Experimental and numerical study on the influence of imperfections on the buckling load of unstiffened CFRP shells

Composite Structures 131 (2015) 128–138

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Experimental and numerical study on the influence of imperfections on the buckling load of unstiffened CFRP shells Conny Schillo a,⇑, Dirk Röstermundt b, Dieter Krause a a b

TUHH, Institute of Product Development and Mechanical Engineering Design, Hamburg, Germany DLR, Institute of Composite Structures and Adaptive Systems, Braunschweig, Germany

a r t i c l e

i n f o

Article history: Available online 25 April 2015 Keywords: Buckling Composite shells Imperfections Model uncertainty

a b s t r a c t Due to a lack of statistical data, unstiffened cylindrical lightweight structures are designed following a knockdown factor (KDF) approach described in the NASA SP-8007 (Weingarten and Seide, 1965). This approach aims at a lower-bound, conservative design. The basis for the proposed KDFs are a number of tests with different, partially unknown boundary conditions and imperfection patterns for metallic cylinders that do not resemble the imperfection patterns achieved by state-of-the-art manufacturing methods for cylinders made of carbon fibre reinforced plastics (CFRP). Within this work, two CFRP tubes are manufactured using the filament winding method and are cut into a total of 12 cylinders from which optical measurements are taken to analyse characteristics of the corresponding imperfection patterns. 11 cylinders are tested in axial compression until buckling occurs and tests are accompanied by high speed displacement measurements, thermography and measurements of occurring load imperfections. Four different FE-models, which differ in the parameters considered, are set up for prediction of the buckling load of each cylinder. Model response showed the highest sensitivity to load imperfections. Negligence of this parameter lead to a 15% increase of the corresponding model bias while geometric imperfections made up for only 3%. Ó 2015 Elsevier Ltd. All rights reserved.

1. Introduction Lightweight unstiffened cylindrical shells under axial compression have been within the focus of research efforts since the 1930s when these structures became interesting for the aerospace industry. The large discrepancy found between theoretical prediction and tests has motivated theoreticians and experimentalists for decades [2]. The isolation and study of certain effects proved to be difficult due to a sensitive structural component and uncertainties considering boundary conditions through the test setup [3]. A large number of buckling tests of unstiffened shells was carried out at different institutes with varying materials, geometries and boundary conditions from the 30s to 60s. Despite the results that became available, comparability was not given and made a statistical analysis of uncertainties difficult. However, a clear tendency towards a reduced relation of measured buckling load to computed perfect shell buckling load with increasing radius ⇑ Corresponding author at: Hamburg University of Technology (TUHH), Institute of Product Development and Mechanical Engineering Design, 21073 Hamburg, Germany. Tel.: +49 (0) 40 42878 2176; fax: +49 (0) 40 42878 2296. E-mail address: [email protected] (C. Schillo). http://dx.doi.org/10.1016/j.compstruct.2015.04.032 0263-8223/Ó 2015 Elsevier Ltd. All rights reserved.

to thickness ratio was observed. In addition to the lack of comparability of available test data, this finding led to the development of an empirically based knockdown factor concept with respect to the theoretical buckling load [1]. The implementation was carried out by Weingarten et al. [4] who fitted a design curve to lie below the test data analysed, resulting in a conservative design criterion. These factors principally also apply for cylinders made of fibre reinforced plastics although the authors indicate that not enough data on these kind of shells was available at the time. Koiter [5] showed that initial geometric imperfections make up for the largest part of the gap between theoretical prediction of buckling load and test result. It became obvious, that the study of initial geometric imperfections is vital for the prediction of buckling loads of axially compressed cylindrical shells. A description of geometrical imperfections for CFRP-cylinders using different layup processes was published in [6]. Striking is the tendency for long-wave imperfection patterns as opposed to patterns of metallic cylinders that tend to short-wave patterns due to the manufacturing process associated [7]. However, only few studies are available that investigate the influence of manufacturing parameters on the resulting geometrical imperfections (see for example [8]).

