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Probabilistic imperfection sensitivity analysis of axially compressed composite cylinders
EngineeringStructures,Vol.
~UTTERWORTH ~r[E
I N E M A
N N
0141-0296(95)00048-8
17, No. 6, pp. 398406, 1995 Copyright © 1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0141--0296/95 $10.00 + 0.00
Probabilistic imperfection sensitivity analysis of axially compressed composite cylinders M a r i o s K.
Chryssanthopoulos
Department of Civil Engineering, Imperial College, London SW7, UK
C a r l o Poggi Department of Structural Engineering, Politecnico di Milano, Milano, Italy (Received August 1993; revised version accepted April 1994)
Buckling analysis of cylinders under axial compression is sensitive to the assumptions made in the modelling of initial imperfections. Normally, imperfection modes are selected solely on the basis of buckling mode considerations and their amplitudes are determined using tolerance specifications in codes or experimentally recorded values. Whilst this approach may be used for metal cylinders with some confidence, due to the many test results available for validation purposes, it is not appropriate for the analysis and design of fibre-reinforced composite cylinders where the test results are limited and the effects of manufacturing on the imperfection characteristics have not yet been studied in detail. This paper presents a methodology for probabilistic buckling strength assessment based on the results of a statistical analysis on imperfections on two groups of composite cylinders manufactured by lay-up. The dominant features are quantified and the effect of fibre orientation on imperfections is examined. Simple models describing the random variability of imperfection modal amplitudes are presented. Using these probabilistic models, characteristic imperfection shapes are developed for fibre-reinforced cylinders and their use in buckling strength prediction and tolerance specification is demonstrated
Keywords: buckling, imperfections, composites, cylinders, axial compression, probabilistic modelling, manufacturing effects Buckling of composite cylinders has been studied in the last two decades, mainly using the Galerkin method to analyse the eigenvalue problem and the postbuckling response 1-4. The analysis can also be performed by means of perturbation procedures, which, in conjunction with the theory proposed by Koiter5, lead to quantitative estimates of the imperfection sensitivity. This method was first applied by Khot and Venkayya6 and, more recently, by A r b o c z 7"8 who included prebuckling nonlinearities. The latter demonstrated that the influence of prebuckling nonlinearities is significant in the postbuckling behaviour of axially compressed composite cylinders, even though critical loads are not substantially affected. Using these methods the effect of fibre orientation on the buckling load and on the postbuckling behaviour as well as the imperfection sensitivity of laminated composite cyl-
inders can be examined9. Their advantage lies in the derivation of exact solutions for the buckling and postbuckling equations leading to compact expressions for the critical load and the postbuckling b-coefficients. It is also possible from imperfection sensitivity curves to define 'knockdown' factors, which are of use in design. The 'knockdown' factor is a key ingredient in many design procedures, since the limit load of the imperfect shell buckling elastically is obtained by multiplying the critical buckling load of the perfect shell by this factor. On the other hand, it is well known that the actual buckling load and associated buckling mode can be different from those obtained from a single mode analysis, due to nonlinearities and interaction phenomena, and that real imperfections may give rise to different knock-down factors, due to multimode effects. For these reasons, and in
398
Probabilistic imperfection sensitivity analysis: M. K. Chryssanthopoulos and C. Poggi spite of the fact that the above methods may be extended to include several modes, knock-down factors are normally
determined from the lower bound curves based on test results. However, large databases of test results exist only for few basic shell geometries and loading conditions, and even in these cases problems with regard to the homogeneity of data (e.g. large and small-scale tests, different manufacturing methods and/or materials) may lead to knockdown factors which are unduly conservative for certain applications. These considerations have prompted the creation of imperfection data banks 1°'11 and the use of probabilistic methods in calculating a reliability function and, hence, characteristic values for the elastic buckling strength of metal shells with random imperfections 12'~3. In related studies, use of statistical techniques on the Fourier coefficients of imperfection modes for nominally identical stringer-stiffened steel cylinders has been made to arrive at characteristic imperfection models TM. The results of the statistical analysis were subsequently incorporated into finite element models in order to quantify the effects of multimode imperfection patterns on elastoplastic buckling strength 1~. Figure 1 summarizes the probabilistic design procedure adopted in these studies and contrasts it with a traditional deterministic design procedure. In the latter, measured imperfections are used to validate strength prediction models (e.g. finite elements) which are subsequently used in wider parametric studies. In these, imperfection modelling becomes a very important factor but the number of cases required to cover all possible modes and spatial distributions is prohibitively large. In the absence of measured imperfection data, one possibility is to consider imperfections related to the critical mode shape obtained from linear eigenvalue analyses, although nonlinear prebuckling and mode interaction weakens the case for selecting imperfection profiles in this m a ~ e r ~6. Geometric ImperfectionI Measurements [
1 Imperfection Data Processing
Buckling Strength Analysi=; of Measured Shells
Statistical Analysis of Fourier Coefficients
Validation of Strength Prediction Models
Characteristic Imperfection
Deterministic Design I Strength Assessment
Probabilistic Design Strength Assessment
l
Surfaces
Figure 1 Schematic diagram illustrating deterministic and pro-
babilistic approaches
399
Probabilistic design strength assessment, which requires that imperfections are recorded on groups of similarly manufactured shells, uses statistical analysis to arrive at characteristic imperfection models associated with a particular manufacturing method. These models form the basis on which the rules for specifying the parameters of multimode imperfection models are devised and, hence, the imperfection sensitivity is assessed. In this paper, this procedure is used to analyze the imperfection sensitivity of composite cylinders under axial compression. The strength analysis is done by nonlinear finite element modelling. The above remarks on imperfection modelling and associated buckling load sensitivity are equally valid for both metallic and composite shells. As is well known, the latter can also suffer from material defects, such as delaminations, porosities etc., and it might be expected that tolerance specifications should also account for these effects. However, as far as buckling load prediction is concerned, geometric imperfections are deemed to play a dominant role 17 and are examined in detail in this paper.
Notation number of half-waves along cylinder length total number of estimated Fourier coefficients in axial direction n number of waves along cylinder circumference total number of estimated Fourier coefficients in nT circumferential direction threshold value of exceedance probability Peer cylinder thickness t tp ply thickness Wo(X,O) cylinder initial imperfection function Wo~(x,O) characteristic initial imperfection function imperfection value at point i after 'best-fit' analywl~p sis WiF imperfection value at point i using Fourier representation co-ordinate along cylinder length X E(A) mean value of random variable A elastic modulus along fibre direction Ell elastic modulus perpendicular to fibre direction E22 shear modulus G~2 N total number of imperfection readings on cylinder surface critical linear buckling axial stress resultant Ncr L cylinder length critical buckling load of perfect cylinder Pcr P. limit load of imperfect cylinder R cylinder radius A* imperfection corresponding to exceedance probability threshold co-ordinate along cylinder circumference 0 o, ply angle Poisson's ratio 1)12 initial imperfection amplitude associated with mode ( m,n ) o'(A ) standard deviation of random variable A p( A,B ) correlation coefficient between random variables A and B phase angle associated with mode (m,n) m mT
Statistical analysis of imperfections In many shell buckling investigations, effort has been directed at detailed measurement of imperfections on test
Probabilistic imperfection sensitivity analysis: M. K. Chryssanthopoulos and C. Poggi
400
specimens, as well as some full-scale components. In most of these studies a standard method of data analysis has been adopted, based on the concept of the 'best-fit' cylinder ~°. This concept has enabled a unified datum to be established for shell imperfections and can be of particular use in comparative studies. Following the 'best-fit' procedure, the resulting imperfections are analysed using two-dimensional harmonic analysis to produce a set of Fourier coefficients, e.g. mT nT
mTrx Wo (x,O)) = ~ ~ ~,,, sin - ~ - sin (nO + tkm,)
(1)
m=l n=O
where 0 -< x --- L and 0 - 0 -< 27r. Hence, each surface is described by a set of modal amplitudes and phase angles, ~m, and ~bm,, respectively, where ~m, -> 0 and --'IT ~
(~rnn ~
"17".
