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Buckling strength of GFRP under-water vehicles
Composites: Part B 32 (2001) 89–101 www.elsevier.com/locate/compositesb
Buckling strength of GFRP under-water vehicles V. Carvelli, N. Panzeri, C. Poggi* Department of Structural Engineering, Polytechnic of Milan, Piazza Leonardo Da Vinci 32, 20133 Milan, Italy Received 19 April 2000; revised 14 July 2000; accepted 28 October 2000
Abstract The paper is concerned with the structural response of a composite shell structure intended as a model of an under-water vehicle for service in sea environment. The main objective of the research is the prediction of the collapse pressure using both analytical expressions and linear or non-linear numerical analysis and the following comparison with the experimental pressure obtained in off-shore tests. The structure is composed of three basic parts with regular geometry: a cylindrical part (with the following geometrical properties: R=t 30:5; L=R 2 being the internal radius 305 mm, the length 610 mm and the thickness 10 mm) and two conical and spherical end-closures with the same thickness. The cylindrical shell was made up of 7 plies of E-glass woven roving with polyester resin. Various structural analyses were conducted before performing the experiment in the sea to verify the reliability of the analytical and numerical tools. Firstly the entire model was analysed to predict the nature of the collapse (material failure or elastic buckling) and it was stated that the collapse was due to elastic buckling of the cylindrical part. Consequently, the attention was focused on this component and approximation formulae for the evaluation of the linear buckling pressure of isotropic and composite cylindrical shells were used together with finite element models. Afterward the study was enlarged to consider the effects of the recorded geometric imperfections into a non-linear buckling analysis. The collapse pressures were compared to the design values derived from the available recommendations and to the experimental result obtained in an off-shore test (1.3 MPa). 䉷 2001 Elsevier Science Ltd. All rights reserved. Keywords: B. Buckling; B. Strength; A. Glass fibres; C. Finite element analysis (FEA); Woven roving
1. Introduction Composite materials have a great potentiality of application in structures subjected primarily to compressive loads such as under-water vehicles where different problems can arise depending on the employed material: weight and magnetism for steel, cost for titanium, corrosion for aluminium alloys [1,2]. Composite materials avoid these drawbacks and have other attractive aspects like the relatively high compressive strength, the good adaptability in fabricating thick composite shells, the low weight and the corrosion resistance. On the other hand the material characterisation and failure evaluation of thick composite materials in compression are still an item of research [3]. Cylindrical, spherical and conical shells, which are typical geometries in submarine structures, are ideal shapes in responding to external pressure since, if the wall thickness to diameter ratios are small, they show a nearly uniform distribution of the strains through the thickness. When compressive stresses generated by the external * Corresponding author. Tel.: ⫹390-2-2399-4362; fax: ⫹390-2-23994369. E-mail address: [email protected] (C. Poggi).
pressure reach elevated levels buckling phenomena become dominant. The research on buckling of shells has been carried out over many years with particular attention to isotropic materials [4,5]. It is well known that the linear buckling load, that can be easily determined both analytically and numerically for some basic geometries, represents only an indication of the real buckling resistance of a shell. This is more evident in the case of axially compressed cylinders where other important factors such as the geometric imperfections and the effects of the boundary conditions must be accounted for. Some design recommendations propose that an estimate of the shell buckling strength can be obtained applying knock-down factors to the linear buckling load [6]. These are usually based on available experimental data and therefore take into consideration various levels of accuracy in the manufacturing of the tested specimens. This philosophy, originally proposed in the field of isotropic shells, is still in use in several codes of practice, but the development of efficient finite element programs and the improvement of the experimental techniques have allowed the validation of numerical tools that can be used in further parametric studies. Because of the costs of shell buckling experiments, the effects of the large number of variables to be considered can be analysed only by a combination of
1359-8368/01/$ - see front matter 䉷 2001 Elsevier Science Ltd. All rights reserved. PII: S1359-836 8(00)00063-9
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Fig. 1. The model of the GFRP under-water vehicle.
experimental and numerical approaches [7]. This implies a preliminary calibration and validation of the numerical model that is possible only by detailed measurements of the fundamental variables during the tests. In particular the mapping of geometric imperfections should be conducted in a regular mesh of points over the shell surface to allow the analytical treatment
of the recorded data by means, for example, of double Fourier analysis [8]. Several authors have carried out both analytical and experimental investigations on the buckling analysis of composite structures but most of the work is dedicated to very thin shells typical for applications in the aerospace industry and the materials used are often carbon–epoxy
d
t
t
H
rc ra
rc ra (a)
(b)
rc ra L
t
(c) Fig. 2. Geometry of the specimen components: (a) hemisphere; (b) cone; (c) cylinder.
V. Carvelli et al. / Composites: Part B 32 (2001) 89–101 Table 1 Specimen geometric characteristics (see Fig. 2) ra 370 mm rc 305 mm d 70 mm H 1122 mm L 610 mm t 10 mm
systems. Only recently experimental and numerical activities have been undertaken on medium thick cylinders subjected to external pressure [9,10] and concentric or eccentric compression [11]. The latter study, based on specimens produced by the marine industry, was oriented towards obtaining experimental data on the interaction between buckling and material failure. The objective of this paper is to present some aspects related to the determination of the buckling strength of medium thick composite shells subjected to external pressure. An investigation based on experimental, analytical and numerical results is illustrated with reference to a specific model of an under-water vehicle. Numerical and analytical studies were firstly performed to determine the nature of a possible collapse considering both material failure and elastic shell buckling. When it was stated that the cylindrical central part was the most critical component subject to possible elastic buckling, additional calculations were performed using refined FE models of this component including the recorded geometric imperfections. The numerical predictions were compared to the result of an off-shore experiment. 2. Specimen geometry and material details The model of the underwater vehicle was composed of a central cylindrical part and two end-closures connected by bolted joints (Fig. 1). The specimen was made from woven
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roving laminates by a manufacturer specialised in marine structures using specific mandrels and the hand lay-up technique. The dimensions of the components are reported in Fig. 2 and Table 1. The three components have thick endflanges to limit the radial displacement along the edges and to avoid local phenomena at the boundaries. The connection was realised by means of 10 mm thick steel rings to fix the flanges with steel bolts (see the sketch in Fig. 3). The material used, denominated ‘Rovimat’, consists of woven glass-fibre roving with polyester resin matrix. Fig. 4 shows the two sides of the fabric. It should be noted that the number of fibres in the weft direction (x2) is doubled with respect to the warp direction (x1) (Fig. 4a). The chopped fibre mat (Fig. 4b) improves the bonding of the plies. The characteristics of the fibres and the matrix are reported in Table 2. The mechanical properties of the laminate were accurately verified after a characterisation process on a series of two-ply flat specimens of type [0⬚]2. Tensile, compression and ILSS tests were carried out on three series of specimens cut in the directions 0, 45 and 90⬚ from a flat laminated panel [12] according to the following standards ASTM: D303995, D695-91 and D2344-84. The nominal and experimental properties of the fabric Rovimat are reported in Table 3 where the indexes ‘1’ and ‘2’ refer to the warp and weft directions respectively; Et, Ec, s t and s c are the tensile and compression elastic moduli and strengths while s s is the shear strength. It should be noted that the average thickness of the specimens used for the characterisation (3.1 mm) was higher than the nominal one (2.8 mm, being 1.4 mm the plythickness). As a consequence the fibre volume fraction is smaller and the mechanical properties results smaller than the nominal ones. It is also known that the mechanical properties of the laminate vary depending on the number of laminae and thickness of the specimens and this should be considered in applying the results to very thick laminates [3]. The cylindrical shell was made up of 7 plies with the main direction coincident with the hoop direction. The conical and hemispherical shells, due to their geometry, were built laying up plies in irregular order. For these components the numerical analyses have been carried out using the elastic global properties of a quasi-isotropic laminate made up of 7 plies [0, 45, ⫺45, 90, ⫺45, 45, 0] and performing a homogenisation procedure using the classical lamination theory [4]. The elastic characteristics obtained are listed in Table 4, where x refers to 0⬚ direction and y refers to 90⬚ direction of the laminate. The strength values listed in Table 4 were obtained by specimens with [0, 90] lamination [12].
