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Stability of curved panels under uniform axial compression
Journal of Constructional Steel Research 69 (2012) 30–38
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Journal of Constructional Steel Research
Stability of curved panels under uniform axial compression Khanh Le Tran a, Laurence Davaine a,⁎, Cyril Douthe b, Karam Sab c a b c
SNCF, Direction de l'Ingénierie, 6 av. F. Mitterrand, 93574 La Plaine St Denis, France Université Paris-Est, IFSTTAR, 58 Bd Lefebvre, 75732 Paris, France Université Paris-Est, Laboratoire Navier (ENPC/IFSTTAR/CNRS), École des Ponts, ParisTech, 6 et 8 av. B. Pascal, 77455 Marne-la-valléé Cedex 2, France
a r t i c l e
i n f o
Article history: Received 26 April 2011 Accepted 29 July 2011 Available online 9 September 2011 Keywords: Cylindrical panel Buckling Capacity curve Steel bridge GMNIA Eurocode 3
a b s t r a c t The use of curved panels for the construction of steel bridges becomes more and more popular. Their design is however made difficult by a lack of specifications, especially in European Standards. The present study aims thus at developing a method for predicting the ultimate strength of cylindrical unstiffened curved panels subjected to uniform axial compression. The methodology used in this study is based on the formal procedure recommended by Eurocode 3 for all types of stability verifications. A series of numerical simulations is first carried out to identify the fundamental characteristics of curved panels' elasto-plastic behaviour. Then on the basis of these numerical results, semi-empirical formulae for predicting the elastic buckling and ultimate strength are derived and illustrated on a practical example. © 2011 Elsevier Ltd. All rights reserved.
1. Introduction 1.1. Engineering context Typical examples of curved panels in the civil engineering domain are orthotropic decks in box-girder bridges and webs of in-plane curved I-girder. When designing such panels, the engineer will search for the best compromise between economy and safety, he will try to reduce as much as possible the thickness of the plate to lighten the structure and reduce its cost. Nevertheless, the stability verification of cylindrical curved members is not exactly covered by European Standards: EN 1993-1-5 [1] gives specifications for flat or slightly curved panels (with the condition b 2/Rt b 1) and EN 1993-1-6 [2] deals with revolution cylindrical shells. It will be seen in the following example that curved panels in bridges have characteristics exactly between these two cases. Fig. 1 shows such a bridge built over the large arm of the river Seine between Boulogne-Billancourt and the Seguin Island (France). The bridge is a steel box girder with a constant web spacing of 6.40 m. The radius of the bottom flange varies from 80 to 120 m. The flange thickness also varies from 14 mm at mid-span to 60 mm at the bridge's supports. These geometrical characteristics give a ratio b2/Rt of 5.7 to 36.8, clearly out-of EN 1993-1-5 scope. EN 1993-1-6 is not applicable neither because these curved flanges are not revolution cylinders. In such a case, the use of finite element modelling (F.E.M.) becomes necessary. However it requires sophisticated models, time-
⁎ Corresponding author. Tel.: + 33 1 41 62 03 56; fax: +33 1 41 62 48 79. E-mail address: [email protected] (L. Davaine). 0143-974X/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.jcsr.2011.07.015
consuming calculations and then careful analysis of results. Consequently, daily practitioners need a reliable and relatively simple hand calculation method which proposal is the aim of this work. 1.2. Scientific context Papers related to the buckling theory of curved panels are not so numerous in comparison to the literature devoted to plate buckling or cylindrical shell buckling. Chronologically, in the first models, it was assumed that, for curved panels like for full revolution cylinders, the ultimate load is equal to the critical buckling load. For instance, the reference expressions developed by Redshaw [3] and Timoshenko [4] rely on this assumption. Stowell [5] used it in its proposition of a modified form of Redsaw's expression taking into account the influence of the boundary conditions. Then other expressions similar to Timoshenko's were achieved by using Donnell's equation [6] or the Schapitz criterion [7]. However, the experimental data obtained by Cox and Clenshaw [8], Crate and Levin [9], Jackson and Hall [10], Welter [11], and Schuette [12] exhibit significant differences with these theoretical values. These differences remain unexplained until the works of Batdorf et al. [13], who attempted to make a synthesis of the test data from Crate and Levin [9] and Cox and Clenshaw [8]. Their modified solutions are presented as abacus, also known as Batdorf's curves, which have been widely applied in the aeronautical fields till now [14]. In a second period, researchers wondered if the critical buckling load really represents the failure load for all curved panels, whatever their radius of curvature, because it could be expected that, at least for panels with small curvature (close to single plate), the ultimate load will increase after buckling. This was indeed verified by the
K.L. Tran et al. / Journal of Constructional Steel Research 69 (2012) 30–38
31
Fig. 1. Renault Bridge — Seguin Island, France.
experimental work of Wenzek [15]. Some expressions including this post-buckling resistance were thus proposed by Pope [16], Sekine and Tamate [17] and Gerard [18]. These expressions are still approximations: from a mathematical point of view, the solution of the underlying differential problem is highly complicated by the shell curvature. During last decades, the development of computers and finite elements codes make it possible to evaluate the stability of complex structures. Recognising a need for updating the N.A.S.A. structural stability monographs, Domb and Leigh [19] proposed an update of the currently used design curves. Other studies in the field of shipbuilding were also conducted [20]. In this last case, the plates are so thick that elastic buckling rarely occurs and that plastic buckling dominates which is exactly the opposite to the aeronautical field where only elastic buckling occurs. One should however notice that, for the ship industry, information on the elastic–plastic behaviour may be useful if a reduced thickness due to corrosion is considered. Further investigations were then conducted by Yumura et al. [21] who studied numerically the buckling/plastic collapse and by Park et al. [22–24] who provided a simplified method to estimate the ultimate strength based on Faulkner's formulae for a single plate with a newly defined slenderness parameter including curvature effects. 1.3. Objective and methodology Despite abundant research, there is today no satisfying analytical expression for the estimation of the ultimate load of cylindrical curved panels. The present study aims thus at the development of such an expression. For simplicity reason, it will be limited to cylindrical, unstiffened square panels subjected to uniform longitudinal compression. Except the stiffening condition, this configuration is the most common one for bottom flanges in box girder bridges. The general framework of this study is that of Eurocode 3 for all kinds of stability verifications [25,26]. Eurocode 3 proposes to use a procedure called χ−λ method (Fig. 2), which consists of three steps: • Calculate the linear elastic critical load (Fcr) for the ideal (imperfection free) panel and its elastic or plastic resistance (Fy) obtained when instability phenomena are absent.
• Calculate the relative slenderness λ of the panel, defined as pffiffiffiffiffiffiffiffiffiffiffiffiffiffi λ = Fy = Fcr . • Evaluate the reduction factor χ which depends on the relative slenderness λ and on the imperfections in the panel (initial geometric imperfections and residual stresses). χ is a semi-empirical parameter. Its mathematical expression comes from an analytical approach using standard strength of materials theory. It includes some numerical parameters which have to be calibrated with the available numerical and/or experimental data. These parameters cover the effects of the geometrical initial imperfections and material imperfections such as residual stresses in the panels. • Calculate the ultimate resistance (Fu) by Fu = χFy. Normally, Fu should be divided by γM. In the present paper, γM is assumed to be equal to 1.0, in order to be able to compare codes value of the resistance with values obtained by numerical simulations. The article's structure will follow the three steps of the procedure mentioned above. The critical buckling will be investigated first and then the ultimate load. Afterwards a semi-empirical expression will be deduced and calibrated with previous numerical results. Finally a practical example will be shown to illustrate the design procedure and to verify the accuracy of the proposed formulae. 2. Finite element modelling In the following, the studied cylindrical panels have square dimensions and uniform curvature (see Fig. 3). They are simply supported on all edges (uR = 0, no radial displacement): along the curved edges AC and BD (the loaded ones) longitudinal displacements are allowed but restricted (uZ = cste) whilst the straight edges AB and CD are free to wave in the circumferential direction. These boundary conditions are similar to that usually used for the study of compressed plates. The external load is introduced by means of an external uniform compression in the longitudinal direction. The numerical studies are conducted with the software ANSys version 11. The shell is modelled with the standard structural shell element, Shell 181 [27]. This element type is a quadrilateral 4-nodes
Fig. 2. The χ−λ design method according to Eurocode 3 recommendations for stability verifications.