Within the current programme of work, a number of composite cylinders have been manufactured by lay-up using an orthogonal Kevlar fabric embedded into an epoxy resin matrix. The first group consisted of 16 nominally identical (i.e. with nominally identical geometric dimensions and material characteristics) cross-ply cylinders with symmetrical lay-up (0°/90°)s, while the second group comprised 14 nominally identical angle-ply cylinders with symmetrical lay-up (+45°)s. Their relevant geometric and material parameters are given in Table 1. As can be seen, the size of the models is such that the manufacturing process and the resulting imperfections are representative of real components. The imperfection surface of each cylinder was recorded using displacement transducers in a regular mesh with an interval of 10 mm axially and 20 mm circumferentially. At each point, measurements were taken on both the inside and the outside cylindrical surface. The apparatus used for this purpose, together with the objectives and the initial experimental results of this project are described elsewhere TM. The overall objective is to develop validated design guidelines for composite cylinders subjected to compression and torsion. After recording the imperfections, 'best-fit' and harmonic analysis was carried out as described above. Equation ( 1 ) represents a half-range sine expansion in the axial direction, thus, imposing zero imperfection values at the two cylinder ends. Although this is not strictly correct, the error introduced is not significant and is confined to the end regions only, provided the number of calculated coefficients is not too small. Both half-range (sine and cosine) as well as full-range expansions were assessed using the following error function 1
e = ~/~
(w~/r - wF)2
i:1
Table 1
R L t tp Ell
Geometric and material parameters 350 m m 700 m m 1.04 m m 0.26 m m
E22 G12
23450 N / m m 2 23450 N / r a m 2 1520 N / m m 2
u12
0.20
(2)
Comparisons were also made at the points of maximum imperfection (in/out). The half-range sine series with mr = 20 and nT = 40 offered the best alternative in terms of accuracy and compactness. Figure 2 shows the imperfection surfaces (after 'bestfit') obtained for typical cylinders in both series A (crossply) and series B (angle-ply). Although the internal surfaces appear to have similar characteristics (i.e. dominance of long imperfection waves in both axial and circumferential directions), the external surface is strongly influenced by the orientation of the individual layers. In fact, the sharp peaks observed on the external surfaces are the result of local thickness variations due to overlapping layers and, hence, should not be treated as initial geometric imperfections. In order to quantify this result, a detailed correlation analysis of the Fourier modal amplitudes was undertaken. It was found that the correlation coefficient between modal amplitudes with identical wavenumbers, obtained for external and internal surfaces for each group of cylinders, D ( e x ~ m n , i n ~ m n ) , is positive and is statistically significant for low wavenumbers in the circumferential direction 19. For n -> 15, this correlation is not statistically significant, demonstrating that short wavelength modes are only present on the external surface due to the localized thickness variations. As a result, internal surface measurements were used in the ensuing analysis to study the characteristics of geometric imperfections. By treating the model amplitudes and associated phase angles, i.e. ~m, and thin,, as random variables, a number of univariate and bivariate (correlation) statistical parameters are calculated. In addition, the fitting of probability distributions is carried out by goodness-of-fit tests (Kolmogorov-Smirnov) and graphical methods. Several probability laws were tested, namely • • • • • •
Uniform Normal Log-normal Rayleigh Exponential Weibull (two-parameter)
Figure 3(a) shows the results of the mean value analysis of series A (cross-ply), clearly indicating that the dominant amplitudes are associated with long wavelengths in the circumferential direction. Modes with n > 15 have negligible influence on the imperfection profiles. Similar results were found for series B (angle-ply) 19. In fact, only about 2% of the total number of modes considered (1-< mr-< 20, 0 -< n~ -< 40) exhibit mean values greater than 10% of the maximum mean value, which for both cross-ply and angleply cylinders is associated with mode ~,2. Due to the monotonic trend observed for n---2, it is possible to fit a simple expression to describe the relationship between the mean model amplitudes and the circumferential wavenumber for any given axial wavenumber. Thus,
E(~,,,) = e '~"~
(3)
where a and /3 are constants given in Table 2 for both cross- and angle-ply cylinders. The mean value of modes with shorter axial wavelength (m > 2) can be approximately described by the following expressions
Probabilistic imperfection sensitivity analysis: M. K. Chryssanthopoulos and C. Poggi
401
"~r~)
-,
Figure2
--q~)
Typical imperfection surfaces of composite cylinders manufactured by lay-up
Table2
~
~o.,o
~o.o5
°
i
-...
........... ! ....................
.
... ....... i ....................
.t1. \...,,", 0.00 0
I 10
° =
=
~=
i
! ...................
E
i ...................
i i
! ...................
i ..................
!
i
i
!
, 20
, 30
, 4O
Series
Modes
a
/3
A A
m = 1, 2 -< n <- 20 m = 2, 2 -< n -< 20 m = 1, 2 -< n -< 20 m = 2, 2 -< n -< 20
-1.54 -3.03
0.5 0.3 0.4 0.3
B B
(a)
Constants for mean value models
Table3
-1.86
-2.82
Constants for variability models
clrc. we've number
o
rn
7 (series A )
7 (series B)
1 2 3
0.65 0.69 0.66
0.72 0.56 0.71 0.56 0.71 0.57
4
0.68
5 6
0.66 0.69
,.o
(b)
8
¢50
10
20
30
40
circ. wove number
Figure3
(a) Mean value analysis of initial (b) variability analysis of initial imperfections
. . . .
(. \~ " "] / = 2m for m = 4 , 6 . . . . E~-~2
imperfections;
Furthermore, as shown in Figure 3(b), the coefficient of variation (standard deviation/mean value) does not exhibit significant variability and this leads to simple expressions, using regression techniques, linking the mean modal amplitude value to its standard deviation, i.e. oy~,..) = 3,E(¢m.)
(4)
(5)
where 7 is a constant given in Table 3. Equations (3)-(5), capturing the important features of imperfection amplitudes, may be helpful in the comparison of alternative manufacturing methods for composite shells, e.g. lay-up versus filament winding. As far as the present
402
Probabilistic imperfection sensitivity analysis: M. K. Chryssanthopoulos and C. Poggi
models are concerned, it can be concluded that fibre orientation (cross-ply versus angle-ply) does not significantly influence the amplitude or spatial distribution of the geometric imperfections. The next task was to explore the correlation structure of the imperfection modal amplitudes. This type of analysis is useful in deciding how to combine modes in a multimode imperfection model. Results for both series reveal strong positive correlation for modes with common circumferential wavenumber and odd axial wavenumber, i.e. P(~],, ~m,) for m = 3,5 . . . . This may be explained by the relatively smooth imperfection profile in the axial direction, as shown in Figure 2 for typical cylinders in both series. In examining modes with common axial wavenumber, it was found that the correlation between P(~12, ~3) was equal to 0.93 for series A and 0.61 for series B. This indicates that the two most dominant imperfection modes are probably introduced by a single factor in the manufacturing process. Distribution fitting using graphical and goodness-of-fit tests revealed that the log-normal model was suitable for a large number of the model amplitudes that were investigated. The relevant distribution parameters and moments for the dominant modes are tabulated in Table 4. It should be noted that unless a causal relationship can be established (as, for example, in the case of additive factors which lead to a normal distribution model), the specification of probability distribution types is semi-empirical and the goodness-of-fit tests should be used in an adjutant role to investigate whether or not a preselected distribution fits the observed date. In the present study, the log-normal and the two-parameter Weibull distribution yielded the best results as far as the Kolmogorov-Smirnov test statistic was concerned. It was decided to adopt the former partly due to its simplicity and also because it has performed well in describing initial imperfection amplitudes in steel cylinders~4. Statistical analysis of phase angles has shown that these variables are best described by a uniform distribution in the range (-Tr, 70. The only significant correlation was between angles with common circumferential wavenumber, similar to the results presented above for modal amplitudes. Thus, a characteristic imperfection model for these cylinders should include several modes of different axial wavelength for any given circumferential wavelength, with both amplitudes and phase angles of modes with m = 2k + 1 highly correlated to the corresponding values of the m = 1 mode. Similarly, modes with m = 2k should be related to the parameters of mode m = 2. Finally, it is interesting to compare the results of the current study with those obtained in previous investigations.
The dominance of ovalization has been a clear feature of imperfection distributions where the manufacturing does not involve the joining (or welding) of a number of panels. Similarly, the inverse proportionality between amplitude and wavenumber (or wavelength) has also been observed in previous studies 1°,]4. Moreover, the significant correlation in modes with a common circumferential wavenumber was also present in the imperfections in steel cylinders 14. The negligible influence of modes with n > 15 is also much more evident in these unstiffened cylinders compared to the stringer-stiffened cylinders, as might be expected.
