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Probabilistic buckling analysis of plates and shells
Thin-Walled Structures Vol. 30, Nos 1–4, pp. 135–157, 1998 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0263-8231/98 $19.00 + 0.00 PII: S0263-8231(97)00035-9
Probabilistic Buckling Analysis of Plates and Shells M. K. Chryssanthopoulos Department of Civil Engineering, Imperial College of Science, Technology and Medicine, London SW7 2BU, U.K.
ABSTRACT For many years, a significant amount of research has been directed towards experimental modelling of thin-walled plates and shells, as well as towards the development of analytical and numerical methods to improve their design against buckling. This paper presents methodologies for probabilistic buckling analysis and reliability assessment of such structural components and examines the link between probabilistic and deterministic studies. In particular, the effect of manufacturing variabilities, such as initial geometric imperfections and residual stresses, on elastoplastic buckling response is investigated through parametric reliability studies of plate panels and cylinders under axial compression. 1998 Elsevier Science Ltd. All rights reserved
NOTATION a b m n pf t x Astif E L N R ␣s 
Panel dimension in the longitudinal direction Panel dimension in the transverse direction Wavenumber along cylinder length Wavenumber along cylinder circumference Probability of failure Shell/plate thickness Coordinate along cylinder length Stiffener area in stringer-stiffened cylinder Elastic modulus Cylinder length between supports or heavy ring stiffeners Number of stiffeners in stringer-stiffened cylinder Cylinder radius Stiffening ratio = NAstif/2Rt FORM/SORM reliability index 135
136
p ␦0 ␦0all cr u 0 r
M. K. Chryssanthopoulos
Plate panel slenderness parameter = (b/t)√(0/E) Initial imperfection amplitude Allowable imperfection (code tolerance) Width of tensile block Coordinate along cylinder circumference Poisson’s ratio Elastic critical buckling stress Ultimate strength Material yield stress Compressive residual stress
1 INTRODUCTION The efficiency of shell structures in resisting in-plane and distributed lateral loads has long been recognised in structural design. Depending on the particular application and type of loading, stiffening of the shell skin by meridional stringers and/or circular rings can further enhance their structural performance. Aerospace structures, as well as submarine and other pressure vessels, are examples where shells have been used for many years. A large variety of storage tanks and silos can be found in onshore applications. More recently, the stiffened cylinder has become a primary structural element in offshore platforms such as semi-submersible and tension leg platforms. On the other hand, stiffened flat plating has been employed extensively in various forms of steel box girder construction typically found in bridge and ship structures. In recent years, improvements in material and fabrication quality, coupled with the need for weight saving, have consistently led to the use of more slender geometries. As a result, design procedures consider in detail the various buckling modes that can occur in plate or shell structures, with emphasis being placed on elasto-plastic response and on the effect of manufacturing parameters, i.e. imperfections and welding residual stresses. Significant steps forward have also been made in specifying appropriate boundary conditions for the analysis of single components. Thus, although codified design is still mainly undertaken at a component level, the effects of continuity and restraint from adjacent members have been incorporated approximately in design, especially insofar as stiffened flat plates are concerned. Furthermore, the distinction between a design model and an assessment model has become much sharper. In design, various analytical difficulties can be overcome by setting allowable limits (tolerances). In assessment of existing structures (or of new construction where non-conformities occur),
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additional information may be available as a result of collecting and analysing data from the actual structure. Hence, more complete strength (and loading) models, which do not impose geometrical or other restrictions, are often required. Such models are also useful in the rational interpretation of physical and numerical experiments, especially if manufacturing parameters are explicitly taken into account, even under idealised assumptions. The above remarks are relevant to deterministic studies, of which a large number has taken place in the last 15 years or so. However, during the same period, significant developments took place in probabilistic modelling and structural reliability theory. Such methods were first used in the early 1980s to illustrate how new designs could be compared in a reliability framework and how the assessment of existing structures could benefit by introducing time-dependent performance criteria, also based on probabilistic concepts. These early studies invariably concentrated on introducing the new methodologies, but were somewhat lacking in terms of the mathematical models used for strength prediction. However, as probabilistic modelling was progressively made more detailed and specific to thin-walled structures, the quantitative treatment of uncertain parameters which have an important influence on buckling response became an important issue in reliability studies. Thus, strength formulations with explicit dependence on manufacturing parameters were required for reliability purposes in order to allow random variability in imperfections and residual stresses to be taken into account. The aim of this paper is to review the application of probabilistic methods and reliability analysis for thin-walled plates and shells, with emphasis on how decisions on codified rules and on design selection have been aided by the introduction of these methods. The reference list is not exhaustive but, hopefully, representative of the broad spectrum of studies undertaken in this area. Throughout the paper, the interaction between deterministic and probabilistic analysis is highlighted. Suggestions about future research needs are also made in the last section. 2 PROBABILISTIC SHELL BUCKLING ANALYSIS 2.1 Imperfection sensitivity The solution to the bifurcation problem of a long cylinder under axial compression can be traced to the work of Lorenz,1 Timoshenko2 and Southwell,3 where the elastic critical buckling stress is found as
cr =
E
t 冑3(1 ⫺ ) R 2
(1)
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assuming a displacement function for the buckling mode of the form w = sin mx/L sin n
(2)
Eqn (1) is valid for different combinations of axial and circumferential wavenumbers, m and n respectively, satisfying the following equation m ¯ + n2/m ¯ = k1/2
(3)
where m ¯ = mR/L and k = [12(1 ⫺ 2)]1/2R/t. Thus, the lowest eigenvalue given by eqn (1) corresponds to a large number of buckling modes obtained from eqn (3). This is shown schematically in Fig. 1, where the point on the horizontal axis corresponds to axisymmetric buckling (n = 0) while all the other modes on the semicircle (known as the ‘Koiter circle’) correspond to asymmetric buckling (n ⬎ 0). The validity of eqn (1) in predicting design loads was soon challenged by experimental results that gave much lower buckling loads. The main reason for these discrepancies is now well understood and lies in the extreme imperfection sensitivity of cylindrical shells under axial compression, highlighted in the classical papers by Donnell4 and Von Karman and Tsien5 and fully explained by Koiter’s work.6 This is brought about by the existence of multiple buckling modes associated with the lowest eigenvalue, as illustrated in Fig. 1. In the presence of small initial imperfections the bifurcation point, cr, is transformed into a limit point, s, which can be found by maximising an expression of the form (1 ⫺ /cr) + a2 + b3 + … = (/cr)¯
(4)
where ¯ represents the amplitude of the initial imperfection assumed to
Fig. 1. Buckling mode locus for an axially compressed long cylinder (Koiter’s circle).
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have a shape similar to one of the allowable buckling displacement functions, see eqn (2), and a and b are constants. For example, by assuming a small axisymmetric imperfection (n = 0), the following imperfection sensitivity curve is obtained 3(s/cr)¯ = 2[3(1 ⫺ 2)] ⫺ 1/2(1 ⫺ s/cr)2
(5)
Starting from expressions similar to eqn (4), many studies have been devoted to the imperfection sensitivity of cylinders buckling in the elastic range. Imperfections have been assumed to occur either in isolated modes (e.g. axisymmetric) or in combinations, in order to identify the ‘most severe’ imperfection sensitivity curve. 2.2 Probabilistic imperfection studies Initial geometric imperfections are introduced during manufacture of the shell. Hence, they can be represented by a random process or, in more simplified models, by random variables. Several investigations have been carried out attempting to quantify the effect of random initial imperfections on elastic buckling. In general, this can be accomplished by expressing the state of the structure as a function of several random variables, M(X1,X2,…,Xn), so that M(X1,X2,…,Xn) = 0
(6)
represents the threshold of moving from a safe state (M ⬎ 0) to a failure state (M ⱕ 0). The probability of failure is then defined as
冕
pf = …
冕
fX1,X2,…,Xn(x1,x2,…,xn) dx1, dx2,…, dxn
(7a)
M(X1,X2,…,Xn) ⱕ 0
where f X1,X2,…,Xn(x1,x2,…,xn) is the joint probability density function of the random variables X1,X2,…,Xn. Bolotin7 analysed a cylinder containing an asymmetric imperfection with random amplitude modelled by a normal distribution. By assuming that all other parameters can be treated as deterministic, it was possible to calculate the probability of failure from eqn (7a). Furthermore, the expectation and variance of the buckling load were evaluated as a function of the stochastic characteristics of the initial imperfection model. Similar studies were carried out by Thompson8 and Roorda and Hansen,9 examining different imperfection modes and probabilistic models for their random amplitude.