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Many authors state the knockdown factors recommended by the NASA SP are too conservative for state of the art composite cylinders [6,7,9,10]. But for a less conservative design approach, the material and manufacturing dependent uncertainties need to be identified. To achieve this, statistical analysis is necessary. Chryssanthopoulos et al. [6] tested 30 Kevlar fabric cylinders with two different lay-ups made by hand layup. For sixteen of them, which consisted of a symmetric ±45° layup, finite element analysis was carried out considering the measured imperfections. The analysis overestimated the buckling load by 13–31%. Hillburger and Starnes [11] investigated six cylindrical CFRP shells with varying layup. Three of the cylinders failed due to buckling, deviation to predicted buckling failure of a corresponding perfect shell varied between 7% and 17%. Bisagni [12] tested 16 composite cylinders with 6 different layups. For one of these cylinders a finite element analysis using two different methods and including the measured imperfections was carried out. The analytical result overestimated the observed testing value by 15–20%. Within the European research project ‘‘Design and Validation of Imperfection-Tolerant Laminated Shells’’ (DEVILS), nine cylindrical shells were tested under axial compression at the German Aerospace Centre (Deutsches Zentrum für Luft- und Raumfahrt, DLR) in Braunschweig [13]. Seven different, asymmetric layups were used and buckling loads computed using the analytical DLR tool Baccus with different boundary conditions as well as a finite element code. Deviations to the measured buckling loads varied between 8% and 30% in the case of Baccus4 and between 16% and 13% for the finite element implementation [14]. Hühne [10] computed buckling loads for the same set of cylinders using Abaqus. His results deviate from the buckling loads published in [14] by between 4% and 30%. In order to study the influence of load imperfection, four different cylinders were tested in axial compression but with shims inserted at the loaded edges at varying circumferential position. Depending on shim size and position, the buckling load reduces by a maximum of 9% and 27% for a 0.4 mm thick shim, which implies a high sensitivity towards load imperfections. Similar was later found by Kriegesmann [15], who assumed a load imperfection consisting of a tilted load introduction, thus introducing an inhomogeneous stress state. He computed the inclination angle for different positions of the cylinder and reached good agreement with test cylinders for an inclination angle of 0.009° which effectively decreased the buckling load by about 17% as compared to no assumed load imperfection. Ten nominally identical CFRP cylinders have been tested at DLR by Degenhardt et al. [9]. Within an ESA funded project the cylinders were tested and showed a coefficient of variance of buckling loads of 5.5%. Broggi [16] computed buckling loads for each of

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these cylinders using non-linear FEM and included geometric imperfections as well as thickness variations. Both models overestimated the buckling load by 15–43%. Despite advanced techniques in measuring the geometrical imperfections and their consideration in finite element models, a considerable discrepancy between test and model results remain. Among the parameters that could explain those discrepancies are the spatial variability of material and structural parameters, the usage of simplified boundary conditions and negligence of load imperfections. In order to develop a new, less conservative design approach, it is necessary to identify the influence of these uncertainties as well as the accuracy of the models used for prediction. Only then can the inherent safety of a design approach be assessed [17]. Within the present study, geometrical imperfection measurements of a set of 12 CFRP cylinders are taken as a basis for statistical analysis of imperfection amplitudes to be expected with the specified manufacturing method. Loading imperfections are measured and implemented in the finite element analysis. Model uncertainties are quantified with respect to loading and geometric imperfections as well as level of detail of the as-built layup. 2. Manufacturing, characterisation and test setup 2.1. Manufacturing of unstiffened cylinders Two cylinders are manufactured at DLR in Braunschweig using the filament winding process. The material used is a 1=4 ’’ slit Hexcel AS7/8552 CFRP-prepreg with a nominal thickness of 135 lm at 60% fibre volume content. A mandrel consisting of a 1.82 m steel pipe and end domes is grinded and used to manufacture a cylinder of corresponding length that is cut to 6 cylinders of 255 mm length each. The diameter ratio of the mandrel to the inner tube is 2 for realisation of geodesic winding angles of the ±30° layers. The mandrel is measured using the GOM-ATOS system at DLR. ATOS uses two cameras to take images of the object and uses photogrammetric methods to compute the position of the measured points in space. Depicted in Fig. 1 is the deviation from its radius of 113.905 mm. The area that is used to wind the cylinders shows a tolerance of ±0.03 mm. The outer edges show slightly higher deviations of up to 0.04 mm. After cleaning the mandrel, applying release agent and referencing the mandrel in the filament winding machine, the layers corresponding to a [90°, ±30°, ±30°, 90°]-layup are wound with a continuously pre-stressed slit-tape (F = 50 N).