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K.L. Tran et al. / Journal of Constructional Steel Research 69 (2012) 30–38
Fig. 3. Cylindrical curved panel under uniform axial compression.
element involving both bending and membrane properties with six degrees of freedom per node. This element is suitable for thin to moderately-thick shell structures. It is necessary to note that the precision of the numerical result may depend on the mesh density. A very fine meshing allows to obtain a good precision but requires long time-calculation. In this study, a mesh convergence analysis has been carried out in order to obtain a “sufficiently” refined mesh. It has been showed that the use of 40 elements within every panel edges gives a very good precision (error less than 1%) without increasing too much the calculation duration. This mesh density has been retained for the next steps of this study. The panels are all made of steel which is assumed to be elasticperfectly plastic for the material non-linear analysis (MNA) and elastic– plastic with linear strain hardening as indicated in EN 1993-1-5 C.6 for the material non-linear second-order analyses with initial imperfections (GMNIA). The Young modulus E and Poisson ratio ν are taken equal to 210 GPa and 0.3 respectively. The steel grade is S355 with a yield strength equal to fy = 355 MPa for a thickness up to 16 mm (for a higher thickness, it is reduced according to EN 1993-1-1 [28]). For the GMNIA, the modelling of the initial imperfections is of great importance, like for many stability problems. The influence of imperfections in curved panels (geometrical defects and residual stresses pattern) has been discussed by Feartherston [29] but, to the authors mind, this point is not yet sufficiently investigated. So, noting that the fabrication process of curved panels is very similar to that of plates, it can be assumed that one can use an equivalent initial imperfection as given by EN 1993-1-5 with the shape of the first critical buckling mode and with a maximum deflection wo = max (a/200 ; b/200) (where a and b are the dimensions of the panel). This equivalent geometric imperfection aims at covering the influence of geometric defects and of residual stresses. Some preliminary tests were hence conducted for various configurations of curved panels with geometric imperfections and residual stresses pattern taken from the Swedish design code for welded sections [30] and they have shown that the equivalent geometric imperfection as defined in EN 1993-1-5 was reasonably conservative in all configurations.
Poisson's ratio). From a physical point of view, this parameter is proportional to the maximum deviation of the panel from its chord (given by b 2/8R) divided by its thickness t. The curvature parameter Z has an influence on the shape of the first buckling mode as one can see in Fig. 4. The case Z equal to zero corresponds to a flat plate with a single centred buckling shape (Fig. 4a). For small values of Z, curved panels buckle with a nearly square wave like plates (Fig. 4b). When Z increases, the shape of the buckling mode progressively evolves, the out-of-plane deformation decreases in the middle of the panel whereas the maximum deflection becomes localised along the two loaded edges of the panel (Fig. 4c and d). To quantify this influence of curvature, the numerical values of the critical buckling load are compared to the reference analytical expressions, namely those of Redshaw [3], of Timoshenko [4], of Stowell [5] and of Domb and Leigh [19]. Batdorf's formula will not be considered because its theoretical background is similar to Timoshenko's formula and because a modification of it has been proposed in the work of Domb and Leigh [19]. To simplify the comparison, all formulae have been rewritten in the following form:
3. Elastic buckling resistance
• Timoshenko made assumption on the form of the displacements so that they satisfy the boundary conditions. Substituting these expressions into the equilibrium equations, he obtained the critical load looking for the value which nullifies the determinant of these equations:
3.1. Numerical investigation In order to investigate the elastic buckling resistance of unstiffened curved panels, a series of linear buckling analyses is performed. The results of these simulations will allow the verification of the accuracy of the theoretical formulae found in the literature. The investigated panels have a constant width b of 3 m, a constant depth a of 3 m and a constant thickness t of 20 mm. Only the radius R varies in order to change the curvature of panels. The non-dimensional parameter Z = b 2/Rt is defined to characterise the curvature of the panels (this parameter is a modified Bartdorf's parameter skipping the effects of
Z
Z
σcr = kc :σE
ð1Þ
where σE is the classic Euler critical stress: σE =
2 2 π E t 2 b 12 1−ν
ð2Þ
and kcZ is a buckling coefficient which includes the curvature effect. This dimensionless coefficient will be used to compare the various expressions which are listed below with some information on the method used for their determination: • Redshaw applied the classical energy approach: 0 Z kc;Redshaw
Z
= 2@1 +
kc;Timoshenko =
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi1 12 1−ν2 2 A 1+ Z : π4
ð3Þ
8 2 > 3 1−ν > > 2 > Z 4+ > > > π4 <
2π4 if Z ≤ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 1−ν 2
pffiffiffi > > > 4 3 > > > > π2 Z :
2π if qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ≤Z 3 1−ν2
4
:
ð4Þ
K.L. Tran et al. / Journal of Constructional Steel Research 69 (2012) 30–38
33
Fig. 4. First elastic buckling modes for different curvatures.
• Stowell proposed a modification of Redshaw's equation to better take into account the boundary conditions:
Z
kc;Stowell
0 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u u B 48 1−ν 2 2 kplate c B 1+u 1+ = 2 Z t @ 2 π4 kplate c
1 C C: A
ð5Þ
• Domb and Leigh's expression is calibrated on a numerical database by some curve fitting method:
Z
kc;Domb &
Leight
=
8 3 > < 10 ∑ c ð logZ Þi i b > :
i=0 d
cð Z Þ
1 ≤ Zb < 23:15
ð6Þ
23:15 ≤ Zb ≤ 200
with Zb = Z(1 − ν 2), co = 0.6021, c1 = 0.005377, c2 = 0.192495, c3 = 0.00267, c = 0.4323 and d = 0.9748.. The results of this comparison are presented in Fig. 5 where the buckling coefficient kcZ is plotted as a function of the curvature parameter Z. It is first noticed that all formulae converge towards the value corresponding to a flat plate (kplate = 4) when Z tends towards 0 which is logical and necessary: for Z = 0, the curved panel is actually a plate. It is remarked then that, for all curves, the buckling coefficient increases with Z which can be explained by the fact that the higher the values of Z, the higher the circumferential membrane stresses which hold up the radial deflection. The asymptotic behaviour of these expressions is however obviously different. The theoretical formulae of Redshaw and Timoshenko converge asymptotically towards a straight line whose slope is equal to the buckling coefficient of a revolution cylinder whilst the formulae of Domb and Leigh, of Stowell
and the numerical results converge towards half this value. The differences of the boundary conditions involved explain this gap: curved cylindrical panels have four free cut edges whereas revolution cylinders have only two, the circumferential membrane stresses develops thus in different ways. The numerical results indicate that the buckling coefficient of curved panels (with constant width b) converge on half the value of the buckling coefficient of the corresponding revolution cylinder (defined by b = 2πR). To clarify this observation, further numerical simulations are carried out by studying cylindrical curved panels with constant length, radius and thickness (a = 9 m; R = 3 m, t = 15 mm) but with a width b varying from 0 to 2πR. The last panel (b = 2πR) is hence a revolution cylinder with a straight cut along its length a. Fig. 6 shows the ratio (σcurved/σcylinder) between the critical buckling stress of the curved panel and that of the full revolution cylinder against the curvature parameter Z. This ratio decreases when Z increases, from infinity for Z = 0 (i.e. b = 0) to 0.5 for Z = 4π2 Rt (Z = 7895). It should be noticed that, in this specific case, the value 0.5 is quasi reached (less than 2% difference) for Z ≥ 40, i.e. b = 1.34 m which is far from the upper value of b (2πR = 18.85 m).
3.2. Buckling coefficient formula for curved panels Although Fig. 5 shows a good agreement of Stowell and Domb and Leigh's expressions with the numerical results, there is still a need for a reasonable expression of the critical buckling mode. The reason is first the complexity and the non-intuitive aspect of Domb and Leigh's expression which includes several coefficients calibrated on numerical simulations. Secondly, the expression of Stowell has not been demonstrated yet. The new expression is
Fig. 5. Comparison of reference analytic buckling coefficients with numerical results.