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The results reproduced qualitatively certain experimental trends (e.g. increasing buckling load variance with decreasing mean imperfection amplitude) but could not be readily extended to more complex imperfection patterns encountered in real shells. A different approach was followed by Amazigo10 by modelling axisymmetric imperfections as an ergodic zero-mean Gaussian random process. As a result, it was possible to calculate an imperfection sensitivity curve similar to that obtained by deterministic analysis (i.e. eqn (5)) but with the imperfection amplitude replaced by the value of the spectral density of the process at a ‘critical’ frequency. Finally, the evaluation of probabilistic buckling response parameters using simulation should be mentioned. Hansen11 concentrated on the relative importance of axisymmetric and asymmetric imperfections across the slenderness range (20 ⬍ R/t ⬍ 4000) assuming zero-mean jointly Gaussian random variables for imperfection amplitudes. In all of the above studies the probabilistic imperfection parameters (e.g. mean value, standard deviation) were largely hypothetical and no comparisons with actual data measured on structures or laboratory specimens were attempted. Arbocz and Elishakoff with their co-workers12,13 were the first to develop a method for obtaining the full probability distribution of the elastic buckling load for random imperfection patterns whose stochastic properties match those of experimentally measured imperfections on groups of similarly manufactured cylinders. Thus, the possibility of linking the manufacturing process to imperfection sensitivity and stochastic buckling load evaluation was introduced and demonstrated through a number of case studies. In the same context, Chryssanthopoulos et al.14,15 and Chryssanthopoulos and Poggi16 presented a methodology for producing characteristic imperfection shapes, again based on measured imperfections on nominally identical shells. These characteristic shapes, which are obtained by considering the probability of exceedance at any point on the shell surface, were used to guide the selection of imperfection modes in finite element models, which is a vital step in calculating buckling strength by numerical analysis. 2.3 Reliability assessment In contrast to the above studies, work linked to reliability assessment of offshore shells is markedly different. This is due to the fact that offshore cylinders are normally governed by elasto-plastic behaviour, in which case material as well as geometric non-linearity is important. As a result of stiffening and relatively stockier proportions, imperfection sensitivity is, in principle, reduced (compared with elastic behaviour). For this class of shells, the random nature of imperfections is still a factor in determining
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buckling response but probabilistic modelling of other design parameters, such as thickness and material properties, must also be taken into account. Because of the higher number of random variables, direct evaluation of eqn (7a) becomes practically impossible and approximate methods must be introduced, of which the most common is the First Order Reliability Method (FORM). Accordingly, the probability of failure is calculated approximately by using the reliability index  from the following expression pf ⬇ ⌽( ⫺ )
(7b)
where ⌽ is the standard normal distribution function. FORM is widely used in structural engineering problems and is described in many textbooks.17 In addition to the probability of failure, the method determines values for the so-called sensitivity factors. These factors, one for each random variable included in the analysis, provide a measure of the relative importance of the uncertainty in each random variable on the computed reliability. Their absolute value ranges between zero and unity and the closer this is to the upper limit, the more significant the influence of the respective random variable is to the reliability (or, equivalently, to the probability of failure). Negative sensitivity factors (in the range ⫺ 1 to 0) are associated with ‘resistance’ variables, i.e. variables for which the reliability increases as their mean value increases, whereas positive values (in the range 0 to 1) are associated with ‘loading’ variables, i.e. variables for which the reliability decreases as their mean value increases. The first offshore-related reliability study of a cylindrical shell was performed for one of the main legs of an offshore jacket.18 Two different strength formulations were used, the second explicitly accounting for random variability in the amplitude of axisymmetric imperfections. The results suggested that yield stress variability was more important than that associated with imperfections for the selected geometry (L/R = 0.25, R/t = 63). Several studies have also been carried out in connection with the Hutton TLP structure. Initially, a FORM analysis was presented for a narrow-panelled stringer-stiffened cylinder using a code formulation.19 The amplitude of column-type initial distortions was modelled probabilistically but its influence on reliability was negligible. Further FORM analyses using a strength formulation based on a discrete stiffener/shell analysis allowing for curvature effects were undertaken for different geometries.20 Insofar as response parameters are concerned, computed reliabilities were found to be most sensitive to variabilities in model uncertainty (a random variable accounting for deviations of predicted response from real behaviour), yield stress and shell thickness. The influence of initial imperfections was relatively unimportant. By using the results of the reliability analysis, partial
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factors were estimated for the design variables, following the procedure which was first used in the calibration of the U.K. bridge design code.17 Reliability studies using FORM have also been undertaken on stiffened cylinders under axial compression based on response surface methodology (RSM). The use of these methods is not suitable for routine applications but offers considerable advantages when accuracy in predicting buckling response is required. The origins of RSM lie in industrial research where a system is influenced by a large number of variables. The objective is to predict system response given a particular set of input variables or to study response sensitivity to changes in those variables. A further objective might be to determine the combination of input variables that lead to an optimum response.21 The concepts of RSM can be introduced in structural reliability calculations. A good example is the reliability of components requiring modelling by non-linear finite element (FE) analysis. In such cases, the computation of reliability would require a large number of FE runs, because of the iterative nature of the FORM algorithm. In order to reduce the computational effort required to a practical level, RSM is used in conjunction with FORM. For example, the reliability of axially compressed stringer-stiffened cylinders failing by local buckling was estimated with a multi-stage approach that combined a crude regression model and an accurate response surface based on non-linear FE analysis in a confined area of interest.22 The latter is identified on the basis of the fact that the major contribution to the failure probability comes from the area close to the FORM ‘design point’. This area is shown shaded in Fig. 2, which illustrates schematically the FORM ‘design point’ in a two-dimensional standard normal space. From an extensive numerical database on stringer-stiffened cylinders, a first estimate to this point is obtained by FORM analysis with a multivariable regression model to predict buckling strength. Finite element analysis for a few combinations of input variables (such as imperfections and material properties) in the vicinity of this point is then undertaken. Subsequently, the FORM analysis is repeated by using a polynomial approximation—i.e. a response surface—accurate within this confined area since it is entirely based on the FE results generated in the previous step. The computed reliability is accepted provided that the buckling strength at the final ‘design point’ is within a small tolerance of that predicted by finite element analysis. Fig. 3 presents a flowchart of the method, whereas Fig. 4 illustrates the response surfaces for a particular case. The buckling strength is plotted against the two most important resistance random variables, i.e. the yield stress and the maximum imperfection amplitude in the critical mode. The latter is expressed as a fraction of the appropriate code specified tolerance, i.e. ⌬ = ␦0max/␦0all, since, in this way, it is possible to use
Probabilistic buckling analysis of plates and shells
Fig. 2. Definition of FORM reliability index in standard normal space.
Fig. 3. The multi-stage approach for FORM reliability analysis.
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Fig. 4. Response surfaces obtained via the multi-stage approach.
measurements on model and full-scale structures to estimate the appropriate probabilistic parameters (e.g. mean value and standard deviation). The important property derived from analysis of measurements is the probability of exceeding the specified tolerance, i.e. the probability that ⌬ ⬎ 1). The multi-stage approach was also used in a design study investigating the effect of stiffening on structural reliability.23 For a particular shell geometry (R/t = 491, L/R = 0.4), the number of stringers was varied from 24 to 48, all other parameters remaining constant. The mean applied axial load that can be sustained for a target reliability index of  = 3 (or, equivalently, for a target failure probability, pf = 1.4 × 10 ⫺ 3) is shown in Fig. 5(a). As shown by the solid line, the optimum design should have about 36 to 40 stringers, since beyond that stiffening level the rate at which the mean load increases reduces markedly. The dotted line on the same figure shows the mean load after dividing it by the factor (1 + ␣s), thus effectively comparing the different cases on an iso-weight basis. Once again, a shell
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Fig. 5. (a) Mean load versus stiffening for a target  = 3; (b) sensitivity factors versus stiffening.
with about 36 to 40 stiffeners would appear to be a good choice. It should be noted, however, that fabrication costs (especially those associated with stringer welding) are not taken into account here. Fig. 5(b) presents the FORM sensitivity factors, whose significance has been outlined above, showing that imperfection amplitude and yield stress uncertainties can both be important depending on panel geometry. In broad panelled cylinders imperfection variability has a significant influence on the reliability, whilst in narrow panelled cylinders the same conclusion applies to yield stress variability. This also reflects the transition from elastic to elasto-plastic buckling, as the shell panel becomes narrower and the stiffening ratio increases. In order to avoid regions where a particular source of strength uncertainty exercises a dominant role, it was suggested23 that the design should be selected so that FORM sensitivity factors for imperfection and yield stress are approximately equal, thus pointing to a cylinder with 40 stringers. As shown in Fig. 5(b), the reliability is highly sensitive to uncertainties associated with the applied load, but in many cases there is little control over this variable.
