Characterization of manufacturing effects for buckling-sensitive composite cylinders

Manujbcluring 6 (1995) 93-101 I(; 1995 Elsevier Science Limited in Great Britain. All rights reserved 0957143:95/$10.00

Composites

Printed

Characterization of manufacturing for buckling-sensitive composite

M.K.

effects cylinders

Chryssanthopoulos”

Department

of Civil Engineering,

and V. Giavottot

and

Imperial College, London

SW7 2BU, UK

C. Poggis

t Department of Aerospace Engineering and $ Department of Structural Engineering, Politecnico di Milano, 32 p.za Leonardo da Vinci, l-20133 Milano, Italy

presents a detailed statistical analysis on geometric imperfections recorded on two series of nominally identical composite cylinders. These defects can be classified in two categories, both due to the particular manufacturing method used: out-of-roundness and change of thickness due to the overlapping of various layers. The statistical analysis is developed for various purposes: to evaluate the common properties of cylinders with different laminations, to build up a characteristic model for the geometric imperfections suitable for probabilistic simulations in buckling analysis and to identify the parameters for quality control processes. The analysis of the change in thickness due to overlapping layers allows evaluation of the stiffening effects of the manufacturing process that, in some cases,could affect the buckling behaviour of composite cylinders. A standard procedure for the characterization and qualification of manufacturing processes for composite shells, with particular attention to the factors that influence their buckling behaviour, is proposed.

This paper

(Keywords:

composite

cylinders;

manufacturing

process;

geometric

INTRODUCTION It is well known that the buckling behaviour of composite cylinders is highly influenced by the presence of geometric imperfections. causing large discrepancies between experimental loads and analytical values’. The effect is particularly severe in axially compressed cylindrical shells, where the presence of almost coincident buckling modes makes these structural elements most sensitive to the presence of geometric imperfections2. The buckling and post-buckling problem can be studied theoretically using different techniques based on single or multimode analysis. In both cases it is also possible to define imperfection sensitivity curves, but ‘knock-down’ factors for design are normally determined from lower bound curves based on test results’. Despite the fact that much research activity has been devoted to this problem, this practice is still common in metallic shells. However, large data bases of test results exist only for a few basic shell geometries and loading conditions, and, even in these cases, problems with regard to the homogeneity of data (such as the use of large- and small-scale tests or of different manufacturing methods * To whom

correspondence

should

be addressed

COMPOSITES

imperfections;

statistical

analysis;

buckling

behaviour)

and materials) may lead to unduly conservative knockdown factors. These considerations have suggested the creation of imperfection data banks where compatible (or, in statistical terms, homogeneous) results are collected and compared on the basis of statistical techniques applied on the Fourier coefficients of imperfection modes for nominally identical specimens4. This allows the definition of characteristic imperfection modes that represent the particular manufacturing method and is helpful in performing numerical calculations to evaluate the effects of multimode imperfection patterns. In this work, the results of a statistical analysis on two groups of nominally identical composite cylinders are presented. The dimensions of the specimens, similar to those of real structures, imply that the recorded geometric imperfections are fully representative of the manufacturing process which in this case was hand lay-up. It is known that composite shells can also suffer from material defects, such as delamination and porosity, but these are not considered in this work. The research is part of a project aimed at improving the understanding of buckling behaviour of composite shells and providing scientific background for a better exploitation of the material properties, together with a control on

MANUFACTURING

Volume 6 Number 2 1995

93

Manufacturing

effects for composite

the influence performance’.