Fig. 1. Mandrel (top), deviation from mean mandrel diameter with histogram (bottom).

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Fig. 2. ±30° crossply.

The nearly gap free and overlap free fibre placement quality achieved in the process is shown exemplary in Fig. 2. The filament winding process is finished with the winding of a peel-ply tape (50 mm width) with an overlap of 50%. Finally the mandrel is placed in an autoclave and the laminate is cured under 7 bar and 180 °C in a cycle according to [18]. After curing, the mandrel is fixed in the filament winding machine and the peel ply is removed. The positions of the edges of the 6 cylinders are marked with an initial saw-cut. After cutting off the domes of the laminate and loosening of the laminate from the mandrel, the laminate can be shifted along the rotational axis of the mandrel. A circular saw is mounted on the head of the filament winding machine and positioned to the end of the cylindrical range of the mandrel. The previously marked cutting edges are aligned to the saw blade and fixed on the mandrel. The specimens are cut by moving the rotating saw blade towards the mandrel. 2.2. Layup identification

Fig. 3. Layup patterns of cylinders Z1 (top) and Z2 (bottom).

Layer 1

2

3 4 voids

5

6

The layup process programmed by the manufacturer resulted in a non-homogeneous layup-sequence that varied in longitudinal as well as circumferential direction for both cylinders manufactured. The two resulting winding-ups are depicted in Fig. 3 with individual cylinders indicated through vertical dotted lines and numbered according to their position on the mandrel.

resinlayer

2.3. Manufacturing and testing of coupon specimens For the preparation of coupon specimens, a symmetric [90, ±30°]S is wound with process parameters identical to the cylinder manufacturing method. The laminate is then cut from the mandrel and peel plies applied to both sides. The sheets are placed between two stainless steel plates and a vacuum setup is prepared. The setup is cured in an autoclave process similar to the cylinder manufacturing process [18]. After cooling, 11 specimens are cut from the plate with dimensions 25 mm  250 mm. All specimens are tested according to DIN EN ISO 527 in a tension testing rig. From that sample size, 10 specimens broke within the free length and have been used for the determination of the Young’s modulus between strain levels 0.05–0.25. The resulting mean Young’s modulus is 46.8 GPa with a standard deviation of 2.3 GPa. 2.4. Material analysis An exemplary polished micrograph section (Fig. 4) shows the six layers from inner contour of the cylinder to outer contour (left to right). This section was cut out of a cylinder at an angle of 20°

resinrichinterface

250 µm

Fig. 4. Polished micrograph section.

with respect to the cylinder axis, resulting in the outer 90° layers being cut under 70° and appearing as very long ellipses. The boundaries of all layers are clearly visible, with a particular separation between layer one and two. The interface here is more pronounced and resin rich than for other interfaces. This zone indicates that during the autoclave process, not all surplus resin from the prepreg bled out. Some voids are also visible as black spots in layer two and four. A distinct contour of the inner 90° layer (layer one) of the cylinder can be seen on the left hand side. This is where the prepreg tape was wound on the mandrel. On the outside 90° layer a wave-like pattern with growing and decreasing thickness of a resin layer is visible. This resin layer strongly varies in thickness approximately from 15 to 100 lm.

