epoxy tubes under combined static loading. Part I: Experimental

Composites Science and Technology 69 (2009) 2241–2247

Contents lists available at ScienceDirect

Composites Science and Technology journal homepage: www.elsevier.com/locate/compscitech

Mechanical behavior of glass/epoxy tubes under combined static loading. Part I: Experimental Alexandros E. Antoniou a, Christoph Kensche b, Theodore P. Philippidis a,* a b

Department of Mechanical Engineering and Aeronautics, University of Patras, P.O. Box 1401, GR 26504 Panepistimioupolis, Rio, Greece Hexion Specialty Chemicals Stuttgart GmbH, Am Ostkai 21/22, D-70327 Stuttgart, Germany

a r t i c l e

i n f o

Article history: Received 10 December 2008 Received in revised form 4 June 2009 Accepted 12 June 2009 Available online 18 June 2009 Keywords: B. Strength B. Non-linear behavior B. Stress/strain curves C. Elastic properties Biaxial testing

a b s t r a c t A series of biaxial static tests of E-glass/epoxy tubular specimens [±45]2, subjected to combined torsion and tension/compression were performed to simulate complex stress states encountered in a wind turbine rotor blade. The failure locus in the effective axial-shear stress plane was derived experimentally while in-plane strain tensor components were measured in the tube outer surface. By means of shell theory and strain measurements in the surface of the specimen, the in-plane shear response of the outer ply was obtained, revealing dependence each time to the combined tube loading. The correlation established between the ratio of transverse normal and in-plane shear stress in the principal coordinate ply system and the elastic shear modulus, suggested a strong dependence, warning on the implications for design and certification procedures. Ó 2009 Elsevier Ltd. All rights reserved.

1. Introduction Testing of cylindrical specimens made of uni- or multi-directional lay-ups produced either by filament winding or layer wrapping has been used since long for composites characterization. Early works by Pagano and Whitney [1] and Whitney [2] were devoted in defining the appropriate tube dimensions along with investigations on the developed stress fields under multi-axial loading. Torsion tests were also performed by Wall and Card [3] on various laminate configurations to characterize shear strength and elastic behavior of glass/epoxy tubes. Specimens made of unidirectional stacking sequence, consisting of fibres in the hoop direction, [90]n, and tested under pure torsion were proposed as test standard for in-plane shear stress–strain characterization [4]. On the other hand, angle-ply lay-ups [±h]n or quasi–isotropic ones [0/±45/90]nS were mostly used by Soden et al. [5–8] and Swanson and Christoforou [9], respectively, in examining load interaction effects in composite tubes subjected to axial load and internal pressure. Besides material characterization, tube testing was also of great assistance in verifying failure theories and progressive damage models as early as in 1970 by Lantz and Foye [10] while an account of work performed in this field was recently edited by Soden and co-workers [11]. Simulation of specific complex stress fields for

* Corresponding author. Tel.: +30 2610 969450/997235; fax: +30 2610 969417. E-mail address: [email protected] (T.P. Philippidis). 0266-3538/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.compscitech.2009.06.009

deriving design allowable characteristics under static or cyclic loading was reported as well by Kensche [12]. The motivation for the present work was also the development of test methods for simulating complex stress fields encountered in various regions of large wind turbine rotor blades. A great amount of tests were designed and conducted in the frame of a comprehensive experimental program [13], aiming to gain insight in material performance under uniaxial and biaxial loading. Prismatic coupons, tubular and cruciform specimens as reported by Philippidis et al. [14], Kensche [15], Smits et al. [16], respectively, were tested to simulate potentially developed complex stress fields in a wind turbine rotor blade. Although several test methods to induce complex stress states in FRP composites were previously proposed [17– 19], the specific procedures adopted in [13] were selected by the research partners based on their experience, existing equipment and property database availability concerning similar materials. Experimental results from the various tests contributed in validating constitutive non-linear material models and progressive damage strategies, developed in parallel and implemented in a numerical FE routine by Philippidis and Antoniou [20]. The above mentioned numerical tool was further validated in the second part of this work [21] by extensive comparison with experimental data from E-glass/epoxy tube testing under combined axial and torsion loads. Test and measurement procedures along with strength results were presented herein. Processing of the strain data measured in the outer tube surface revealed interesting trends of the interaction of transverse to the fibre normal stress and shear strain and the corresponding influence on