3. Thickness variations and geometric imperfections
Fig. 3. Detail of the link between the components of the specimen.
The geometric imperfections that are present in cylindrical shells made with the hand lay-up method, are not only limited to the out-of-roundness but may include significant
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x2
x1 (a)
(b) Fig. 4. Two sides of the woven fabric: (a) roving side; (b) mat side.
thickness variations particularly close to the overlapping of the layers. The measurement of the actual thickness is usually carried out using specially designed equipment in a regular grid of points. A detailed measurement of the thickness variation was performed on nominally identical specimens produced by the same manufacturer but with Table 2 Nominal characteristics for fibres and matrix
Elastic modulus Density Tensile strength
Fibre
Matrix
69 GPa 2.5 g/m 2 2.4 GPa
4.0 GPa 1.2 g/m 2 73 MPa
different thickness [11]. The results confirmed that the coefficient of variation of the thickness in the normal zones (not including an overlap) varied between 5 and 10% while the measured total thickness can be up to 20% higher than the nominal value. This increase is associated with an excess of the resin during the manufacturing and does not affect significantly the material properties. Since the main objective of the numerical analyses was not the simulation of the experiment but the prediction of the collapse pressure, the nominal thickness of the shell was considered in all the calculations and the overlapping were neglected. In order to evaluate the influence of the geometric imperfections on the collapse pressure of the under-water model, detailed measurements of the out-of-roundness were undertaken on the cylindrical and conical parts separately. The hemisphere was not analysed since it was not expected to be a critical part for the collapse. Automatic laser scanning techniques are currently used to obtain the imperfection surface of shell specimens [11] but this is possible only when the specimens are positioned into specific rigs and the experiments are performed in a testing laboratory. Since the experimental activity was carried out in the sea, as requested by the producer of the specimen, the measurement of the geometric imperfections was undertaken using a simple device designed to record the imperfections of the external surface (Fig. 5). This is composed of a vertical bar fixed on a basement and a LVDT transducer running along the bar to record imperfections of the meridians. The specimens were positioned on a rotating plate to cover a selected number of meridian lines. The imperfections of the external surface were measured on a mesh of points as reported in Table 5. A comparison of the measured external circumference with the perfect shape of the specimen at a level of 875 mm from the basis of the conical component is depicted in Fig. 6. The data recorded in a scanning process (‘raw’ imperfections) may be non-representative of the real geometric imperfections and must be post-processed to consider possible misalignments of the specimen axis with the acquisition frame [8]. In fact an imperfect positioning of the specimen centre on the rotating plate or the misalignment between the axis of the cylinder and the vertical bar used as support of the transducer can cause systematic errors in the measurements. Therefore, the recorded data of the cylindrical part were manipulated according to a best-fit procedure [8]. The external imperfection surface of the cylindrical component, as obtained after the best-fit procedure, is drawn in Fig. 7. The recorded imperfections are in the range ⫺3.5 to 5.2 mm, where positive values indicate the outward radial imperfection. The radius of the reference best-fit cylinder is 308.4 mm. After the above described post-processing the imperfection data can be elaborated to facilitate the comparison of initial imperfections generated by different manufacturing techniques. This can be obtained performing a double
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Table 3 Nominal and experimental mechanical properties of the laminate Nominal
Experimental
Et1 16,100 MPa Ec1 17,000 MPa G12 3480 MPa s t1 205 MPa s c1 215 MPa s s 80 MPa
Et2 21,500 MPa Ec2 18,100 MPa n 12 0.14 s t2 238 MPa s c2 235 MPa
Et1 13,100 MPa Ec1 13,300 MPa G12 3500 MPa s t1 178 MPa s c1 217 MPa
4. Buckling and collapse strength analyses of the vehicle model
Fourier analysis as follows: w 0 x; q
mpx j mn sin sin nq ⫹ fmn L n0
mT X nT X m1
Et2 14,700 MPa Ec2 14,400 MPa n 12 0.26 s t2 200 MPa s c2 250 MPa
1
where 0 ⱕ x ⱕ L (L being the cylinder axial length) and 0 ⱕ u ⱕ 2p; w0 is the initial imperfection function on the cylinder surface, m is the number of axial half waves, n is the number of circumferential waves and j mn and f mn are the initial imperfection amplitude and phase angle associated with the mode (m, n). It should be noted that Eq. (1) represents a half sine expansion in the axial direction. This assumption imposes artificial zero imperfection values at the two edges of the cylinder but the error introduced is limited to the end regions of the shell. This method allows us to describe each imperfection surface by means of two sets of coefficients j mn, f mn and therefore to have an analytical representation of the imperfection surface. Furthermore, the method facilitates comparative studies on the maximum amplitude and different distribution of the geometric imperfections. It should be remembered that the imperfections data were recorded on the external surface and were influenced by the thickness of the paint layer and by the roughness of the surface that do not represent real imperfections of the shell middle surface. These local imperfections influence the Fourier’s coefficients corresponding to modes with a high number of hoop waves and can be disregarded through a selection of the most significant modes (filtering process). In our case, the imperfection surfaces were reconstructed using only Fourier’s modes with a reduced number of waves. Fig. 8 shows the imperfection surface of the cylinder obtained using only the first five modes. Four circumferential waves are dominant in this surface as in the real imperfection surface (Fig. 7). Table 4 Mechanical properties of the conical and spherical components (a) Homogenisation procedure
(b) Experimental investigations [12]
Ex 13,953 MPa Ey 14,093 MPa Gxy 5445 MPa n xy 0.41
s tx 184 MPa s cx 255 MPa s ty 165 MPa s cy 242 MPa s s 80 MPa
The first step in the determination of the strength properties of the under-water vehicle included a numerical analysis on the entire structure to detect if the collapse was due to material failure or to elastic buckling of a shell part. A finite element model was prepared to perform strength and buckling analyses using a commercial FE package. The finite element mesh, consisting of approximately 6800 shell elements, is represented in Fig. 9. It should be noted that also the steel joints between the components of the vehicle were modelled to simulate their stiffening effects on the boundary conditions. Since the gradient of the underwater pressure between the top and the bottom of the specimen was negligible, the model was supposed to be subjected to constant pressure on the external surface. The nominal values of the mechanical properties (Table 3) were used for the cylindrical shell of the model while the experimental and numerical values (Table 4) were adopted to simulate the lamination of the conical and spherical parts. Two separate analyses were performed. The first was aimed at determining the component of the structure more prone to buckling and the relevant buckling pressure under the hypothesis of elastic material behaviour. In the second analysis, the pressure corresponding to the material failure was established using Tsai–Hill criterion, namely [13]: 2 s 12 s 22 s 12 s s ⫹ ⫹ ⫺ 122 1 2 2 2 F1 F2 F12 F1
where ( F1
s t1 if s 1 ⬎ 0 s c1 if s 1 ⬍ 0
( F2
s t2 if s 2 ⬎ 0 s c2 if s 2 ⬍ 0
2
F12 s s 3
The material strength values (s t1 ; s c1 ; s t2 ; s c2 ; s s ;) in Eq. (3), are listed in Table 3 and Table 4. The second numerical analysis, through scaling of the stress field, provided a limit pressure corresponding to a global material failure equal to 4.38 MPa. The numerical failure occurred in the points of the cylindrical component near the end-flanges. The first numerical analysis gives a buckling pressure of 1.8 MPa. The buckling mode shape is depicted in Fig. 9. It is evident how the cylindrical part is buckled with five circumferential waves and one half-wave in the axial direction. The results
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Fig. 5. Experimental apparatus for imperfection recording.
tion, to show how classical shell theories can become inadequate [10]. In particular it was shown how, for certain ranges of length, orthotropic shells under external pressure could be imperfection-sensitive. On the basis of the above considerations it is evident that a non-linear buckling analysis including the effects of geometric imperfections and thickness variations, is necessary in the design of marine shells like underwater vehicles. Nevertheless, the evaluation of the linear buckling load is a fundamental step to have a first estimate of the strength of the shell and may constitute the basis for a design procedure. An estimate of the linear buckling pressure for composite cylinders may be achieved both using finite element models or through simplified theoretical expressions that, for preliminary predictions, could be even referred to isotropic cylinders [4]. Analytical solutions are easily available for basic geometries and isotropic materials while are limited for composite shells in terms of laminate construction and boundary conditions. When theoretical solutions are adopted, the imperfection sensitivity of the shells is accounted for on the basis of correction factors based on experimental results. On the contrary, the finite element models allow to carry out fully non-linear analyses including the effects of the geometric imperfections and of the real boundary conditions.
clearly show that the model is more prone to elastic buckling in the cylindrical component. In fact, the ratio between the buckling pressure and the collapse pressure is equal to 0.41.