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This epoxy-layer results from the application of the peel-ply and the processing under heat and pressure. Epoxy gets effectively pressed out of the layers (which is intended to increase the fibre volume fraction). However, when removing the peel-ply, an epoxy-layer remains on the surface. Due to the existence of the outer resin layer, the optical measurements carried out to determine the thickness of the cylinders were not used. Instead, 9 microscopic specimens are taken from cross sections of cylinders Z1.1, Z1.2, Z2.1 and Z2.4. 20 thickness measures are taken from each sample to evaluate the overall thickness of the load carrying 6 prepreg layers as well as the thickness of the outer resin layer. From these, a mean thickness of 800 lm is evaluated with coefficient of variance (cov) of 0.51%. The mean resin layer thickness is 33 lm (cov = 57%). Fig. 6. Cylinder Z2.1 prior (left) and after mounting (right).

2.5. Clamping of cylinders A schematic drawing of the clamping arrangement is depicted in Fig. 5. It consists of a circular ground plate on which the inner ring and outer ring are attached. Both rings are divided into two parts to allow for easy decomposition. The inner contour of the outer ring is inclined at an angle of 60° with respect to the top surface and is clear off the cylinder. The outer diameter of the inner ring corresponds to the nominal diameter of the cylinder in the top section. Here, the cylinder fits tight with tightness varying due to manufacturing tolerances of the cylinders. From there the contour descends into a small recess. Before the cylinder is placed into the mounting, the recess area is filled with epoxy. The cylinder is then slowly lowered into the matrix-bed so surplus epoxy can escape through the gap between outer ring and cylinder. The matrix is then cured for several hours at around 30 °C. After cooling, the procedure is repeated for the other side. Due to matrix shrinkage, small amounts of epoxy are repeatedly injected into the gap until the epoxy levels with the top of the outer ring. 2.6. Geometric imperfections All cylinders are measured prior to testing using the ATOS-system. The cylinders are measured after they are glued into the mountings in order to account for the influence of the mounting on the shape during testing. Set two of the cylinders has additionally been measured prior to its mounted state (Ref. 2.6.1). Contour measurements provided by ATOS are analysed using the software GOM Inspect V5R7.2. From here, the point data and the radius of best-fit cylinders are exported. Deviations of the measured points to the best fit cylinders are computed using Matlab. 2.6.1. Comparison of imperfections prior and after mounting Cylinders of set two have been measured prior and after mounting. The short-waved stripes visible in circumferential direction result from peel-ply fibre leftovers at overlapping lines that could not be removed. As is clearly visible on Fig. 6 (left), cylinders show a clear tendency towards forming two-half waves in axial direction and four in circumferential direction with maximum deflections of around

0.7 mm. When mounted, the inner steel rings force the cylinder into an ovalization mode (Figs. 6 (right) and 7 (right)). The amplitude of the imperfection is significantly reduced with a maximum of less than 0.2 mm. 2.6.2. Characterisation of geometric imperfections The winding-up of the cylinder surface with its shell-wall mid-surface imperfection on the z-axis are plotted for cylinders Z1.1 and Z1.2 in Fig. 7. The tendency for long wave modes in axial and circumferential direction is clearly visible on all plots. Furthermore, there is an influence of the mounting sequence visible: at the lower edge with z = 0 mm, which has been put on the steel mounting and glued first, the mid-surface imperfections are significantly lower than at the upper edge. This is due to the fact that the cylinder could deform more freely as compared to the second mounting step when the lower edge is already constrained. These patterns are hence indicating a certain pre-stress state. 2.6.3. Power spectral density (PSD) The characteristics of the shell-wall mid-surface imperfection can be considered as signal information with varying amplitude over the cylinders circumference or length. A common way to transform this information from a length-domain to a frequency domain (with unit 1/mm) is to use a Fourier transformation and decompose the imperfection pattern into its sine or cosine contents. In the context of geometric imperfections, the frequency axis is commonly scaled to represent the number of half-waves fitting into the considered length of the cylinder (compare e.g. Refs. [19–21]). Since the Fourier transform assumes next to stationary also ergodic signals, a more general way to analyse a random signal is to consider its power spectral density (PSD), which has been done by e.g. [22–24]. A truncated Fourier transform F(x) is used, which decomposes a zero-mean random field f(x) of finite length L into its sine and cosine contributions as a function of frequency.

