FAILURE CRITERIA FOR NON-METALLIC MATERIALS PART I: FIBRE REINFORCED PLASTICS

Transcription

1 FAILURE CRITERIA FOR NON-METALLIC MATERIALS PART I: FIBRE REINFORCED PLASTICS C. Kaiser, E. Kuhnel and A. Obst 3 HPS GmbH, Christian-Pommer-Str. 34, 38 Braunschweig, Germany IKV, RWTH Aachen, Pontstr. 49, 506 Aachen, Germany 3 ESA/ESTEC, Keplerlaan, 00 AG Noordwijk ZH, The Netherlands ABSTRACT Within the rame o the study Failure Criteria or Non-Metallic Materials -Part I unded by ESA, detailed guidelines or the proper use o ailure criteria during the dierent stages o composites spacecrat structures design have been elaborated or ibre reinorced plastics (FRP). Thereore, one important part o this work was the evaluation o the results which the World Wide Failure Exercise (WWFE) has delivered. The test results and the predictions o the dierent authors given in the WWFE did allow a benchmarking o ailure criteria that was necessary to evaluate the enormous amount o existing ailure criteria. One conclusion o this study was that more reliable test results must be gathered in order to be able to better benchmark the existing criteria. An overview o the results and a comparison with data rom literature is given. A unique high level o biaxial inplane compressive stress was reached compared to test results o other authors but ar below o the Tsai-Wucriterion predictions. However, the expected strength level was not reached as the ailure o the specimens was initiated by undesired ailures so ar. On the theoretical side Puck s action plane racture criteria are presented in some detail which seem to be the best tool or describing the brittle racture in FRP with polymer matrix.. INTRODUCTION Within the rame o the ESA study Failure Criteria or Non-Metallic Materials detailed guidelines or the proper use o ailure criteria during the dierent stages o composites spacecrat structures design have been elaborated or non-metallic materials. For the engineers it is o interest what kind o ailure will happen and at which load level initial ailure or ultimate ailure occurs leading to corresponding margin o saety. Part I is ocussing on Fibre Reinorced Plastics (FRP) and Part II on Ceramic Matrix Composites []. With respect to ailure criteria o ibre reinorced plastics (FRP) a signiicant amount o work has already been perormed by universities and industry. Many models exist and a considerable amount o test data is available. The World Wide Failure Exercise (WWFE) has helped to have now a kind o evaluated overview on ailure criteria. This Part I shall give a comprehensive overview on ailure criteria rom the engineering point o view with ocus on the most used one o Tsai-Wu and the best one according to the results rom the WWFE. When analysing composite structures it is very important to deine in advance the kind o ailure which has to be assessed. The deinition o ailure is in act an underestimated issue during the ailure analysis. Also within the World Wide Failure Exercise it was noticed, that especially initial ailure has to be deined in the right way. For instance leakage o a pressurised composite tube oten has nothing to do with the initial ailure o single lamina within the tube s wall. A look on the classiication o ailure criteria regarding their original aim has to be made beore application. Failure criteria can be classiied according to: analysis range (initial or ultimate ailure), physical structure model (micro, macro or component level) and mathematical approach (limit, polynomial tensor or physically based). It has to be reminded, that experimental data inluenced by the material and manuacturing process on the one side and the test specimen and test itsel on the other side and also by the physical structure level selected. To ensure a proper design o the composite structure experimental data consistent with the selected ailure criteria has to be used.