5. Linear buckling pressure of the cylindrical shell On the basis of these results it was decided to perform more refined analyses on the cylindrical part of the model to evaluate the efficiency of the numerical methods and of the available design recommendations in predicting both the linear buckling pressure and the effects of geometric imperfections. It is known that the critical load of shell structures determined using classical linear theories is often non-adequate and the influence of geometric imperfections and other factors as the boundary conditions must be considered [4]. The non-linear buckling behaviour can be studied using asymptotic theories and Koiter’s general post-buckling theory, according to which the imperfection sensitivity is defined by means of a coefficient representative of the post-buckling behaviour. This theory has been recently applied also to moderately thick orthotropic cylinders, with geometries similar to the specimens under consideraTable 5 Grid dimensions for imperfection measurements Axial direction
Cylinder Cone
Circumferential direction
No. of acquisitions
Distance (mm)
No. of acquisitions
Distance (mm)
6 9
120 120
16 15
120 33–128
V. Carvelli et al. / Composites: Part B 32 (2001) 89–101
angle 0˚ 315˚
measured exact
45˚
270˚
90˚ 0
10
20
30
radius [cm]
225˚
135˚ 180˚
Fig. 6. Measured and perfect external circumference at level 875 mm from the basis of the conical component.
The particular lamination and the mechanical properties of the specimen under consideration allowed us to produce a preliminary prediction of the buckling pressure using analytical expressions for isotropic cylinders. In fact the elastic stiffness and the strength of each lamina in the two orthogonal directions are very similar (see Table 3) and the particular lamination of the conical and spherical parts makes the shells quasi-isotropic. Various codes in the field of steel structures provide approximate formulas for buckling design of cylinders, cones and spheres. The well-known recommendations for buckling of shells and the Eurocode part related to shells are summarised in Appendix A. Recently a large project, including experimental and analytical activities, has been dedicated to buckling behaviour of composite shells to produce a first draft of guidelines, which are summarised in Appendix A [14]. As pointed
95
out in these guidelines, in the case of composite cylinders subject to external pressure the detrimental effects due to the presence of geometrical imperfections are less important than for axial compression. In the latter case many imperfection modes with different axial and circumferential waves should be included in a numerical non-linear analysis while in the case of externally pressurised shells, the out-ofroundness and the relevant modes play the main role. In Ref. [14] it is stated that the maximum recorded imperfection should be limited by the relation: Dmax ⫺ Dmin ⱕ 0:02 Dnom
4
where Dmax, Dmin and Dnom are the maximum, the minimum and the nominal diameter of the cylinder, respectively. This factor was equal 0.014 for the experimental model here considered. In Table 6 the analytical results obtained for an externally pressurised cylinder with the geometry of Fig. 2 and Table 1 are presented. The cylinder is subjected to external pressure acting also on the two end-closures. The design values of Table 6 are obtained applying the knock-down factors suggested by the codes. The characteristic strength of the material has been set equal to 66% of the ultimate strength in compression in the circumferential direction. The discrepancies, in terms of external buckling pressure, between Eurocode 3 [15] and the guidelines for laminated shells [14] is negligible (see Table 6). The ECCS recommendations [6] provide a higher value of the buckling pressure because no reduction of the limit pressure for the presence of axial stresses is considered in the case of hydrostatic pressure.
6. Non-linear buckling of the cylindrical component Since the amplitude of the measured geometric
Fig. 7. Imperfections of the external surface of the cylindrical component after the best-fit procedure.
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Fig. 8. External surface of the cylindrical component using the first five Fourier modes.
imperfections were significant and the analyses outlined above showed that the cylindrical part of shell presents the lowest buckling pressure, the non-linear buckling analysis was limited to this part of the structure. The effects of the pressure acting on the conical and hemispherical components were considered applying an equivalent axial compression at the two edges of the cylinder. The FE mesh was modified and reduced to 900 shell elements and 3700 nodes (Fig. 10b) to fit the recorded imperfections. The imperfections were modelled in two different ways. In the first non-linear analysis (NA1) the imperfections obtained after the best-fit procedure (Fig. 7) were assumed while in the second (NA2) the imperfections resulting from the first five Fourier modes were considered (Fig. 8) assuming the initial imperfection amplitudes j mn and the phase angles f mn listed in Table 7.
The results of the non-linear buckling analysis (NA1) performed assuming the best-fit imperfections, are reported in Fig. 10a in terms of pressure and shortening of the cylinder. The maximum pressure is 1.38 MPa and the relevant shape of the imperfect cylinder is drawn in Fig. 10b. The results of the non-linear analysis (NA2) adopting an imperfect surface obtained from a selection of Fourier modes are reported in Fig. 11. In this case the limit pressure is 1.15 MPa.
7. Experimental investigations An experimental investigation was carried out on the specimen in Fig. 1. The experience was conducted off-shore in the Mediterranean Sea about three miles from the coast in front of La Spezia (Italy). The sea depth of nearly 200 m was
Fig. 9. Buckling mode of the specimen (linear elastic analysis).
V. Carvelli et al. / Composites: Part B 32 (2001) 89–101
The procedure was based on the following steps:
Table 6 Buckling pressure (pb) and design pressure (pbd) from different recommendations and codes
ECCS [6] Eurocode 3 [15] Devils [14]
pb (MPa)
pbd (MPa)
1.70 1.34 1.26
0.85 0.76 0.61
chosen in accordance with the numerical prediction of the collapse pressure (Table 6). The specimen was set up into a supporting steel frame using flexible cables. A camcorder, a lamp and two microphones were fixed to the frame to monitor the experiment (Fig. 12). A ballast, sufficient for the sinking of the apparatus, was positioned at the bottom of the frame. The sinking of the specimen was controlled at a speed of 0.2 m/s to ensure a regular recording of the noise and of the images. The camcorder and the microphones were useful to monitor the specimen during the descent in the sea and allowed to detect with reasonable accuracy the collapse depth. The specimen collapsed suddenly and producing a considerable noise, at the depth of 130 m which correspond to a pressure of 1.3 MPa. Fig. 13 shows two details of the specimen during the experiment. In Fig. 13a it reached nearly 120 m where some considerable deformations occurred in the cylindrical component. Fig. 13b captured the collapse instant. The equipment and the staff for the submersion were provided by the manufacturer of the specimens in the framework of a joint research project. 8. Conclusions The main objective of the present work was to present a procedure to predict the collapse pressure of composite shells. A particular application for underwater vehicles was examined and the results of off-shore experiments were presented.
(i) Recording of geometric imperfections of the shell components. (ii) Numerical analyses of the entire specimen to predict the nature of the collapse (material failure or elastic buckling) and to localise the component where it occurs. (iii) Non-linear buckling analyses limited to the weakest component including the recorded imperfections processed according to a best-fit procedure and a Fourier analysis. (iv) Comparison of the numerical limit pressures with design approximation formulae (based on linear buckling and reduction factors) and with the experimental result (when available). The following conclusions can be summarised from the above study: • The effects of geometric imperfections could be significant when the maximum amplitude is close to the acceptable limit of Eq. (4). In these cases, a non-linear analysis including the effects of geometric imperfections is essential. • Numerical analyses showed that the weakest part of the shell structure under consideration was the cylindrical component while the conical shells and the hemisphere presented higher buckling pressure. • The numerical non-linear analyses performed assuming the geometric imperfection distribution of the best-fit procedure provide an overestimation of the experimental limit pressure (6.1%). On the other hand the non-linear analysis assuming the Fourier distribution of the imperfections underestimates the experimental limit pressure (⫺11%). These results may be regarded as satisfactory (see Table 8) considering the simplifications of the numerical models and the approximations of the in-situ recorded imperfections. • The design buckling pressures obtained with two codes for isotropic shells and with the recent guidelines for
1.50
Pressure [MPa]
1.25 1.00 0.75 0.50 0.25 0.00 0
0.1 0.2 0.3 Shortening [mm]
(a)
97
0.4
(b)
Fig. 10. NA1: (a) pressure vs. axial shortening; and (b) collapse mode of the cylindrical component.