2242

A.E. Antoniou et al. / Composites Science and Technology 69 (2009) 2241–2247

non-linear shear stress–strain response. By means of analytical expressions derived using either classical lamination plate or shell theory and the measured strain field at the outer tube surface it was possible to derive simple expressions for the determination of non-linear ply shear behavior. The dependence of the in-plane elastic shear modulus of the lamina, as derived by the various tube tests under different loading conditions, on the developed ply complex stress state was thoroughly investigated. 2. Experimental 2.1. Tube material and construction Tube design was developed in the frame of OPTIMAT BLADES project [13] by DLR, although the majority of the static tests presented here was conducted in IFB (Institut fuer Flugzeugbau) of Stuttgart University. The glass fiber reinforcement was a biaxial, stitch bonded, [±45] fabric made of glass roving from PPG. The two layers of fibers at +45° and 45°, 400 g/m2 each were stitched together with a polyester yarn. An epoxy resin with a slow hardener from SP Systems, Prime 20, was used as a binder. Detailed information on constituent materials was given by Jacobsen [22]. The dry glass fabric was wrapped two times around a steel mandrel, external diameter of 28 mm. At the gripping area of each tube two or three more layers of the same fabric were added to reinforce the load introduction points, forming an outer diameter of 32 mm for specimens tested at DLR and 33 mm for those tested at IFB due to restrictions of the test rigs. Specimens were impregnated using vacuum assisted resin transfer molding (VARTM) technique. Layers were cut perpendicular at their ends resulting in steep overlap drop of 24–30 mm, Fig. 1, or an arc of up to 150° After initial curing at room temperature, tubes were post-cured at 80 °C for 4 h. Then they were filled with a foam core to avoid buckling either in pure torsion or combined with axial load, based on previous experience at DLR with glass/epoxy tube testing. Material processing, tube manufacturing and preparation were performed by LM Glasfiber. Tube external diameter, see Table 1, was measured in two perpendicular directions, one including the overlap area denoted as Dmax and the other Dmin. Wall thickness in the gauge length region was calculated by taking into account the mandrel diameter of 28 mm and the mean value of Dmin, Dmax. ID number for each entry in Table 1 corresponds to the last two digits of the respective OptiDAT [13] record, a public access database. For example, #06 stands for GEV215-S0232–06. Exceptionally, specimens 01 and 03 refer to GEV215-S0233-# and 02 to ‘‘Tube 0200-002”. Specimen length was equal to 170 mm; gauge length of 100 mm. Geometry is shown in Fig. 2. Fibre volume fraction was measured on typical samples from different tube manufacturing batches [15]. Specimens tested statically, presented herein, were of two batches of 63.60% and 63.25% fibre volume fraction, respectively. This low variation re-

Table 1 Specimen external diameter and wall thickness. Tube#

Dmin (mm)

Dmax (mm)

h (mm)

01 02 03 06 07 08 09 11 12 13 14 15

30.03 30.01 30.14 30.06 30.12 30.10 30.07 30.18 30.11 30.17 30.13 30.09

30.58 30.60 30.90 30.73 31.15 30.73 30.78 30.73 30.84 30.76 30.78 30.65

1.15 1.15 1.26 1.20 1.32 1.21 1.21 1.23 1.24 1.23 1.23 1.19

flects also on the thickness measurements of Table 1 for which coefficient of variation (COV) was less than 4%. The cured UD ply was of nominal thickness equal to 0.3 mm and its elastic properties in the principal coordinate system were given by:

E1 ¼ 39:04 GPa; G12 ¼ 4:24 GPa E2 ¼ 14:08 GPa;