of processing

SPECIMEN

cylinders:

conditions

M.K. Chryssanthopoulos

on product

CHARACTERISTICS

AS has been highlighted above, a statistically significant study of the characteristics of geometric imperfections and other manufacturing defects may be obtained only on specimens with realistic geometric dimensions and made using the same manufacturing process as real components. In this work cylindrical specimens have been considered both because of the relative ease in measuring geometric imperfections on this elementary shape and because of the known high imperfection sensitivity of the axially compressed cylinder. In this respect, it is considered the most interesting buckling case to be studied. The available specimens have the following geometric properties: length = 700 mm, diameter = 700 mm, thickness = 1.04 or 2.08 mm. Two thicker parts (reinforced toes) at the top and bottom of each cylinder facilitate the fixing of the specimen into the testing rig. As a consequence, the actual shell length is limited to 560 mm (Figure 1). For all the specimens, each ply is made with Kevlar cross-weaved fibres embedded into epoxy resin matrix and has the mechanical properties reported in Table 1. The indices refer to the orthogonal in-plane axes x1 and x2. The lay-up orientation of the available specimens is reported in Table 2. It is evident that in both series A and B, the number of specimens is sufficient to perform a statistical analysis on manufacturing effects and to find the variability of buckling strength through experimental Imperfection

measurements -

7oOmm

20 mm -

#1omm

IiWfi

El

8mm

Table

2

et al.

Specimen

lay-up

Lay-up

No.

A B c D E

16 14 4 4 4

(O”/90”)s (45”/-45”)s (45”/0”/90”/45”)s (45”/-45”/O”/90”)s (O”/90”/45”/-45”)s

of specimens

Thickness 1.04 1.04 2.08 2.08 2.08

results. Series C, D and E are intended to provide information on some other lay-up configurations which are currently being used by the aerospace industry.

GEOMETRIC

IMPERFECTION

MEASUREMENT

The geometric imperfections were recorded using the apparatus depicted schematically in Figure 2. The cylindrical specimen is mounted on a rigid circular base that can rotate about the C-C axis by means of a step-bystep motor (no. 1). Two linear voltage displacement transducers (LVDTs) measure the imperfections of the inner and outer surfaces. Their vertical movement is controlled by another step-by-step motor (no. 2) while the meridional and circumferential positions are detected by two encoders. The frequency of data acquisition and the space interval can be fixed by computer-controlled equipment. The acquisition path can be either circumferential with subsequent axial movements at fixed steps or the opposite. As a result, the mapping of the geometric imperfections can be as accurate as needed and a convergency procedure can eventually be performed. The geometric imperfections were recorded in a regular mesh of points with an interval of 1Omm axially and 20mm circumferentially (Figure I), resulting in 46 measurements along any meridian and 110 measurements along any circumference. Due to practical reasons, the imperfection readings were limited to the central part of the cylinder surface. The covered area has a length of 450 mm and therefore is shorter than the actual meridian length (560 mm). Nevertheless, these measurements represent a significant data base on the nature of the

.I .

Steo-bv-steu

: Cylindrical Figure 1 Cylindrical specimen imperfections were recorded

and mesh

of points

where

specimeni

1

Material

urooerties

23450 23450 1520 1520 1520 0.20 1.32 x 1O-6 0.26

VI2

Density, p (kgmnm3) Thickness, t (mm)

COMPOSITES

-+

LVDT

i

/

i

\&?

i F

of the lamina

41 OfPa) ~722 (MW (712@@‘a) G13NW G23 OfPa)

94

motor No.2

the geometric

Cvlinderirotation Table

MANUFACTURING

(mm)

Figure

2

Apparatus

Volume 6 Number 2 1995

for imperfection

measurements

movements

Manufacturing

effects for composite

cylinders: M.K. Chryssanthopoulos

et

al.