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V. Carvelli et al. / Composites: Part B 32 (2001) 89–101
Table 7 The initial imperfection amplitudes j and the phase angles f associated to the first five Fourier modes (m 1; n 1; …; 5) employed in the nonlinear analysis NA2 m
n
j
f
1 1 1 1 1
1 2 3 4 5
0.753 0.047 1.242 2.663 0.853
⫺1.571 0.300 2.172 2.084 ⫺2.454
Table 8 Numerical predictions of the buckling pressure compared to the experimental result pb (MPa) Experiment Numerical predictions
1.30 1.38 1.15 1.80
Imperfect model NA1 Imperfect model NA2 Perfect model
laminated cylinders (see Table 6) are very conservative but may be considered reasonable as design approximated values. Acknowledgements The work presented in this paper was partially sponsored by the European Commission under a Brite-Euram project (BE-7550: Design and Validation of Imperfection Tolerant Shell Structures). The authors wish to acknowledge the support of INTERMARINE SpA (Sarzana-La Spezia, Italy) for providing the specimens and for the assistance in performing the off-shore experiments.
It is assumed that the cylinder is subjected to constant external pressure and the buckling resistance is determined on the basis of the compressive stress in the circumferential direction r A1 sw p t where r and t are the radius and thickness of the cylinder, while p is the external pressure. Since the pressure acts on the end closures, the shell is also subjected to a longitudinal compressive stress s x s w =2: For short or medium length shells subjected to external pressure the stiffening effects of the end supports must be also considered. A.1. ECCS recommendations for steel shells [6] The ECCS Recommendations provide a procedure valid for cylinders with a maximum circular imperfection within 0.5% of the radius measured from the true centre. The critical pressure of a simply supported elastic cylindrical shell may be calculated through the expression: t pcr E bmin r
A2
where E is the elastic modulus and b min is a function of L/r and t/r. Its value can be read directly from a diagram or evaluated from an expression that is a function of the number of circumferential waves. The ECCS recommendations do not impose a reduction of the limit pressure due to the presence of the longitudinal stresses in the case of hydrostatic pressure. The ECCS reduction factor to account for the presence of geometric imperfections is equal to 0.5. A.2. Eurocode 3 — supplementary rules for strength and stability of shell structures [15]
Appendix A The following codes and guidelines provide approximation formulas for the buckling pressure of cylinders.
This code allows us to determine the critical circumferential buckling stress of shells with different boundary conditions.
1.25
Pressure [MPa]
1.00 0.75 0.50 0.25 0.00 0
0.15 0.3 0.45 Shortening [mm]
(a)
0.6
(b)
Fig. 11. NA2: (a) pressure vs. axial shortening; and (b) collapse mode of the cylindrical component.
V. Carvelli et al. / Composites: Part B 32 (2001) 89–101
99
Fig. 12. The specimen in the steel frame during the experiment in the sea.
The length of the shell segment is characterised in terms of the parameter: p rt v L
A3
For medium-length cylinders with 20 ⱕ
1 r ⱕ 1:63 Cq v t
A4
the critical circumferential buckling stress should be
obtained from:
s qRc 0:92ECq v
t r
A5
The factor Cq depends on the boundary conditions as detailed in Ref. [15]. Since the pressure acts also on the end closures the cylinder is subjected to an axial compression s x s w =2: The interaction between axial and circumferential stresses has to be accounted for. For medium length cylinders with 1:7 ⱕ
1 r ⱕ 0:5 v t
A6
100
V. Carvelli et al. / Composites: Part B 32 (2001) 89–101
Fig. 13. The specimen during the experiment: (a) ⫺120 m; (b) collapse ⫺130 m.
steel yield stress)
the critical meridian buckling stress shall be taken as:
s xRc 0:605E
t r
A7
Although buckling is not a purely stress-initiated failure phenomena, the buckling limit state is defined by limiting the design values of membrane stress. As suggested in Ref. [15], the interaction is set by: kq kt s xSc kx s t ⫹ qSc ⫹ Sc ⱕ 1 A8 s xRc s qRc tRc with kx ku 1:25 and kt 2: In order to obtain the maximum design load, the critical stress has to be reduced by a reduction factor. This depends on the shell quality class. The parameter used to determine the quality class for any cylinder is ur
Dmax ⫺ Dmin Dnom
A9
In our case the cylinder is of class C and the elastic imperfection reduction factor a u is equal to 0.5, while a x has to be obtained by mean of a formula reported in Ref. [15]. For circumferential and meridian buckling the squash limit slenderness is assumed as l q0 0:4 and l x0 0:2; respectively. If the conditions (being fyk the characteristic
r ⱕ t
s E 23fyk
r E ⱕ t 25fyk
A10
are not satisfied the following investigation must be carried out. The stability reduction factors x u and x x:
a xq q2 lq
l x ⫺ l x0 xx 1 ⫺ 0:6 l p ⫺ l x0
A11
have to be evaluated assuming: s s p f fyk yk l q ; l qp 2:5aq ; l x ; s qRc s xRc p l qp 2:5ax The characteristic buckling stresses are:
s qRk xq fyk
s xRk xx fyk
A12
The interaction between axial and circumferential actions is accounted for in Eq. (A8).
V. Carvelli et al. / Composites: Part B 32 (2001) 89–101
A.3. Guidelines for analysis of laminated shells against buckling collapse [14] Closed form solutions and recommendations for carrying out numerical analyses for the calculation of the critical buckling load of composite shells have been recently suggested in Ref. [14]. In the case of orthotropic shells subject to external pressure and axial compression, the buckling pressure may be evaluated by means of the following expressions where the components of the stiffness matrix of the laminate are introduced. The critical axial compressive load is: C11 C12 C13 C21 C22 C23 2 L C31 C32 C33 A13 Nxcr C11 C12 mp C 21 C22 where Cij i; j 1; 2; 3 are functions of the stiffness matrix components and of the number of mode waves in the axial and circumferential directions. The expressions may be found in Ref. [14]. The critical external pressure is given by: C11 C12 C13 C21 C22 C23 r C31 C32 C33 A14 pcr C11 C12 n2 C C22 21 The interaction between axial compression and external pressure is Nx;c p ⫹ 1 Nxcr pcr
A15
where Nx;c P= 2pr (P being the applied axial compressive load) and p the applied external pressure. Buckling occurs when the value on the left hand side of Eq. (A15) is greater then or equal to one. The use of semi-empirical knock-down factor, currently
101
adopted in design of metal shells, is proposed also for composite shells. The design pressure presented in Table 6 is obtained by means of the minimum between the knock-down factors suggested for axial compression and external pressure.