3 PROBABILISTIC PLATE BUCKLING ANALYSIS Whereas shell buckling research is dominated by studies on geometric imperfections both in the elastic and the elasto-plastic range, plates under in-plane loads have been studied under more general and realistic manufacturing conditions. This is partly due to the stable post-buckling response of plates, which implies that, at least in the elastic range, initial imperfections are primarily associated with a stiffness degradation rather than a strength limitation, but also reflects a stronger interaction between design and manufacturing in plated construction. Since the mid 1970s, the effect of imperfec-
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tions in welded plates has been assessed together with the influence of residual stresses. The deterministic studies are far too numerous to outline here (appropriate review papers may be consulted24,25) but their overall objective has been to predict the ultimate strength and the deformation characteristics of isolated plate panels, as well as plate assemblages, subjected to combined loading. 3.1 Strength and stiffness of plate panels in compression Much of the work in this area has been undertaken by using numerical (finite difference or finite element) analysis, which enables a systematic evaluation of the important parameters such as initial imperfections, residual stresses and boundary conditions, albeit based on idealised (preferably conservative) models. For example, most initial imperfection models have been based on the critical mode shape determined from linear buckling analysis (the ‘square mode’ for flexurally supported plate panels in compression) and parametric studies have been primarily concerned with variable amplitude but fixed shape. Similarly, welding residual stresses have been most often represented by the Dwight and Moxham model,26 which is, of course, a simplification of the real distribution. Furthermore, this model has been used for modelling residual stresses in both longitudinal and transverse directions, whereas it has been validated primarily for the former. Finally, although the importance of specifying boundary conditions that are appropriate to the position of the panel within a plate assembly was recognised early on, numerical analysis is, in most cases, undertaken on single panel models. Hence, only approximate edge conditions can generally be introduced (e.g. boundaries of internal plate panels in a stiffened flange are assumed to remain straight, or for multi-bay continuous stiffened flanges zero deflections are assumed at cross-frames). The results of numerical analysis were often incorporated into design formulae, either through simple curve-fitting exercises or, more frequently, by calibrating a mechanics-based strength or stiffness formulation. The design formula predictions were also compared against experimental points in order to assess the validity of the model, to calculate its model uncertainty and to identify its limitations with respect to actual behaviour. For example, the ultimate longitudinal strength of plate panels in compression may be predicted by equations of the form a1 a2 a3 u = a0 + + + 0 p 2p 3p where p is the plate slenderness parameter.
(8)
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Different values for the coefficients, ai, can be found, ranging from the well-known Faulkner formula27 for which a0 = 0; a1 = 2; a2 = ⫺ 1; a3 = 0 to a finite element calibrated formula28 for which a0 = 0.23; a1 = 1.16; a2 = ⫺ 0.48; a3 = 0.09 The former has been validated against a large body of experimental results, whereas the latter is more tightly calibrated by considering the results of non-linear FE analysis for a range of plate slenderness values, assuming that imperfections occur in a square mode, that their maximum amplitude is equal to 0.1b2 and that compressive residual stress levels are represented by the Dwight and Moxham model with a level equal to 20% of the yield stress. Bearing in mind assessment of existing structures or comparisons with laboratory tests, formulations with explicit dependence on the imperfection amplitude and the level of compressive residual stresses have also been proposed. Usually, the fundamental slenderness-related term given by eqn (8) is modified by factors that are functions of the imperfections and residual stresses.29,30 As previously mentioned, the behaviour of plate panels is often required in relation to strength and deformation properties of a larger structure. In general, different components of a structure will attain maximum resistance at different strain levels, and because the compressive resistance diminishes thereafter, a proper estimation of overall strength on the basis of component models requires assumptions to be made regarding the load–displacement response of the components. Thus, beyond the pure strength models highlighted above, some investigations have provided more extensive information about the load–end shortening response of welded imperfect plates.31,32 Recently, a model that derives complete load–end shortening curves for welded plates with any given level of imperfections (in a square mode) and residual stresses (following Dwight’s model) has been proposed.33 The curves are constructed from a knowledge of the ordinates and secant stiffnesses of four points, denoted O, A, C and B in Fig. 6. Point O denotes the equilibrium stress state within an imperfect plate prior to loading, point A defines the point of first membrane yield at the plate edges, point C defines the peak load and its corresponding strain, and point B represents the point where the characteristic load–displacement curve of a plate panel with residual stresses meets the corresponding curve of a panel without residual stresses. Intermediate points are then calculated through linear
Fig. 6. Schematic load–end shortening diagram.
148 M. K. Chryssanthopoulos
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interpolation. Use of earlier results31,32 is made and further extensions are introduced in order to develop the full model, which has been calibrated against about 200 numerically derived plots covering a slenderness range of 0 ⱕ p ⱕ 12, as well as wide ranges of imperfections and residual stresses (Fig. 7 presents a typical comparison between the model and a FEbased curve). The purpose of deriving this model was twofold: first, to use it in a reliability study on unstiffened plates in compression and, second, to introduce it into an inelastic analysis numerical tool for longitudinally stiffened plates. Both strength and reliability assessments of stiffened plates were then carried out. 3.2 Reliability assessment of plate panels In one of the earliest probabilistic studies specifically aimed at plate buckling, Ivanov and Rousev,34 by using a strength formula with explicit dependence on imperfection amplitude, developed the expressions for the statistical parameters of the allowable axial load, in a manner similar to Bolotin.7 Uncertainty analysis leading to appropriate values for the probabilistic parameters of variables influencing plate response have been presented by Faulkner,27 Antoniou et al.35 and Fukumoto and Itoh36 regarding imperfection amplitudes and shapes, and by Faulkner27 and Bonello et al.37 on residual stresses; the latter37 also attempt to unify various data sets on manufacturing variability and to propose some consistent probabilistic parameter
Fig. 7. Comparison of load–end shortening curves.
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values for geometrical imperfections and welding residual stresses examined concurrently. Thus, with reference to the Dwight and Moxham model,26 which relates the level of compressive residual stress to the width of the tensile block, i.e.
r 2 = 0 b ⫺ 2 t
冋
册
(9)
it was found that the variability in can be modelled by
= R + ⌬
(10)
Here, R is the mean value of obtained by regression analysis
R = 1·20 + 0.06(b/t)
(11)
and ⌬ is a normal zero-mean random variable with a standard deviation given by
⌬兩b/t = 0.04(b/t)
(12)
This implies a coefficient of variation of which varies with plate slenderness, a conclusion supported by the data collected in previous studies. As for initial geometric imperfections, a similar probabilistic model was constructed on the basis of the ratio ␦0/b, so that
冋册 冋册
␦0 ␦0 = b b
␦0 b
+⌬
R
(13)
where [␦0/b]R is the mean line given by
冋册 ␦0 b
= 0.12
R
b 0 t E
(14)
according to Faulkner’s proposal,27 and ⌬[␦0/b] is a zero-mean random variable accounting for the variation about the mean line with a standard deviation given by
⌬[␦0/b]兩b/t =
冋册 ␦0 b
R
c.o.v.
冋册 ␦0 b
(15)
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where
冋册
c.o.v.
␦0 = 0.675 ⫺ 0.004(b/t) b
(16)
It is worth noting that eqns (15) and (16) imply a conditional variance which, as for residual stresses, increases with panel slenderness, b/t. The above models are based on data measured on panels with b/t ⱕ 120, but with the majority being in the range 20 ⱕ b/t ⬍ 100. Model uncertainty analysis has been a feature of many studies, but the results are, of course, pertinent to the particular response (strength or stiffness) model used. Guedes Soares30 presents a comprehensive review of model uncertainty depending on the completeness of the strength model and the context within which it is used (e.g. to analyse specific structures or to assess future designs). A FORM analysis of plate panels under general in-plane loading (biaxial compression and shear) and lateral pressure has also been performed.38 The buckling strength equations allowed for the initial imperfection amplitude to be considered as a random variable but welding stresses were not included. It was shown that, although environmental loading uncertainties were most influential, response uncertainty (mainly due to thickness and yield strength variability) were also significant (this conclusion is similar to that reached in connection with the reliability-based cylinder study outlined above). Broadly similar conclusions are also reported elsewhere39 but the emphasis here was in comparing reliability results obtained by FORM with those of an earlier study30 based on mean-value methods. For the cases examined, only small differences between the two methods were revealed, but FORM is generally the preferred approximate reliability method. Bonello et al.37 performed a parametric study of three design codes40–42 in order to compare, for each code, the implied reliability over the practical plate slenderness range, and also in order to quantify the relative contribution of uncertainties. Fig. 8(a) shows the variation of the reliability index with respect to all three codes examined, and it can be seen that the trends are similar, with a dip occurring in the so-called intermediate slenderness range (where both material and manufacturing parameters are important) and significantly higher reliabilities computed for slender plates. It is worth noting that the U.K. bridge code40 is associated with the smallest range of values, which may be attributed to the reliability calibration studies which were performed during development of this code and whose objective was to ensure, as far as possible, a uniform reliability for the full range of possible code designs. On the other hand, it should be emphasised that in reliability assessment studies which concentrate on the resistance uncertainty (such as the one reviewed here), comparisons of the reliability indices
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Fig. 8. (a) Variation of  across slenderness range; (b) variation of sensitivity factors.