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K.L. Tran et al. / Journal of Constructional Steel Research 69 (2012) 30–38
Fig. 6. Convergence of the asymptotic behaviour of cylindrical curved panels.
hence looked for assuming a mathematical formula inspired by Stowell (see Eq. (5)): qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z kc = A 1 + 1 + B⋅ Z 2
ð7Þ
A and B are two parameters which will be identified with respect to the limit behaviours: • For Z = 0, curved panels behave like flat plates, Z
plate
k c j Z = 0 = kc
plate
⇒A =
kc : 2
ð8Þ
• For Z → ∞, curved panels behave like cut revolution cylinders, 2 48 1−υ 1 cylinder Z ⇒B = lim kc = kc 2 : Z→∞ 2 π4 kplate c
Fu = Fy = b:t:fy
ð9Þ
ð10Þ
We obtain exactly Stowell's expression but, let us say, in a more pedagogical manner. This formula exhibits slight differences with the numerical results especially in the transition zone from plates to full revolution cylinders (Fig. 5) for which it was calibrated. This is not of big importance here because this approximation of the critical buckling load is only the first step in the whole hand-calculation procedure which is developed for the determination of the ultimate strength. These differences between Eq. (10) and numerical results will be partly recovered during the calibration process of the reduction factor χ (cf. Section 4). The expression (Eq. (1)) with the buckling coefficient given by Eq. (10) is thus satisfactory and it will be used in the following for the evaluation of critical buckling stresses of curved panels under uniform longitudinal compression. 4. Ultimate strength For stocky curved panels under uniform compression (i.e. small values of the slenderness parameter λ), no buckling phenomenon
ð11Þ
For more slender curved panels, the slenderness parameter λ increases, the failure mode becomes a shell buckling and the ultimate strength can be written as a fraction of the yield resistance characterised by the reduction factor chi (by definition χ ≤ 1): Fu = χFy :
The proposed formula for the buckling coefficient is thus: 1 0 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u plate 2 u C B 48 1−υ k Z 2C 1+u kc = c B t1 + 4 plate 2 Z A: 2 @ π kc
occurs and the ultimate strength is governed by the yielding of the whole cross-section and the ultimate load is equal to the yield resistance (if required, this yield resistance can be estimated numerically with a material non-linear analysis and an elasticperfectly plastic material law):
ð12Þ
The ultimate strength Fu of the panel is evaluated numerically using non-linear analysis including geometrical and material imperfections (GMNIA). Practically, these numerical simulations are conducted using the arc-length method due to the complexity in the load–displacement response when entering into the post-buckling regime. For such an iterative method, it is necessary to select suitable criteria convergence for the termination of the iteration process. Ansys's solution control uses L2-norm of force (and moment) tolerance equal to 0.5%, a setting that is appropriate for most engineering applications. The reduction factor χ is then determined from the ratio between the maximum load Fu and the yield resistance Fy. 4.1. Numerical simulations A series of simulations (GMNIA) is thus carried out for several curved panel configurations with a curvature parameter Z varying from 1 to 100. These geometrical configurations have been chosen in order to cover the whole range of curved panels in steel bridges. The numerical results are presented in Fig. 7 where the reduction factor χ is plotted against the slenderness parameter λ. Fig. 7 also shows the reduction factors coming from EN 1993-1-5 for plate buckling and from EN 1993-1-6 for revolution shell buckling, the highest and lowest curves respectively. The numerical results show a clear influence of the curvature parameter Z on the ultimate strength Fu. For a given slenderness λ, the highest the panel curvature, the lowest the ultimate strength. The variations of the ultimate strength with the slenderness λ show however approximately the same scheme for all the curvatures Z. (This all gives an additional justification to the choice of the Z
K.L. Tran et al. / Journal of Constructional Steel Research 69 (2012) 30–38
35
Fig. 7. Ultimate strength of curved panels under uniform axial compression.
parameter for representing the variability of this problem.) It is then noticed that all the marks are located between the two reduction factors coming from the European Standards. For small values of Z, curved panels behave like plates (EN 1993-1-5) and for high valuesof Z, the behaviour turns to that of a cylinder (EN 1993-1-6). The clause 1.1(2) of EN 1993-1-5 which says that the reduction factor for plates under uniform compression can be applied for relatively small curvatures (Z ≤ 1) is verified: the distribution of marks for Z = 1 follows more or less the curve from EN 1993-1-5. For higher curvature, the use of EN 1993-1-5 would lead to a significant overestimation of the ultimate strength of the curved panel. For high values of Z (Z ≥ 40) and for small slenderness (λ≤1:0), numerical results show that the reduction factor of EN 1993-1-6 provides a good approximation of the ultimate strength for curved panels. However a non-negligible gap is noticed when the slenderness becomes high where EN 1993-1-6 becomes too conservative. This gap is mainly due to the post-critical strength reserve of curved panels which leads to a critical buckling load smaller than the ultimate load. This strength reserve is also related to the different boundary conditions of the curved panel edges (four free edges) and of the revolution cylinder (only two free edges). For high slenderness values (λ N 3), the collapse behaviour is governed by the elastic buckling. The influence of imperfections becomes thus smaller. Fig. 8 zooms in Fig. 7 on these high values of λ
and shows that the curves vary as a functions of 1 = λ. A proportionality coefficient β can thus be calibrated on the basis of the numerical results. For example β is equal to 1 for Z = 0 (plate behaviour) and equal to 0.5 when Z = 75. β decreases when the curvature parameter increases mainly because of the interaction between the out-of-plane bending stresses due to the curvature and the membrane stresses. The higher the curvature, the higher this effect. The parameter β can thus be interpreted as an imperfection factor representing the curvature effect. 4.2. New capacity curves The previous observations lead to the search for a new expression of the reduction factor χ of Eq. (12) allowing to fill in the gap between EN 1993-1-5 and EN 1993-1-6. This new expression will obviously depend on the curvature parameter Z. Its mathematical form will come from an analytical approach based on the theory of strength of materials, similar to the ones already used for the development of the European buckling curves in EN 1993-1-1 [31] or for the plate buckling phenomenon during bridge launching [32]. It is hence proposed that the reduction factor χ is defined by: ð1−χ Þ β−χ⋅ λ = η⋅χ
Fig. 8. Behaviour of curved panels in the elastic buckling zone (high values of λ).
ð13Þ
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K.L. Tran et al. / Journal of Constructional Steel Research 69 (2012) 30–38
where η and β are numerical parameters to be calibrated with the numerical results. As for the standard European buckling curves, η represents the generalised imperfection which is assumed to have the following form: η = χ⋅ αZ λ−λ0
ð14Þ
where αZ is the elastic imperfection factor, λ0 represents the length of the yield plateau (associated with the value χ = 1) where no buckling occurs. Solving Eq. (13) after introduction of Eq. (14), one obtains: χ=
β+λ+
2β qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 : β + λ −4 λ−αZ λ−λ0
imperfection parameter αZ vary in a smooth manner. So, in order to make things easier for practitioners, it is proposed to reduce the number of characteristic values of αZ (see Table 1) and to interpolate linearly the values in-between. The quality of this approximation is tested in next section.
ð15Þ
5. Validation and summary of the proposed hand calculation method To illustrate and verify the accuracy of the proposed approach, an example of curved panel verification is presented. This panel is 1.5 m long and wide, 20 mm thick and has a 4.5 m radius of curvature. The hand-calculation design method goes as follows: • Evaluate the curvature parameter Z:
Eq. (15) represents a series of curves indexed with the imperfection parameter αZ. This series of curves admits two upper bounds, χ = 1 for full yielding (λ ≤ λ0 ) and χ = a = λ for asymptotic elastic buckling (λ→∞).