Characteristic imperfection On the basis of the entire statistical analysis, a characteristic imperfection model for both cross-ply and angle-ply cylinders is proposed for the particular manufacturing method 7
8
w~(x,O) = ~ ~ ~,,,sin TmTrx sin (nO + qbm,)
(6)
m = l n=2
where ~,n, and Ohm,are random variables described by lognormal and uniform distributions, respectively. Modes with up to seven axial half-waves and eight circumferential full waves are included since the preceding analysis has revealed that the amplitude of modes with even shorter wavelengths is very small (see equations (3) and (4)). Using the relevant distribution parameters and correlation structure, simulation studies can be undertaken in order to evaluate the probabilistic properties of the characteristic model. In particular, the mean value, E(wf~), and the standard deviation, or(w3), are estimated at any point on the cylinder surface. In addition, extreme value properties are determined either by considering a fixed threshold value, e.g. A = 2t, and calculating the probability of exceedance, P(wf~ > A), or by specifying a desired probability level, e.g. Pecr----0.05, and calculating the corresponding imperfection value, A*, that gives P(w~ > A*) = p .... The latter can be of more direct use in imperfection sensitivity studies since it gives rise to a characteristic shape that is associated with a constant exceedance probability at any point. The simulation procedure adopted for creating random samples of w6 makes use of the mean vector and the variance-covariance matrix of the random variables, as estimated from the statistical analysis of measured imperfections summarized above. In order to use well-known methods for simulating multivariate normal distributions 2°, it is first necessary to transform the random variables, ~,,,
Table 4 Lognormal distribution parameters and moments of dominant modal amplitudes Distribution parameters
Distribution moments
Mode
/~
o-z
Mean
Std. deviation
~:12 ~:13 ~:14 ~:22 ~32 ~:33 ~s2
-2.279 -3.463 -3.232 -3.504 -3.401 -4.542 -3.908
0.352 1.319 0.253 0.473 0.364 1.282 0.353
O. 121 0.054 0.044 0.037 0.040 0.018 0.023
0.074 0.052 0.021 0.024 0.025 0.018 0.015
403
Probabilistic imperfection sensitivity analysis: M. K. Chryssanthopoulos and C. Poggi
G
-L
411.,
0
Figure 4 Typical simulated imperfection surface and qb,~, into normally distributed variables, ~m. and qSm., and re-calculate their corresponding mean vector, A, and variance-covariance matrix, C. It is then possible to obtain a vector X, representing a random realization of the imperfection process from X = LU + A
(7)
where LL r= C and the components of U are random values obtained from independent standard normal distributions. The required transformations are given by ~',,., = In ~,,~
(8) ~bm.= (/)-l (~b,,J27r + 1/2) where (I)-1 (.) is the inverse of the standard normal distribution function. Figure 4 shows one of the randomly generated imperfection surfaces, whereas Figure 5 presents typical results for the characteristic imperfection, w~, obtained by simulation at various locations on the cylinder surface. An independent sample of 50 000 cases has been generated for each of these locations. The sample size was determined by considering sets of independent samples of a certain size and estimating the sampling error. This process was repeated with increasing sample sizes until a coefficient of variation of less than 0.05 on extreme value estimates was obtained. As can be seen, the histograms are symmetric and exhibit a high kurtosis (which may qualitatively be postulated from the peak ordinate, cf. 0.4 for the standard normal density). The mean value, E(w~), was, in general, found to be close to zero while the standard deviation, (r(w~), varied between 0.15 and 0.30, where the higher values were obtained for points close to the cylinder ends. In addition, extreme value properties have been estimated by specifying a desired probability level, p .... and calculating the corresponding imperfection value, A*, that f(W¢,c)
--"
5% exceedance probability
gives P(wf) > A*)=Pecr" Figure 6 shows the characteristic imperfection shape corresponding to Pecr= 0.05. It is of interest to note that the shape indicates that the imperfection process is stationary in the circumferential direction and that the threshold levels attain higher values close to the cylinder ends than in central region. Qualitatively these remarks are confirmed by Figure 7, which shows the location of the maximum (outwards) and minimum (inwards) imperfection of the manufactured cylindrical shells. As can be seen, the probability of the extreme imperfection occurring either in the region 0 <--x/L <--0.25 or in the region 0.75 -< x/L -< 1.0 is about three times higher than the corresponding value in the region 0.25 < x/L < 0.75. Buckling
i
5.0
a
0.40
L . . . .
L . . . . . . . .
•
i
L . . . . . . . . .
I I I
p
0.30
', ', ', * : • Outwards ,~. . . . ,~__~_~e__ _,~__~_. . . . . . ', ~Inwards
.....
i
0.20
i i
L . . . .
. . . .
. . . .
0.00
0
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*
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I
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i
. . . . . . .
L
el
i
0.10
. . . . . . . . .
L i
"
.........
..........0 ,3 01
,:' Woc 05
t
q
L . . . .
L . . . . . . . .
L . . . .
L . . . .
I
,,
,,
:
,
,
i .....
i .....
; ~
; .....
; .....
60
~ .....
120 180 240 300 360
Figure 7 Location of extreme imperfections on nominally identical test specimens ~ffi172.8 , x/Lffi.4737
.............................................
~
5.0
~.......-
_ /...~.,/--,-_..~ 0.0 5 03 01
i i
~
. . . .
6.0 i
"
prediction
x/L o.~o . . . . . :-z-:-. . . . . . . . . . : . . . . . . . . ,,
4 . 0
3.04"02.0"1.0 -
strength
The results presented so far can be used in tolerance specification and in the evaluation of alternative manufacturing
7 . 0
6.0
¢
Figure6 Characteristic imperfection shape associated with a
f(Woc)
dffiT.2, x/L=.2105
7.0
die,
.
.
.
.
. .
.
.
.
.
.
.
.
3.0 2.0
1.0 o.o
- 2""'""2"'"="2s oa o 1 ' o l . ..........oa. .........o51 ='
Figure 5 Histograms of characteristic imperfection at two different locations
Woc
Probabilistic imperfection sensitivity analysis: M. K. Chryssanthopoulos and C. Poggi
404 •4 ~ - v
2) NCrR/E22Lz
seen that the reduction in buckling load due to nonlinear prebuckling is about 13%. It is also worth noting that the numerically computed eigenvalue is in excellent agreement with that predicted by the theoretical model 9 shown in Fig-
1.20
1.00
ure 8. 0.80
0.60
0.40
0.20
. . . . . . . . ' . . . . . . . . . . . . . . . . L. . . . . . . . . . . i
i: 0.00
....
0
, ....
5
::
I .........
10
I .........
15
20
25
i op
I .........
30
35
I,,,
40
45
Figure 8 Nondimensional buckling load of composite cylinders
processes but, equally, they may help in establishing the imperfection sensitivity of the cylinders manufactured by this particular method. For the latter, they must be combined with a strength analysis method and, in the following study, nonlinear finite element modelling is employed zl. However, in order to gain an insight into the effect of fibre orientation on the buckling response of axially compressed cylinders with overall geometric parameters and material properties as given in Table 1, the linear buckling loads for a variety of fibre angles are plotted in Figure 8. An analytical method using symbolic manipulation on Donnell-type equations assuming membrane prebuckling has been used 9. This figure is reported here to show that in the region 0 ° - 0p - 22.5 ° the critical buckling mode is asymmetric, whereas for 22.50-< 0p -< 45 ° it is axisymmetric. The near coincidence of critical loads corresponding to different buckling modes is also worth noting. Since the angle-ply cylinders are prone to axisymmetric buckling, an appropriate finite element model is generated and both eigenvalue and incremental analyses are performed for several cases (Table 5). By comparing the first two cases, which correspond to perfect geometries, it may be
In the absence of the statistical imperfection analysis presented above, one may consider various amplitudes for the critical mode(s) in order to construct an imperfection sensitivity diagram, cases 3(a)-(d) and 4(a)-(d) for m = 13 and m = 15, respectively, these being the modes corresponding to the two lowest eigenvalues. A conservative assumption, frequently made in deterministic design, is to consider that the maximum imperfection occurs in the critical mode. On the basis of the experimentally recorded values, this would imply a value of ~max~ 0.4--0.5 mm. As can be seen from Table 5, taking an average value of 0.45 mm, the associated knock-down factors are 0.71 for m = 13 and 0.69 for m=15. On the other hand, one may use the statistical imperfection analysis together with buckling mode considerations. The need to combine the information from these two different sources is clearly demonstrated below. Consider cases 5(a)-(b), for which only axisymmetric modes (m = 1. . . . 20) have been included in the finite element model. The first case corresponds to the modal amplitudes being equal to their mean value, while in the second all amplitudes have been increased by twice their corresponding standard deviation. As can be seen, due to the small modal amplitudes associated with axisymmetric imperfections in this manufacturing process, the knockdown factor is approximately equal to that for case 2 (perfect geometry, nonlinear prebuckling). It cannot be accepted for design purposes since, as is well known, the overall imperfection, which comprises both axisymmetric and asymmetric components, influences the limit load, even for structures which buckle axisymmetrically. It is, thus, proposed to use classical buckling analysis to identify the critical imperfection modes and the characteristic shape corresponding to 5% exceedance probability to define an allowable maximum amplitude for the imperfection. One typical cross-section of the 'tunnel' shape shown
Table 5 Finite element results for angle-ply (_+45°)s cylinders under axial compression
Case
Imperfection modes
Maximum amplitude ~:max (mm)
Critical/limit load (kN)
1 2
---
---
36.97 32.33
3(a) 3(b) 3(c) 3(d)
m= m= m = m =
13 13 13 13
0.13 0.26 0.39 0.45
29.79 28.25 26.99 26.29
4(a) 4(b) 4(c) 4(d)
m = m = rn = m=
15 15 15 15
0.13 0.26 0.39 0.45
29.20 27.27 25.69 25.41
5(a) 5(b)
E(~o) E(~:mo) + 2 ~ o )
.012 .031
32.36 32.18
6(a) 6(b) 6(c) 6(d) 6(e)
rn = m = m= m= m =
0.26 0.26 0.26 0.26 0.26
28.25 27.27 28.18 28.40 27.98
13 15 13, 15 (1:1) 13, 15 (3:1) 13, 15 (1:3)
Knock-down factor I
0.87 0.81
0.76 0.73 0.71 0.79 0.74 0.69 0.69
0.87 0.87 0.76 0.74 0.76 0.77 0.76
Probabilistic imperfection sensitivity analysis: M. K. Chryssanthopoulos and C. Poggi
x/L 1.00
.....