obtained for different codes are to be treated cautiously. This is because the overall code reliability is also strongly affected by loading uncertainties and their associated partial factors. In this respect, the results of Fig. 8(a) should not be interpreted as one code having higher target reliability than another. Fig. 8(b) presents the distribution of FORM sensitivity factors across the
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slenderness range for the DnV offshore code. Material (0) and model (XmB) uncertainty are the dominant random variables, followed by residual stresses (⌬) and imperfections (␦o/b), the latter decreasing significantly as the slenderness increases beyond b/t = 60. At about b/t = 45 the combined sensitivity factor for manufacturing variabilities (imperfections and residual stresses) is at its highest level but still well below the yield stress sensitivity factor. This is in accordance with observations made elsewhere.30 On this basis, it may be argued that plate panel imperfection tolerances could be relaxed, especially since in this study (as in many others) the effect of imperfections has been included under most unfavourable assumptions (i.e. that measured data can be assumed to represent the amplitude of the critical mode). 3.3 Reliability assessment of longitudinally stiffened plates A popular design method for longitudinally stiffened plates in compression is the so-called ‘column’ approach.40–42 The main feature of this approach is that a single longitudinal stiffener with an associated width of plating is considered as representative of the behaviour of the longitudinally stiffened panel. This approach is particularly suited to the design of wide stiffened flanges where the effect of the transverse rigidity of the plate diminishes. The column approach caters for inter-frame buckling, i.e. overall buckling of stiffened plating between cross-frames. Local plate buckling is normally taken into account by an effective width concept. The column approach does not normally allow for lateral torsional buckling and/or tripping of stiffener outstands. Such stiffener behaviour is suppressed by imposing conservative limits on the geometry of the stiffener cross-section. Reliability assessment of longitudinally stiffened plates failing by interframe buckling has been the subject of two studies.43,44 In the former43 reliability is viewed as a potential tool used in code development. Specifically, the reliability of alternative stiffened plate geometries representing the deck and bottom panels of an oil tanker is estimated by FORM analysis, with the strength being modelled by a codified version of the column approach (similar to the one given in the U.K. bridge code40 for stiffened compression flanges), a variety of construction and operating conditions (e.g. with varying degrees of steel mill rolling control, corrosion control, cargo loading control). From the scenarios considered, effective corrosion control appears to have the largest influence on the estimated reliability of stiffened plates in compression. In the latter,44 a reliability study on stiffened plates similar in scope to its counterpart on unstiffened plate panels37 is presented. The strength analysis tool was based on the column approach but plate behaviour was modelled more accurately by introducing into the inelastic column analysis
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the complete load–end shortening panel curves. Furthermore, the effect of longitudinal continuity was considered by modelling three bays of a continuous stiffened plate. Thus, the strength analysis tool allowed a more detailed probabilistic model to be used, which included both plate variabilities, as outlined in the previous section, and stiffened plate variables, such as column imperfections. In addition to a parametric investigation of sensitivities across the column slenderness range, system reliability concepts were used in order to estimate the system reliability for different failure modes, depending on the spatial distribution of the imperfection profile. However, in contrast to the earlier studies on unstiffened plates, it was found that code imperfection tolerances in a column mode could not be relaxed without a significant effect on the estimated reliability.
4 CONCLUDING REMARKS This paper has reviewed probabilistic concepts and their application to thinwalled plates and shells susceptible to buckling failure. FORM reliability analysis has been employed in a variety of problems dealing with partial factors, rationalisation of imperfection tolerances, code calibration and assessment, as well as more specific design-orientated studies. It was shown that the methods of strength analysis used in conjunction with FORM can be as advanced and detailed as required by the particular problem in hand, and certainly on a par with the methods used in deterministic studies. Furthermore, probabilistic modelling of relevant random variables is nowadays undertaken with greater confidence as a result of studies on data collection and interpretation. This is not to say that further data are not required, and one of the advantages of a reliability approach is that formal updating procedures exist and could be used as new data became available. Further work is also required in spatial modelling of random variables, such as initial imperfections, in order to improve on traditional conservative assumptions, e.g. imperfection affinity to the critical mode. Finally, as more work is undertaken on stiffened plates and other assemblages it would be rational to examine the various possible buckling modes by using system concepts, with a view of undertaking reliability-based optimisation studies.
ACKNOWLEDGEMENTS The probabilistic buckling studies carried out in the Department of Civil Engineering at Imperial College were funded by the Marine Technology Directorate of EPSRC, A.S. Veritas Research and CONOCO Inc. The author is grateful to Professors Patrick Dowling and Michael Baker for
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their inspiration and support over many years and to Dr Marc Bonello, whose excellent work on probabilistic plate buckling forms the basis for the second part of this paper.
REFERENCES 1. Lorenz, R., Die nicht achsensymmetrische knickung dunnwandigen hohlzzylinder. Physik. Z., 13 (1911) 241–260. 2. Timoshenko, S., Achsensymmetrische form des ausbeulels einer zylindrischen schale. Z. Math. Physik., 58 (1910) 378–385. 3. Southwell, R. V., On the general theory of elastic stability. Phil. Trans. R. Soc. London, Series A, 213 (1914) 187–244. 4. Donnell, L. H., A new theory for the buckling of thin cylinders under axial compression and bending. Trans. ASME, 56 (1934) 795–806. 5. Von Karman, Th. and Tsien, H. S., The buckling of thin cylindrical shells under axial compression. J. Aero. Sci., 8(8) (1941) 303–312. 6. Koiter, W. T., Elastic stability and post-buckling behavior. In Nonlinear Problems, ed. R. E. Langer. Proc. Symp. of Math. Res. Center, US Army at the University of Wisconsin, Madison, 1962, pp. 257–275. 7. Bolotin, V. V., Statistical Methods in Structural Mechanics. Holden-Day, San Francisco, 1969. 8. Thompson, J. M. T., Towards a general statistical theory of imperfection sensitivity in elastic post-buckling. J. Mech. Phys. Solids, 15 (1967) 413–417. 9. Roorda, J. and Hansen, J. S., Random buckling behavior in axially loaded cylindrical shells with axisymmetric imperfections. J. Spacecraft and Rockets, 9(2) (1972) 88–91. 10. Amazigo, J. C., Buckling under axial compression of long cylindrical shells with random axisymmetric imperfections. Quart. Appl. Math., 26(4) (1969) 537–566. 11. Hansen, J. S., General random imperfections in the buckling of axially loaded cylindrical shells. AIAA J., 15(9) (1977) 1250–1256. 12. Elishakoff, I. and Arbocz, J., Reliability of axially compressed cylindrical shells with general nonsymmetric imperfections. J. Appl. Mech., 52 (1985) 122–128. 13. Arbocz, J. and Hol, J. M. A. M., Collapse of axailly compressed cylindrical shells with random imperfections. Thin-Walled Struct., 23(1-4) (1995) 131– 158. 14. Chryssanthopoulos, M. K., Baker, M. J. and Dowling, P. J., Statistical analysis of imperfections in stiffened cylinders. J. Struct. Eng., ASCE, 117(7) (1991) 1979–1997. 15. Chryssanthopoulos, M. K., Baker, M. J. and Dowling, P. J., Imperfection modelling for buckling analysis of stiffened cylinders. J. Struct. Eng., ASCE, 117(7) (1991) 1998–2017. 16. Chryssanthopoulos, M. K. and Poggi, C., Stochastic imperfection modelling in shell buckling studies. Thin-Walled Struct., 23(1-4) (1995) 179–200. 17. Thoft-Christensen, P. and Baker, M. J., Structural Reliability Theory and its Applications. Springer-Verlag, Berlin, 1982. 18. Baker, M. J. and Ramachandran, K., Reliability design of fixed offshore struc-
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24. 25. 26. 27. 28.