The parameters β and λ0 correspond to two different collapse modes which are clearly identified in Fig. 7. For a given value of Z, the coefficient λ0 is determined by the intersection of the numerical curve with the horizontal line χ = 1 corresponding to pure yielding of the panel. Then, the coefficient β is identified from the asymptotic behaviour for high value of λ where the reduction factor tends towards β = λ. The two parameters evidently depend on the curvature parameter Z and can be approximated by the following expressions, where the limit cases (Z = 0 and Z → ∞) are determined by EN 1993-1-5 and EN 1993-1-6 respectively:
β=
1 + 0:97Z : 2
b2 = 25: Rt
ð18Þ
• Calculate the Euler critical stress σE and the buckling coefficient kcZ:
4.3. Calibration of parameters
Z λ0 = 0:2 + 0:473 0:95
Z=
ð16Þ
ð17Þ
The last parameter αZ is then calibrated in order to minimise the gap between the values given by Eq. (15) and the values obtained numerically. This gap is measured by the sum of the squares of the differences between those values. Its variations with αZ and with Z are shown in Fig. 9. It is noted that for high values of the curvature parameter (Z ≥ 40), αZ is almost constant and generally that the
σE =
2 π2 E t = 33:74 MPa 12 1−ν 2 b vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi1 u 2 u 48 1−υ u B 2C B1 + u1 + 2 Z C t A = 10:61 2 @ π4 kplate c 0
Z
kc =
kplate c
with kcplate = 4 for the case of simple supported plate. Z : • Calculate the linear elastic critical stress σcr Z
Z
σcr = kc⋅ σE = 357:87 MPa:
ð19Þ
• Evaluate the reduced slenderness λ: λ=
sffiffiffiffiffiffiffi fy σcrZ
= 0:98:
• Identify the parameters β, λ0 and αZ as a function of Z: Z λ0 = 0:2 + 0:473 0:95 = 0:33
Fig. 9. Calibration of the αZ parameter.
ð20Þ
K.L. Tran et al. / Journal of Constructional Steel Research 69 (2012) 30–38 Table 1 Value of αZ. Z
0
10
20
30
≥ 40
αZ
0.28
0.38
0.33
0.21
0.13
β=
1 + 0:97 = 0:73 2
Z
• Determine the reduction factor χ: 2β qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 = 0:59: β+λ+ β + λ −4 λ−αZ λ−λ0
ð21Þ
• Calculate the ultimate strength σu: σu = χ: fy = 196:65 MPa:
(RMS) are close to 1. The low standard deviation (STDEV = 0.00452) also indicates that the data points tend to be very close to this mean value. The observed error between the predicted value by the proposal and the corresponding numerical one is not significant, because the mean absolute percentage errors (MAPE) are only 3.42% and the maximum error (in absolute value) is lower than 0.07. It is thus possible to conclude that the proposed analytical method fits very well with the numerical results. 6. Conclusion and further work
αZ = 0:27ðevaluated from Table 1Þ:
χ=
37
ð22Þ
This process is illustrated in the flowchart of Fig. 10. From the comparison of the proposed method with existing recommendations and numerical analysis, some remarks can be made out: • Although the theoretical value for critical buckling stress is a bit higher Z than the numerical one (σcrZ = 357.87 MPa and σcr, F. E. = 320.91 MPa), the predicted value for the ultimate strength is as good as the numeric alone (σu = σu, F. E. = 196.65 MPa). As indicated in Section 3.2, the error in the approximation of the critical stress is recovered by the calibration process of the reduction factor χ. • The ultimate strength given by EN 1993-1-5 is significantly higher than the real resistance of the panel (χEN1993 − 1 − 5 =0.79N χF. E =0.59), the use of EN 1993-1-5 leads to an overestimation of the panel strength which is against security. • The exact value for αZ in this case is 0.24 which is not far from the interpolated value (αZ = 0.27). The assumption of linear interpolation is thus acceptable. Beyond this practical example, the accuracy of the proposed method has been checked with about 524 numerical simulations, for a range of Z from 0 to 100. Fig. 11 shows that the proposed formula fits well the numerical database. Both the mean value and the root mean square
In this paper, the critical buckling and the ultimate strength of cylindrical curved panels have been investigated analytically and by means of linear and non-linear finite elements analyses. It appeared that the behaviour of cylindrical curved panels usually depends on its curvature, its slenderness and its imperfections. These results can be summarised as follows: • The elastic buckling load of cylindrical curved panels increases with the curvature parameter Z = b 2/Rt and it is best approximated by Stowell's semi-empirical formula. • The ultimate strength is very sensitive to the panel imperfections which shape can reasonably be based on the shape of the first buckling mode with a maximum deflection wo of max(a/200; b/200) as given in EN 1993-1-5 for plate buckling. • For slender panels where the collapse behaviour is governed by the elastic buckling, the reduction factor χ can be assumed to behave as β = λ. β represents the influence of the curvature on the value of the imperfection factor. It decreases with the curvature Z and it is equal to 1 for Z = 0. • A reduction factor χ has been developed for predicting the ultimate strength of curved panels. Its mathematical expression has been derived in the same way as existing European buckling curves. The various parameters have been calibrated using numerical results especially built for covering the range of steel bridges (i.e. Z b 100). • Although the finite element method is widely recognised as a reliable tool for stability verifications, the accuracy of results is sensitive to initial geometric and material imperfections. It would thus be interesting to validate the proposed methodology by experimental measurements on cylindrical curved panels under uniform axial compression. This study has been carried out in the framework of a running research project essentially targeted for bridge engineering applications.
Fig. 10. Practical example of the proposed hand-calculation method.
38
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Fig. 11. Correlation of the proposed formula with numerical reduction factor.
The study of unstiffened curved panels under uniform axial longitudinal compression represents the first step of this research project. In the second step, the different loading conditions (transverse compression and shear stress) and also the stiffened curved panels will be investigated. References [1] EN 1993-1-5: design of steel structures, part 1–5: plated structural elements. Brussels: CEN; 2007. [2] EN 1993-1-6: design of steel structures, part 1–6: strength and stability of shell structures. Brussels: CEN; 2007. [3] Redshaw SC. The elastic instability of a thin curved panel subjected to an axial thrust, its axial and circumferential edges being simply supported. Tech. Rep. Bristish aeronautical research council; R&M-1565, 1933. [4] Timoshenko S, Gere J. Theory of elastic stability. New York: McGraw-Hill; 1961. [5] Stowell EZ. Critical compressive stress for curved sheet supported along all edges and elastically restrained against rotation along the unloaded edges. National Advisory Committee for Aeronautics; 1943. p. 99–109. [6] Donell L. A new theory for the buckling of thin cylinders under axial compression and bending. Trans ASME 1934;56:795–806. [7] Spier E, Smith G, Sullins R. Manual for structural stability analysis of sandwich plates and shells. Tech. Rep. NASA CR-1457, 1969. [8] Cox H, Clenshaw W. Compression tests on curved plates of thin sheet duralumin. HM Stationery Off; 1941.