;
-
,
~'~
t
'
y
.
t
1" ~" ~ - ; . r = .
~
i
* i
/,
,'
A' for .~.~-Pecr=O'05,.~ - ~ m = 1 3 ~=,==0.26
----
I
,
I
-0.6
:
....
,' ~---=-., , --~:'=-=-L._
0.60
ooo
~
..~-~ J. _
i
0.60
-
..i~- ~'~
,
-0.3
-0.1
0.'1
'
0.3
'6.5
Figure 9 Use of characteristic imperfection shape in axisymmetric analysis
in Figure 6 is reproduced in Figure 9. Various imperfection profiles are considered consisting either of single modes, cases 6(a)-(b), or two-mode combinations, cases 6(c)-(e), with their maximum amplitude determined by the limits imposed by the characteristic shape (~,,~ = 0.26 mm in the centre region by examining the shape over the entire cylinder surface, Figure 6). In the latter, a decision has to be made for the ratio of the modal amplitudes, which on the basis of the statistical analysis should be approximately taken as unity, but different ratios are also considered. As can be seen from Table 5, the knock-down factor for all five cases varies between 0.74 and 0.77, which is about 8% higher than that obtained using conservative deterministic assumptions (i.e. setting 1theamplitude of the critical imperfection mode equal to the maximum recorded imperfection). The comparison between deterministic and probabilistic knock-down factors is also illustrated in Figure 10, which contains the relevant imperfection sensitivity curves and the appropriate points at which knock-down factors are calculated. Thus, in this specific case, probabilistic assessment has led to somewhat less severe knock-down factors but, more significantly, these are related to the manufacturing process used. In this respect, the quality of the particular process is not unduly penalized.
Conclusions This paper has presented the results of a statistical imperfection analysis on two series of composite cylinders and has outlined a methodology for buckling strength prediction accounting for random imperfections. The objective is to Pu/Pcr
0.75
i 0.25 0.00
I
0.00
I
],
0
m=13 ]
•
m=15 I
I
[
0.10
i
~m 0
,
,
,
I
I
0.20
I
I
I
i
0.30
I
I
I
I
]
0.40
I
l
,
I
0.50
Figure 10 Imperfection sensitivity curves for angle-ply composite cylinders
4@5
estimate knock-down factors for design by combining probabilistic imperfection modelling with finite element-based strength prediction. It is demonstrated that the probabilistic analysis can lead to characteristic imperfection models which can then be used in conjunction with classical imperfection sensitivity analysis to arrive at knock-down factors. In the particular example investigated, the proposed methodology resulted in less severe knock-down factors, but this conclusion cannot be generalized. It is believed that the methodology provides a useful framework for specifying the parameters of multimode imperfection models, which are necessary for quantifying shell buckling response, whilst taking into account the actual manufacturing process used. Further work will concentrate on extended geometry and imperfection models to accommodate asymmetric modes. With regard to the present specimens, this is essential for the analysis of the cross-ply cylinders but will also be used for the angle-ply cylinders in order to explore fully their imperfection sensitivity and associated knock-down factors.
Acknowledgments The invaluable contribution to the project by Professor V Giavotto of Politecnico di Milano is gratefully acknowledged. The authors wish to thank Marie Ciavarella and Nicola Agostoni, former students in the Department of Aerospace Engineering at Politecnico di Milano, who carried out the probabilistic imperfection analysis as part of their dissertations for the Laurea.
References 1 Tennyson, R. C. 'Buckling of fiber-reinforced circular cylinders under axial compression', AFFDL Report TR 72-102, 1972 2 Simitses, G. J., Shaw, D. and Sbeinman, I. 'Imperfection sensitivity of laminated cylindrical shells in torsion and axial compression', Comput. Struct. 1985, 4, 335-360 3 Iu, V. P. and Chia, C. Y. 'Non-linear vibration and post-buckling of unsymmetric cross-ply circular cylindrical shells', Int. J. Solids Struct. 1988, 24 (2), 195-210 4 Simitses, G. J. and Anastasiadis, J. A. 'Buckling of axially-loaded, moderately thick cylindrical laminated shells', Composites Engng 1991, 1 (6), 375-391 5 Koiter, W. T. 'On the stability of elastic equilibrium', PhD thesis, Delft University. H. J. Pads, Amsterdam, 1945 (in Dutch); NASA TT-F10, 1967 (English translation) 6 Khot, N. S. and Venkayya, V. B. 'Effect of fiber orientation on initial post-buckling behaviour and imperfection sensitivity of composite cylindrical shells', AFFDL TR-70-125, 1970 7 Arbocz, J. and Hol, J. M. A. M. 'Koiter's stability theory in a computer aided engineering environment' Int. J. Solids Struct. 1990, 26 (9/10), 945-973 8 Arbocz, J. 'The effect of initial imperfections on shell stability - a review', Report LR-618, Delft University, 1991 9 Poggi, C., Taliercio, A. and Capsoni, A. 'Fibre orientation effects on the buckling behaviour of imperfect composite cylinders', in Buckling of shell structures, on land, in the sea and in the air (J. F. Jullien, Ed.), Elsevier, 1991, pp 114-123 10 Arbocz, J. 'Shell stability analysis: theory and practice', in Collapse:the buckling of structures in theory and in practice (J. M. T. Thompson and G. W. Hunt, Eds), Cambridge University Press, 1983, pp 43-74 11 Singer, J., Abramovich, H. and Yaffe, R. 'Evaluation of stiffened shell characteristics from imperfection measurements', J. Strain Analysis 1987, 22 (1), 17-23 12 Elishakoff, I. and Arbocz, J. 'Reliability of axially compressed cylindrical shells with general nonsymmetric imperfections', J. Appl. Mech. 1985, 52, 122-128 13 Elishakoff, I., van Manen, S., Vermeulen, P. G. and Arbocz, J. 'Firstorder second-moment analysis of the buckling of shells with random imperfections', AIAA J. 1987, 25, 1113-1117
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Probabilistic imperfection sensitivity analysis: M. K. Chryssanthopoulos and C. Poggi
14 Chryssanthopoulos, M. K., Baker, M. J. and Dowling, P. J. 'Statistical analysis of imperfections in stiffened cylinders', J. Struct. Engng, ASCE 1991, 117 (7), 1979-1997 15 Chryssanthopoulos, M. K., Baker, M. J. and Dowling, P. J. 'Imperfection modelling for buckling analysis of stiffened cylinders', J. Struct. Engng, ASCE 1991, 117(7), 1998-2017. 16 Chryssanthopoulos, M. K., Kim, K. D. and Dowling, P. J. 'Buckling behaviour of composite GFRP cylindrical panels under axial compression', in Proc. 6th Int. Conf. Behaviour of Offshore Structures (M. H. Patel and R. Gibbins, Eds), London, 1992, pp 733-743 17 Krishnakumar, S. and Tennyson, R. C. 'Effect of impact delaminations on axial buckling of composite cylinders', in Buckling of shell
structures, on land, in the sea and in the air (J. F. Jullien, Ed.), Elsevier, 1991, pp 61-70 18 Giavotto, V., Poggi, C., Dowling, P. J. and Chryssanthopoulos, M. K. 'Buckling behaviour of composite shells under combined loading', ibid pp 53-60 19 Chryssanthopoulos, M. K., Giavotto, V. and Poggi, C. 'Statistical imperfection models for buckling analysis of composite shells', ibm pp 43-52 20 Kendall, M. G. and Stuart, A. The advanced theory of statistics, Vol. 1 (3rd edn) Griffin, London, 1965 21 ABAQUS, Theory and user's manual, Hibbit, Karlsson and Sorensen, Providence, RI, 1989
~UTTERWORTH ~r[E
I N E M A
N N
0141-0296(95)00048-8
17, No. 6, pp. 398406, 1995 Copyright © 1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0141--0296/95 $10.00 + 0.00
Probabilistic imperfection sensitivity analysis of axially compressed composite cylinders M a r i o s K.