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tures. In Integrity of Offshore Structures, ed. D. Faulkner et al. Applied Science Publishers, London, 1981, pp. 135–153. Das, P. K., Frieze, P. A. and Faulkner, D., Reliability of stiffened steel cylinders to resist extreme loads. In Behaviour of Offshore Structures, Hemisphere Publishing Company, New York, Vol. 2, 1982, pp. 769–783. Frieze, P. A., Das, P. K. and Faulkner, D., Partial safety factors for stringer stiffened cylinders under extreme compressive loads. In Proc. 2nd Int. Symp. on Practical Design in Shipbuilding, Tokyo, 1983, pp. 475–482. Faravelli, L., Response surface approach for reliability analysis. J. Eng. Mech., ASCE, 115(2) (1989) 2763–2781. Chryssanthopoulos, M. K., Baker, M. J. and Dowling P. J., A reliability approach to the local buckling of stringer-stiffened cylinders. In Proc. 5th Offshore Mechanics and Arctic Engineering Symposium, ASME, Tokyo, 1986. Chryssanthopoulos, M. K., Baker, M. J. and Dowling P. J., Reliability-based design of stringer-stiffened cylinders under axial compression. In Proc. 6th Offshore Mechanics and Arctic Engineering Symposium, ASME, Houston, 1987. Walker, A. C., A brief review of plate buckling research. In Conf. on Behaviour of Thin-Walled Structures, Strathclyde University, ed. J. Rhodes and J. Spence, 1984, Elsevier Applied Science, London, pp. 375–397. Dowling, P. J. and Burgan, B. A., Plated structures—onshore and offshore. Steel Construction Today, 2 (1988) 175–188. Dwight, J. B. and Moxham, K. E., Welded steel plates in compression. The Struct. Engineer, 47(2), (1969) 49–66. Faulkner, D., A review of effective plating for use in the analysis of stiffened plating in bending and compression. J. Ship Res., 19(1) (1975) 1–17. Chapman, J. C., Smith, C. S., Davidson, P. C. and Dowling, P. J., Recent developments in the design of stiffened plate structures. In Advances in Marine Structures—2, ed. C. S. Smith and R. S. Dow. 1991, Elsevier Applied Science, London, pp. 529–548. Carlsen, C. A., Simplified collapse analysis of stiffened plates. Norwegian Maritime Res., 4 (1977) 20–36. Guedes Soares, C., Uncertainty modelling in plate buckling. Struct. Safety, 5 (1988) 17–34. Rhodes, J., On the approximate predictions of elasto-plastic plate behaviour. Proc. ICE, 71 (1981) 165–183. Vilnay, O. and Rockey, K. C., A generalised effective width method for plates loaded in compression. J. Constr. Steel Res., 1(3) (1980) 3–12. Bonello M. A., Reliability assessment and design of stiffened compression flanges. Ph.D. Thesis, Imperial College, University of London, 1992. Ivanov, L. D. and Roussev, S. G., Statistical estimation of reduction coefficient of ship’s hull plates with initial deflections. The Naval Architect, 4 (1979) 158–160. Antoniou, A. C., Lavidas, M. and Karvounis, G., On the shape of post-welding deformations of plate panels in newly built ships. J. Ship Res., 24(1) (1984) 1–10. Fukumoto, Y. and Itoh, Y., Basic compressive strength of steel plates from test data. Trans. Japan Soc. Civil Eng., 344/I-1 (1984) 129–139. Bonello, M. A., Chryssanthopoulos, M. K. and Dowling, P. J., Probabilistic strength modelling of unstiffened plates under axial compression. In Proc.
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10th Offshore Mechanics and Arctic Engineering Symposium, ASME, Norway, 1991. Moghtaderi-Zadeh, M. and Madsen, H., Probabilistic analysis of buckling and collapse of plates in marine structures. In Proc. 6th Offshore Mechanics and Arctic Engineering Symposium, ASME, Houston, 1986. Guedes Soares, C. and Silva, A. G., Reliability of unstiffened plate elements under in-plane combined loading. In Proc. 10th Offshore Mechanics and Arctic Engineering Symposium, ASME, Norway, 1991. BS5400: Steel, Concrete and Composite Bridges; Part 3: Code of Practice for Design of Steel Bridges. British Standards Institution, London, 1982. European Recommendations for the Design of Longitudinally Stiffened Webs and of Stiffened Compression Flanges. European Convention for Constructional Steelwork, ECCS Technical Working Group 8.3, Brussels, 1990. Buckling Strength Analysis of Mobile Offshore Units, Classification Note 30.1. Det norske Veritas, Oslo, 1992. Hart, D. K., Rutherford, S. E. and Wickham, A. H. S., Structural reliability analysis of stiffened panels. Trans. RINA, paper 9 (1985) 293–310. Bonello, M. A. and Chryssanthopoulos, M. K., Buckling analysis of plated structures using system reliability concepts. In Proc. 12th Offshore Mechanics and Arctic Engineering Symposium, ASME, Glasgow, 1993.
Probabilistic Buckling Analysis of Plates and Shells M. K. Chryssanthopoulos Department of Civil Engineering, Imperial College of Science, Technology and Medicine, London SW7 2BU, U.K.
ABSTRACT For many years, a significant amount of research has been directed towards experimental modelling of thin-walled plates and shells, as well as towards the development of analytical and numerical methods to improve their design against buckling. This paper presents methodologies for probabilistic buckling analysis and reliability assessment of such structural components and examines the link between probabilistic and deterministic studies. In particular, the effect of manufacturing variabilities, such as initial geometric imperfections and residual stresses, on elastoplastic buckling response is investigated through parametric reliability studies of plate panels and cylinders under axial compression. 1998 Elsevier Science Ltd. All rights reserved
NOTATION a b m n pf t x Astif E L N R ␣s 
Panel dimension in the longitudinal direction Panel dimension in the transverse direction Wavenumber along cylinder length Wavenumber along cylinder circumference Probability of failure Shell/plate thickness Coordinate along cylinder length Stiffener area in stringer-stiffened cylinder Elastic modulus Cylinder length between supports or heavy ring stiffeners Number of stiffeners in stringer-stiffened cylinder Cylinder radius Stiffening ratio = NAstif/2Rt FORM/SORM reliability index 135
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p ␦0 ␦0all cr u 0 r
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Plate panel slenderness parameter = (b/t)√(0/E) Initial imperfection amplitude Allowable imperfection (code tolerance) Width of tensile block Coordinate along cylinder circumference Poisson’s ratio Elastic critical buckling stress Ultimate strength Material yield stress Compressive residual stress
1 INTRODUCTION The efficiency of shell structures in resisting in-plane and distributed lateral loads has long been recognised in structural design. Depending on the particular application and type of loading, stiffening of the shell skin by meridional stringers and/or circular rings can further enhance their structural performance. Aerospace structures, as well as submarine and other pressure vessels, are examples where shells have been used for many years. A large variety of storage tanks and silos can be found in onshore applications. More recently, the stiffened cylinder has become a primary structural element in offshore platforms such as semi-submersible and tension leg platforms. On the other hand, stiffened flat plating has been employed extensively in various forms of steel box girder construction typically found in bridge and ship structures. In recent years, improvements in material and fabrication quality, coupled with the need for weight saving, have consistently led to the use of more slender geometries. As a result, design procedures consider in detail the various buckling modes that can occur in plate or shell structures, with emphasis being placed on elasto-plastic response and on the effect of manufacturing parameters, i.e. imperfections and welding residual stresses. Significant steps forward have also been made in specifying appropriate boundary conditions for the analysis of single components. Thus, although codified design is still mainly undertaken at a component level, the effects of continuity and restraint from adjacent members have been incorporated approximately in design, especially insofar as stiffened flat plates are concerned. Furthermore, the distinction between a design model and an assessment model has become much sharper. In design, various analytical difficulties can be overcome by setting allowable limits (tolerances). In assessment of existing structures (or of new construction where non-conformities occur),
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additional information may be available as a result of collecting and analysing data from the actual structure. Hence, more complete strength (and loading) models, which do not impose geometrical or other restrictions, are often required. Such models are also useful in the rational interpretation of physical and numerical experiments, especially if manufacturing parameters are explicitly taken into account, even under idealised assumptions. The above remarks are relevant to deterministic studies, of which a large number has taken place in the last 15 years or so. However, during the same period, significant developments took place in probabilistic modelling and structural reliability theory. Such methods were first used in the early 1980s to illustrate how new designs could be compared in a reliability framework and how the assessment of existing structures could benefit by introducing time-dependent performance criteria, also based on probabilistic concepts. These early studies invariably concentrated on introducing the new methodologies, but were somewhat lacking in terms of the mathematical models used for strength prediction. However, as probabilistic modelling was progressively made more detailed and specific to thin-walled structures, the quantitative treatment of uncertain parameters which have an important influence on buckling response became an important issue in reliability studies. Thus, strength formulations with explicit dependence on manufacturing parameters were required for reliability purposes in order to allow random variability in imperfections and residual stresses to be taken into account. The aim of this paper is to review the application of probabilistic methods and reliability analysis for thin-walled plates and shells, with emphasis on how decisions on codified rules and on design selection have been aided by the introduction of these methods. The reference list is not exhaustive but, hopefully, representative of the broad spectrum of studies undertaken in this area. Throughout the paper, the interaction between deterministic and probabilistic analysis is highlighted. Suggestions about future research needs are also made in the last section. 2 PROBABILISTIC SHELL BUCKLING ANALYSIS 2.1 Imperfection sensitivity The solution to the bifurcation problem of a long cylinder under axial compression can be traced to the work of Lorenz,1 Timoshenko2 and Southwell,3 where the elastic critical buckling stress is found as
cr =
E
t 冑3(1 ⫺ ) R 2
(1)
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assuming a displacement function for the buckling mode of the form w = sin mx/L sin n
(2)
Eqn (1) is valid for different combinations of axial and circumferential wavenumbers, m and n respectively, satisfying the following equation m ¯ + n2/m ¯ = k1/2
(3)
where m ¯ = mR/L and k = [12(1 ⫺ 2)]1/2R/t. Thus, the lowest eigenvalue given by eqn (1) corresponds to a large number of buckling modes obtained from eqn (3). This is shown schematically in Fig. 1, where the point on the horizontal axis corresponds to axisymmetric buckling (n = 0) while all the other modes on the semicircle (known as the ‘Koiter circle’) correspond to asymmetric buckling (n ⬎ 0). The validity of eqn (1) in predicting design loads was soon challenged by experimental results that gave much lower buckling loads. The main reason for these discrepancies is now well understood and lies in the extreme imperfection sensitivity of cylindrical shells under axial compression, highlighted in the classical papers by Donnell4 and Von Karman and Tsien5 and fully explained by Koiter’s work.6 This is brought about by the existence of multiple buckling modes associated with the lowest eigenvalue, as illustrated in Fig. 1. In the presence of small initial imperfections the bifurcation point, cr, is transformed into a limit point, s, which can be found by maximising an expression of the form (1 ⫺ /cr) + a2 + b3 + … = (/cr)¯
(4)
where ¯ represents the amplitude of the initial imperfection assumed to
Fig. 1. Buckling mode locus for an axially compressed long cylinder (Koiter’s circle).