[9] Crate H, Levin L. Data on buckling strength of curved sheet in compression. Tech. Rep. NACA WR-557, 1943. [10] Jackson KB, Hall AH. Curved plates in compression. Tech. Rep. National Research Council of Canada Aeronautical; 1947. [11] Welter G. The effect of the radius of curvature and preliminary artificial eccentricities on buckling loads of curved thin aluminum-alloy sheets for monocoque constructions. J Aeronaut Sci 1946;13:593–6. [12] Schuette E. Buckling of curved sheet in compression and its relation to the secant modulus. J Aeronaut Sci 1948;15(1):18–22. [13] Batdorf SS, Schildcrout M, Stein M. Critical combinations of shear and longitudinal direct stress for long plates with transverse curvature. National Advisory Committee for Aeronautics; TN-1347; 1947. [14] Bruhn EF. Analysis and design of flight vehicle strutures. Jacobs Pub; 1973. [15] Wenzek W. The effective of the curved sheet after buckling. Luftfahrtforschung 1938;15:340–3. [16] Pope GG. On the axial compression of long, slightly curved panel. Tech. Rep. Bristish aeronautical research council; R&M-3392; 1963. [17] Sekine H, Tamate O. Postbuckling behavior of thin curved panels under axial compression. Jpn Soc Mech Eng 1969;12:415–20. [18] Gerard G, Becker H. Handbook of structural stability. Part III: buckling of curved plates and shells. Tech. Rep.New York University; 1957. [19] Domb MM, Leigh BR. Refined design curves for compressive buckling of curved panels using nonlinear finite element analysis. 42nd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference; 2001. p. 449–57. [20] Maeno Y, Yamaguchi H, Fujii Y, Yao T. Buckling/plastic collapse behaviour and strength of bilge circle and its contribution to ultimate longitudinal strength of ship's hull girder. Proc. of the fourteenth International Offshore and Polar Engineering Conference, Toulon, France; 2004. p. 296–302. [21] Yumura K, Katsura S, Iijima K, Yao T. Simulation of buckling collapse behaviour of cylindrically curved plates under axial compression. The nineteenth Asian Technical Exchange and Advisory Meeting on Marine Structures, Singapore; 2005. p. 482–9. [22] Park JS, Iijima K, Yao T. Characteristics of buckling and ultimate strength and collapse behaviour of cylindrically curved plates subjected to axial compression. Adv Mater Res 2008;33:1195–200. [23] Park JS, Fujikubo M, Iijima K, Yao T. Prediction of the secondary buckling strength and ultimate strength of cylindrically curved plate under axial compression. Proc. the nineteenth International Offshore and Polar Engineering Conference, Osaka, Japan; 2009. p. 740–7. [24] Park JS, Yumura K, Katsura S, Iilima K, Yao T. Buckling and post-buckling behaviour of cylindrically curved plates under axial compression. Conference Proceedings. Japan Society of Naval Architects and Ocean Engineers; 2006. p. 191–4. [25] Maquoi R, Skaloud M. Stability of plates and plated structures: general report. J Constr Steel Res 2000;55:45–68. [26] Schmidt H. Stability of steel shell structures: general report. J Constr Steel Res 2000;55:159–81. [27] ANSYS user's theory manual V11. ANSYS Inc.; 2007. [28] EN 1993-1-1: design of steel structures, part 1–1: general rules and rules for buildings. Brussels: CEN; 2005. [29] Featherston C. Imperfection sensitivity of curve panels under combined compression and shear. Int J Non Linear Mech 2003;38:225–38. [30] BSK99. Swedish design rules for steel structures; 1999. [31] Maquoi R, Rondal J. Mise en équation des nouvelles courbes Européennes de flambement. Rev Constr Métallique 1978;1:17–30. [32] L. Davaine. Formulations de la résistance au lancement d'une âme métallique de pont raidie longitudinalement. Ph.D. thesis; INSA de Rennes; 2005.
Contents lists available at ScienceDirect
Journal of Constructional Steel Research
Stability of curved panels under uniform axial compression Khanh Le Tran a, Laurence Davaine a,⁎, Cyril Douthe b, Karam Sab c a b c
SNCF, Direction de l'Ingénierie, 6 av. F. Mitterrand, 93574 La Plaine St Denis, France Université Paris-Est, IFSTTAR, 58 Bd Lefebvre, 75732 Paris, France Université Paris-Est, Laboratoire Navier (ENPC/IFSTTAR/CNRS), École des Ponts, ParisTech, 6 et 8 av. B. Pascal, 77455 Marne-la-valléé Cedex 2, France
a r t i c l e
i n f o
Article history: Received 26 April 2011 Accepted 29 July 2011 Available online 9 September 2011 Keywords: Cylindrical panel Buckling Capacity curve Steel bridge GMNIA Eurocode 3
a b s t r a c t The use of curved panels for the construction of steel bridges becomes more and more popular. Their design is however made difficult by a lack of specifications, especially in European Standards. The present study aims thus at developing a method for predicting the ultimate strength of cylindrical unstiffened curved panels subjected to uniform axial compression. The methodology used in this study is based on the formal procedure recommended by Eurocode 3 for all types of stability verifications. A series of numerical simulations is first carried out to identify the fundamental characteristics of curved panels' elasto-plastic behaviour. Then on the basis of these numerical results, semi-empirical formulae for predicting the elastic buckling and ultimate strength are derived and illustrated on a practical example. © 2011 Elsevier Ltd. All rights reserved.
1. Introduction 1.1. Engineering context Typical examples of curved panels in the civil engineering domain are orthotropic decks in box-girder bridges and webs of in-plane curved I-girder. When designing such panels, the engineer will search for the best compromise between economy and safety, he will try to reduce as much as possible the thickness of the plate to lighten the structure and reduce its cost. Nevertheless, the stability verification of cylindrical curved members is not exactly covered by European Standards: EN 1993-1-5 [1] gives specifications for flat or slightly curved panels (with the condition b 2/Rt b 1) and EN 1993-1-6 [2] deals with revolution cylindrical shells. It will be seen in the following example that curved panels in bridges have characteristics exactly between these two cases. Fig. 1 shows such a bridge built over the large arm of the river Seine between Boulogne-Billancourt and the Seguin Island (France). The bridge is a steel box girder with a constant web spacing of 6.40 m. The radius of the bottom flange varies from 80 to 120 m. The flange thickness also varies from 14 mm at mid-span to 60 mm at the bridge's supports. These geometrical characteristics give a ratio b2/Rt of 5.7 to 36.8, clearly out-of EN 1993-1-5 scope. EN 1993-1-6 is not applicable neither because these curved flanges are not revolution cylinders. In such a case, the use of finite element modelling (F.E.M.) becomes necessary. However it requires sophisticated models, time-
⁎ Corresponding author. Tel.: + 33 1 41 62 03 56; fax: +33 1 41 62 48 79. E-mail address: [email protected] (L. Davaine). 0143-974X/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.jcsr.2011.07.015
consuming calculations and then careful analysis of results. Consequently, daily practitioners need a reliable and relatively simple hand calculation method which proposal is the aim of this work. 1.2. Scientific context Papers related to the buckling theory of curved panels are not so numerous in comparison to the literature devoted to plate buckling or cylindrical shell buckling. Chronologically, in the first models, it was assumed that, for curved panels like for full revolution cylinders, the ultimate load is equal to the critical buckling load. For instance, the reference expressions developed by Redshaw [3] and Timoshenko [4] rely on this assumption. Stowell [5] used it in its proposition of a modified form of Redsaw's expression taking into account the influence of the boundary conditions. Then other expressions similar to Timoshenko's were achieved by using Donnell's equation [6] or the Schapitz criterion [7]. However, the experimental data obtained by Cox and Clenshaw [8], Crate and Levin [9], Jackson and Hall [10], Welter [11], and Schuette [12] exhibit significant differences with these theoretical values. These differences remain unexplained until the works of Batdorf et al. [13], who attempted to make a synthesis of the test data from Crate and Levin [9] and Cox and Clenshaw [8]. Their modified solutions are presented as abacus, also known as Batdorf's curves, which have been widely applied in the aeronautical fields till now [14]. In a second period, researchers wondered if the critical buckling load really represents the failure load for all curved panels, whatever their radius of curvature, because it could be expected that, at least for panels with small curvature (close to single plate), the ultimate load will increase after buckling. This was indeed verified by the
K.L. Tran et al. / Journal of Constructional Steel Research 69 (2012) 30–38
31
Fig. 1. Renault Bridge — Seguin Island, France.
experimental work of Wenzek [15]. Some expressions including this post-buckling resistance were thus proposed by Pope [16], Sekine and Tamate [17] and Gerard [18]. These expressions are still approximations: from a mathematical point of view, the solution of the underlying differential problem is highly complicated by the shell curvature. During last decades, the development of computers and finite elements codes make it possible to evaluate the stability of complex structures. Recognising a need for updating the N.A.S.A. structural stability monographs, Domb and Leigh [19] proposed an update of the currently used design curves. Other studies in the field of shipbuilding were also conducted [20]. In this last case, the plates are so thick that elastic buckling rarely occurs and that plastic buckling dominates which is exactly the opposite to the aeronautical field where only elastic buckling occurs. One should however notice that, for the ship industry, information on the elastic–plastic behaviour may be useful if a reduced thickness due to corrosion is considered. Further investigations were then conducted by Yumura et al. [21] who studied numerically the buckling/plastic collapse and by Park et al. [22–24] who provided a simplified method to estimate the ultimate strength based on Faulkner's formulae for a single plate with a newly defined slenderness parameter including curvature effects. 1.3. Objective and methodology Despite abundant research, there is today no satisfying analytical expression for the estimation of the ultimate load of cylindrical curved panels. The present study aims thus at the development of such an expression. For simplicity reason, it will be limited to cylindrical, unstiffened square panels subjected to uniform longitudinal compression. Except the stiffening condition, this configuration is the most common one for bottom flanges in box girder bridges. The general framework of this study is that of Eurocode 3 for all kinds of stability verifications [25,26]. Eurocode 3 proposes to use a procedure called χ−λ method (Fig. 2), which consists of three steps: • Calculate the linear elastic critical load (Fcr) for the ideal (imperfection free) panel and its elastic or plastic resistance (Fy) obtained when instability phenomena are absent.