Chryssanthopoulos
Department of Civil Engineering, Imperial College, London SW7, UK
C a r l o Poggi Department of Structural Engineering, Politecnico di Milano, Milano, Italy (Received August 1993; revised version accepted April 1994)
Buckling analysis of cylinders under axial compression is sensitive to the assumptions made in the modelling of initial imperfections. Normally, imperfection modes are selected solely on the basis of buckling mode considerations and their amplitudes are determined using tolerance specifications in codes or experimentally recorded values. Whilst this approach may be used for metal cylinders with some confidence, due to the many test results available for validation purposes, it is not appropriate for the analysis and design of fibre-reinforced composite cylinders where the test results are limited and the effects of manufacturing on the imperfection characteristics have not yet been studied in detail. This paper presents a methodology for probabilistic buckling strength assessment based on the results of a statistical analysis on imperfections on two groups of composite cylinders manufactured by lay-up. The dominant features are quantified and the effect of fibre orientation on imperfections is examined. Simple models describing the random variability of imperfection modal amplitudes are presented. Using these probabilistic models, characteristic imperfection shapes are developed for fibre-reinforced cylinders and their use in buckling strength prediction and tolerance specification is demonstrated
Keywords: buckling, imperfections, composites, cylinders, axial compression, probabilistic modelling, manufacturing effects Buckling of composite cylinders has been studied in the last two decades, mainly using the Galerkin method to analyse the eigenvalue problem and the postbuckling response 1-4. The analysis can also be performed by means of perturbation procedures, which, in conjunction with the theory proposed by Koiter5, lead to quantitative estimates of the imperfection sensitivity. This method was first applied by Khot and Venkayya6 and, more recently, by A r b o c z 7"8 who included prebuckling nonlinearities. The latter demonstrated that the influence of prebuckling nonlinearities is significant in the postbuckling behaviour of axially compressed composite cylinders, even though critical loads are not substantially affected. Using these methods the effect of fibre orientation on the buckling load and on the postbuckling behaviour as well as the imperfection sensitivity of laminated composite cyl-
inders can be examined9. Their advantage lies in the derivation of exact solutions for the buckling and postbuckling equations leading to compact expressions for the critical load and the postbuckling b-coefficients. It is also possible from imperfection sensitivity curves to define 'knockdown' factors, which are of use in design. The 'knockdown' factor is a key ingredient in many design procedures, since the limit load of the imperfect shell buckling elastically is obtained by multiplying the critical buckling load of the perfect shell by this factor. On the other hand, it is well known that the actual buckling load and associated buckling mode can be different from those obtained from a single mode analysis, due to nonlinearities and interaction phenomena, and that real imperfections may give rise to different knock-down factors, due to multimode effects. For these reasons, and in
398
Probabilistic imperfection sensitivity analysis: M. K. Chryssanthopoulos and C. Poggi spite of the fact that the above methods may be extended to include several modes, knock-down factors are normally
determined from the lower bound curves based on test results. However, large databases of test results exist only for few basic shell geometries and loading conditions, and even in these cases problems with regard to the homogeneity of data (e.g. large and small-scale tests, different manufacturing methods and/or materials) may lead to knockdown factors which are unduly conservative for certain applications. These considerations have prompted the creation of imperfection data banks 1°'11 and the use of probabilistic methods in calculating a reliability function and, hence, characteristic values for the elastic buckling strength of metal shells with random imperfections 12'~3. In related studies, use of statistical techniques on the Fourier coefficients of imperfection modes for nominally identical stringer-stiffened steel cylinders has been made to arrive at characteristic imperfection models TM. The results of the statistical analysis were subsequently incorporated into finite element models in order to quantify the effects of multimode imperfection patterns on elastoplastic buckling strength 1~. Figure 1 summarizes the probabilistic design procedure adopted in these studies and contrasts it with a traditional deterministic design procedure. In the latter, measured imperfections are used to validate strength prediction models (e.g. finite elements) which are subsequently used in wider parametric studies. In these, imperfection modelling becomes a very important factor but the number of cases required to cover all possible modes and spatial distributions is prohibitively large. In the absence of measured imperfection data, one possibility is to consider imperfections related to the critical mode shape obtained from linear eigenvalue analyses, although nonlinear prebuckling and mode interaction weakens the case for selecting imperfection profiles in this m a ~ e r ~6. Geometric ImperfectionI Measurements [
1 Imperfection Data Processing
Buckling Strength Analysi=; of Measured Shells
Statistical Analysis of Fourier Coefficients
Validation of Strength Prediction Models
Characteristic Imperfection
Deterministic Design I Strength Assessment
Probabilistic Design Strength Assessment
l
Surfaces
Figure 1 Schematic diagram illustrating deterministic and pro-
babilistic approaches
399
Probabilistic design strength assessment, which requires that imperfections are recorded on groups of similarly manufactured shells, uses statistical analysis to arrive at characteristic imperfection models associated with a particular manufacturing method. These models form the basis on which the rules for specifying the parameters of multimode imperfection models are devised and, hence, the imperfection sensitivity is assessed. In this paper, this procedure is used to analyze the imperfection sensitivity of composite cylinders under axial compression. The strength analysis is done by nonlinear finite element modelling. The above remarks on imperfection modelling and associated buckling load sensitivity are equally valid for both metallic and composite shells. As is well known, the latter can also suffer from material defects, such as delaminations, porosities etc., and it might be expected that tolerance specifications should also account for these effects. However, as far as buckling load prediction is concerned, geometric imperfections are deemed to play a dominant role 17 and are examined in detail in this paper.
Notation number of half-waves along cylinder length total number of estimated Fourier coefficients in axial direction n number of waves along cylinder circumference total number of estimated Fourier coefficients in nT circumferential direction threshold value of exceedance probability Peer cylinder thickness t tp ply thickness Wo(X,O) cylinder initial imperfection function Wo~(x,O) characteristic initial imperfection function imperfection value at point i after 'best-fit' analywl~p sis WiF imperfection value at point i using Fourier representation co-ordinate along cylinder length X E(A) mean value of random variable A elastic modulus along fibre direction Ell elastic modulus perpendicular to fibre direction E22 shear modulus G~2 N total number of imperfection readings on cylinder surface critical linear buckling axial stress resultant Ncr L cylinder length critical buckling load of perfect cylinder Pcr P. limit load of imperfect cylinder R cylinder radius A* imperfection corresponding to exceedance probability threshold co-ordinate along cylinder circumference 0 o, ply angle Poisson's ratio 1)12 initial imperfection amplitude associated with mode ( m,n ) o'(A ) standard deviation of random variable A p( A,B ) correlation coefficient between random variables A and B phase angle associated with mode (m,n) m mT
Statistical analysis of imperfections In many shell buckling investigations, effort has been directed at detailed measurement of imperfections on test
Probabilistic imperfection sensitivity analysis: M. K. Chryssanthopoulos and C. Poggi
400
specimens, as well as some full-scale components. In most of these studies a standard method of data analysis has been adopted, based on the concept of the 'best-fit' cylinder ~°. This concept has enabled a unified datum to be established for shell imperfections and can be of particular use in comparative studies. Following the 'best-fit' procedure, the resulting imperfections are analysed using two-dimensional harmonic analysis to produce a set of Fourier coefficients, e.g. mT nT
mTrx Wo (x,O)) = ~ ~ ~,,, sin - ~ - sin (nO + tkm,)
(1)
m=l n=O
where 0 -< x --- L and 0 - 0 -< 27r. Hence, each surface is described by a set of modal amplitudes and phase angles, ~m, and ~bm,, respectively, where ~m, -> 0 and --'IT ~
(~rnn ~
"17".