FðxÞ ¼

1 2p

Z

L

f ðxÞ  elxx dx

ð1Þ

0

The PSD is then described through [23]

h i Z E jFðxÞj2 ¼

L

Sðx; xÞdx

ð2Þ

0

In this case, the well-known Welch method [25] is used to estimate the PSD. It consists of the following steps:

Fig. 5. Clamping arrangement, (1) outer ring, (2) inner ring.

1. Dividing the data sequence into segments. 2. Multiplying a segment by a window function (here: turkey window). 3. Taking the Fourier transform of the product.

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Fig. 7. Shell-wall mid-surface imperfection for Z1.1 (left) and Z1.2 (right).

4. Multiplying procedure 3 by its conjugate to obtain the spectral density of the segment. 5. Repeat procedures 2 through 4 for each segment so that the average of these periodogram estimates produce the power spectral density estimate. The resulting plots are depicted in Fig. 8 for cylinders Z1.1 (left) and Z1.2 (right). They illustrate the strong narrow-bandedness of the information analysed, which is represented solely through very low frequencies with their power increasing towards the top of the cylinder. For easier comparison of stochastic properties with other published cylinders, the following representation is chosen (Fig. 9). From the PSD the y-axis of the plots is scaled down to represent the amplitude of the mid-surface imperfection. On the x-axis, instead of frequencies, the more intuitive unit of number of half-waves fitting into the considered direction of the cylinder is depicted. For each half wave number occurring, the median (red bar), standard deviation (blue box) and 5–95% percentile (black bars) are depicted. Appearances of outliers are marked with red crosses. In contrast to the dominance of the ovalization mode (4 half-waves) of the unconstrained cylinder, it is now the mode consisting of two half-waves that show the largest amplitude contribution, with the mean lying below 5% of the cylinder thickness. From the PSD the root mean square (rms) of the amplitude can directly be calculated as the square root of the area under the spectrum vs. frequency curve corresponding to certain height of the cylinders. So for each z-position on the cylinder the rms-value is computed and depicted for cylinders of set 1 and set 2 in Fig. 10. Although some information about stochastic properties is lost, this representation offers a very direct way to compare an

important characteristic regarding the amplitude of the geometric imperfection. However, the strong deviation of rms values at the bottom of the cylinders at z = 0 mm as compared to the top of the free length at z = 215 mm indicate that for these cylinders, taking only the mean rms over the whole cylinder surface may be misleading in terms of a characteristic of geometric quality achieved. 2.7. Testing of specimen 11 CFRP-cylinders were tested on the Hexapod testing facility at the Hamburg University of Technology (Fig. 11). Six hydraulic cylinders move a 2.5t ring allowing for load introduction using all translational and rotational degrees of freedom (for further details Ref. to [26]). The maximum applicable load in axial direction is 500 kN. All cylinders are tested in uniaxial compression only. Any load imperfections are measured using a custom made 6 dof-load cell with a maximum loading capacity in axial direction of 200 kN and 150 kN in transverse directions. The load is applied displacement controlled and load speed is reduced after 50 s from 1.6 mm to 0.6 mm per minute. 2.8. Load introduction For the buckling tests the load is introduced through an additional uniaxial load cell that is placed centrically between the hexapod-ring and the cylinder attachment (Fig. 12). A ball socket connection is chosen for the load transfer between load cell and cylinder attachment. The upper face of the load cell lays plane against the hexapod attachment plate.

Fig. 8. Power spectrum density of Z1.1 (left) and Z1.2 (right).

C. Schillo et al. / Composite Structures 131 (2015) 128–138

Fig. 9. Statistical information modes underlying geometrical imperfections, Z1.1 (left), Z1.2 (right).

Fig. 10. Root mean square values over cylinder height for set 1 (left) and set 2 (right).

Fig. 11. Hexapod test rig.

Fig. 12. Test setup, (1) 6 DOF load cell, (2) load introduction.

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Table 1 Buckling loads.