(4

Figure

33

--F-a, e?+.&

External surface before “best-fit” procedure

External surface after “best-fit” procedure Figure 3 (b) after

Imperfection surfaces the best-fit procedure

of an angle-ply

cylinder

(a) before

and

4

Possible

xp,

l

l

of axes A--A,

B-B

and C-C

Hence a better choice is to select a perfect cylindrical shape that is positioned in such a way as to minimize all the readings. It must also be considered that if the three axes A-A, B-B and C-C are not perfectly aligned (Figure 4) the mapping results in an artificial conicity that must be evaluated and removed from the LVDT readings. Figure 4 shows the possible measurements in the case of misalignment of the CC axis with the A-A and B-B axes whose parallelism has been verified. This possibility is accounted for in the ‘best-fit’ procedure by considering an artificial conicity of the cylinder and, therefore, assuming a conical shell as the reference perfect surface (Figure 5). The position of the centre of the upper circle of the cylinder is indicated by A and is located at x0 and .ro with z = 0. The relevant radius of the best-fit cylinder is indicated by RA. The coordinates of any point on the cylindrical surface are

geometric imperfections and are sufficient to characterize the particular manufacturing process used. A typical mapping of the imperfections recorded on a particular specimen is reported in Figure 3a, where the external surface of an angle-ply cylinder is shown. At first sight, the following features are evident: l

misalignments

a strong conicity can be identified due to the slope of the imperfection surface along the axial direction; ovalization can be detected in the circumferential direction; and a series of peaks oriented at about 45” can be seen running the full length of the specimen.

The detected conicity is not due to a real geometric defect of the cylinders but due to small, practically unavoidable defects in the particular acquisition process adopted. Measurements are in fact influenced by rigid-body motions and misalignments. These can be filtered by means of an analytical procedure of minimization commonly known as ‘best-fit cylinder’. This procedure was originally proposed by Arbocz and Babcock4 and has since been adopted in several experimental programmes6”. The principles of the method and its implementation in the present work are outlined below. It is worth noting that, as described in the following, it has been adapted to cater also for measurement-induced limitations. The basic idea is that imperfection measurements have to be taken from a reference surface. One possibility is to use the surface corresponding to the nominal geometry of the specimen. However, this does not account for possible misalignments and other defects in the measuring process.

COMPOSITES

= x, - x0 -

/3,2;

RP; = (.Y& + y;;)“’

(14

= [(x; - x0 - p,-z,)’

RB, = RA +z;tanp

WI

Therefore the actual geometric imperfection the difference between RPI and RBi

is given by

H’i = Rpi - RB; = [(x; - x0 - I&=;)~ + (J’~- y. - &zi)2]1’2 - RA - z, tan p

(3)

where all the symbols are defined in Figure 5. The minimization process is performed on the following error function, defined as the summation of the squared deviations of all the measured points from the best-fit surface F, = 5 w’ = 51,(x; i= I i=l

- x0 - fi,.+

+ (J, - y. - &z,)~]“’

- RA - z, tan p}’

(4)

where N is the total number of imperfection measurements. Minimizing equation (4) with respect to the six parameters

XO, JO,

A?,, &

dF’-() 8x0-

and

Q

5 -2(Rp, - R,,,)z i= I

!z0+ 5 3Yo i=

MANUFACTURING

RA

-2(R,; I

- RBi) F

= 0

(5a)

= 0

(5b)

Pl

Volume 6 Number 2 1995

95

Manufacturing

effects for composite

cylinders: M.K. Chryssanthopoulos

: point of intemecthm

A

coneaxlswlth (x,. y,)

between of A

8

: point of lnteroectlon

R.

: radius

eoneaxlswlth~

between

the’kst-IIt’

: distance

P

B

=0

(5c)

= 0

(5d)

PI

dF 2 = 0 + 2 -2(Rpi - RBi) = 0 aRA i=l dF

+)+$2(Rpi-RBi)~= i=l

cos2

p

0

COMPOSITES

MANUFACTURING

pmJeeUons

reading

z-axis

of the on the

at 2 = 5

from P, to ‘best-lit’

cone axis

angle of the C0tlC : nomlnal send-vertex angle to account for the lncllncd posiUOn : 0f Ihe ‘best-w cone

(angle-ply) and are mixed in the case of quasi-isotropic lamination. These peaks should not be treated as geometric imperfections since they represent a thickness variation rather than a deviation from the perfect geometry. However, their presence influences the structural behaviour of the shell, both in the linear behaviour as well as in the post-buckling range, and some of the conclusions are reported in the following sections.