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have a shape similar to one of the allowable buckling displacement functions, see eqn (2), and a and b are constants. For example, by assuming a small axisymmetric imperfection (n = 0), the following imperfection sensitivity curve is obtained 3(s/cr)¯ = 2[3(1 ⫺ 2)] ⫺ 1/2(1 ⫺ s/cr)2
(5)
Starting from expressions similar to eqn (4), many studies have been devoted to the imperfection sensitivity of cylinders buckling in the elastic range. Imperfections have been assumed to occur either in isolated modes (e.g. axisymmetric) or in combinations, in order to identify the ‘most severe’ imperfection sensitivity curve. 2.2 Probabilistic imperfection studies Initial geometric imperfections are introduced during manufacture of the shell. Hence, they can be represented by a random process or, in more simplified models, by random variables. Several investigations have been carried out attempting to quantify the effect of random initial imperfections on elastic buckling. In general, this can be accomplished by expressing the state of the structure as a function of several random variables, M(X1,X2,…,Xn), so that M(X1,X2,…,Xn) = 0
(6)
represents the threshold of moving from a safe state (M ⬎ 0) to a failure state (M ⱕ 0). The probability of failure is then defined as
冕
pf = …
冕
fX1,X2,…,Xn(x1,x2,…,xn) dx1, dx2,…, dxn
(7a)
M(X1,X2,…,Xn) ⱕ 0
where f X1,X2,…,Xn(x1,x2,…,xn) is the joint probability density function of the random variables X1,X2,…,Xn. Bolotin7 analysed a cylinder containing an asymmetric imperfection with random amplitude modelled by a normal distribution. By assuming that all other parameters can be treated as deterministic, it was possible to calculate the probability of failure from eqn (7a). Furthermore, the expectation and variance of the buckling load were evaluated as a function of the stochastic characteristics of the initial imperfection model. Similar studies were carried out by Thompson8 and Roorda and Hansen,9 examining different imperfection modes and probabilistic models for their random amplitude.
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The results reproduced qualitatively certain experimental trends (e.g. increasing buckling load variance with decreasing mean imperfection amplitude) but could not be readily extended to more complex imperfection patterns encountered in real shells. A different approach was followed by Amazigo10 by modelling axisymmetric imperfections as an ergodic zero-mean Gaussian random process. As a result, it was possible to calculate an imperfection sensitivity curve similar to that obtained by deterministic analysis (i.e. eqn (5)) but with the imperfection amplitude replaced by the value of the spectral density of the process at a ‘critical’ frequency. Finally, the evaluation of probabilistic buckling response parameters using simulation should be mentioned. Hansen11 concentrated on the relative importance of axisymmetric and asymmetric imperfections across the slenderness range (20 ⬍ R/t ⬍ 4000) assuming zero-mean jointly Gaussian random variables for imperfection amplitudes. In all of the above studies the probabilistic imperfection parameters (e.g. mean value, standard deviation) were largely hypothetical and no comparisons with actual data measured on structures or laboratory specimens were attempted. Arbocz and Elishakoff with their co-workers12,13 were the first to develop a method for obtaining the full probability distribution of the elastic buckling load for random imperfection patterns whose stochastic properties match those of experimentally measured imperfections on groups of similarly manufactured cylinders. Thus, the possibility of linking the manufacturing process to imperfection sensitivity and stochastic buckling load evaluation was introduced and demonstrated through a number of case studies. In the same context, Chryssanthopoulos et al.14,15 and Chryssanthopoulos and Poggi16 presented a methodology for producing characteristic imperfection shapes, again based on measured imperfections on nominally identical shells. These characteristic shapes, which are obtained by considering the probability of exceedance at any point on the shell surface, were used to guide the selection of imperfection modes in finite element models, which is a vital step in calculating buckling strength by numerical analysis. 2.3 Reliability assessment In contrast to the above studies, work linked to reliability assessment of offshore shells is markedly different. This is due to the fact that offshore cylinders are normally governed by elasto-plastic behaviour, in which case material as well as geometric non-linearity is important. As a result of stiffening and relatively stockier proportions, imperfection sensitivity is, in principle, reduced (compared with elastic behaviour). For this class of shells, the random nature of imperfections is still a factor in determining
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buckling response but probabilistic modelling of other design parameters, such as thickness and material properties, must also be taken into account. Because of the higher number of random variables, direct evaluation of eqn (7a) becomes practically impossible and approximate methods must be introduced, of which the most common is the First Order Reliability Method (FORM). Accordingly, the probability of failure is calculated approximately by using the reliability index  from the following expression pf ⬇ ⌽( ⫺ )
(7b)
where ⌽ is the standard normal distribution function. FORM is widely used in structural engineering problems and is described in many textbooks.17 In addition to the probability of failure, the method determines values for the so-called sensitivity factors. These factors, one for each random variable included in the analysis, provide a measure of the relative importance of the uncertainty in each random variable on the computed reliability. Their absolute value ranges between zero and unity and the closer this is to the upper limit, the more significant the influence of the respective random variable is to the reliability (or, equivalently, to the probability of failure). Negative sensitivity factors (in the range ⫺ 1 to 0) are associated with ‘resistance’ variables, i.e. variables for which the reliability increases as their mean value increases, whereas positive values (in the range 0 to 1) are associated with ‘loading’ variables, i.e. variables for which the reliability decreases as their mean value increases. The first offshore-related reliability study of a cylindrical shell was performed for one of the main legs of an offshore jacket.18 Two different strength formulations were used, the second explicitly accounting for random variability in the amplitude of axisymmetric imperfections. The results suggested that yield stress variability was more important than that associated with imperfections for the selected geometry (L/R = 0.25, R/t = 63). Several studies have also been carried out in connection with the Hutton TLP structure. Initially, a FORM analysis was presented for a narrow-panelled stringer-stiffened cylinder using a code formulation.19 The amplitude of column-type initial distortions was modelled probabilistically but its influence on reliability was negligible. Further FORM analyses using a strength formulation based on a discrete stiffener/shell analysis allowing for curvature effects were undertaken for different geometries.20 Insofar as response parameters are concerned, computed reliabilities were found to be most sensitive to variabilities in model uncertainty (a random variable accounting for deviations of predicted response from real behaviour), yield stress and shell thickness. The influence of initial imperfections was relatively unimportant. By using the results of the reliability analysis, partial
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factors were estimated for the design variables, following the procedure which was first used in the calibration of the U.K. bridge design code.17 Reliability studies using FORM have also been undertaken on stiffened cylinders under axial compression based on response surface methodology (RSM). The use of these methods is not suitable for routine applications but offers considerable advantages when accuracy in predicting buckling response is required. The origins of RSM lie in industrial research where a system is influenced by a large number of variables. The objective is to predict system response given a particular set of input variables or to study response sensitivity to changes in those variables. A further objective might be to determine the combination of input variables that lead to an optimum response.21 The concepts of RSM can be introduced in structural reliability calculations. A good example is the reliability of components requiring modelling by non-linear finite element (FE) analysis. In such cases, the computation of reliability would require a large number of FE runs, because of the iterative nature of the FORM algorithm. In order to reduce the computational effort required to a practical level, RSM is used in conjunction with FORM. For example, the reliability of axially compressed stringer-stiffened cylinders failing by local buckling was estimated with a multi-stage approach that combined a crude regression model and an accurate response surface based on non-linear FE analysis in a confined area of interest.22 The latter is identified on the basis of the fact that the major contribution to the failure probability comes from the area close to the FORM ‘design point’. This area is shown shaded in Fig. 2, which illustrates schematically the FORM ‘design point’ in a two-dimensional standard normal space. From an extensive numerical database on stringer-stiffened cylinders, a first estimate to this point is obtained by FORM analysis with a multivariable regression model to predict buckling strength. Finite element analysis for a few combinations of input variables (such as imperfections and material properties) in the vicinity of this point is then undertaken. Subsequently, the FORM analysis is repeated by using a polynomial approximation—i.e. a response surface—accurate within this confined area since it is entirely based on the FE results generated in the previous step. The computed reliability is accepted provided that the buckling strength at the final ‘design point’ is within a small tolerance of that predicted by finite element analysis. Fig. 3 presents a flowchart of the method, whereas Fig. 4 illustrates the response surfaces for a particular case. The buckling strength is plotted against the two most important resistance random variables, i.e. the yield stress and the maximum imperfection amplitude in the critical mode. The latter is expressed as a fraction of the appropriate code specified tolerance, i.e. ⌬ = ␦0max/␦0all, since, in this way, it is possible to use
Probabilistic buckling analysis of plates and shells
Fig. 2. Definition of FORM reliability index in standard normal space.