Within the current programme of work, a number of composite cylinders have been manufactured by lay-up using an orthogonal Kevlar fabric embedded into an epoxy resin matrix. The first group consisted of 16 nominally identical (i.e. with nominally identical geometric dimensions and material characteristics) cross-ply cylinders with symmetrical lay-up (0°/90°)s, while the second group comprised 14 nominally identical angle-ply cylinders with symmetrical lay-up (+45°)s. Their relevant geometric and material parameters are given in Table 1. As can be seen, the size of the models is such that the manufacturing process and the resulting imperfections are representative of real components. The imperfection surface of each cylinder was recorded using displacement transducers in a regular mesh with an interval of 10 mm axially and 20 mm circumferentially. At each point, measurements were taken on both the inside and the outside cylindrical surface. The apparatus used for this purpose, together with the objectives and the initial experimental results of this project are described elsewhere TM. The overall objective is to develop validated design guidelines for composite cylinders subjected to compression and torsion. After recording the imperfections, 'best-fit' and harmonic analysis was carried out as described above. Equation ( 1 ) represents a half-range sine expansion in the axial direction, thus, imposing zero imperfection values at the two cylinder ends. Although this is not strictly correct, the error introduced is not significant and is confined to the end regions only, provided the number of calculated coefficients is not too small. Both half-range (sine and cosine) as well as full-range expansions were assessed using the following error function 1
e = ~/~
(w~/r - wF)2
i:1
Table 1
R L t tp Ell
Geometric and material parameters 350 m m 700 m m 1.04 m m 0.26 m m
E22 G12
23450 N / m m 2 23450 N / r a m 2 1520 N / m m 2
u12
0.20
(2)
Comparisons were also made at the points of maximum imperfection (in/out). The half-range sine series with mr = 20 and nT = 40 offered the best alternative in terms of accuracy and compactness. Figure 2 shows the imperfection surfaces (after 'bestfit') obtained for typical cylinders in both series A (crossply) and series B (angle-ply). Although the internal surfaces appear to have similar characteristics (i.e. dominance of long imperfection waves in both axial and circumferential directions), the external surface is strongly influenced by the orientation of the individual layers. In fact, the sharp peaks observed on the external surfaces are the result of local thickness variations due to overlapping layers and, hence, should not be treated as initial geometric imperfections. In order to quantify this result, a detailed correlation analysis of the Fourier modal amplitudes was undertaken. It was found that the correlation coefficient between modal amplitudes with identical wavenumbers, obtained for external and internal surfaces for each group of cylinders, D ( e x ~ m n , i n ~ m n ) , is positive and is statistically significant for low wavenumbers in the circumferential direction 19. For n -> 15, this correlation is not statistically significant, demonstrating that short wavelength modes are only present on the external surface due to the localized thickness variations. As a result, internal surface measurements were used in the ensuing analysis to study the characteristics of geometric imperfections. By treating the model amplitudes and associated phase angles, i.e. ~m, and thin,, as random variables, a number of univariate and bivariate (correlation) statistical parameters are calculated. In addition, the fitting of probability distributions is carried out by goodness-of-fit tests (Kolmogorov-Smirnov) and graphical methods. Several probability laws were tested, namely • • • • • •
Uniform Normal Log-normal Rayleigh Exponential Weibull (two-parameter)
Figure 3(a) shows the results of the mean value analysis of series A (cross-ply), clearly indicating that the dominant amplitudes are associated with long wavelengths in the circumferential direction. Modes with n > 15 have negligible influence on the imperfection profiles. Similar results were found for series B (angle-ply) 19. In fact, only about 2% of the total number of modes considered (1-< mr-< 20, 0 -< n~ -< 40) exhibit mean values greater than 10% of the maximum mean value, which for both cross-ply and angleply cylinders is associated with mode ~,2. Due to the monotonic trend observed for n---2, it is possible to fit a simple expression to describe the relationship between the mean model amplitudes and the circumferential wavenumber for any given axial wavenumber. Thus,
E(~,,,) = e '~"~
(3)
where a and /3 are constants given in Table 2 for both cross- and angle-ply cylinders. The mean value of modes with shorter axial wavelength (m > 2) can be approximately described by the following expressions
Probabilistic imperfection sensitivity analysis: M. K. Chryssanthopoulos and C. Poggi
401
"~r~)
-,
Figure2
--q~)
Typical imperfection surfaces of composite cylinders manufactured by lay-up
Table2
~
~o.,o
~o.o5
°
i
-...
........... ! ....................
.
... ....... i ....................
.t1. \...,,", 0.00 0
I 10
° =
=
~=
i
! ...................
E
i ...................
i i
! ...................
i ..................
!
i
i
!
, 20
, 30
, 4O
Series
Modes
a
/3
A A
m = 1, 2 -< n <- 20 m = 2, 2 -< n -< 20 m = 1, 2 -< n -< 20 m = 2, 2 -< n -< 20
-1.54 -3.03
0.5 0.3 0.4 0.3
B B
(a)
Constants for mean value models
Table3
-1.86
-2.82
Constants for variability models
clrc. we've number
o
rn
7 (series A )
7 (series B)
1 2 3
0.65 0.69 0.66
0.72 0.56 0.71 0.56 0.71 0.57
4
0.68
5 6
0.66 0.69
,.o
(b)
8
¢50
10
20
30
40
circ. wove number
Figure3
(a) Mean value analysis of initial (b) variability analysis of initial imperfections
. . . .
(. \~ " "] / = 2m for m = 4 , 6 . . . . E~-~2
imperfections;
Furthermore, as shown in Figure 3(b), the coefficient of variation (standard deviation/mean value) does not exhibit significant variability and this leads to simple expressions, using regression techniques, linking the mean modal amplitude value to its standard deviation, i.e. oy~,..) = 3,E(¢m.)
(4)
(5)
where 7 is a constant given in Table 3. Equations (3)-(5), capturing the important features of imperfection amplitudes, may be helpful in the comparison of alternative manufacturing methods for composite shells, e.g. lay-up versus filament winding. As far as the present
402
Probabilistic imperfection sensitivity analysis: M. K. Chryssanthopoulos and C. Poggi
models are concerned, it can be concluded that fibre orientation (cross-ply versus angle-ply) does not significantly influence the amplitude or spatial distribution of the geometric imperfections. The next task was to explore the correlation structure of the imperfection modal amplitudes. This type of analysis is useful in deciding how to combine modes in a multimode imperfection model. Results for both series reveal strong positive correlation for modes with common circumferential wavenumber and odd axial wavenumber, i.e. P(~],, ~m,) for m = 3,5 . . . . This may be explained by the relatively smooth imperfection profile in the axial direction, as shown in Figure 2 for typical cylinders in both series. In examining modes with common axial wavenumber, it was found that the correlation between P(~12, ~3) was equal to 0.93 for series A and 0.61 for series B. This indicates that the two most dominant imperfection modes are probably introduced by a single factor in the manufacturing process. Distribution fitting using graphical and goodness-of-fit tests revealed that the log-normal model was suitable for a large number of the model amplitudes that were investigated. The relevant distribution parameters and moments for the dominant modes are tabulated in Table 4. It should be noted that unless a causal relationship can be established (as, for example, in the case of additive factors which lead to a normal distribution model), the specification of probability distribution types is semi-empirical and the goodness-of-fit tests should be used in an adjutant role to investigate whether or not a preselected distribution fits the observed date. In the present study, the log-normal and the two-parameter Weibull distribution yielded the best results as far as the Kolmogorov-Smirnov test statistic was concerned. It was decided to adopt the former partly due to its simplicity and also because it has performed well in describing initial imperfection amplitudes in steel cylinders~4. Statistical analysis of phase angles has shown that these variables are best described by a uniform distribution in the range (-Tr, 70. The only significant correlation was between angles with common circumferential wavenumber, similar to the results presented above for modal amplitudes. Thus, a characteristic imperfection model for these cylinders should include several modes of different axial wavelength for any given circumferential wavelength, with both amplitudes and phase angles of modes with m = 2k + 1 highly correlated to the corresponding values of the m = 1 mode. Similarly, modes with m = 2k should be related to the parameters of mode m = 2. Finally, it is interesting to compare the results of the current study with those obtained in previous investigations.
The dominance of ovalization has been a clear feature of imperfection distributions where the manufacturing does not involve the joining (or welding) of a number of panels. Similarly, the inverse proportionality between amplitude and wavenumber (or wavelength) has also been observed in previous studies 1°,]4. Moreover, the significant correlation in modes with a common circumferential wavenumber was also present in the imperfections in steel cylinders 14. The negligible influence of modes with n > 15 is also much more evident in these unstiffened cylinders compared to the stringer-stiffened cylinders, as might be expected.