References [1] Dvorak GJ, Prochazka P, Srinivas MV. Design and fabrication of submerged cylindrical laminates — I. Int J Solids Struct 1999;36:3917–43. [2] Srinivas MV, Dvorak GJ, Prochazka P. Design and fabrication of submerged cylindrical laminates — II. Effects of fiber pre-stress. Int J Solids Struct 1999;36:3945–76. [3] Zhou G, Davies GAO. Characterisation of thick glass woven roving/ polyester laminates: tension, compression and shear. Composites 1995;26:579–86. [4] Vinson JR. The behaviour of shells composed of isotropic and composite materials. Netherlands: Kluwer Academic Publishers, 1993. [5] Samuelson LA, Eggwertz S. Shell stability handbook. London: Elsevier, 1992. [6] ECCS. Buckling of Steel Shells European Recommendations. Fourth edition, 1988. [7] Chryssanthopoulos M, Poggi C. Stochastic imperfection modelling in shell buckling studies. Thin Walled Struct 1995;23:179–200. [8] Chryssanthopoulos M, Giavotto V, Poggi C. Characterization of manufacturing effects for buckling sensitive composite cylinders. Compos Manufact 1995;6:93–101. [9] Kardomateas GA, Philobox MS. Buckling of thick orthotropic cylindrical shells under combined external pressure and axial compression. AIAA J 1995;33:1946–53. [10] Kardomateas GA. Koiter-based solution for the initial post-buckling behaviour of moderately thick orthotropic and shear deformable cylindrical shells under external pressure. J Appl Mech 1997;64:885–96. [11] Chryssanthopoulos MK, Elghazouli AY, Esong IE. Compression tests on anti-symmetric two-ply GFRP cylinders. Compos Part B 1999;30:335–50. [12] Poggi C. Characterisation of woven GFRP laminates. Report WP04.DR/PM. Politecnico di Milano, Italy, 1997. [13] Tsai SW, Hahn HT. Introduction to composite materials. Westport: Technomic, 1989. [14] Guidelines for analysis and design of laminated shells against buckling collapse. Document prepared by the members of the Brite Euram project “Devils”, July 1999. [15] Eurocode 3. Design of steel structures, part 1–6: supplementary rules for the strength and stability of shell structures. ENV 1993-1-6, June 1998.
Buckling strength of GFRP under-water vehicles V. Carvelli, N. Panzeri, C. Poggi* Department of Structural Engineering, Polytechnic of Milan, Piazza Leonardo Da Vinci 32, 20133 Milan, Italy Received 19 April 2000; revised 14 July 2000; accepted 28 October 2000
Abstract The paper is concerned with the structural response of a composite shell structure intended as a model of an under-water vehicle for service in sea environment. The main objective of the research is the prediction of the collapse pressure using both analytical expressions and linear or non-linear numerical analysis and the following comparison with the experimental pressure obtained in off-shore tests. The structure is composed of three basic parts with regular geometry: a cylindrical part (with the following geometrical properties: R=t 30:5; L=R 2 being the internal radius 305 mm, the length 610 mm and the thickness 10 mm) and two conical and spherical end-closures with the same thickness. The cylindrical shell was made up of 7 plies of E-glass woven roving with polyester resin. Various structural analyses were conducted before performing the experiment in the sea to verify the reliability of the analytical and numerical tools. Firstly the entire model was analysed to predict the nature of the collapse (material failure or elastic buckling) and it was stated that the collapse was due to elastic buckling of the cylindrical part. Consequently, the attention was focused on this component and approximation formulae for the evaluation of the linear buckling pressure of isotropic and composite cylindrical shells were used together with finite element models. Afterward the study was enlarged to consider the effects of the recorded geometric imperfections into a non-linear buckling analysis. The collapse pressures were compared to the design values derived from the available recommendations and to the experimental result obtained in an off-shore test (1.3 MPa). 䉷 2001 Elsevier Science Ltd. All rights reserved. Keywords: B. Buckling; B. Strength; A. Glass fibres; C. Finite element analysis (FEA); Woven roving
1. Introduction Composite materials have a great potentiality of application in structures subjected primarily to compressive loads such as under-water vehicles where different problems can arise depending on the employed material: weight and magnetism for steel, cost for titanium, corrosion for aluminium alloys [1,2]. Composite materials avoid these drawbacks and have other attractive aspects like the relatively high compressive strength, the good adaptability in fabricating thick composite shells, the low weight and the corrosion resistance. On the other hand the material characterisation and failure evaluation of thick composite materials in compression are still an item of research [3]. Cylindrical, spherical and conical shells, which are typical geometries in submarine structures, are ideal shapes in responding to external pressure since, if the wall thickness to diameter ratios are small, they show a nearly uniform distribution of the strains through the thickness. When compressive stresses generated by the external * Corresponding author. Tel.: ⫹390-2-2399-4362; fax: ⫹390-2-23994369. E-mail address: [email protected] (C. Poggi).
pressure reach elevated levels buckling phenomena become dominant. The research on buckling of shells has been carried out over many years with particular attention to isotropic materials [4,5]. It is well known that the linear buckling load, that can be easily determined both analytically and numerically for some basic geometries, represents only an indication of the real buckling resistance of a shell. This is more evident in the case of axially compressed cylinders where other important factors such as the geometric imperfections and the effects of the boundary conditions must be accounted for. Some design recommendations propose that an estimate of the shell buckling strength can be obtained applying knock-down factors to the linear buckling load [6]. These are usually based on available experimental data and therefore take into consideration various levels of accuracy in the manufacturing of the tested specimens. This philosophy, originally proposed in the field of isotropic shells, is still in use in several codes of practice, but the development of efficient finite element programs and the improvement of the experimental techniques have allowed the validation of numerical tools that can be used in further parametric studies. Because of the costs of shell buckling experiments, the effects of the large number of variables to be considered can be analysed only by a combination of
1359-8368/01/$ - see front matter 䉷 2001 Elsevier Science Ltd. All rights reserved. PII: S1359-836 8(00)00063-9
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Fig. 1. The model of the GFRP under-water vehicle.
experimental and numerical approaches [7]. This implies a preliminary calibration and validation of the numerical model that is possible only by detailed measurements of the fundamental variables during the tests. In particular the mapping of geometric imperfections should be conducted in a regular mesh of points over the shell surface to allow the analytical treatment
of the recorded data by means, for example, of double Fourier analysis [8]. Several authors have carried out both analytical and experimental investigations on the buckling analysis of composite structures but most of the work is dedicated to very thin shells typical for applications in the aerospace industry and the materials used are often carbon–epoxy
d
t
t
H
rc ra
rc ra (a)
(b)
rc ra L
t
(c) Fig. 2. Geometry of the specimen components: (a) hemisphere; (b) cone; (c) cylinder.
V. Carvelli et al. / Composites: Part B 32 (2001) 89–101 Table 1 Specimen geometric characteristics (see Fig. 2) ra 370 mm rc 305 mm d 70 mm H 1122 mm L 610 mm t 10 mm
systems. Only recently experimental and numerical activities have been undertaken on medium thick cylinders subjected to external pressure [9,10] and concentric or eccentric compression [11]. The latter study, based on specimens produced by the marine industry, was oriented towards obtaining experimental data on the interaction between buckling and material failure. The objective of this paper is to present some aspects related to the determination of the buckling strength of medium thick composite shells subjected to external pressure. An investigation based on experimental, analytical and numerical results is illustrated with reference to a specific model of an under-water vehicle. Numerical and analytical studies were firstly performed to determine the nature of a possible collapse considering both material failure and elastic shell buckling. When it was stated that the cylindrical central part was the most critical component subject to possible elastic buckling, additional calculations were performed using refined FE models of this component including the recorded geometric imperfections. The numerical predictions were compared to the result of an off-shore experiment. 2. Specimen geometry and material details The model of the underwater vehicle was composed of a central cylindrical part and two end-closures connected by bolted joints (Fig. 1). The specimen was made from woven
91
roving laminates by a manufacturer specialised in marine structures using specific mandrels and the hand lay-up technique. The dimensions of the components are reported in Fig. 2 and Table 1. The three components have thick endflanges to limit the radial displacement along the edges and to avoid local phenomena at the boundaries. The connection was realised by means of 10 mm thick steel rings to fix the flanges with steel bolts (see the sketch in Fig. 3). The material used, denominated ‘Rovimat’, consists of woven glass-fibre roving with polyester resin matrix. Fig. 4 shows the two sides of the fabric. It should be noted that the number of fibres in the weft direction (x2) is doubled with respect to the warp direction (x1) (Fig. 4a). The chopped fibre mat (Fig. 4b) improves the bonding of the plies. The characteristics of the fibres and the matrix are reported in Table 2. The mechanical properties of the laminate were accurately verified after a characterisation process on a series of two-ply flat specimens of type [0⬚]2. Tensile, compression and ILSS tests were carried out on three series of specimens cut in the directions 0, 45 and 90⬚ from a flat laminated panel [12] according to the following standards ASTM: D303995, D695-91 and D2344-84. The nominal and experimental properties of the fabric Rovimat are reported in Table 3 where the indexes ‘1’ and ‘2’ refer to the warp and weft directions respectively; Et, Ec, s t and s c are the tensile and compression elastic moduli and strengths while s s is the shear strength. It should be noted that the average thickness of the specimens used for the characterisation (3.1 mm) was higher than the nominal one (2.8 mm, being 1.4 mm the plythickness). As a consequence the fibre volume fraction is smaller and the mechanical properties results smaller than the nominal ones. It is also known that the mechanical properties of the laminate vary depending on the number of laminae and thickness of the specimens and this should be considered in applying the results to very thick laminates [3]. The cylindrical shell was made up of 7 plies with the main direction coincident with the hoop direction. The conical and hemispherical shells, due to their geometry, were built laying up plies in irregular order. For these components the numerical analyses have been carried out using the elastic global properties of a quasi-isotropic laminate made up of 7 plies [0, 45, ⫺45, 90, ⫺45, 45, 0] and performing a homogenisation procedure using the classical lamination theory [4]. The elastic characteristics obtained are listed in Table 4, where x refers to 0⬚ direction and y refers to 90⬚ direction of the laminate. The strength values listed in Table 4 were obtained by specimens with [0, 90] lamination [12].