(50

produces a system of equations from which the parameters of the best-fit cylinder are determined. In Figures 3a and b the imperfections of the external surface of an angle-ply cylinder are reported before and after application of the best-fit procedure, respectively. The pronounced effect of a fictitious conicity, evident in Figure 3a, disappears in Figure 3b where the spurious effects due to the misalignments of axes A-A, B-B and C-C (Figure 4) and the rigid-body motions have been eliminated. However, the ovalization still remains as this is actually part of the imperfection pattern observed. Finally, the sharp peaks evident on the external surface are due to overlapping of layers, which produces local thickness variations. These local thickness variations are clearly seen on the external surface maps of all the specimens (see Figures 6-8). In fact, the peaks have a different orientation in each series: they are directed meridionally in series A (cross-ply), have an inclination of about 45” in series B

96

yz planes

R, a

PI

i=l

XL and

the

axis and the

5 Best-fit procedure applied on the artificial cone

0 -+ 5 -2(Rp, - RBi)zi F -

cone

A

cone at polnl

al;,- 0 -+ 2 -2(Rp, - R&2 a,& i=l w,

between

‘best-IIt’

: Imperkctlon

= 5 plane

of ‘best-At

: the an@

4

cone al point

Figure

aF,-

(& ,fi,)

2 =Oplane

: The aordlnales

: mdhrsof ‘kst-nt’

4

the Ybest-AC

et al.

BI-DIMENSIONAL FOURIER ANALYSIS GEOMETRIC IMPERFECTIONS

OF

Following the best-fit analysis, a two-dimensional Fourier analysis was undertaken. This procedure, often adopted on metal shell specimens4 as well as on some full-scale components’, is of particular use in comparative studies to evaluate the dominant characteristics of the imperfections. Evaluation of the standard deviation provides an important factor for quality control procedures. In the current study, the following expression has been adopted for the bi-dimensional analysis ws(x,13) = 71 T
(6)

m=ln=O

where 0 < x < L, 0 I 19< 27r, I,,, is the modal amplitude of the geometric imperfection and 4, is the phase angle associated with the mode (m,n). In this way the imperfections of each cylinder are described by a set of

Volume 6 Number 2 1995

Manufacturing

( Angle-ply

cylinder

effects

for composite

cylinders:

et al.

M. K. Chryssanthopoulos

Cross-ply cylinder 1

)

External surface

a

Internal surface Figure

7

External

and internal

surfaces

for a cross-ply

cylinder

b Figure

6

(a) External

and (b) internal

surfaces

for an angle-ply

cylinder

coefficients &,,, and c$,,,~with mT = 20 and nT = 40. The above expression is represented by a half-range sinus expansion both in the axial and circumferential directions. This is not strictly correct in the axial direction since the series imposes zero imperfection values at the top and bottom ends of the measured area which do not coincide with the cylinder ends and therefore their radial coordinates could be different from the ideal radius. The possibility of using a half-range cosinus series has been also investigated and compared with the half-range sinus using the following error function (7) E:F

where M’~ and \$‘I; represent the imperfection values at point i after the best-fit analysis and using the Fourier representation, respectively. In addition, comparisons were made at points of maximum amplitude (inwards/ outwards) and it was concluded that the half-range sinus series was the best alternative as the error introduced is

COMPOSITES

Internal surface Figure

8

External

and internal

surfaces

for a quasi-isotropic

cylinder

limited to the end regions and, provided the number of terms considered is not too small, is not significant. A correlation analysis of the Fourier mode amplitudes of the external and internal surfaces was made to ensure that the internal surface is the appropriate one to measure the actual deviation of the cylinder from the perfect geometry. It was found that the correlation coefficient between modal amplitudes with identical wavenumbers (m, n), obtained for external and internal surfaces for each