• Calculate the relative slenderness λ of the panel, defined as pffiffiffiffiffiffiffiffiffiffiffiffiffiffi λ = Fy = Fcr . • Evaluate the reduction factor χ which depends on the relative slenderness λ and on the imperfections in the panel (initial geometric imperfections and residual stresses). χ is a semi-empirical parameter. Its mathematical expression comes from an analytical approach using standard strength of materials theory. It includes some numerical parameters which have to be calibrated with the available numerical and/or experimental data. These parameters cover the effects of the geometrical initial imperfections and material imperfections such as residual stresses in the panels. • Calculate the ultimate resistance (Fu) by Fu = χFy. Normally, Fu should be divided by γM. In the present paper, γM is assumed to be equal to 1.0, in order to be able to compare codes value of the resistance with values obtained by numerical simulations. The article's structure will follow the three steps of the procedure mentioned above. The critical buckling will be investigated first and then the ultimate load. Afterwards a semi-empirical expression will be deduced and calibrated with previous numerical results. Finally a practical example will be shown to illustrate the design procedure and to verify the accuracy of the proposed formulae. 2. Finite element modelling In the following, the studied cylindrical panels have square dimensions and uniform curvature (see Fig. 3). They are simply supported on all edges (uR = 0, no radial displacement): along the curved edges AC and BD (the loaded ones) longitudinal displacements are allowed but restricted (uZ = cste) whilst the straight edges AB and CD are free to wave in the circumferential direction. These boundary conditions are similar to that usually used for the study of compressed plates. The external load is introduced by means of an external uniform compression in the longitudinal direction. The numerical studies are conducted with the software ANSys version 11. The shell is modelled with the standard structural shell element, Shell 181 [27]. This element type is a quadrilateral 4-nodes
Fig. 2. The χ−λ design method according to Eurocode 3 recommendations for stability verifications.
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K.L. Tran et al. / Journal of Constructional Steel Research 69 (2012) 30–38
Fig. 3. Cylindrical curved panel under uniform axial compression.
element involving both bending and membrane properties with six degrees of freedom per node. This element is suitable for thin to moderately-thick shell structures. It is necessary to note that the precision of the numerical result may depend on the mesh density. A very fine meshing allows to obtain a good precision but requires long time-calculation. In this study, a mesh convergence analysis has been carried out in order to obtain a “sufficiently” refined mesh. It has been showed that the use of 40 elements within every panel edges gives a very good precision (error less than 1%) without increasing too much the calculation duration. This mesh density has been retained for the next steps of this study. The panels are all made of steel which is assumed to be elasticperfectly plastic for the material non-linear analysis (MNA) and elastic– plastic with linear strain hardening as indicated in EN 1993-1-5 C.6 for the material non-linear second-order analyses with initial imperfections (GMNIA). The Young modulus E and Poisson ratio ν are taken equal to 210 GPa and 0.3 respectively. The steel grade is S355 with a yield strength equal to fy = 355 MPa for a thickness up to 16 mm (for a higher thickness, it is reduced according to EN 1993-1-1 [28]). For the GMNIA, the modelling of the initial imperfections is of great importance, like for many stability problems. The influence of imperfections in curved panels (geometrical defects and residual stresses pattern) has been discussed by Feartherston [29] but, to the authors mind, this point is not yet sufficiently investigated. So, noting that the fabrication process of curved panels is very similar to that of plates, it can be assumed that one can use an equivalent initial imperfection as given by EN 1993-1-5 with the shape of the first critical buckling mode and with a maximum deflection wo = max (a/200 ; b/200) (where a and b are the dimensions of the panel). This equivalent geometric imperfection aims at covering the influence of geometric defects and of residual stresses. Some preliminary tests were hence conducted for various configurations of curved panels with geometric imperfections and residual stresses pattern taken from the Swedish design code for welded sections [30] and they have shown that the equivalent geometric imperfection as defined in EN 1993-1-5 was reasonably conservative in all configurations.
Poisson's ratio). From a physical point of view, this parameter is proportional to the maximum deviation of the panel from its chord (given by b 2/8R) divided by its thickness t. The curvature parameter Z has an influence on the shape of the first buckling mode as one can see in Fig. 4. The case Z equal to zero corresponds to a flat plate with a single centred buckling shape (Fig. 4a). For small values of Z, curved panels buckle with a nearly square wave like plates (Fig. 4b). When Z increases, the shape of the buckling mode progressively evolves, the out-of-plane deformation decreases in the middle of the panel whereas the maximum deflection becomes localised along the two loaded edges of the panel (Fig. 4c and d). To quantify this influence of curvature, the numerical values of the critical buckling load are compared to the reference analytical expressions, namely those of Redshaw [3], of Timoshenko [4], of Stowell [5] and of Domb and Leigh [19]. Batdorf's formula will not be considered because its theoretical background is similar to Timoshenko's formula and because a modification of it has been proposed in the work of Domb and Leigh [19]. To simplify the comparison, all formulae have been rewritten in the following form:
3. Elastic buckling resistance
• Timoshenko made assumption on the form of the displacements so that they satisfy the boundary conditions. Substituting these expressions into the equilibrium equations, he obtained the critical load looking for the value which nullifies the determinant of these equations:
3.1. Numerical investigation In order to investigate the elastic buckling resistance of unstiffened curved panels, a series of linear buckling analyses is performed. The results of these simulations will allow the verification of the accuracy of the theoretical formulae found in the literature. The investigated panels have a constant width b of 3 m, a constant depth a of 3 m and a constant thickness t of 20 mm. Only the radius R varies in order to change the curvature of panels. The non-dimensional parameter Z = b 2/Rt is defined to characterise the curvature of the panels (this parameter is a modified Bartdorf's parameter skipping the effects of
Z
Z
σcr = kc :σE
ð1Þ
where σE is the classic Euler critical stress: σE =
2 2 π E t 2 b 12 1−ν
ð2Þ
and kcZ is a buckling coefficient which includes the curvature effect. This dimensionless coefficient will be used to compare the various expressions which are listed below with some information on the method used for their determination: • Redshaw applied the classical energy approach: 0 Z kc;Redshaw
Z
= 2@1 +
kc;Timoshenko =
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi1 12 1−ν2 2 A 1+ Z : π4
ð3Þ
8 2 > 3 1−ν > > 2 > Z 4+ > > > π4 <
2π4 if Z ≤ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 1−ν 2
pffiffiffi > > > 4 3 > > > > π2 Z :
2π if qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ≤Z 3 1−ν2
4
:
ð4Þ
K.L. Tran et al. / Journal of Constructional Steel Research 69 (2012) 30–38
33
Fig. 4. First elastic buckling modes for different curvatures.