3. Thickness variations and geometric imperfections
Fig. 3. Detail of the link between the components of the specimen.
The geometric imperfections that are present in cylindrical shells made with the hand lay-up method, are not only limited to the out-of-roundness but may include significant
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x2
x1 (a)
(b) Fig. 4. Two sides of the woven fabric: (a) roving side; (b) mat side.
thickness variations particularly close to the overlapping of the layers. The measurement of the actual thickness is usually carried out using specially designed equipment in a regular grid of points. A detailed measurement of the thickness variation was performed on nominally identical specimens produced by the same manufacturer but with Table 2 Nominal characteristics for fibres and matrix
Elastic modulus Density Tensile strength
Fibre
Matrix
69 GPa 2.5 g/m 2 2.4 GPa
4.0 GPa 1.2 g/m 2 73 MPa
different thickness [11]. The results confirmed that the coefficient of variation of the thickness in the normal zones (not including an overlap) varied between 5 and 10% while the measured total thickness can be up to 20% higher than the nominal value. This increase is associated with an excess of the resin during the manufacturing and does not affect significantly the material properties. Since the main objective of the numerical analyses was not the simulation of the experiment but the prediction of the collapse pressure, the nominal thickness of the shell was considered in all the calculations and the overlapping were neglected. In order to evaluate the influence of the geometric imperfections on the collapse pressure of the under-water model, detailed measurements of the out-of-roundness were undertaken on the cylindrical and conical parts separately. The hemisphere was not analysed since it was not expected to be a critical part for the collapse. Automatic laser scanning techniques are currently used to obtain the imperfection surface of shell specimens [11] but this is possible only when the specimens are positioned into specific rigs and the experiments are performed in a testing laboratory. Since the experimental activity was carried out in the sea, as requested by the producer of the specimen, the measurement of the geometric imperfections was undertaken using a simple device designed to record the imperfections of the external surface (Fig. 5). This is composed of a vertical bar fixed on a basement and a LVDT transducer running along the bar to record imperfections of the meridians. The specimens were positioned on a rotating plate to cover a selected number of meridian lines. The imperfections of the external surface were measured on a mesh of points as reported in Table 5. A comparison of the measured external circumference with the perfect shape of the specimen at a level of 875 mm from the basis of the conical component is depicted in Fig. 6. The data recorded in a scanning process (‘raw’ imperfections) may be non-representative of the real geometric imperfections and must be post-processed to consider possible misalignments of the specimen axis with the acquisition frame [8]. In fact an imperfect positioning of the specimen centre on the rotating plate or the misalignment between the axis of the cylinder and the vertical bar used as support of the transducer can cause systematic errors in the measurements. Therefore, the recorded data of the cylindrical part were manipulated according to a best-fit procedure [8]. The external imperfection surface of the cylindrical component, as obtained after the best-fit procedure, is drawn in Fig. 7. The recorded imperfections are in the range ⫺3.5 to 5.2 mm, where positive values indicate the outward radial imperfection. The radius of the reference best-fit cylinder is 308.4 mm. After the above described post-processing the imperfection data can be elaborated to facilitate the comparison of initial imperfections generated by different manufacturing techniques. This can be obtained performing a double
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Table 3 Nominal and experimental mechanical properties of the laminate Nominal
Experimental
Et1 16,100 MPa Ec1 17,000 MPa G12 3480 MPa s t1 205 MPa s c1 215 MPa s s 80 MPa
Et2 21,500 MPa Ec2 18,100 MPa n 12 0.14 s t2 238 MPa s c2 235 MPa
Et1 13,100 MPa Ec1 13,300 MPa G12 3500 MPa s t1 178 MPa s c1 217 MPa
4. Buckling and collapse strength analyses of the vehicle model
Fourier analysis as follows: w 0 x; q
mpx j mn sin sin nq ⫹ fmn L n0
mT X nT X m1
Et2 14,700 MPa Ec2 14,400 MPa n 12 0.26 s t2 200 MPa s c2 250 MPa
1
where 0 ⱕ x ⱕ L (L being the cylinder axial length) and 0 ⱕ u ⱕ 2p; w0 is the initial imperfection function on the cylinder surface, m is the number of axial half waves, n is the number of circumferential waves and j mn and f mn are the initial imperfection amplitude and phase angle associated with the mode (m, n). It should be noted that Eq. (1) represents a half sine expansion in the axial direction. This assumption imposes artificial zero imperfection values at the two edges of the cylinder but the error introduced is limited to the end regions of the shell. This method allows us to describe each imperfection surface by means of two sets of coefficients j mn, f mn and therefore to have an analytical representation of the imperfection surface. Furthermore, the method facilitates comparative studies on the maximum amplitude and different distribution of the geometric imperfections. It should be remembered that the imperfections data were recorded on the external surface and were influenced by the thickness of the paint layer and by the roughness of the surface that do not represent real imperfections of the shell middle surface. These local imperfections influence the Fourier’s coefficients corresponding to modes with a high number of hoop waves and can be disregarded through a selection of the most significant modes (filtering process). In our case, the imperfection surfaces were reconstructed using only Fourier’s modes with a reduced number of waves. Fig. 8 shows the imperfection surface of the cylinder obtained using only the first five modes. Four circumferential waves are dominant in this surface as in the real imperfection surface (Fig. 7). Table 4 Mechanical properties of the conical and spherical components (a) Homogenisation procedure
(b) Experimental investigations [12]
Ex 13,953 MPa Ey 14,093 MPa Gxy 5445 MPa n xy 0.41
s tx 184 MPa s cx 255 MPa s ty 165 MPa s cy 242 MPa s s 80 MPa
The first step in the determination of the strength properties of the under-water vehicle included a numerical analysis on the entire structure to detect if the collapse was due to material failure or to elastic buckling of a shell part. A finite element model was prepared to perform strength and buckling analyses using a commercial FE package. The finite element mesh, consisting of approximately 6800 shell elements, is represented in Fig. 9. It should be noted that also the steel joints between the components of the vehicle were modelled to simulate their stiffening effects on the boundary conditions. Since the gradient of the underwater pressure between the top and the bottom of the specimen was negligible, the model was supposed to be subjected to constant pressure on the external surface. The nominal values of the mechanical properties (Table 3) were used for the cylindrical shell of the model while the experimental and numerical values (Table 4) were adopted to simulate the lamination of the conical and spherical parts. Two separate analyses were performed. The first was aimed at determining the component of the structure more prone to buckling and the relevant buckling pressure under the hypothesis of elastic material behaviour. In the second analysis, the pressure corresponding to the material failure was established using Tsai–Hill criterion, namely [13]: 2 s 12 s 22 s 12 s s ⫹ ⫹ ⫺ 122 1 2 2 2 F1 F2 F12 F1
where ( F1
s t1 if s 1 ⬎ 0 s c1 if s 1 ⬍ 0
( F2
s t2 if s 2 ⬎ 0 s c2 if s 2 ⬍ 0
2
F12 s s 3
The material strength values (s t1 ; s c1 ; s t2 ; s c2 ; s s ;) in Eq. (3), are listed in Table 3 and Table 4. The second numerical analysis, through scaling of the stress field, provided a limit pressure corresponding to a global material failure equal to 4.38 MPa. The numerical failure occurred in the points of the cylindrical component near the end-flanges. The first numerical analysis gives a buckling pressure of 1.8 MPa. The buckling mode shape is depicted in Fig. 9. It is evident how the cylindrical part is buckled with five circumferential waves and one half-wave in the axial direction. The results
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V. Carvelli et al. / Composites: Part B 32 (2001) 89–101
Fig. 5. Experimental apparatus for imperfection recording.