Fig. 3. The multi-stage approach for FORM reliability analysis.
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Fig. 4. Response surfaces obtained via the multi-stage approach.
measurements on model and full-scale structures to estimate the appropriate probabilistic parameters (e.g. mean value and standard deviation). The important property derived from analysis of measurements is the probability of exceeding the specified tolerance, i.e. the probability that ⌬ ⬎ 1). The multi-stage approach was also used in a design study investigating the effect of stiffening on structural reliability.23 For a particular shell geometry (R/t = 491, L/R = 0.4), the number of stringers was varied from 24 to 48, all other parameters remaining constant. The mean applied axial load that can be sustained for a target reliability index of  = 3 (or, equivalently, for a target failure probability, pf = 1.4 × 10 ⫺ 3) is shown in Fig. 5(a). As shown by the solid line, the optimum design should have about 36 to 40 stringers, since beyond that stiffening level the rate at which the mean load increases reduces markedly. The dotted line on the same figure shows the mean load after dividing it by the factor (1 + ␣s), thus effectively comparing the different cases on an iso-weight basis. Once again, a shell
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Fig. 5. (a) Mean load versus stiffening for a target  = 3; (b) sensitivity factors versus stiffening.
with about 36 to 40 stiffeners would appear to be a good choice. It should be noted, however, that fabrication costs (especially those associated with stringer welding) are not taken into account here. Fig. 5(b) presents the FORM sensitivity factors, whose significance has been outlined above, showing that imperfection amplitude and yield stress uncertainties can both be important depending on panel geometry. In broad panelled cylinders imperfection variability has a significant influence on the reliability, whilst in narrow panelled cylinders the same conclusion applies to yield stress variability. This also reflects the transition from elastic to elasto-plastic buckling, as the shell panel becomes narrower and the stiffening ratio increases. In order to avoid regions where a particular source of strength uncertainty exercises a dominant role, it was suggested23 that the design should be selected so that FORM sensitivity factors for imperfection and yield stress are approximately equal, thus pointing to a cylinder with 40 stringers. As shown in Fig. 5(b), the reliability is highly sensitive to uncertainties associated with the applied load, but in many cases there is little control over this variable.
3 PROBABILISTIC PLATE BUCKLING ANALYSIS Whereas shell buckling research is dominated by studies on geometric imperfections both in the elastic and the elasto-plastic range, plates under in-plane loads have been studied under more general and realistic manufacturing conditions. This is partly due to the stable post-buckling response of plates, which implies that, at least in the elastic range, initial imperfections are primarily associated with a stiffness degradation rather than a strength limitation, but also reflects a stronger interaction between design and manufacturing in plated construction. Since the mid 1970s, the effect of imperfec-
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tions in welded plates has been assessed together with the influence of residual stresses. The deterministic studies are far too numerous to outline here (appropriate review papers may be consulted24,25) but their overall objective has been to predict the ultimate strength and the deformation characteristics of isolated plate panels, as well as plate assemblages, subjected to combined loading. 3.1 Strength and stiffness of plate panels in compression Much of the work in this area has been undertaken by using numerical (finite difference or finite element) analysis, which enables a systematic evaluation of the important parameters such as initial imperfections, residual stresses and boundary conditions, albeit based on idealised (preferably conservative) models. For example, most initial imperfection models have been based on the critical mode shape determined from linear buckling analysis (the ‘square mode’ for flexurally supported plate panels in compression) and parametric studies have been primarily concerned with variable amplitude but fixed shape. Similarly, welding residual stresses have been most often represented by the Dwight and Moxham model,26 which is, of course, a simplification of the real distribution. Furthermore, this model has been used for modelling residual stresses in both longitudinal and transverse directions, whereas it has been validated primarily for the former. Finally, although the importance of specifying boundary conditions that are appropriate to the position of the panel within a plate assembly was recognised early on, numerical analysis is, in most cases, undertaken on single panel models. Hence, only approximate edge conditions can generally be introduced (e.g. boundaries of internal plate panels in a stiffened flange are assumed to remain straight, or for multi-bay continuous stiffened flanges zero deflections are assumed at cross-frames). The results of numerical analysis were often incorporated into design formulae, either through simple curve-fitting exercises or, more frequently, by calibrating a mechanics-based strength or stiffness formulation. The design formula predictions were also compared against experimental points in order to assess the validity of the model, to calculate its model uncertainty and to identify its limitations with respect to actual behaviour. For example, the ultimate longitudinal strength of plate panels in compression may be predicted by equations of the form a1 a2 a3 u = a0 + + + 0 p 2p 3p where p is the plate slenderness parameter.
(8)
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Different values for the coefficients, ai, can be found, ranging from the well-known Faulkner formula27 for which a0 = 0; a1 = 2; a2 = ⫺ 1; a3 = 0 to a finite element calibrated formula28 for which a0 = 0.23; a1 = 1.16; a2 = ⫺ 0.48; a3 = 0.09 The former has been validated against a large body of experimental results, whereas the latter is more tightly calibrated by considering the results of non-linear FE analysis for a range of plate slenderness values, assuming that imperfections occur in a square mode, that their maximum amplitude is equal to 0.1b2 and that compressive residual stress levels are represented by the Dwight and Moxham model with a level equal to 20% of the yield stress. Bearing in mind assessment of existing structures or comparisons with laboratory tests, formulations with explicit dependence on the imperfection amplitude and the level of compressive residual stresses have also been proposed. Usually, the fundamental slenderness-related term given by eqn (8) is modified by factors that are functions of the imperfections and residual stresses.29,30 As previously mentioned, the behaviour of plate panels is often required in relation to strength and deformation properties of a larger structure. In general, different components of a structure will attain maximum resistance at different strain levels, and because the compressive resistance diminishes thereafter, a proper estimation of overall strength on the basis of component models requires assumptions to be made regarding the load–displacement response of the components. Thus, beyond the pure strength models highlighted above, some investigations have provided more extensive information about the load–end shortening response of welded imperfect plates.31,32 Recently, a model that derives complete load–end shortening curves for welded plates with any given level of imperfections (in a square mode) and residual stresses (following Dwight’s model) has been proposed.33 The curves are constructed from a knowledge of the ordinates and secant stiffnesses of four points, denoted O, A, C and B in Fig. 6. Point O denotes the equilibrium stress state within an imperfect plate prior to loading, point A defines the point of first membrane yield at the plate edges, point C defines the peak load and its corresponding strain, and point B represents the point where the characteristic load–displacement curve of a plate panel with residual stresses meets the corresponding curve of a panel without residual stresses. Intermediate points are then calculated through linear
Fig. 6. Schematic load–end shortening diagram.