• Stowell proposed a modification of Redshaw's equation to better take into account the boundary conditions:
Z
kc;Stowell
0 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u u B 48 1−ν 2 2 kplate c B 1+u 1+ = 2 Z t @ 2 π4 kplate c
1 C C: A
ð5Þ
• Domb and Leigh's expression is calibrated on a numerical database by some curve fitting method:
Z
kc;Domb &
Leight
=
8 3 > < 10 ∑ c ð logZ Þi i b > :
i=0 d
cð Z Þ
1 ≤ Zb < 23:15
ð6Þ
23:15 ≤ Zb ≤ 200
with Zb = Z(1 − ν 2), co = 0.6021, c1 = 0.005377, c2 = 0.192495, c3 = 0.00267, c = 0.4323 and d = 0.9748.. The results of this comparison are presented in Fig. 5 where the buckling coefficient kcZ is plotted as a function of the curvature parameter Z. It is first noticed that all formulae converge towards the value corresponding to a flat plate (kplate = 4) when Z tends towards 0 which is logical and necessary: for Z = 0, the curved panel is actually a plate. It is remarked then that, for all curves, the buckling coefficient increases with Z which can be explained by the fact that the higher the values of Z, the higher the circumferential membrane stresses which hold up the radial deflection. The asymptotic behaviour of these expressions is however obviously different. The theoretical formulae of Redshaw and Timoshenko converge asymptotically towards a straight line whose slope is equal to the buckling coefficient of a revolution cylinder whilst the formulae of Domb and Leigh, of Stowell
and the numerical results converge towards half this value. The differences of the boundary conditions involved explain this gap: curved cylindrical panels have four free cut edges whereas revolution cylinders have only two, the circumferential membrane stresses develops thus in different ways. The numerical results indicate that the buckling coefficient of curved panels (with constant width b) converge on half the value of the buckling coefficient of the corresponding revolution cylinder (defined by b = 2πR). To clarify this observation, further numerical simulations are carried out by studying cylindrical curved panels with constant length, radius and thickness (a = 9 m; R = 3 m, t = 15 mm) but with a width b varying from 0 to 2πR. The last panel (b = 2πR) is hence a revolution cylinder with a straight cut along its length a. Fig. 6 shows the ratio (σcurved/σcylinder) between the critical buckling stress of the curved panel and that of the full revolution cylinder against the curvature parameter Z. This ratio decreases when Z increases, from infinity for Z = 0 (i.e. b = 0) to 0.5 for Z = 4π2 Rt (Z = 7895). It should be noticed that, in this specific case, the value 0.5 is quasi reached (less than 2% difference) for Z ≥ 40, i.e. b = 1.34 m which is far from the upper value of b (2πR = 18.85 m).
3.2. Buckling coefficient formula for curved panels Although Fig. 5 shows a good agreement of Stowell and Domb and Leigh's expressions with the numerical results, there is still a need for a reasonable expression of the critical buckling mode. The reason is first the complexity and the non-intuitive aspect of Domb and Leigh's expression which includes several coefficients calibrated on numerical simulations. Secondly, the expression of Stowell has not been demonstrated yet. The new expression is
Fig. 5. Comparison of reference analytic buckling coefficients with numerical results.
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K.L. Tran et al. / Journal of Constructional Steel Research 69 (2012) 30–38
Fig. 6. Convergence of the asymptotic behaviour of cylindrical curved panels.
hence looked for assuming a mathematical formula inspired by Stowell (see Eq. (5)): qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z kc = A 1 + 1 + B⋅ Z 2
ð7Þ
A and B are two parameters which will be identified with respect to the limit behaviours: • For Z = 0, curved panels behave like flat plates, Z
plate
k c j Z = 0 = kc
plate
⇒A =
kc : 2
ð8Þ
• For Z → ∞, curved panels behave like cut revolution cylinders, 2 48 1−υ 1 cylinder Z ⇒B = lim kc = kc 2 : Z→∞ 2 π4 kplate c
Fu = Fy = b:t:fy
ð9Þ
ð10Þ
We obtain exactly Stowell's expression but, let us say, in a more pedagogical manner. This formula exhibits slight differences with the numerical results especially in the transition zone from plates to full revolution cylinders (Fig. 5) for which it was calibrated. This is not of big importance here because this approximation of the critical buckling load is only the first step in the whole hand-calculation procedure which is developed for the determination of the ultimate strength. These differences between Eq. (10) and numerical results will be partly recovered during the calibration process of the reduction factor χ (cf. Section 4). The expression (Eq. (1)) with the buckling coefficient given by Eq. (10) is thus satisfactory and it will be used in the following for the evaluation of critical buckling stresses of curved panels under uniform longitudinal compression. 4. Ultimate strength For stocky curved panels under uniform compression (i.e. small values of the slenderness parameter λ), no buckling phenomenon
ð11Þ
For more slender curved panels, the slenderness parameter λ increases, the failure mode becomes a shell buckling and the ultimate strength can be written as a fraction of the yield resistance characterised by the reduction factor chi (by definition χ ≤ 1): Fu = χFy :
The proposed formula for the buckling coefficient is thus: 1 0 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u plate 2 u C B 48 1−υ k Z 2C 1+u kc = c B t1 + 4 plate 2 Z A: 2 @ π kc
occurs and the ultimate strength is governed by the yielding of the whole cross-section and the ultimate load is equal to the yield resistance (if required, this yield resistance can be estimated numerically with a material non-linear analysis and an elasticperfectly plastic material law):
ð12Þ
The ultimate strength Fu of the panel is evaluated numerically using non-linear analysis including geometrical and material imperfections (GMNIA). Practically, these numerical simulations are conducted using the arc-length method due to the complexity in the load–displacement response when entering into the post-buckling regime. For such an iterative method, it is necessary to select suitable criteria convergence for the termination of the iteration process. Ansys's solution control uses L2-norm of force (and moment) tolerance equal to 0.5%, a setting that is appropriate for most engineering applications. The reduction factor χ is then determined from the ratio between the maximum load Fu and the yield resistance Fy. 4.1. Numerical simulations A series of simulations (GMNIA) is thus carried out for several curved panel configurations with a curvature parameter Z varying from 1 to 100. These geometrical configurations have been chosen in order to cover the whole range of curved panels in steel bridges. The numerical results are presented in Fig. 7 where the reduction factor χ is plotted against the slenderness parameter λ. Fig. 7 also shows the reduction factors coming from EN 1993-1-5 for plate buckling and from EN 1993-1-6 for revolution shell buckling, the highest and lowest curves respectively. The numerical results show a clear influence of the curvature parameter Z on the ultimate strength Fu. For a given slenderness λ, the highest the panel curvature, the lowest the ultimate strength. The variations of the ultimate strength with the slenderness λ show however approximately the same scheme for all the curvatures Z. (This all gives an additional justification to the choice of the Z
K.L. Tran et al. / Journal of Constructional Steel Research 69 (2012) 30–38
35
Fig. 7. Ultimate strength of curved panels under uniform axial compression.
parameter for representing the variability of this problem.) It is then noticed that all the marks are located between the two reduction factors coming from the European Standards. For small values of Z, curved panels behave like plates (EN 1993-1-5) and for high valuesof Z, the behaviour turns to that of a cylinder (EN 1993-1-6). The clause 1.1(2) of EN 1993-1-5 which says that the reduction factor for plates under uniform compression can be applied for relatively small curvatures (Z ≤ 1) is verified: the distribution of marks for Z = 1 follows more or less the curve from EN 1993-1-5. For higher curvature, the use of EN 1993-1-5 would lead to a significant overestimation of the ultimate strength of the curved panel. For high values of Z (Z ≥ 40) and for small slenderness (λ≤1:0), numerical results show that the reduction factor of EN 1993-1-6 provides a good approximation of the ultimate strength for curved panels. However a non-negligible gap is noticed when the slenderness becomes high where EN 1993-1-6 becomes too conservative. This gap is mainly due to the post-critical strength reserve of curved panels which leads to a critical buckling load smaller than the ultimate load. This strength reserve is also related to the different boundary conditions of the curved panel edges (four free edges) and of the revolution cylinder (only two free edges). For high slenderness values (λ N 3), the collapse behaviour is governed by the elastic buckling. The influence of imperfections becomes thus smaller. Fig. 8 zooms in Fig. 7 on these high values of λ
and shows that the curves vary as a functions of 1 = λ. A proportionality coefficient β can thus be calibrated on the basis of the numerical results. For example β is equal to 1 for Z = 0 (plate behaviour) and equal to 0.5 when Z = 75. β decreases when the curvature parameter increases mainly because of the interaction between the out-of-plane bending stresses due to the curvature and the membrane stresses. The higher the curvature, the higher this effect. The parameter β can thus be interpreted as an imperfection factor representing the curvature effect. 4.2. New capacity curves The previous observations lead to the search for a new expression of the reduction factor χ of Eq. (12) allowing to fill in the gap between EN 1993-1-5 and EN 1993-1-6. This new expression will obviously depend on the curvature parameter Z. Its mathematical form will come from an analytical approach based on the theory of strength of materials, similar to the ones already used for the development of the European buckling curves in EN 1993-1-1 [31] or for the plate buckling phenomenon during bridge launching [32]. It is hence proposed that the reduction factor χ is defined by: ð1−χ Þ β−χ⋅ λ = η⋅χ
Fig. 8. Behaviour of curved panels in the elastic buckling zone (high values of λ).