MANUFACTURING

Volume 6 Number 2 19%

97

Manufacturing Table

3

Matrix

effects for composite of non-dimensional

m=l

mean

cylinders: M.K. Chryssanthopoulos

values

of dominant

m=2

modal

amplitudes

m=3

et al.

for series A and B

m=4

m=5

m=6

m=7

n

A

B

A

B

A

B

A

B

A

B

A

B

A

B

2 3 4 5 6 7 8

1.00 0.44 0.36 0.14 0.12 0.15 -

1.oo 0.51 0.44 0.26 0.16 0.19 0.16

0.31 0.14 0.11

0.34 0.19 0.18

0.33 0.15 0.12

0.34 0.17 0.14

0.15

0.17

0.20

0.20

0.10

0.11

0.14

0.14

group of cylinders, is positive and statistically significant for low wavenumbers (n < 15) in the circumferential direction. Short wavelength modes are present only in the external surface in order to reproduce the sharp peaks mentioned above.

STATISTICAL

Table

4

Constants

for mean

Cross-ply 0.25 0,15---A+

composite

-..‘.-’

..

0

10

cylinders

+

Max

.

Mean Min

20

; .:

30

40

a composite ; +

Series

Mode

a

P

A A B B

m=l,2
-1.54 -3.03 -1.86 -2.82

0.5 0.3 0.4 0.3

-

..

L -

+

cylinders .~ Max

:

ivfeao .-..

i

statistical dependencies or correlations between the variables. Through an extensive analysis, all the important statistical parameters needed to describe the random process wo(x, 0) have been obtained for both series A (cross-ply) and series B (angle-ply). Because the number of nominally identical specimens in series C, D and E is too small, they are not amenable to statistical analysis. The results of this statistical analysis are summarized in what follows, while further details may be found elsewhere’)‘. Modal amplitude Mean value analysis. The mean value analysis of the variables &,,,,, reported in Figure 9, shows that the dominant amplitudes are associated with long wavelengths in the circumferential direction. In the two plots, referring to series A and B, the mean values together with the maximum and minimum values are reported. In both cases the extreme values associated with each mode are in agreement with the mean value trends. The mean amplitudes of the dominant modes (see Table 3, which contains only the results for modes with an amplitude greater than 10% of the maximum mean amplitude, i.e. m = 1, n = 2) are comparable for the two series examined. These dominant modes constitute only 2% of the total number of modes considered. In both cases, modes with n > 15 have a negligible influence on the imperfection profiles. The monotonic trend observed for n > 2 allows us to propose a simple expression to describe the relationship between the mean modal amplitudes and the circumferential wavenumber for any given axial wavenumber. Thus,

E(L,) 10

0

20

30

40

b

98

models

ANALYSIS

Referring to the equation given previously to describe the imperfection surface, it is evident that since ws(x, 0) will vary from one specimen to another due to the random nature of imperfections, it is possible to capture the features of this variability by modelling the modal amplitudes and phase angles, i.e. &,,,, and $,,, as random variables. Thus, the objective of .the statistical analysis is to provide information on the mean and variance of these variables as well as on suitable probability distributions. Furthermore, since all these variables are used in combination to describe the imperfection surface, it is important to reveal any

Figure 9 Mean (b) series B

value

value

analysis

COMPOSITES

of mode

amplitudes

MANUFACTURING

for: (a) series A and

= exp (an’)

(8)

where (Yand p are constants evaluated from sample mean values (Table 4). The mean value of modes with shorter axial wavelength (m > 2) can be approximately described by the

Volume 6 Number 2 1995

Manufacturing

0

20

10 circ.

wave

30

effects for composite

40

number

Correlation analysis. The correlation structure needs to be explored to enable rational combination of modes in a characteristic model and in any subsequent buckling analysis. Results for both series reveal high correlation coefficients for modes with common circumferential and odd axial wavenumber, i.e. p(<,,, &,,,) G 1 form = 3.5, 0

10

20

30

Phase angle

40

circ. wave number

b Figure 10 Variability

Table 5

analysis of imperfection (a) series A and (b) series B

Variability

et al.