Buckling load [kN]

1.1

1.2

1.3

1.5

1.6

2.1

2.2

2.3

2.4

2.5

2.6

l

Cov [%]

60.2

57.4

62.1

61.9

60.4

58.8

61.7

60.5

56.2

57.7

55.4

59.3

3.96

of magnitude as the ten nominal identical cylinders tested by Degenhardt [9], which showed a cov of 5.5%. 3.2. Load imperfections The shear loads acting on the cylinder at the moment of its respective highest load capacity are depicted in Fig. 13. A comparison of the resulting shear vectors leads to a mean shear load of 3.1 kN with a standard deviation of 0.4 kN. A reason for the occurring load imperfection might be that drill holes of the attachment rings are centred with some tolerance, so that the position of the ball socket connection also has some tolerance. Since less screws have been used for the attachment of set 2 of the cylinders, this could explain the larger scatter of load imperfections. 3.3. High-speed camera

3. Test results

Buckling test measurements were complemented by recording with a CamRecord 5000 high speed camera at a frame rate of 8000 f/s. The videos reveal that almost all buckling failures initiate with a single buckle in close vicinity to the lower mounting. The buckles then quickly propagate until the post-buckling pattern is visible at mid-height of the cylinders (Fig. 14).

3.1. Buckling loads

3.4. Thermography

Maximum loads that the structures sustained are read from the measured load displacement curves and are summarised in Table 1. The cov is 3.96% is quite low and within a similar order

Pulse Phase Thermography is an NDT technique in which a thermal pulse is applied to a specimen and the thermal response is measured by an infrared camera. Any subsurface defects will

Fig. 13. Measured load imperfections at buckling.

The lower attachment of the cylinder is screwed to the 6 dof-load cell.

1

2

3 second buckle first buckle

4

5 propagation of buckling

6 postbuckling pattern

Fig. 14. Onset of buckling, observed via high speed camera (Z2.2).

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disturb the heat flow and will appear as local hot or cold areas on the resulting temperature images. Images from eight different angles of cylinders of set 2 have been taken prior and after buckling tests. For set 1 thermography was applied after the final buckling test only. The equidistant horizontal lines visible on all images result from peel-ply leftovers. Typical appearances of the thermography analyses are depicted in Fig. 15. Almost no defects are observed in cylinders of set one (Z1.1, Fig. 15, left). In contrast, thermography of set 2 shows multiple white lines oriented at +30° (Z2.1, Fig. 15, right). After discussion with the manufacturer it became clear that these lines indicate longitudinal folding of slit-tape segments. Thus, there are local thickness increases and epoxy-rich areas adjacent to it. Pictures taken after buckling indicate that for cylinders 2.1 and 2.4–2.6, the observed white spots, indicating delaminations, occur in direct proximity or between these flaws, as depicted in Fig. 16 for Z2.5. The appearance of these delaminations between flaws correlates with lower buckling loads below the mean. Thermography analysis of the cylinders that achieved buckling loads above the mean showed no initial flaws and in case of two cylinders (Z1.1, Z2.3) no delaminations after buckling at all. Regarding the position of delaminations, comparison with the buckling pattern recorded by Aramis and depicted in Fig. 16 for

cylinder 2.5 show the delaminations to occur at points of reflexion of radial displacement curves, the sharp kinks between minimum and maximum deflection. Cylinder Z1.2, the only one out of the first set with buckling load below 60 kN showed no pronounced fibre lines that might indicate flaws. However, a very large delamination area is detected oriented

Fig. 17. Z1.2 delaminated area.