More details on thermo-mechanical material properties can be found in various reports and the database OPTIDAT [13]. 2.2. Test equipment At IFB, a hydraulic Tension/Compression–Torsion Instron 8502 machine with maximum torsion capacity Mmax = 2000 N m and maximum tensile/compressive load Fmax = ± 100 kN was used. Data sampling was carried out with Hottinger UPM60 and Spider 8 acquisition boards. At DLR, a servo-hydraulic 1000 N m Schenck torsion machine of type 88603 was used while data were acquired at a frequency of 2 Hz with Hottinger UPM60 and UPM100 boards. TML – Tokyo Sokki Kenkyujo, Ltd., 3-element strain gauge rosettes of 6 mm gauge length were glued on the diametric side of the tube overlap. Gauge elements were oriented along and at ±45° to the tube axis. 2.3. Specimen loading Tubes were subjected to biaxial loading by applying axial force, P, along the cylinder axis and torque, T. Positive force and moment were shown in Fig. 3. In an infinitesimal surface element of the thin-walled tube, the load combination induced a biaxial field with axial Nx and shear Ns, membrane stress resultants, acting on the laminate. The resulting plane stress field in the principal coordinate system of a single ply, also shown in Fig. 3, was complex as well. Monotonic tests in the servo-hydraulic test rigs were performed under load control. The ratio of axial force to torque was unique for each of the nine tubes tested statically and kept constant during the experiment. Its values spanned all four quadrants of the force–moment plane. Three more tubular specimens were subjected to pure torsion. Stress resultants, subject to thin-wall theory assumptions, were given by:

Nx ¼ Fig. 1. Steep drop overlap (dimensions in mm).

m12 ¼ 0:29

P ; 2pRm

Ns ¼

T 2pR2m

where Rm the mean wall radius was denoted.

ð1Þ

2243

A.E. Antoniou et al. / Composites Science and Technology 69 (2009) 2241–2247

+45o -45o 35

100

35

28 32/33

170

Fig. 2. Geometry of tubular specimens (dimensions in mm, light grey is for grip material, dark grey for gauge length thickness).

x

P

σ2

σ6

σ1

T

P

ply

T σ1

σ6

σ2

Ns

laminate

y Nx

Nx

1

2

x

Ns Fig. 3. Induced stress field from biaxial loading.

ðkÞ ðkÞ ðkÞ ðkÞ ðkÞ ðkÞ rðkÞ x ¼ ðQ xx bxx þ Q xy byx þ Q xs bsx ÞN x þ ðQ xx bxs þ Q xy bys þ Q xs bss ÞN s

3. Stress field in a laminated tube Implementing shell theory formulation, Whitney and co-workers [2,23], derived analytical expressions for the stress field in each ply of a laminated tube under complex loading. For the biaxial loading case examined in this work, see Fig. 4, the stresses in the local physical coordinate system of the kth ply were determined by:

þ

ðkÞ ðkÞ ðkÞ ðkÞ ðkÞ ðkÞ rðkÞ y ¼ ðQ xy bxx þ Q yy byx þ Q ys bsx ÞN x þ ðQ xy bxs þ Q yy bys þ Q ys bss ÞN s

þ

400

In the local ply and laminate coordinate system, see Fig. 3, x-axis is parallel to the global x-axis of the cylinder, y-axis coincides with the hoop direction, z-axis is in the thickness direction while by rs the ðkÞ in-plane shear stress component was denoted. Q ij stands for the reduced stiffness matrix of the kth ply in the physical coordinate system. For the non-symmetric, balanced lay-up [±45]2 of interest, matrix bij was given by:

200

06

Ns/h [MPa]

15

07

100

08 0

-100

-50

0

50

100

150

-100

13

12 -200

i z h ðkÞ ðkÞ ðkÞ ðQ ss bsx  Q ðkÞ ys byx ÞN x þ ðQ ss bss  Q ys bys ÞN s Rm

ð2Þ 1

300

-150

i z h ðkÞ ðkÞ ðkÞ ðQ ys bsx  Q ðkÞ yy byx ÞN x þ ðQ ys bss  Q yy bys ÞN s Rm

ðkÞ ðkÞ ðkÞ ðkÞ ðkÞ ðkÞ rðkÞ s ¼ ðQ xs bxx þ Q ys byx þ Q ss bsx ÞN x þ ðQ xs bxs þ Q ys bys þ Q ss bss ÞN s

þ

14

i z h ðkÞ ðkÞ ðkÞ ðQ xs bsx  Q ðkÞ xy byx ÞN x þ ðQ xs bss  Q xy bys ÞN s Rm

11

3

09

2 -300

Nx/h [MPa] Fig. 4. Experimental failure envelope of [±45]2 laminated tubes in effective axialshear stress plane.