ð13Þ
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K.L. Tran et al. / Journal of Constructional Steel Research 69 (2012) 30–38
where η and β are numerical parameters to be calibrated with the numerical results. As for the standard European buckling curves, η represents the generalised imperfection which is assumed to have the following form: η = χ⋅ αZ λ−λ0
ð14Þ
where αZ is the elastic imperfection factor, λ0 represents the length of the yield plateau (associated with the value χ = 1) where no buckling occurs. Solving Eq. (13) after introduction of Eq. (14), one obtains: χ=
β+λ+
2β qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 : β + λ −4 λ−αZ λ−λ0
imperfection parameter αZ vary in a smooth manner. So, in order to make things easier for practitioners, it is proposed to reduce the number of characteristic values of αZ (see Table 1) and to interpolate linearly the values in-between. The quality of this approximation is tested in next section.
ð15Þ
5. Validation and summary of the proposed hand calculation method To illustrate and verify the accuracy of the proposed approach, an example of curved panel verification is presented. This panel is 1.5 m long and wide, 20 mm thick and has a 4.5 m radius of curvature. The hand-calculation design method goes as follows: • Evaluate the curvature parameter Z:
Eq. (15) represents a series of curves indexed with the imperfection parameter αZ. This series of curves admits two upper bounds, χ = 1 for full yielding (λ ≤ λ0 ) and χ = a = λ for asymptotic elastic buckling (λ→∞).
The parameters β and λ0 correspond to two different collapse modes which are clearly identified in Fig. 7. For a given value of Z, the coefficient λ0 is determined by the intersection of the numerical curve with the horizontal line χ = 1 corresponding to pure yielding of the panel. Then, the coefficient β is identified from the asymptotic behaviour for high value of λ where the reduction factor tends towards β = λ. The two parameters evidently depend on the curvature parameter Z and can be approximated by the following expressions, where the limit cases (Z = 0 and Z → ∞) are determined by EN 1993-1-5 and EN 1993-1-6 respectively:
β=
1 + 0:97Z : 2
b2 = 25: Rt
ð18Þ
• Calculate the Euler critical stress σE and the buckling coefficient kcZ:
4.3. Calibration of parameters
Z λ0 = 0:2 + 0:473 0:95
Z=
ð16Þ
ð17Þ
The last parameter αZ is then calibrated in order to minimise the gap between the values given by Eq. (15) and the values obtained numerically. This gap is measured by the sum of the squares of the differences between those values. Its variations with αZ and with Z are shown in Fig. 9. It is noted that for high values of the curvature parameter (Z ≥ 40), αZ is almost constant and generally that the
σE =
2 π2 E t = 33:74 MPa 12 1−ν 2 b vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi1 u 2 u 48 1−υ u B 2C B1 + u1 + 2 Z C t A = 10:61 2 @ π4 kplate c 0
Z
kc =
kplate c
with kcplate = 4 for the case of simple supported plate. Z : • Calculate the linear elastic critical stress σcr Z
Z
σcr = kc⋅ σE = 357:87 MPa:
ð19Þ
• Evaluate the reduced slenderness λ: λ=
sffiffiffiffiffiffiffi fy σcrZ
= 0:98:
• Identify the parameters β, λ0 and αZ as a function of Z: Z λ0 = 0:2 + 0:473 0:95 = 0:33
Fig. 9. Calibration of the αZ parameter.
ð20Þ
K.L. Tran et al. / Journal of Constructional Steel Research 69 (2012) 30–38 Table 1 Value of αZ. Z
0
10
20
30
≥ 40
αZ
0.28
0.38
0.33
0.21
0.13
β=
1 + 0:97 = 0:73 2
Z
• Determine the reduction factor χ: 2β qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 = 0:59: β+λ+ β + λ −4 λ−αZ λ−λ0
ð21Þ
• Calculate the ultimate strength σu: σu = χ: fy = 196:65 MPa:
(RMS) are close to 1. The low standard deviation (STDEV = 0.00452) also indicates that the data points tend to be very close to this mean value. The observed error between the predicted value by the proposal and the corresponding numerical one is not significant, because the mean absolute percentage errors (MAPE) are only 3.42% and the maximum error (in absolute value) is lower than 0.07. It is thus possible to conclude that the proposed analytical method fits very well with the numerical results. 6. Conclusion and further work
αZ = 0:27ðevaluated from Table 1Þ:
χ=
37
ð22Þ
This process is illustrated in the flowchart of Fig. 10. From the comparison of the proposed method with existing recommendations and numerical analysis, some remarks can be made out: • Although the theoretical value for critical buckling stress is a bit higher Z than the numerical one (σcrZ = 357.87 MPa and σcr, F. E. = 320.91 MPa), the predicted value for the ultimate strength is as good as the numeric alone (σu = σu, F. E. = 196.65 MPa). As indicated in Section 3.2, the error in the approximation of the critical stress is recovered by the calibration process of the reduction factor χ. • The ultimate strength given by EN 1993-1-5 is significantly higher than the real resistance of the panel (χEN1993 − 1 − 5 =0.79N χF. E =0.59), the use of EN 1993-1-5 leads to an overestimation of the panel strength which is against security. • The exact value for αZ in this case is 0.24 which is not far from the interpolated value (αZ = 0.27). The assumption of linear interpolation is thus acceptable. Beyond this practical example, the accuracy of the proposed method has been checked with about 524 numerical simulations, for a range of Z from 0 to 100. Fig. 11 shows that the proposed formula fits well the numerical database. Both the mean value and the root mean square
In this paper, the critical buckling and the ultimate strength of cylindrical curved panels have been investigated analytically and by means of linear and non-linear finite elements analyses. It appeared that the behaviour of cylindrical curved panels usually depends on its curvature, its slenderness and its imperfections. These results can be summarised as follows: • The elastic buckling load of cylindrical curved panels increases with the curvature parameter Z = b 2/Rt and it is best approximated by Stowell's semi-empirical formula. • The ultimate strength is very sensitive to the panel imperfections which shape can reasonably be based on the shape of the first buckling mode with a maximum deflection wo of max(a/200; b/200) as given in EN 1993-1-5 for plate buckling. • For slender panels where the collapse behaviour is governed by the elastic buckling, the reduction factor χ can be assumed to behave as β = λ. β represents the influence of the curvature on the value of the imperfection factor. It decreases with the curvature Z and it is equal to 1 for Z = 0. • A reduction factor χ has been developed for predicting the ultimate strength of curved panels. Its mathematical expression has been derived in the same way as existing European buckling curves. The various parameters have been calibrated using numerical results especially built for covering the range of steel bridges (i.e. Z b 100). • Although the finite element method is widely recognised as a reliable tool for stability verifications, the accuracy of results is sensitive to initial geometric and material imperfections. It would thus be interesting to validate the proposed methodology by experimental measurements on cylindrical curved panels under uniform axial compression. This study has been carried out in the framework of a running research project essentially targeted for bridge engineering applications.
Fig. 10. Practical example of the proposed hand-calculation method.
38
K.L. Tran et al. / Journal of Constructional Steel Research 69 (2012) 30–38
Fig. 11. Correlation of the proposed formula with numerical reduction factor.
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