Characteristic imperfection On the basis of the entire statistical analysis, a characteristic imperfection model for both cross-ply and angle-ply cylinders is proposed for the particular manufacturing method 7
8
w~(x,O) = ~ ~ ~,,,sin TmTrx sin (nO + qbm,)
(6)
m = l n=2
where ~,n, and Ohm,are random variables described by lognormal and uniform distributions, respectively. Modes with up to seven axial half-waves and eight circumferential full waves are included since the preceding analysis has revealed that the amplitude of modes with even shorter wavelengths is very small (see equations (3) and (4)). Using the relevant distribution parameters and correlation structure, simulation studies can be undertaken in order to evaluate the probabilistic properties of the characteristic model. In particular, the mean value, E(wf~), and the standard deviation, or(w3), are estimated at any point on the cylinder surface. In addition, extreme value properties are determined either by considering a fixed threshold value, e.g. A = 2t, and calculating the probability of exceedance, P(wf~ > A), or by specifying a desired probability level, e.g. Pecr----0.05, and calculating the corresponding imperfection value, A*, that gives P(w~ > A*) = p .... The latter can be of more direct use in imperfection sensitivity studies since it gives rise to a characteristic shape that is associated with a constant exceedance probability at any point. The simulation procedure adopted for creating random samples of w6 makes use of the mean vector and the variance-covariance matrix of the random variables, as estimated from the statistical analysis of measured imperfections summarized above. In order to use well-known methods for simulating multivariate normal distributions 2°, it is first necessary to transform the random variables, ~,,,
Table 4 Lognormal distribution parameters and moments of dominant modal amplitudes Distribution parameters
Distribution moments
Mode
/~
o-z
Mean
Std. deviation
~:12 ~:13 ~:14 ~:22 ~32 ~:33 ~s2
-2.279 -3.463 -3.232 -3.504 -3.401 -4.542 -3.908
0.352 1.319 0.253 0.473 0.364 1.282 0.353
O. 121 0.054 0.044 0.037 0.040 0.018 0.023
0.074 0.052 0.021 0.024 0.025 0.018 0.015
403
Probabilistic imperfection sensitivity analysis: M. K. Chryssanthopoulos and C. Poggi
G
-L
411.,
0
Figure 4 Typical simulated imperfection surface and qb,~, into normally distributed variables, ~m. and qSm., and re-calculate their corresponding mean vector, A, and variance-covariance matrix, C. It is then possible to obtain a vector X, representing a random realization of the imperfection process from X = LU + A
(7)
where LL r= C and the components of U are random values obtained from independent standard normal distributions. The required transformations are given by ~',,., = In ~,,~
(8) ~bm.= (/)-l (~b,,J27r + 1/2) where (I)-1 (.) is the inverse of the standard normal distribution function. Figure 4 shows one of the randomly generated imperfection surfaces, whereas Figure 5 presents typical results for the characteristic imperfection, w~, obtained by simulation at various locations on the cylinder surface. An independent sample of 50 000 cases has been generated for each of these locations. The sample size was determined by considering sets of independent samples of a certain size and estimating the sampling error. This process was repeated with increasing sample sizes until a coefficient of variation of less than 0.05 on extreme value estimates was obtained. As can be seen, the histograms are symmetric and exhibit a high kurtosis (which may qualitatively be postulated from the peak ordinate, cf. 0.4 for the standard normal density). The mean value, E(w~), was, in general, found to be close to zero while the standard deviation, (r(w~), varied between 0.15 and 0.30, where the higher values were obtained for points close to the cylinder ends. In addition, extreme value properties have been estimated by specifying a desired probability level, p .... and calculating the corresponding imperfection value, A*, that f(W¢,c)
--"
5% exceedance probability
gives P(wf) > A*)=Pecr" Figure 6 shows the characteristic imperfection shape corresponding to Pecr= 0.05. It is of interest to note that the shape indicates that the imperfection process is stationary in the circumferential direction and that the threshold levels attain higher values close to the cylinder ends than in central region. Qualitatively these remarks are confirmed by Figure 7, which shows the location of the maximum (outwards) and minimum (inwards) imperfection of the manufactured cylindrical shells. As can be seen, the probability of the extreme imperfection occurring either in the region 0 <--x/L <--0.25 or in the region 0.75 -< x/L -< 1.0 is about three times higher than the corresponding value in the region 0.25 < x/L < 0.75. Buckling
i
5.0
a
0.40
L . . . .
L . . . . . . . .
•
i
L . . . . . . . . .
I I I
p
0.30
', ', ', * : • Outwards ,~. . . . ,~__~_~e__ _,~__~_. . . . . . ', ~Inwards
.....
i
0.20
i i
L . . . .
. . . .
. . . .
0.00
0
.....
*
L-
I
•
i
. . . . . . .
L
el
i
0.10
. . . . . . . . .
L i
"
.........
..........0 ,3 01
,:' Woc 05
t
q
L . . . .
L . . . . . . . .
L . . . .
L . . . .
I
,,
,,
:
,
,
i .....
i .....
; ~
; .....
; .....
60
~ .....
120 180 240 300 360
Figure 7 Location of extreme imperfections on nominally identical test specimens ~ffi172.8 , x/Lffi.4737
.............................................
~
5.0
~.......-
_ /...~.,/--,-_..~ 0.0 5 03 01
i i
~
. . . .
6.0 i
"
prediction
x/L o.~o . . . . . :-z-:-. . . . . . . . . . : . . . . . . . . ,,
4 . 0
3.04"02.0"1.0 -
strength
The results presented so far can be used in tolerance specification and in the evaluation of alternative manufacturing
7 . 0
6.0
¢
Figure6 Characteristic imperfection shape associated with a
f(Woc)
dffiT.2, x/L=.2105
7.0
die,
.
.
.
.
. .
.
.
.
.
.
.
.
3.0 2.0
1.0 o.o
- 2""'""2"'"="2s oa o 1 ' o l . ..........oa. .........o51 ='
Figure 5 Histograms of characteristic imperfection at two different locations
Woc
Probabilistic imperfection sensitivity analysis: M. K. Chryssanthopoulos and C. Poggi
404 •4 ~ - v
2) NCrR/E22Lz
seen that the reduction in buckling load due to nonlinear prebuckling is about 13%. It is also worth noting that the numerically computed eigenvalue is in excellent agreement with that predicted by the theoretical model 9 shown in Fig-
1.20
1.00
ure 8. 0.80
0.60
0.40
0.20
. . . . . . . . ' . . . . . . . . . . . . . . . . L. . . . . . . . . . . i
i: 0.00
....
0
, ....
5
::
I .........
10
I .........
15
20
25
i op
I .........