2

Axx Ass þ

6 6 c1 6 bij ¼ 6 6 Axy Ass 6 c1 4 Bxs Axy R

c1

m

B2 xs R2 m



 Axy Ass þ

c1

B2 xs R2 m



3 BRxs m

c2

Axx Ass

BRxs m

Axx BRxs

Axx þAxy

c1

c2

c1

m

c2

7 7 7 7; 7 7 5

i; j ¼ x; y; s

ð3Þ

where Aij, Bij the components of the extensional and coupling stiffness matrix of the laminate were denoted. Constants c1, c2 appearing in Eq. (3) were defined by:

2244

A.E. Antoniou et al. / Composites Science and Technology 69 (2009) 2241–2247

c2 ¼ Ass ðAxx þ Axy Þ þ

B2xs R2m

;

Table 3 Strains at failure.

c1 ¼ ðAxx  Axy Þc2

In the principal coordinate system of the outer [45] ply, the in-plane shear stress component, r6, was calculated by:

1 2   1 z z byx Nx þ bys Ns ¼ ðQ xx  Q xy Þ ðbxx  byx ÞNx þ 2 Rm Rm

r6 ¼ ðrx  ry Þ

ð4Þ

For the glass/epoxy material and tube geometry investigated, terms containing the ratio z/Rm are negligible and Eq. (4) reads with great accuracy as:

1 2

r6 ¼ ðQ xx  Q xy Þðbxx  byx ÞNx 1 Nx P ¼ 2 h 2pRm h

ð6Þ

h denoting the total laminate thickness. Eq. (6) suggests that the inplane shear stress in the principal coordinate system of the outer ply depends only on the applied axial force and is directly measurable, a fact reminiscent of the axial tensile test of a [±45] flat coupon. However, the other two in-plane stress components in the principal coordinate system will depend on the applied torque and thus the complex stress field will be different from that of the flat coupon and further from one tube to the other, depending on the loading conditions. 4. Results and discussion 4.1. Biaxial strength test results Monotonic tests on tubes under combined axial force and torsion were performed for several loading conditions. Maximum force and torque values which were also the failure loads were summarized in Table 2 while in Table 3 failure strains at the physical coordinate system were presented. Corresponding axial and shear stress resultants, normalized over the laminate thickness, Nx /h, Ns /h, were derived by means of Eq. (1) and were plotted in Fig. 4. Labels denoting coupon ID number, along with a best fit quadratic approximation, as per Eq. (7), were also included.

a11

ex % axial

ey % hoop

es % shear

01 02 03 06 07 08 09 11 12 13 14 15

0 0 0 0.85 2.41 2.83 3.49 1.18 0.59 1.30 1.16 0.87

0 0 0 0.43 1.36 1.66 2.27 0.88 0.31 0.88 0.80 0.51

2.46 2.14 1.96 1.45 0.59 0.14 1.11 1.43 1.19 1.18 0.96 1.56

ð5Þ

By means of Eq. (3) it can be further proved that:

r6 ¼

Tube#

 2  2        Nx Ns Nx Ns Nx Ns þ a22 þ a12 þ a1 þ a2 1 ¼0 h h h h h h

ations were observed for negative shear stress resultants where test data give the impression that at an almost constant level of negative torque, axial load at failure exhibits considerable scatter. However, for tubes #09, 11 in the 4th quadrant and #12, 13 in the 3rd respectively, the loading paths, i.e. the ratio of axial force to torque, were different and it was observed that axial strength was substantially reduced as the ratio Nx/Ns decreased. Similar trend, albeit of a less pronounced effect was also noticed for positive torque. The difference lied in the fact that under negative torque the stress field at the principal material axes of the outer [+45] ply consisted of compression along the fiber and tensile stress in the transverse direction. Tensile matrix cracks weaken fiber support, promoting microbuckling. At the same time, the [45] ply underneath, was stressed in compression transversely to the fibers which were pulled in tension. Matrix cracking at that layer was of an explosive nature, favoring delamination in the interface of the two outer plies and leading finally at local buckling of the external [+45] ply. The above failure mode, induced by the instability, was prominent under pure negative torque, see left picture of Fig. 5. Note the buckled fibers of the delaminated ply area and the detonated bands at both coupon ends driven by the underlying burst matrix damage mode. On the other hand, tube strength was enhanced under positive torque due to fiber stretching of the outermost ply that failed in a compressive matrix mode. Therefore it was concluded, that loading ratio influenced the magnitude of ultimate loads but not the failure mechanisms. The latter, were drastically affected by the sign of applied torque. 4.2. Lamina in-plane shear response

ð7Þ This experimental failure locus for the tubular specimen in the N x N s ; h plane was satisfactory for the first two quadrants while devih Table 2 Failure loads. Tube#

k = Nx/Ns

T (N m)

P (N)

01 02 03 06 07 08 09 11 12 13 14 15

0 0 0 0.541 0.971 2.459a 0.965 0.542 0.542 0.971 0.971 0.539

446.00 398.90 364.50 263.08 193.72 79.10 183.90 220.70 230.60 208.94 160.70 230.72

0 0 0 9758 12,834 13,322 12,153 8187 8557 13,882 10,676 8526

a

Due to load control loss, biaxial load ratio during this test was not constant.

By analytical calculations, using linear elastic classical lamination plate (CPT) and shell theory, see Section 3, but also FEA numerical results, it was proved that shear stresses in the material coordinate system were constant throughout lamina thickness, changing sign for the [+45] and [45] plies while depending only on the applied axial force and not on the superimposed torsion. FEM results were derived by means of a non-linear, incremental stress–strain model described in detail in [21], however, data compared in Fig. 6 were predicted at low load values where material behavior was still in the linear range. As an example, the stress distribution in the thickness direction of the tube wall was presented in Fig. 6. Computations were performed at an early load stage, 13.5% of the failure load, for a loading condition similar to that for tube #09, see Table 2. Shell theory and FEM calculations were in excellent agreement for all stress components while the three prediction methods yield the same result for the shear stress in the principal coordinate system of each ply. The discrepancy observed for the normal stress values predicted by CPT is due to the neglect of curvature and

2245

A.E. Antoniou et al. / Composites Science and Technology 69 (2009) 2241–2247

Fig. 5. Distinct failure modes under pure negative (left) and positive (right) torque.

CPT

2

FEM

1

0 0

20

40

60

z/ho

z/ho

1

-20

2

SHELL

0

-10

-5

-1

0

5

10

15

20

-1

-2

-2

σ1 [MPa]

σ2 [MPa] 2

z/ho

1 0 -10

-5

0

5

10

-1 -2

σ6 [MPa] Fig. 6. Stress distributions along the wall thickness in the principal coordinate system of each ply.

80

e6 ¼ 2SG2  ðSG1 þ SG3 Þ

70 60

Nx/h [MPa]

mostly to the asymmetric lay-up of the tube. For symmetric stacking sequences, all three methods yield identical results. According to Eq. (6), the shear stress is equal to half the axial stress resultant. Then, since the strain field is measured in the external tube surface it was possible to derive the shear stress– strain response in the principal coordinate system of the outer ply. The corresponding shear strain in terms of the three measuring strain gauges, SG1, SG2, SG3 in the +45°, axial and 45° direction respectively was given by:

ð8Þ

Typical test results of axial stress vs. strain from all three strain gauges were presented in Fig. 7 for tube #11. The strong non-linearity in the axial direction, dominated by the matrix behavior is in contrast to the almost linear response measured in the ±45 fiber directions SG1 and SG3, respectively. The r6–e6 curves were calculated for all tests shown in Table 2 by means of Eqs. (6) and (8) and were presented in Fig. 8. Tube wall thickness, h, and mean radius (Rm = 14 + h/2 in mm) used in Eq. (6) were according to Table 1. In the same figure, the respective experimental data derived using the Iosipescu, V-notched beam (VNB) shear test method [24] and the ISO 14129 tensile test of [±45]S cou-

