¼ ts E kp π2

ð25Þ

The global and local critical stresses, normalized with respect to fy, where σ o ¼ σ Co =f y and σ l ¼ σ Cl =f y respectively, define the elastic buckling curve, thus:  −2 σ o ¼ λo ;

 −2 σ l ¼ λl :

ð26Þ

The idealized buckling design curve is plotted in terms of the nondimensional stress and slenderness, as sketched in Fig. 12. It is clear that when λx N1, where x = {o, l}, the respective elements are relatively slender and elastic buckling dominates. However, for λx b1, elements are stocky and plasticity dominates. With imperfections, however, the situation is less distinct but there is an imperfection sensitive region in the neighbourhood of λx = 1. When λx ≫1, the ultimate load depends primarily on the elastic post-buckling characteristics with the effects of plasticity being of secondary importance. In the following section, a parametric study is presented to determine the ultimate strength of stiffened plates for a given range of lengths L as well as stiffener heights h1. The bounds defined in Fig. 11 are found and the corresponding normalized slendernesses are calculated using Eqs. (24) and (25) respectively, which all turn out to be in the slender range for the studied examples. 4.5. Numerical results Initially, numerical continuations are performed in Auto for the stiffened plate with a pinned joint between the stiffener and the main plate (i.e. cp = 0). It is observed in Fig. 13 that, for the particular section properties given in Table 1, interactive buckling occurs between stiffened plate lengths Ll = 800 mm and Lo = 3500 mm. The critical strut length Lc is found to be approximately 1235 mm by both theoretical calculation and the numerical solution, whereas the length Ls ≈ 1000 mm represents the length where the first snap-back occurs on the post-buckling path while the mechanical stiffness is still positive. When L = [Ls, Lc], the first snap-back occurs when the post-buckling stiffness is negative, whereas for L = [Ll, Ls], the same occurs but when the post-buckling stiffness is positive. Fig. 14 shows the results in terms of the classic mode interaction curve that was originally devised by van der Neut [35]. The central graph shows the normalized ultimate load versus the ratio of the global to local critical buckling loads while varying the stiffened plate length L. With the section properties given in Table 1 and L being fixed to Lc (1235 mm) and hence hc = 38 mm (from Table 1), the results while varying the stiffener height h1 from hc are also computed. Fig. 15 corresponds to Fig. 13 for varying h1. It is observed in Fig. 15 that, for the example properties, stiffened plates with stiffener heights between ho = 27 mm and hl = 46 mm are most vulnerable to interactive buckling. There is no technical reason why there would not be a distinct value h1 = hs (corresponding to L = Ls shown in Fig. 13), but this was found to be difficult to pinpoint numerically. However, it may be assumed that hs would lie between hc and hl. Fig. 16 shows the equivalent