tion, to show how classical shell theories can become inadequate [10]. In particular it was shown how, for certain ranges of length, orthotropic shells under external pressure could be imperfection-sensitive. On the basis of the above considerations it is evident that a non-linear buckling analysis including the effects of geometric imperfections and thickness variations, is necessary in the design of marine shells like underwater vehicles. Nevertheless, the evaluation of the linear buckling load is a fundamental step to have a first estimate of the strength of the shell and may constitute the basis for a design procedure. An estimate of the linear buckling pressure for composite cylinders may be achieved both using finite element models or through simplified theoretical expressions that, for preliminary predictions, could be even referred to isotropic cylinders [4]. Analytical solutions are easily available for basic geometries and isotropic materials while are limited for composite shells in terms of laminate construction and boundary conditions. When theoretical solutions are adopted, the imperfection sensitivity of the shells is accounted for on the basis of correction factors based on experimental results. On the contrary, the finite element models allow to carry out fully non-linear analyses including the effects of the geometric imperfections and of the real boundary conditions.
clearly show that the model is more prone to elastic buckling in the cylindrical component. In fact, the ratio between the buckling pressure and the collapse pressure is equal to 0.41.
5. Linear buckling pressure of the cylindrical shell On the basis of these results it was decided to perform more refined analyses on the cylindrical part of the model to evaluate the efficiency of the numerical methods and of the available design recommendations in predicting both the linear buckling pressure and the effects of geometric imperfections. It is known that the critical load of shell structures determined using classical linear theories is often non-adequate and the influence of geometric imperfections and other factors as the boundary conditions must be considered [4]. The non-linear buckling behaviour can be studied using asymptotic theories and Koiter’s general post-buckling theory, according to which the imperfection sensitivity is defined by means of a coefficient representative of the post-buckling behaviour. This theory has been recently applied also to moderately thick orthotropic cylinders, with geometries similar to the specimens under consideraTable 5 Grid dimensions for imperfection measurements Axial direction
Cylinder Cone
Circumferential direction
No. of acquisitions
Distance (mm)
No. of acquisitions
Distance (mm)
6 9
120 120
16 15
120 33–128
V. Carvelli et al. / Composites: Part B 32 (2001) 89–101
angle 0˚ 315˚
measured exact
45˚
270˚
90˚ 0
10
20
30
radius [cm]
225˚
135˚ 180˚
Fig. 6. Measured and perfect external circumference at level 875 mm from the basis of the conical component.
The particular lamination and the mechanical properties of the specimen under consideration allowed us to produce a preliminary prediction of the buckling pressure using analytical expressions for isotropic cylinders. In fact the elastic stiffness and the strength of each lamina in the two orthogonal directions are very similar (see Table 3) and the particular lamination of the conical and spherical parts makes the shells quasi-isotropic. Various codes in the field of steel structures provide approximate formulas for buckling design of cylinders, cones and spheres. The well-known recommendations for buckling of shells and the Eurocode part related to shells are summarised in Appendix A. Recently a large project, including experimental and analytical activities, has been dedicated to buckling behaviour of composite shells to produce a first draft of guidelines, which are summarised in Appendix A [14]. As pointed
95
out in these guidelines, in the case of composite cylinders subject to external pressure the detrimental effects due to the presence of geometrical imperfections are less important than for axial compression. In the latter case many imperfection modes with different axial and circumferential waves should be included in a numerical non-linear analysis while in the case of externally pressurised shells, the out-ofroundness and the relevant modes play the main role. In Ref. [14] it is stated that the maximum recorded imperfection should be limited by the relation: Dmax ⫺ Dmin ⱕ 0:02 Dnom
4
where Dmax, Dmin and Dnom are the maximum, the minimum and the nominal diameter of the cylinder, respectively. This factor was equal 0.014 for the experimental model here considered. In Table 6 the analytical results obtained for an externally pressurised cylinder with the geometry of Fig. 2 and Table 1 are presented. The cylinder is subjected to external pressure acting also on the two end-closures. The design values of Table 6 are obtained applying the knock-down factors suggested by the codes. The characteristic strength of the material has been set equal to 66% of the ultimate strength in compression in the circumferential direction. The discrepancies, in terms of external buckling pressure, between Eurocode 3 [15] and the guidelines for laminated shells [14] is negligible (see Table 6). The ECCS recommendations [6] provide a higher value of the buckling pressure because no reduction of the limit pressure for the presence of axial stresses is considered in the case of hydrostatic pressure.
6. Non-linear buckling of the cylindrical component Since the amplitude of the measured geometric
Fig. 7. Imperfections of the external surface of the cylindrical component after the best-fit procedure.
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Fig. 8. External surface of the cylindrical component using the first five Fourier modes.
imperfections were significant and the analyses outlined above showed that the cylindrical part of shell presents the lowest buckling pressure, the non-linear buckling analysis was limited to this part of the structure. The effects of the pressure acting on the conical and hemispherical components were considered applying an equivalent axial compression at the two edges of the cylinder. The FE mesh was modified and reduced to 900 shell elements and 3700 nodes (Fig. 10b) to fit the recorded imperfections. The imperfections were modelled in two different ways. In the first non-linear analysis (NA1) the imperfections obtained after the best-fit procedure (Fig. 7) were assumed while in the second (NA2) the imperfections resulting from the first five Fourier modes were considered (Fig. 8) assuming the initial imperfection amplitudes j mn and the phase angles f mn listed in Table 7.
The results of the non-linear buckling analysis (NA1) performed assuming the best-fit imperfections, are reported in Fig. 10a in terms of pressure and shortening of the cylinder. The maximum pressure is 1.38 MPa and the relevant shape of the imperfect cylinder is drawn in Fig. 10b. The results of the non-linear analysis (NA2) adopting an imperfect surface obtained from a selection of Fourier modes are reported in Fig. 11. In this case the limit pressure is 1.15 MPa.
7. Experimental investigations An experimental investigation was carried out on the specimen in Fig. 1. The experience was conducted off-shore in the Mediterranean Sea about three miles from the coast in front of La Spezia (Italy). The sea depth of nearly 200 m was
Fig. 9. Buckling mode of the specimen (linear elastic analysis).
V. Carvelli et al. / Composites: Part B 32 (2001) 89–101
The procedure was based on the following steps:
Table 6 Buckling pressure (pb) and design pressure (pbd) from different recommendations and codes
ECCS [6] Eurocode 3 [15] Devils [14]
pb (MPa)
pbd (MPa)
1.70 1.34 1.26
0.85 0.76 0.61
chosen in accordance with the numerical prediction of the collapse pressure (Table 6). The specimen was set up into a supporting steel frame using flexible cables. A camcorder, a lamp and two microphones were fixed to the frame to monitor the experiment (Fig. 12). A ballast, sufficient for the sinking of the apparatus, was positioned at the bottom of the frame. The sinking of the specimen was controlled at a speed of 0.2 m/s to ensure a regular recording of the noise and of the images. The camcorder and the microphones were useful to monitor the specimen during the descent in the sea and allowed to detect with reasonable accuracy the collapse depth. The specimen collapsed suddenly and producing a considerable noise, at the depth of 130 m which correspond to a pressure of 1.3 MPa. Fig. 13 shows two details of the specimen during the experiment. In Fig. 13a it reached nearly 120 m where some considerable deformations occurred in the cylindrical component. Fig. 13b captured the collapse instant. The equipment and the staff for the submersion were provided by the manufacturer of the specimens in the framework of a joint research project. 8. Conclusions The main objective of the present work was to present a procedure to predict the collapse pressure of composite shells. A particular application for underwater vehicles was examined and the results of off-shore experiments were presented.