50

SG1 SG2

40

SG3

30 20 10 0

-0.008 -0.006 -0.004 -0.002

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

ε Fig. 7. Axial stress resultant vs. strain in all three measuring directions. Tube #11.

pons [25] were also included. Labels with numbers for each curve conform to tube ID number of Table 2. Also note that end points of shear stress–strain curves correspond to tube failure.

2246

A.E. Antoniou et al. / Composites Science and Technology 69 (2009) 2241–2247 100

Table 4 In-plane shear modulus and r2/|r6| values. VNB

80

Tube# (P > 0)

60

ISO

40

σ6 [MPa]

07

09

0.04

0.06

06 11

20 0 -0.04

-0.02

0

0.02

0.08

0.1

06 07 09 11 12 13 14 15

jr 6 j

r2

G12 (GPa)

1.237 0.406 1.736 2.582 1.240 0.401 1.732 2.589

4.59 4.04 4.29 3.48 4.64 5.15 4.38 3.96

-20 15

12 -40

14 13

6.0

-60

VNB

-80

Fig. 8. Experimental in-plane shear stress–strain curves for glass/epoxy UD ply.

All tubes were subjected to combined torsion and axial load. However, the induced shear stress sign in the principal material system of the outer ply was only dependent to the force direction. Thus it was negative for compressive axial load, i.e. tubes #12–15, and positive, respectively for tensile one. The prevailing lamina stress field in all the above mentioned tests was complex and different for each case with the exception of the VNB test in which pure shear stress field was developed. This is clearly shown in Fig. 8 where the VNB curve is an upper envelope for all the others, even if data from tubes 12–15 were plotted in the positive quadrant. Evidently, the in-plane shear behavior of the UD ply was highly affected by the presence of at least the transverse normal stress, if not also to a certain extent by the longitudinal axial stress along the fibers. Up to ±(12–15) MPa, shear stress–strain response was almost linear, however, beyond that threshold non-linearity was dominant and divergence between lamina behavior was demonstrated. It should be stated that all tests were performed at such strain rates so as to induce tube failure approximately after 1.5 min and therefore reduce strength variance at different load ratios due to viscoelastic behavior. Then, tubes 6, 11, 12 and 15 were tested at load rates for the applied axial force and torque of 110 N s1 and 2.97 N m s1, respectively, while the corresponding values for tubes 7, 9, 13 and 14 were slightly different and equal to 146 N s1 and 2.23 N m s1. The discrepancy observed in shear stress–strain curves of Fig. 8, e.g. among specimens #6 and #11, tested at same loading rates but different torque to axial force ratios, should be then attributed to the effect of the superposition of the transverse normal stress. External loads induced complex stress state in each lamina that affected in a unique way shear non-linearity but also conventional elastic shear modulus determination. In a preliminary attempt to explain this variation, the measured shear modulus, G12, according to ISO 14129 in the strain range of 1000–5000 le was correlated to the ratio of transverse normal to shear stress, r2/|r6|, derived by means of shell theory, Eq. (2), in the principal coordinate system of the outermost layer where strain measurements were conducted. Calculations were performed for the different tube loading configurations using material properties defined in Section 2.1 of [21]. Results were summarized in Table 4. Table 4 data along with the values from the VNB (0, 5.3), ISO (0.61, 4.24) and results from 30° off-axis tensile tests for in-plane shear response determination [26], (0.577, 4.894), respectively, were plotted in Fig. 9, revealing shear modulus linear dependence

G12 [GPa]

ε6

14

15

13

4.5

06

off-30o 12

ISO

9

07 11

3.0

1.5

0.0 -3.0

-2.0

-1.0

0.0

1.0

2.0

3.0

σ 2/|σ 6| Fig. 9. Shear modulus dependence on r2/|r6| ratio.