185

plot shown in Fig. 14 for varying h1 and examples of the actual equilibrium paths for different ranges of the stiffener height. For the rigid main plate–stiffener joint case (cp→∞), Figs. 17–20 correspond to Figs. 13–16 respectively. The length bounds of Ll and Lo reduce to 650 mm and 2100 mm respectively with Lc reducing to 720 mm. Fixing the length to this value of Lc, the stiffener height is then varied, hence hc = 38 mm and the height bounds are found to be ho = 27 mm and hl = 39 mm. Figs. 18 and 20 show a very similar trend as presented in Figs. 14 and 16 respectively. Note that there is no Ls or hs in the rigid case since there are no snap-backs in the postbuckling paths. 4.6. Interactive buckling zone The regions where interactive buckling is significant are summarized in Table 2 by describing the features found in terms of the parametric ranges for the case where the main plate–stiffener joint is pinned. The distinction in the current work as opposed to previous work is the identification of cellular buckling occurring in all cases where interactive buckling is significant (L = [Ll, Lo] and h1 = [ho, hl]). For the rigid case, the only qualitative difference is that there are no snap-backs in the post-buckling paths and hence the distinction between the behaviours defined by L = Ls and h1 = hs does not exist. Otherwise the trends between the pinned and rigid joint cases are remarkably similar with the rigid joint case graphs shown in Figs. 18 and 20 being more compacted versions of Figs. 14 and 16 respectively since the limiting local buckling displacements are defined to be smaller. However, clear changes in local buckling wavelengths are found in the post-buckling mode showing that the effect that inherently causes the cellular behaviour remains [19,20]. It is clear that by increasing the rigidity of the main plate–stiffener joint, the vulnerability to interactive buckling is reduced. Table 3 summarizes the values of the global and the local normalized slendernesses, λo and λl, the values of which are calculated using Eq. (24). These correspond directly to the values of the stiffened plate lengths and the stiffener heights given in the Numerical results section. Note that the normalized global and the local slendernesses, for each critical stiffened plate length Lc and the critical stiffener height hc, are equal, and are therefore denoted using a single piece of notation λc . Moreover, when L is varied, only the normalized global slenderness changes and therefore λc ≈ λl . However, as h1 is varied the normalized local slenderness λl changes together with the normalized global slenderness λo . Therefore, since the critical length Lc is different for the pinned and the rigid cases, λc differs from both λo and λl ; it also explains why when varying   h1, it is found that: λc b λl ; λo , as demonstrated in Table 3. Fig. 21 shows the idealized strength curves (see Fig. 12) corresponding to the values of global and the local normalized slendernesses given in Table 3. With imperfections, of course, the reduction from the idealized strength curve is likely to be the greatest where λ ¼ λc , which is of course well known [35,12]. Moreover, the work presented in Section 3 provides a methodology to determine the strength reductions in the parametric range where the vulnerability to mode interaction is most significant. 5. Concluding remarks Imperfection sensitivity and parametric studies were performed for an example series of thin-walled stiffened plates with two limiting cases, where the main plate–stiffener connection was considered either to be pinned or rigid. Initially, imperfection sensitivity studies were conducted for the cases with the presence of the global and local imperfections only. The highly imperfection sensitive nature of the stiffened plates that are susceptible to cellular buckling was identified. Significant reduction in the load-carrying capacity was observed for the struts with

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small initial global or local imperfections, with capacity reductions of the order of 10–20% for realistic imperfection sizes from manufacturing (global out-of-straightness: L/2000 and local out-of-straightness: ts/100). It was also found that for a given size of local imperfection, a localized or modulated profile was always the worst case in terms of minimizing the load-carrying capacity. The study on different forms of global and local imperfections indicates the need for care in numerical assessments during the design process for actual thin-walled stiffened plates that undergo interactive buckling behaviour. This is because the common practice of using affine imperfections to trigger postbuckling responses may in fact overestimate the true load carrying capacity significantly. The investigation also focused on changing the global and the local slendernesses by varying the stiffened plate length and the stiffener height. A simple deflection based criterion was defined such that the parametric range where mode interaction is most significant may be determined. For pinned main plate–stiffener joints, all examples, i.e. where the global mode or the local mode was critical, within the identified interactive buckling region exhibit unstable cellular buckling at some stage in the post-buckling range. For cases with fully rigid main plate–stiffener joints within the identified interactive buckling region, unstable post-buckling responses are observed at some stage but without the sharp snap-backs found in classical cellular behaviour. However, the local mode wavelength change is still observed, which implies that a remnant of the cellular buckling response is present. The model and results presented a fresh insight into the behaviour of a very common structural element by identifying the potential occurrence of the phenomenon of cellular buckling, a highly dangerous form of instability. This should sound a note of caution to structural analysts and designers in that subtle, yet hazardous, phenomena may be being missed by standard analysis techniques and software. Analytical approaches still provide powerful, rigorous yet practically important methods for understanding complex instability problems. References [1] Koiter WT, Pignataro M. A general theory for the interaction between local and overall buckling of stiffened panels. Tech. Rep. WTHD 83. Delft, The Netherlands: Delft University of Technology; 1976. [2] Fok WC, Rhodes J, Walker AC. Local buckling of outstands in stiffened plates. Aeronaut Q 1976;27:277–91. [3] Budiansky B, editor. Buckling of structures. Berlin Heidelberg: Springer; 1976 [IUTAM symposium]. [4] Thompson JMT, Hunt GW. Elastic instability phenomena. London: Wiley; 1984. [5] Ronalds BF. Torsional buckling and tripping strength of slender flat-bar stiffeners in steel plating. Proc Inst Civ Eng 1989;87:583–604. [6] Murray NW. Buckling of stiffened panels loaded axially and in bending. Struct Eng 1973;51(8):285–300. [7] Butler R, Lillico M, Hunt GW, McDonald NJ. Experiments on interactive buckling in optimized stiffened panels. Struct Multidiscip Optim 2000;23(1):40–8.

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