(i) Recording of geometric imperfections of the shell components. (ii) Numerical analyses of the entire specimen to predict the nature of the collapse (material failure or elastic buckling) and to localise the component where it occurs. (iii) Non-linear buckling analyses limited to the weakest component including the recorded imperfections processed according to a best-fit procedure and a Fourier analysis. (iv) Comparison of the numerical limit pressures with design approximation formulae (based on linear buckling and reduction factors) and with the experimental result (when available). The following conclusions can be summarised from the above study: • The effects of geometric imperfections could be significant when the maximum amplitude is close to the acceptable limit of Eq. (4). In these cases, a non-linear analysis including the effects of geometric imperfections is essential. • Numerical analyses showed that the weakest part of the shell structure under consideration was the cylindrical component while the conical shells and the hemisphere presented higher buckling pressure. • The numerical non-linear analyses performed assuming the geometric imperfection distribution of the best-fit procedure provide an overestimation of the experimental limit pressure (6.1%). On the other hand the non-linear analysis assuming the Fourier distribution of the imperfections underestimates the experimental limit pressure (⫺11%). These results may be regarded as satisfactory (see Table 8) considering the simplifications of the numerical models and the approximations of the in-situ recorded imperfections. • The design buckling pressures obtained with two codes for isotropic shells and with the recent guidelines for
1.50
Pressure [MPa]
1.25 1.00 0.75 0.50 0.25 0.00 0
0.1 0.2 0.3 Shortening [mm]
(a)
97
0.4
(b)
Fig. 10. NA1: (a) pressure vs. axial shortening; and (b) collapse mode of the cylindrical component.
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V. Carvelli et al. / Composites: Part B 32 (2001) 89–101
Table 7 The initial imperfection amplitudes j and the phase angles f associated to the first five Fourier modes (m 1; n 1; …; 5) employed in the nonlinear analysis NA2 m
n
j
f
1 1 1 1 1
1 2 3 4 5
0.753 0.047 1.242 2.663 0.853
⫺1.571 0.300 2.172 2.084 ⫺2.454
Table 8 Numerical predictions of the buckling pressure compared to the experimental result pb (MPa) Experiment Numerical predictions
1.30 1.38 1.15 1.80
Imperfect model NA1 Imperfect model NA2 Perfect model
laminated cylinders (see Table 6) are very conservative but may be considered reasonable as design approximated values. Acknowledgements The work presented in this paper was partially sponsored by the European Commission under a Brite-Euram project (BE-7550: Design and Validation of Imperfection Tolerant Shell Structures). The authors wish to acknowledge the support of INTERMARINE SpA (Sarzana-La Spezia, Italy) for providing the specimens and for the assistance in performing the off-shore experiments.
It is assumed that the cylinder is subjected to constant external pressure and the buckling resistance is determined on the basis of the compressive stress in the circumferential direction r A1 sw p t where r and t are the radius and thickness of the cylinder, while p is the external pressure. Since the pressure acts on the end closures, the shell is also subjected to a longitudinal compressive stress s x s w =2: For short or medium length shells subjected to external pressure the stiffening effects of the end supports must be also considered. A.1. ECCS recommendations for steel shells [6] The ECCS Recommendations provide a procedure valid for cylinders with a maximum circular imperfection within 0.5% of the radius measured from the true centre. The critical pressure of a simply supported elastic cylindrical shell may be calculated through the expression: t pcr E bmin r
A2
where E is the elastic modulus and b min is a function of L/r and t/r. Its value can be read directly from a diagram or evaluated from an expression that is a function of the number of circumferential waves. The ECCS recommendations do not impose a reduction of the limit pressure due to the presence of the longitudinal stresses in the case of hydrostatic pressure. The ECCS reduction factor to account for the presence of geometric imperfections is equal to 0.5. A.2. Eurocode 3 — supplementary rules for strength and stability of shell structures [15]
Appendix A The following codes and guidelines provide approximation formulas for the buckling pressure of cylinders.
This code allows us to determine the critical circumferential buckling stress of shells with different boundary conditions.
1.25
Pressure [MPa]
1.00 0.75 0.50 0.25 0.00 0
0.15 0.3 0.45 Shortening [mm]
(a)
0.6
(b)
Fig. 11. NA2: (a) pressure vs. axial shortening; and (b) collapse mode of the cylindrical component.
V. Carvelli et al. / Composites: Part B 32 (2001) 89–101
99
Fig. 12. The specimen in the steel frame during the experiment in the sea.
The length of the shell segment is characterised in terms of the parameter: p rt v L
A3
For medium-length cylinders with 20 ⱕ
1 r ⱕ 1:63 Cq v t
A4
the critical circumferential buckling stress should be
obtained from:
s qRc 0:92ECq v
t r
A5
The factor Cq depends on the boundary conditions as detailed in Ref. [15]. Since the pressure acts also on the end closures the cylinder is subjected to an axial compression s x s w =2: The interaction between axial and circumferential stresses has to be accounted for. For medium length cylinders with 1:7 ⱕ
1 r ⱕ 0:5 v t
A6
100
V. Carvelli et al. / Composites: Part B 32 (2001) 89–101
Fig. 13. The specimen during the experiment: (a) ⫺120 m; (b) collapse ⫺130 m.
steel yield stress)
the critical meridian buckling stress shall be taken as:
s xRc 0:605E
t r
A7
Although buckling is not a purely stress-initiated failure phenomena, the buckling limit state is defined by limiting the design values of membrane stress. As suggested in Ref. [15], the interaction is set by: kq kt s xSc kx s t ⫹ qSc ⫹ Sc ⱕ 1 A8 s xRc s qRc tRc with kx ku 1:25 and kt 2: In order to obtain the maximum design load, the critical stress has to be reduced by a reduction factor. This depends on the shell quality class. The parameter used to determine the quality class for any cylinder is ur
Dmax ⫺ Dmin Dnom
A9
In our case the cylinder is of class C and the elastic imperfection reduction factor a u is equal to 0.5, while a x has to be obtained by mean of a formula reported in Ref. [15]. For circumferential and meridian buckling the squash limit slenderness is assumed as l q0 0:4 and l x0 0:2; respectively. If the conditions (being fyk the characteristic
r ⱕ t
s E 23fyk
r E ⱕ t 25fyk
A10
are not satisfied the following investigation must be carried out. The stability reduction factors x u and x x:
a xq q2 lq
l x ⫺ l x0 xx 1 ⫺ 0:6 l p ⫺ l x0
A11
have to be evaluated assuming: s s p f fyk yk l q ; l qp 2:5aq ; l x ; s qRc s xRc p l qp 2:5ax The characteristic buckling stresses are:
s qRk xq fyk
s xRk xx fyk
A12
The interaction between axial and circumferential actions is accounted for in Eq. (A8).
V. Carvelli et al. / Composites: Part B 32 (2001) 89–101
A.3. Guidelines for analysis of laminated shells against buckling collapse [14] Closed form solutions and recommendations for carrying out numerical analyses for the calculation of the critical buckling load of composite shells have been recently suggested in Ref. [14]. In the case of orthotropic shells subject to external pressure and axial compression, the buckling pressure may be evaluated by means of the following expressions where the components of the stiffness matrix of the laminate are introduced. The critical axial compressive load is: C11 C12 C13 C21 C22 C23 2 L C31 C32 C33 A13 Nxcr C11 C12 mp C 21 C22 where Cij i; j 1; 2; 3 are functions of the stiffness matrix components and of the number of mode waves in the axial and circumferential directions. The expressions may be found in Ref. [14]. The critical external pressure is given by: C11 C12 C13 C21 C22 C23 r C31 C32 C33 A14 pcr C11 C12 n2 C C22 21 The interaction between axial compression and external pressure is Nx;c p ⫹ 1 Nxcr pcr
A15
where Nx;c P= 2pr (P being the applied axial compressive load) and p the applied external pressure. Buckling occurs when the value on the left hand side of Eq. (A15) is greater then or equal to one. The use of semi-empirical knock-down factor, currently
101
adopted in design of metal shells, is proposed also for composite shells. The design pressure presented in Table 6 is obtained by means of the minimum between the knock-down factors suggested for axial compression and external pressure.
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