on the r2/|r6| in both quadrants. Excluding a few data points, material performance could be described with two linear functions, each for every quadrant. Shear modulus is decreasing as |r2/r6| increases. The slope of G12 degradation is steeper when tensile normal stresses are combined with in-plane shear stresses, while these tend to open the developed micro-cracks parallel to the fiber direction, contributing in further modulus reduction. Although these preliminary test results should be further discussed, possibly duplicated by other researchers for similar glass/ epoxy materials and thoroughly examined, Fig. 9 highlights that the conventional linear shear modulus used for design purposes should be considered with caution. Differences in measured shear modulus, depending on the ratio of r2 over r6, amount to ca. 33% and this is certainly important in elastic stability analyses or determination of the stress/strain field itself. The effect of transverse stress r2 on in-plane shear response was initially addressed by Hashin et al. [27] and further discussed by Kaddour et al. [8] where the same trend of G12 reduction with increasing r2 was revealed. Puck and Mannigel [28] have investigated a similar interaction effect and considered further the dependence of elastic modulus E2 transverse to the fibers on the shear stress r6 acting simultaneously. However, their theoretical treatment, influenced by the experimental data of the materials considered, was only for the non-linear part of the in-plane shear response, corresponding to rather large strains in contrast to the analysis presented herein, valid for strain intervals in the linear region as suggested by ISO 14129. 5. Conclusions Experimental results were presented from a biaxial loading test series performed in the frame of a comprehensive experimental program aiming in complete material characterization under complex stress fields. The failure locus for tubular specimens of a [±45]2 laminate in the axial-shear stress resultants plane was

A.E. Antoniou et al. / Composites Science and Technology 69 (2009) 2241–2247

determined for loading conditions simulating stress fields developed in the shear webs of wind turbine rotor blades. Processing of strain gauge measurement data on tube outer surface provided a simple method for determining the in-plane shear response in the principal coordinate system of the building ply, albeit under the simultaneous action of transverse normal stress. A strong dependence of in-plane shear response on normal stress r2 was observed. It was further revealed that determination of shear modulus according to ISO 14129 procedures from shear stress–strain curves derived from tube tests yields shear modulus values with strong dependence on ratio |r2/r6|. The higher its value, the lower the respective value of the shear modulus. This must be taken into consideration by design offices and certification bodies when a deterministic shear modulus value has to be chosen. The question is not whether an acceptable test method has been used for in-plane shear characterization but the determination of adverse complex stress fields to be encountered in service. Results from this study suggest that further biaxial testing of tubular specimens, with fibers oriented only in the hoop direction, is required to better understand the effect of complex stress field and assess methods and engineering models for shear modulus dependence on the simultaneous action of transverse normal stress. Acknowledgements Research was funded in part by the European Commission in the framework of the research programmes ‘‘Reliable Optimal Use of Materials for Wind Turbine Rotor Blades” (OPTIMAT BLADES), Contract No. ENK6-CT-2001-00552 and ‘‘Integrated Wind Turbine Design (UPWIND), Contract No. 019945, SES6. Partial funding was also provided by the General Secretariat of Research and Technology (GSRT) of the Greek Ministry of Development, Contract Nos. F.K. 6660 and C037. References [1] Pagano NJ, Whitney JM. Geometric design of composite cylindrical characterization specimens. J Compos Mater 1970;4(3):360–78. [2] Whitney JM. On the use of shell theory for determining stresses in composite cylinders. J Compos Mater 1971;5(3):340–53. [3] Wall LD, Card MF. Torsional shear strength of filament-wound glass–epoxy tubes. NASA TN D-6140; 1971. [4] ASTM D-5448M-93. Standard test method for in-plane shear properties of hoop wound polymer matrix composite cylinders. [5] Soden PD, Leadbetter D, Griggs PR, Eckold GC. The strength of filament wound composite under biaxial loading. Composites 1978;9(4):247–50. [6] Soden PD, Kitching R, Tse PC. Experimental failure stresses for ±55o filament wound glass fibre reinforced plastic tubes under biaxial loads. Composites 1989;20(2):125–35.