Figure II shows typical results obtained for two mode amplitudes, Et2 and
a

(m = 1) for:

cylinders: M.K. Chryssanthopoulos

analysis

modal

amplitudes

results

m

y (series

I 2 3 4 5 6

0.65 0.69 0.66 0.68 0.66 0.69

A)

y (series

B)

0.72 0.56 0.71 0.56 0.71 0.57

All the statistical quantities determined enable the development of characteristic imperfection models as briefly discussed below. At the same time, as demonstrated by Arbocz”, such simple expressions enable comparisons between different manufacturing methods to be undertaken. In fact, within the current programme, a second phase is planned where specimens of similar dimensions will be manufactured by filament winding. Measurements on filament-wound cylinders, obtained independently by Hahn and Claus”, indicate that different imperfection patterns are obtained.

following expressions

&!( ) JIln

m

for m = 3,5,.

and for m = 4,6

Statistical analysis of the phase angles has shown that these variables are described by a uniform distribution in the range (--/r, 7r). Phase angles with common circumferential wavenumbers had a significant correlation, similar to the results of modal amplitudes. As a consequence, a characteristic imperfection model for these particular cylinders should include several modes of different axial wavelength for any given circumferential wavelength.

Pb) CHARACTERISTIC

Standard deviation. The dispersion of imperfection amplitudes is represented in Figure 10 in terms of the coefficient of variation (standard deviation/mean value). It is interesting that it does not exhibit significant variability and, therefore, the standard deviation can be easily linked to the mean values by the expression

~(Lnn):= “I~(Fwl?l)

(10)

where y is the coefficient of variation given in Table 5 for both cross- and angle-ply cylinders. Distributionfitting. Fitting of appropriate probability distributions was examined by goodness-of-fit tests (Kolmogorov-Smirnov) and graphical methods. The following probability laws were tested: (1) uniform; (2) normal; (3) log-normal; (4) exponential; and (5) Weibull (two parameter).

COMPOSITES

IMPERFECTION

MODELS

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:

w;(x, 0) = 2 5 &,, sin? m=ln=?

sin (no + 4h)

(11)

where tmmnand &, are now random variables described by log-normal and uniform distributions, respectively. Modes with up to seven axial half-waves (m = 7) and eight circumferential full waves (n = 8) are included since it was revealed that the amplitude of modes with even shorter wavelengths is very small. Using the relevant distribution parameters and correlation structure, simulation studies were then undertaken to evaluate the probabilistic properties of the characteristic model. In

MANUFACTURING

Volume 6 Number 2 1995

99

Manufacturing

effects for composite

cylinders: M.K. Chryssanthopoulos

et al.

3

0

6

a

b

I 0

I

/ 0.1

11

1

i 0.2

$1

5 12 1 0.3

0.b4

0.68

a

0.b4

0.08

0.12

c Figure

11

Typical

distribution

particular, the mean value, E(wg), and the standard deviation, a(wg), are estimated at any point on the cylinder surface. In addition, extreme value properties could be determined by either: (1) considering a fixed threshold value, e.g. A = 2t, and. calculating the probability of exceedance, P(wE > A); or (2) specifying a desired probability level, e.g. peer = 0.05, and calculating the corresponding imperfection value, A*, that gives P(w; > A*) =pecr. 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. A typical example of the use of characteristic models in determining buckling strength is reported in ref. 12.