148 M. K. Chryssanthopoulos
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interpolation. Use of earlier results31,32 is made and further extensions are introduced in order to develop the full model, which has been calibrated against about 200 numerically derived plots covering a slenderness range of 0 ⱕ p ⱕ 12, as well as wide ranges of imperfections and residual stresses (Fig. 7 presents a typical comparison between the model and a FEbased curve). The purpose of deriving this model was twofold: first, to use it in a reliability study on unstiffened plates in compression and, second, to introduce it into an inelastic analysis numerical tool for longitudinally stiffened plates. Both strength and reliability assessments of stiffened plates were then carried out. 3.2 Reliability assessment of plate panels In one of the earliest probabilistic studies specifically aimed at plate buckling, Ivanov and Rousev,34 by using a strength formula with explicit dependence on imperfection amplitude, developed the expressions for the statistical parameters of the allowable axial load, in a manner similar to Bolotin.7 Uncertainty analysis leading to appropriate values for the probabilistic parameters of variables influencing plate response have been presented by Faulkner,27 Antoniou et al.35 and Fukumoto and Itoh36 regarding imperfection amplitudes and shapes, and by Faulkner27 and Bonello et al.37 on residual stresses; the latter37 also attempt to unify various data sets on manufacturing variability and to propose some consistent probabilistic parameter
Fig. 7. Comparison of load–end shortening curves.
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values for geometrical imperfections and welding residual stresses examined concurrently. Thus, with reference to the Dwight and Moxham model,26 which relates the level of compressive residual stress to the width of the tensile block, i.e.
r 2 = 0 b ⫺ 2 t
冋
册
(9)
it was found that the variability in can be modelled by
= R + ⌬
(10)
Here, R is the mean value of obtained by regression analysis
R = 1·20 + 0.06(b/t)
(11)
and ⌬ is a normal zero-mean random variable with a standard deviation given by
⌬兩b/t = 0.04(b/t)
(12)
This implies a coefficient of variation of which varies with plate slenderness, a conclusion supported by the data collected in previous studies. As for initial geometric imperfections, a similar probabilistic model was constructed on the basis of the ratio ␦0/b, so that
冋册 冋册
␦0 ␦0 = b b
␦0 b
+⌬
R
(13)
where [␦0/b]R is the mean line given by
冋册 ␦0 b
= 0.12
R
b 0 t E
(14)
according to Faulkner’s proposal,27 and ⌬[␦0/b] is a zero-mean random variable accounting for the variation about the mean line with a standard deviation given by
⌬[␦0/b]兩b/t =
冋册 ␦0 b
R
c.o.v.
冋册 ␦0 b
(15)
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where
冋册
c.o.v.
␦0 = 0.675 ⫺ 0.004(b/t) b
(16)
It is worth noting that eqns (15) and (16) imply a conditional variance which, as for residual stresses, increases with panel slenderness, b/t. The above models are based on data measured on panels with b/t ⱕ 120, but with the majority being in the range 20 ⱕ b/t ⬍ 100. Model uncertainty analysis has been a feature of many studies, but the results are, of course, pertinent to the particular response (strength or stiffness) model used. Guedes Soares30 presents a comprehensive review of model uncertainty depending on the completeness of the strength model and the context within which it is used (e.g. to analyse specific structures or to assess future designs). A FORM analysis of plate panels under general in-plane loading (biaxial compression and shear) and lateral pressure has also been performed.38 The buckling strength equations allowed for the initial imperfection amplitude to be considered as a random variable but welding stresses were not included. It was shown that, although environmental loading uncertainties were most influential, response uncertainty (mainly due to thickness and yield strength variability) were also significant (this conclusion is similar to that reached in connection with the reliability-based cylinder study outlined above). Broadly similar conclusions are also reported elsewhere39 but the emphasis here was in comparing reliability results obtained by FORM with those of an earlier study30 based on mean-value methods. For the cases examined, only small differences between the two methods were revealed, but FORM is generally the preferred approximate reliability method. Bonello et al.37 performed a parametric study of three design codes40–42 in order to compare, for each code, the implied reliability over the practical plate slenderness range, and also in order to quantify the relative contribution of uncertainties. Fig. 8(a) shows the variation of the reliability index with respect to all three codes examined, and it can be seen that the trends are similar, with a dip occurring in the so-called intermediate slenderness range (where both material and manufacturing parameters are important) and significantly higher reliabilities computed for slender plates. It is worth noting that the U.K. bridge code40 is associated with the smallest range of values, which may be attributed to the reliability calibration studies which were performed during development of this code and whose objective was to ensure, as far as possible, a uniform reliability for the full range of possible code designs. On the other hand, it should be emphasised that in reliability assessment studies which concentrate on the resistance uncertainty (such as the one reviewed here), comparisons of the reliability indices
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Fig. 8. (a) Variation of  across slenderness range; (b) variation of sensitivity factors.
obtained for different codes are to be treated cautiously. This is because the overall code reliability is also strongly affected by loading uncertainties and their associated partial factors. In this respect, the results of Fig. 8(a) should not be interpreted as one code having higher target reliability than another. Fig. 8(b) presents the distribution of FORM sensitivity factors across the
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slenderness range for the DnV offshore code. Material (0) and model (XmB) uncertainty are the dominant random variables, followed by residual stresses (⌬) and imperfections (␦o/b), the latter decreasing significantly as the slenderness increases beyond b/t = 60. At about b/t = 45 the combined sensitivity factor for manufacturing variabilities (imperfections and residual stresses) is at its highest level but still well below the yield stress sensitivity factor. This is in accordance with observations made elsewhere.30 On this basis, it may be argued that plate panel imperfection tolerances could be relaxed, especially since in this study (as in many others) the effect of imperfections has been included under most unfavourable assumptions (i.e. that measured data can be assumed to represent the amplitude of the critical mode). 3.3 Reliability assessment of longitudinally stiffened plates A popular design method for longitudinally stiffened plates in compression is the so-called ‘column’ approach.40–42 The main feature of this approach is that a single longitudinal stiffener with an associated width of plating is considered as representative of the behaviour of the longitudinally stiffened panel. This approach is particularly suited to the design of wide stiffened flanges where the effect of the transverse rigidity of the plate diminishes. The column approach caters for inter-frame buckling, i.e. overall buckling of stiffened plating between cross-frames. Local plate buckling is normally taken into account by an effective width concept. The column approach does not normally allow for lateral torsional buckling and/or tripping of stiffener outstands. Such stiffener behaviour is suppressed by imposing conservative limits on the geometry of the stiffener cross-section. Reliability assessment of longitudinally stiffened plates failing by interframe buckling has been the subject of two studies.43,44 In the former43 reliability is viewed as a potential tool used in code development. Specifically, the reliability of alternative stiffened plate geometries representing the deck and bottom panels of an oil tanker is estimated by FORM analysis, with the strength being modelled by a codified version of the column approach (similar to the one given in the U.K. bridge code40 for stiffened compression flanges), a variety of construction and operating conditions (e.g. with varying degrees of steel mill rolling control, corrosion control, cargo loading control). From the scenarios considered, effective corrosion control appears to have the largest influence on the estimated reliability of stiffened plates in compression. In the latter,44 a reliability study on stiffened plates similar in scope to its counterpart on unstiffened plate panels37 is presented. The strength analysis tool was based on the column approach but plate behaviour was modelled more accurately by introducing into the inelastic column analysis
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the complete load–end shortening panel curves. Furthermore, the effect of longitudinal continuity was considered by modelling three bays of a continuous stiffened plate. Thus, the strength analysis tool allowed a more detailed probabilistic model to be used, which included both plate variabilities, as outlined in the previous section, and stiffened plate variables, such as column imperfections. In addition to a parametric investigation of sensitivities across the column slenderness range, system reliability concepts were used in order to estimate the system reliability for different failure modes, depending on the spatial distribution of the imperfection profile. However, in contrast to the earlier studies on unstiffened plates, it was found that code imperfection tolerances in a column mode could not be relaxed without a significant effect on the estimated reliability.
4 CONCLUDING REMARKS This paper has reviewed probabilistic concepts and their application to thinwalled plates and shells susceptible to buckling failure. FORM reliability analysis has been employed in a variety of problems dealing with partial factors, rationalisation of imperfection tolerances, code calibration and assessment, as well as more specific design-orientated studies. It was shown that the methods of strength analysis used in conjunction with FORM can be as advanced and detailed as required by the particular problem in hand, and certainly on a par with the methods used in deterministic studies. Furthermore, probabilistic modelling of relevant random variables is nowadays undertaken with greater confidence as a result of studies on data collection and interpretation. This is not to say that further data are not required, and one of the advantages of a reliability approach is that formal updating procedures exist and could be used as new data became available. Further work is also required in spatial modelling of random variables, such as initial imperfections, in order to improve on traditional conservative assumptions, e.g. imperfection affinity to the critical mode. Finally, as more work is undertaken on stiffened plates and other assemblages it would be rational to examine the various possible buckling modes by using system concepts, with a view of undertaking reliability-based optimisation studies.
ACKNOWLEDGEMENTS The probabilistic buckling studies carried out in the Department of Civil Engineering at Imperial College were funded by the Marine Technology Directorate of EPSRC, A.S. Veritas Research and CONOCO Inc. The author is grateful to Professors Patrick Dowling and Michael Baker for
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their inspiration and support over many years and to Dr Marc Bonello, whose excellent work on probabilistic plate buckling forms the basis for the second part of this paper.
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