2247

[7] Soden PD, Kitching R, Tse PC, Tsavalas Y, Hinton MJ. Influence of winding angle on the strength and deformation of filament-wound composite tubes subjected to uniaxial and biaxial loads. Compos Sci Tech 1993;46(4):363–78. [8] Kaddour AS, Hinton MJ, Soden PD. Behaviour of ±45o glass/epoxy filament wound composite tubes under quasi-static equal biaxial tension–compression loading: experimental results. Composites: Part B 2003;34(8):689–704. [9] Swanson SR, Christoforou AP. Response of quasi-isotropic carbon/epoxy laminates to biaxial stress. J Compos Mater 1986;20(5):457–71. [10] Lantz RB, Foye RL. Post-yielding behavior of torsionally loaded composite tubes. Analysis of the test methods for high modulus fibers and composites, ASTM STP 521. Am Soc Test Mater; 1973, p. 293–308. [11] Hinton MJ, Kaddour AS, Soden PD. A comparison of the predictive capabilities of current failure theories for composite laminates, judged against experimental evidence. Compos Sci Tech 2002;62(12–13):1725–97. [12] Kensche CW. Fatigue of composites for wind turbines. Int J Fatigue 2006;28(10):1363–74. [13] OPTIMAT BLADES, Reliable optimal use of materials for wind turbine rotor blades, contract ENK6-CT-2001-00552; (2001–2006). . [14] Philippidis TP, Antoniou AE, Assimakopoulou TT, Passipoularidis VA. Static tests on the standard OB UD and MD off-axis coupons, OB-TG2-R022; 2005 (). [15] Kensche CW. Final report on tube testing, OB_TG2_R038; 2006 (). [16] Smits A, van Hemelrijck D, Philippidis TP, Cardon A. Design of a cruciform specimen for biaxial testing of fiber reinforced composite laminates. Compos Sci Tech 2006;66(7–8):964–75. [17] Arcan M, Hashin Z, Voloshin A. A method to produce uniform plane-stress states with applications to fiber-reinforced materials. Exp Mech 1978;18(4): 141–6. [18] Swanson SR, Messick M, Toombes GR. Comparison of torsion tube and Iosipescu in-plane shear test results for a carbon fibre-reinforced epoxy composite. Composites 1985;16(3):220–4. [19] Lee S, Munro M. Evaluation of in-plane shear test methods for advanced composite materials by the decision analysis technique. Composites 1986;17(1):13–22. [20] Philippidis TP, Antoniou AE. A progressive damage FEA model for glass/epoxy shell structures. J Compos Mater, submitted for publication. [21] Antoniou AE, Kensche CW, Philippidis TP. Mechanical behavior of glass/epoxy tubes under combined static loading. Part II: Validation of FEA progressive damage model. Compos Sci Technol 2009;69(13):2248–55. [22] Jacobsen TK. Reference material (OPTIMAT). glass–epoxy, OB_SC_R001_LM; 2002. http://www.kc-wmc.nl/public_docs/index.htm. [23] Whitney JM, Pagano NJ, Pipes RB. Design and fabrication of tubular specimens for composite characterization in composite materials: testing and design (second conference), ASTM STP 497, American Society for Testing and Materials; 1971. p. 52–67. [24] Megnis M, Brondsted P. Measurements of in-plane shear properties of GEV206 at ambient room conditions using V-notched beam test specimen, OB-TG3R009; 2003. . [25] Philippidis TP, Assimakopoulou TT, Passipoularidis VA, Antoniou AE, Static and fatigue tests on ISO standard ±45o coupons, OB-TG2-R020-UP; 2004. . [26] Smits A, van Hemelrijck D. Determination of in-plane shear properties of UD reference material. A comparison of the results obtained with different techniques, OB-TG2-R023-VUB; 2005. . [27] Hashin Z, Bagchi D, Rosen BW. Nonlinear behavior of fiber composite laminates. NASA CR-2313; April 1974. [28] Puck A, Mannigel M. Physically based non-linear stress–strain relations for the inter-fibre fracture analysis of FRP laminates. Compos Sci Tech 2007;67(9):1955–64.