Fig. 15. thermographic comparison of Z1.1 and Z2.1 (at position 90°).

prior to buckling

after buckling

pos. 180° pos. 270° pos. 270°

prior to buckling

pos. 270°

after buckling

pos. 270° pos. 0°

pos. 0°

Fig. 16. Thermography and postbuckling pattern of Z2.5.

pos. 0°

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in 30°-direction along the fibre direction, thus indicating an area of low cohesion (Fig. 17). 4. Numerical analysis 4.1. Model setup Four different models are set up to compute the resistance of the measured cylinders against buckling and to compare results against the buckling loads from tests. For all four models, the finite element code ABAQUS/Standard with a four node shell element with reduced integration (S4R) is used. A convergence study war carried out to determine the number of elements to be used. 214 elements in axial direction and 722 in circumferential direction were used for discretization. For realisation of the boundary conditions, bottom node line and top node line have been coupled rigidly to two master nodes, respectively. The master node controlling the bottom nodes is constrained in all six degrees of freedom. The master node for the top nodes is not constrained to allow for load imperfections. However, through the rigid connections to the master node, independent local rotations of the top nodes are effectively restricted. Load is applied through the master node. A nonlinear, load-controlled quasi-static analysis is carried out using the well-known Newton–Raphson method. The computation

Table 2 Nominal and mean measured data.

a

Radius [mm]

Thickness [mm]

Layup

Free Length [mm]

Total length [mm]

EL [GPa]

Nominal

115.00

0.81

215

255

58.7a

Measured mean

114.86

0.80

[90/+30/ 30]S As built; Ref. 2.1.2

215

255

46.8

From classical lamination and theory and [27,18].

Table 3 Parameters considered within different models. Model number

Geometric imperfection

Load imperfection

Layup as built

1 2 3 4

x x x –

x x – x

– x – –

fails to converge at the buckling point where no further load increase is possible. Cross check calculations were carried out using displacement controlled procedure to compute into the postbuckling regime and compare results, but the postbuckling regime itself was not further investigated. Dimensions and material properties used for the FE-models are the mean measured values as summarised in Table 2, line two. The measured mean radius is only used for one model 4, all others consider the measured geometric imperfections. The four models differ in the number of parameters considered (Table 3). The uncertainties related to each type of model will be analysed in 4.3. 4.2. Implementation of imperfections A plot of the resulting shell-wall mid-surface imperfection is depicted in Fig. 7 with the issue of peel ply leftovers visible. To filter these artificial thickness increases for the FE-model, the measured imperfection data has been Fourier transformed using a double cosine representation. Following a convergence study, a representation with 8 half-waves in axial and 24 in circumferential direction was chosen. The resulting imperfection pattern of the unwound cylinder is plotted in Fig. 18. Using these functions, a mesh for implementation in the finite element program is created. Load imperfections considered in FE-models 1, 2 and 4 are applied according to their proportion with respect to the measured axial buckling load. 4.3. Results and discussion The computational results for all 11 cylinders tested are depicted in Fig. 19 as computed buckling load over tested buckling load. In case of agreement, the results would lie on the blue diagonal line representing a linear resistance model with zero error associated to it. Model 3 (no consideration of load imperfections) computes very similar results for all cylinders, indicating that the considered geometric imperfections have only minor influence on the scatter of the results. Model 4, which neglects geometric imperfections but incorporates load imperfections, shows a high degree of scatter with certain offset from the reference diagonal. Models 1 and 2 scatter around the diagonal. According to EN 1990 [28], annex D, the model uncertainty of each type of model can now be further quantified by analysing the mean value correction factor, denoted b, and the variance of the error term, Vd. b can be interpreted as the least squares best-fit to the slope of the resistance model, given by

b¼

X

re rt

.X

r2t

Fig. 18. Approximated imperfection patterns of Z1.1 (left) and Z1.2 (right).

ð3Þ

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Within this work, the mean shear force appearing at buckling that is measured, causing additional moments, is about 5% of the mean buckling load but making up for 15% of model bias. To make statements about prediction models for unstiffened cylinders, it is hence very important to consider this parameter. 5. Conclusion

Fig. 19. Model uncertainty.