30
35
I,,,
40
45
Figure 8 Nondimensional buckling load of composite cylinders
processes but, equally, they may help in establishing the imperfection sensitivity of the cylinders manufactured by this particular method. For the latter, they must be combined with a strength analysis method and, in the following study, nonlinear finite element modelling is employed zl. However, in order to gain an insight into the effect of fibre orientation on the buckling response of axially compressed cylinders with overall geometric parameters and material properties as given in Table 1, the linear buckling loads for a variety of fibre angles are plotted in Figure 8. An analytical method using symbolic manipulation on Donnell-type equations assuming membrane prebuckling has been used 9. This figure is reported here to show that in the region 0 ° - 0p - 22.5 ° the critical buckling mode is asymmetric, whereas for 22.50-< 0p -< 45 ° it is axisymmetric. The near coincidence of critical loads corresponding to different buckling modes is also worth noting. Since the angle-ply cylinders are prone to axisymmetric buckling, an appropriate finite element model is generated and both eigenvalue and incremental analyses are performed for several cases (Table 5). By comparing the first two cases, which correspond to perfect geometries, it may be
In the absence of the statistical imperfection analysis presented above, one may consider various amplitudes for the critical mode(s) in order to construct an imperfection sensitivity diagram, cases 3(a)-(d) and 4(a)-(d) for m = 13 and m = 15, respectively, these being the modes corresponding to the two lowest eigenvalues. A conservative assumption, frequently made in deterministic design, is to consider that the maximum imperfection occurs in the critical mode. On the basis of the experimentally recorded values, this would imply a value of ~max~ 0.4--0.5 mm. As can be seen from Table 5, taking an average value of 0.45 mm, the associated knock-down factors are 0.71 for m = 13 and 0.69 for m=15. On the other hand, one may use the statistical imperfection analysis together with buckling mode considerations. The need to combine the information from these two different sources is clearly demonstrated below. Consider cases 5(a)-(b), for which only axisymmetric modes (m = 1. . . . 20) have been included in the finite element model. The first case corresponds to the modal amplitudes being equal to their mean value, while in the second all amplitudes have been increased by twice their corresponding standard deviation. As can be seen, due to the small modal amplitudes associated with axisymmetric imperfections in this manufacturing process, the knockdown factor is approximately equal to that for case 2 (perfect geometry, nonlinear prebuckling). It cannot be accepted for design purposes since, as is well known, the overall imperfection, which comprises both axisymmetric and asymmetric components, influences the limit load, even for structures which buckle axisymmetrically. It is, thus, proposed to use classical buckling analysis to identify the critical imperfection modes and the characteristic shape corresponding to 5% exceedance probability to define an allowable maximum amplitude for the imperfection. One typical cross-section of the 'tunnel' shape shown
Table 5 Finite element results for angle-ply (_+45°)s cylinders under axial compression
Case
Imperfection modes
Maximum amplitude ~:max (mm)
Critical/limit load (kN)
1 2
---
---
36.97 32.33
3(a) 3(b) 3(c) 3(d)
m= m= m = m =
13 13 13 13
0.13 0.26 0.39 0.45
29.79 28.25 26.99 26.29
4(a) 4(b) 4(c) 4(d)
m = m = rn = m=
15 15 15 15
0.13 0.26 0.39 0.45
29.20 27.27 25.69 25.41
5(a) 5(b)
E(~o) E(~:mo) + 2 ~ o )
.012 .031
32.36 32.18
6(a) 6(b) 6(c) 6(d) 6(e)
rn = m = m= m= m =
0.26 0.26 0.26 0.26 0.26
28.25 27.27 28.18 28.40 27.98
13 15 13, 15 (1:1) 13, 15 (3:1) 13, 15 (1:3)
Knock-down factor I
0.87 0.81
0.76 0.73 0.71 0.79 0.74 0.69 0.69
0.87 0.87 0.76 0.74 0.76 0.77 0.76
Probabilistic imperfection sensitivity analysis: M. K. Chryssanthopoulos and C. Poggi
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Figure 9 Use of characteristic imperfection shape in axisymmetric analysis
in Figure 6 is reproduced in Figure 9. Various imperfection profiles are considered consisting either of single modes, cases 6(a)-(b), or two-mode combinations, cases 6(c)-(e), with their maximum amplitude determined by the limits imposed by the characteristic shape (~,,~ = 0.26 mm in the centre region by examining the shape over the entire cylinder surface, Figure 6). In the latter, a decision has to be made for the ratio of the modal amplitudes, which on the basis of the statistical analysis should be approximately taken as unity, but different ratios are also considered. As can be seen from Table 5, the knock-down factor for all five cases varies between 0.74 and 0.77, which is about 8% higher than that obtained using conservative deterministic assumptions (i.e. setting 1theamplitude of the critical imperfection mode equal to the maximum recorded imperfection). The comparison between deterministic and probabilistic knock-down factors is also illustrated in Figure 10, which contains the relevant imperfection sensitivity curves and the appropriate points at which knock-down factors are calculated. Thus, in this specific case, probabilistic assessment has led to somewhat less severe knock-down factors but, more significantly, these are related to the manufacturing process used. In this respect, the quality of the particular process is not unduly penalized.
Conclusions This paper has presented the results of a statistical imperfection analysis on two series of composite cylinders and has outlined a methodology for buckling strength prediction accounting for random imperfections. The objective is to Pu/Pcr
0.75
i 0.25 0.00
I
0.00
I
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m=13 ]
•
m=15 I
I
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0.10
i
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,
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I
0.20
I
I
I
i
0.30
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l
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Figure 10 Imperfection sensitivity curves for angle-ply composite cylinders
4@5
estimate knock-down factors for design by combining probabilistic imperfection modelling with finite element-based strength prediction. It is demonstrated that the probabilistic analysis can lead to characteristic imperfection models which can then be used in conjunction with classical imperfection sensitivity analysis to arrive at knock-down factors. In the particular example investigated, the proposed methodology resulted in less severe knock-down factors, but this conclusion cannot be generalized. It is believed that the methodology provides a useful framework for specifying the parameters of multimode imperfection models, which are necessary for quantifying shell buckling response, whilst taking into account the actual manufacturing process used. Further work will concentrate on extended geometry and imperfection models to accommodate asymmetric modes. With regard to the present specimens, this is essential for the analysis of the cross-ply cylinders but will also be used for the angle-ply cylinders in order to explore fully their imperfection sensitivity and associated knock-down factors.
Acknowledgments The invaluable contribution to the project by Professor V Giavotto of Politecnico di Milano is gratefully acknowledged. The authors wish to thank Marie Ciavarella and Nicola Agostoni, former students in the Department of Aerospace Engineering at Politecnico di Milano, who carried out the probabilistic imperfection analysis as part of their dissertations for the Laurea.
References 1 Tennyson, R. C. 'Buckling of fiber-reinforced circular cylinders under axial compression', AFFDL Report TR 72-102, 1972 2 Simitses, G. J., Shaw, D. and Sbeinman, I. 'Imperfection sensitivity of laminated cylindrical shells in torsion and axial compression', Comput. Struct. 1985, 4, 335-360 3 Iu, V. P. and Chia, C. Y. 'Non-linear vibration and post-buckling of unsymmetric cross-ply circular cylindrical shells', Int. J. Solids Struct. 1988, 24 (2), 195-210 4 Simitses, G. J. and Anastasiadis, J. A. 'Buckling of axially-loaded, moderately thick cylindrical laminated shells', Composites Engng 1991, 1 (6), 375-391 5 Koiter, W. T. 'On the stability of elastic equilibrium', PhD thesis, Delft University. H. J. Pads, Amsterdam, 1945 (in Dutch); NASA TT-F10, 1967 (English translation) 6 Khot, N. S. and Venkayya, V. B. 'Effect of fiber orientation on initial post-buckling behaviour and imperfection sensitivity of composite cylindrical shells', AFFDL TR-70-125, 1970 7 Arbocz, J. and Hol, J. M. A. M. 'Koiter's stability theory in a computer aided engineering environment' Int. J. Solids Struct. 1990, 26 (9/10), 945-973 8 Arbocz, J. 'The effect of initial imperfections on shell stability - a review', Report LR-618, Delft University, 1991 9 Poggi, C., Taliercio, A. and Capsoni, A. 'Fibre orientation effects on the buckling behaviour of imperfect composite cylinders', in Buckling of shell structures, on land, in the sea and in the air (J. F. Jullien, Ed.), Elsevier, 1991, pp 114-123 10 Arbocz, J. 'Shell stability analysis: theory and practice', in Collapse:the buckling of structures in theory and in practice (J. M. T. Thompson and G. W. Hunt, Eds), Cambridge University Press, 1983, pp 43-74 11 Singer, J., Abramovich, H. and Yaffe, R. 'Evaluation of stiffened shell characteristics from imperfection measurements', J. Strain Analysis 1987, 22 (1), 17-23 12 Elishakoff, I. and Arbocz, J. 'Reliability of axially compressed cylindrical shells with general nonsymmetric imperfections', J. Appl. Mech. 1985, 52, 122-128 13 Elishakoff, I., van Manen, S., Vermeulen, P. G. and Arbocz, J. 'Firstorder second-moment analysis of the buckling of shells with random imperfections', AIAA J. 1987, 25, 1113-1117
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Probabilistic imperfection sensitivity analysis: M. K. Chryssanthopoulos and C. Poggi
14 Chryssanthopoulos, M. K., Baker, M. J. and Dowling, P. J. 'Statistical analysis of imperfections in stiffened cylinders', J. Struct. Engng, ASCE 1991, 117 (7), 1979-1997 15 Chryssanthopoulos, M. K., Baker, M. J. and Dowling, P. J. 'Imperfection modelling for buckling analysis of stiffened cylinders', J. Struct. Engng, ASCE 1991, 117(7), 1998-2017. 16 Chryssanthopoulos, M. K., Kim, K. D. and Dowling, P. J. 'Buckling behaviour of composite GFRP cylindrical panels under axial compression', in Proc. 6th Int. Conf. Behaviour of Offshore Structures (M. H. Patel and R. Gibbins, Eds), London, 1992, pp 733-743 17 Krishnakumar, S. and Tennyson, R. C. 'Effect of impact delaminations on axial buckling of composite cylinders', in Buckling of shell
structures, on land, in the sea and in the air (J. F. Jullien, Ed.), Elsevier, 1991, pp 61-70 18 Giavotto, V., Poggi, C., Dowling, P. J. and Chryssanthopoulos, M. K. 'Buckling behaviour of composite shells under combined loading', ibid pp 53-60 19 Chryssanthopoulos, M. K., Giavotto, V. and Poggi, C. 'Statistical imperfection models for buckling analysis of composite shells', ibm pp 43-52 20 Kendall, M. G. and Stuart, A. The advanced theory of statistics, Vol. 1 (3rd edn) Griffin, London, 1965 21 ABAQUS, Theory and user's manual, Hibbit, Karlsson and Sorensen, Providence, RI, 1989