OVERLAPPING

EFFECTS

As previously mentioned, the cylinders were made by hand lay-up. Successive plies were wrapped around a cylindrical mandrel, especially manufactured to suit the overall dimensions given above. In the case of cross-ply

100

COMPOSITES

MANUFACTURING

fitting

results

cylinders, each ply was made up of two pieces of fabric of width slightly larger than xR and length L. A similar procedure is used for other ply configurations but the dimensions and orientation of each piece are modified to suit the particular lay-up under consideration. Plies in the form of a parallelogram were used for angle-ply cylinders while the combination of the two models formed the quasi-isotropic lamination. To avoid excessive thickness variations, successive plies were joined together at different circumferential positions. After all the plies were glued together, uniform pressure was applied by fitting an external cylindrical sleeve around the specimen. The whole assembly was then placed into an autoclave and then the external sleeve and mandrel were carefully removed to produce the final specimen. In cross-ply cylinders, each ply gives rise to two overlappings in diametrically opposite positions. With reference to Figure 12, the thickness variations introduced by the first ply are indicated as AA while other thickness variations (indicated as BB, CC and DD) correspond to subsequent plies. In the case of series A (cross-ply) all these thickness variations are oriented along a meridian. After a detailed study of all available specimens it has

Volume 6 Number 2 1995

Manufacturing

.D 1 +bmnm . I ,i

A-

I,___

:

effects

FB ,/ +A

t =0.26mm

B/,.X / ‘, C

overlapping width

‘D overlapping position along the circumference (cross-ply series) Figure

12

Overlapping

width

and positions

along

the circumference

probability 0.16 r---I

0

200

400

600

Overlapping Figure 13 overlappings

Statistical

analysis

of

800

1000

distance the

average

distance

between

been verified that the overlappings are distinct and cause a thickness variation of 0.26mm, corresponding to the thickness of one ply, with a width of about 20mm. The distribution of these overlappings, which are two by two diametrically opposite, is important from the structural point of view because different buckling behaviour can be activated, together with eccentricity effects, in the case where there is a concentration of several overlappings in the same area. The optimum distribution of these thickness variations would be the regular one; i.e. in the case of series A and B, the eight overlappings should be at an interval of 275mm, since the total circumference is equal to 2200mm. A statistical analysis of their position in the specimens has allowed us to estimate that the average distance is 296 mm, with a standard deviation of 188 (see Figure 13). On average the overlapping distribution is regular but the standard deviation is quite large. As a consequence, in some cases the manufacturing process could cause significant eccentricity effects in the structural behaviour of the shell, especially when the cylinder is axially compressed. Further studies are dedicated to the influence of the presence of the overlappings on the buckling load and

COMPOSITES

for composite

cylinders:

M.K.

Chryssanthopoulos

et al.

mode shape. Such analyses can be performed only by means of numerical models such as finite elements and must take into account the effects of the prebuckling non-linearities. In a simplified way, these thickness variations could be thought of as longitudinal stiffeners of low rigidity and, hence, their influence is expected to be noticeable only if their spacing and stiffness force the cylinder into a buckling mode different from that corresponding to their counterpart of uniform thickness. CONCLUSIONS A statistical analysis on geometric imperfections recorded on two series of nominally identical composite cylinders has been presented. The imperfections were classified into two categories, both due to the particular manufacturing method used: out-of-roundness and change of thickness. The results relevant to the first type of imperfections revealed that several common trends exist in the imperfection patterns of the two series. This has enabled a common characteristic imperfection surface to be proposed that can be used in analytical and numerical parametric studies aimed at providing design recommendations for buckling of composite cylinders under combined loading. The analysis of the change of thickness due to overlapping layers allows their stiffening effects to be evaluated and, in the case of their non-regular distribution, significant eccentricity effects in the structural behaviour of the shell to be determined. The proposed standard procedure can be used in the qualification of manufacturing processes for composite shells with particular attention on the factors that influence the buckling behaviour. REFERENCES 1 2 3 4 5

6 7 8 9

10

11

12

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