The relation between resistance model r, b, the theoretical resistance rt and the error term d is r ¼ br t d. The results are summarised in Table 4. The variation of the error terms does not vary largely between models. With values between 3.88% for model 3 and 4.81% for model 1, the variance associated with the model uncertainty is in the range of the measured buckling load scatter which showed a coefficient of variance of 3.96% (Ref. Table 1). The analysis shows that for the cylinders investigated, the highest loss in resistance prediction accuracy occurs for model 3 and is due to the negligence of load imperfections occurring during the cylinder tests. Here, the mean value correction for the resistance is 15% lower than for model 1. In comparison, the influence of geometric imperfections, illustrated by the difference of results between model 4 and 1, is small with only 3%. The consideration of the layup changes the model bias by only 1% and reduces the variance of the error term slightly. Applying this method to the test results of Degenhardt et al. [9] and computations made by Broggi [16] it is found that the introduction of thickness imperfections leads only to minor reduction of the model bias. Hühne [10] reached the maximum b-factor when including geometric imperfections but also high variance of the error term, which might indicate this model could also be improved by consideration of additional parameters. Comparison of these results with non-linear computations for the perfect geometry show that the influence of the geometric imperfections measured has a much larger influence than for the cylinders investigated within this study. This is expected due to the lower radius to thickness ratio as well as the lower imperfections associated with the cylinders. EN 1990 [28] offers a transparent way of quantifying and comparing these influences for different studies, whereat the results always have to be assessed in the context of the geometries and manufacturing methods used.

The objective of this work was to test and analyse a statistical meaningful number of unstiffened cylinders in terms of their material and structural properties. Thus, a contribution is made to the assessment of manufacturing quality of state of the art winding technology and the significance of resulting geometric imperfections for a certain radius to thickness relation. 12 nominal identical cylinders were built at DLR and optically measured to determine shell mid-surface imperfections. The power spectral density shows an extreme concentration of the energy of the analysed imperfection data at long-waved modes with mean root mean squared values in the range of 6–12% of the laminate thickness. However, the order of applying the cylinder fixtures have a significant influence with a clear trend of higher values at the second fixture applied. 11 cylinders are tested on the Hexapod-test rig at TU Hamburg under axial compression. A 6-dof load cell allows for measuring of additionally appearing load imperfections, namely shear loads and moments. Although only part of the cylinders could be observed, high-speed observations confirm the observations of Esslinger et al. [29] that buckling starts with a single buckle followed by a quick propagation of buckling pattern. The cylinders did not buckle elastically, but delaminations could be found using thermography measures at some of the corners of the buckles after collapse. Thermography also reveals the occurrence of manufacturing flaws in terms of individual folded tape stripes adjacent to which elliptical delaminations were found oriented at 90° to the fibre angle. These flaws were apparent in all cylinders of set 2 but not in set 1. On average, buckling loads of set 2 are 2 kN below buckling loads of set 1 and show a slightly increased coefficient of variance (4.2% as compared to 3.1% for set 1). Predictions for the resistance of all cylinders tested are made using four different FE-models with varying number of parameters considered. While the consideration of geometrical imperfections only increased the model prediction agreement by 3%, the largest contribution comes from the introduction of load imperfections which made up for 15% of the model bias. Despite the fact that no material inherent flaws and the simplification of rigidly constrained cylinder edges are considered, the prediction agreement for all models considering load imperfections was very high. However, further research should investigate the sensitivity of the models with respect to the thickness variation that has only been included deterministically in these models, as well the appropriate representation of the cylinder stiffness from coupon tests. Furthermore, the r/t value of the cylinders investigated is low compared to the cylinders referenced in the introduction. A general assessment of the sensitivity of unstiffened cylinders towards geometric and load imperfections should include a wider range of r/t values and different manufacturing methods.

Table 4 Model uncertainty characterisation. Model type

b Vd [%]

Schillo 11 nominal identical cylinders

Degenhardt [9], Broggi [16] 10 nominal identical cylinders

Hühne [10] 8 cylinders, 5 different layups

1

2

3

4

Geom. imp

Geom. imp., thickness imp.

Geom. imp.

Perfect geom.

0.99 4.81

0.98 4.53

0.84 3.88

0.96 4.22

0.75 7.72

0.76 7.59

1.00 10.3

0.88 7.89

138

C. Schillo et al. / Composite Structures 131 (2015) 128–138

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