2 . FAILURE CRITERIA STATE OF THE ART In the early days o FRP-science some work was done to describe strengths o laminates with dierent lay-ups. It was realised that a reasonable way to analyse laminates could only be accomplished on the level o the single lamina o the laminate []. The stress analysis or this layer-by-layer ailure analysis was perormed by the Classical Laminate Theory (CLT). Only using this concept on the level o single unidirectional lamina allows applying ailure criteria generally. In the past three decades many kinds o diering theories have been developed leading to such a variety o criteria that it is not manageable anymore even by the experts. The design engineer has no interest in and can not accomplish the task o checking all existing criteria to detect the one which he eels is best suited or his problem. For this reason the dierent ailure criteria need to be veriied and benchmarked by independent persons in order to assure that the resulting ranking and recommendations are credible and accepted by the design engineers as well as by the scientiic community. The organisers o the WWFE tackled this incredible task in 99, probably not realizing that they were heading or a decade o work [3]. Now, twelve years later, the results have been published [4], but unortunately a ranking was used in which some very dierent criteria attain quite good marks. The major reason or this ranking is undoubtedly the act that the organisers o the WWFE could not ind enough reliable test results that would have allowed better evaluation and judging the dierent criteria. Thus, once again, the design engineers let with a choice o criteria. Also, as long as there is no suicient experimental database a lot o work will possibly be wasted and progress remains slow because everyone will continue in pursuing his own preerences. One aim o the ESA-study was to create some indubitable reerence data against which one can benchmark existing criteria. Ater a thorough literature study it was decided to design tests measuring the ibre parallel compressive strength o a unidirectional (UD) carbon ibre lamina when high transverse compressive stresses are acting simultaneously with the longitudinal compressive stress. For this case there is almost no experimental data available and it is promising very important results because it belongs to a stress space in which the most widely known criterion (Tsai-Wu) predicts strengths up to three times higher than any other theory which took part in the WWFE. Fig.. Strengths o FRP measured in the (, τ )-stress space. Failure criteria are used to predict the strength o a material or any arbitrary state o multiaxial stress with the preliminary knowledge o only some easy to measure strengths. Thus, to benchmark dierent criteria multiaxial experimental results should be used. In Fig.

3 test results obtained in a large German research project [5] in the (, τ )-stress space and predictions using Puck s action plane criteria are shown or carbon (squares) and glass ibre reinorced plastics (circles). Apart rom Tsai and Puck most o the other authors who participated in the WWFE use a so called maximum stress ailure condition or the unidirectional layer. That means (+) (+) =, (-) ( ) = - R, τ =. Hence their ailure envelope or (, τ )-combinations is given by the dashed lines in Fig.. This is in clear contradiction to the well known interaction o and τ as it can be seen rom Fig.. Further Fig. shows that the true ailure envelope is not symmetric to any line parallel to the τ -axis. The Tsai-Wu Criterion describes an interaction o τ and but uses a symmetric curve which is an ellipse that is shited on the axis due to the dierent uniaxial transverse (+) ( ) strengths and which must lie on the ailure curve. The only ailure envelope in the WWFE that gives an unsymmetrical curve or the prediction or the strengths in the (, τ )- stress area is Puck s Criterion as can be seen in Fig.. The Tsai-Wu Criterion is an interpolation-polynomial without any physical background. It allows no clear dierentiation o ailure types in a lamina but it is easy to use. Sometimes it delivers good results, but or the load cases selected or the experimental work it seems to be extremely unconservative! The problem here is that no user or design engineer really knows where it is unconservative and where not. The most important advantages and details necessary to correctly use the Puck Action Plane Criteria are outlined in chapter 4. One part o the presented study was dedicated to the implementation o Puck s physically based Action Plane Criterion in order to be able to use Puck s very realistic criteria with the design sotware ESACOMP [6] and to make it available or composite designers. 3. THE TSAI-WU CRITERION The quadratic equation that deines the racture envelope o the Tsai-Wu Criterion under plane stress conditions is given in Eq. () [7]. F + F + F τ + F + F + F () 66 = and are the stresses o the UD-lamina parallel and transverse to the ibre direction and τ is the in-plane shear stress. The coeicients F ij and F i are deined by points on the -, - and τ -axes through which the ailure envelope must pass. ; ; F = F = F = F ( ) ( ) ( ) ( ) ( ) = F ( + ) + + ( + ) 66= R R R R R R R R R () (6) The interaction term F has a very strong eect on the (, )-racture curve in the * compressive /compressive quadrant. In the WWFE Tsai used = -0,5 in the ormula F * F F F ; =. (7) F ; A signiicant risk in using this criterion is the prediction o extremely high strengths or biaxial inplane compression as it exists in cases o cylindrical vessels with external pressure as or instance in deep submerge applications. 3

4 4. PUCK S ACTION PLANE CRITERIA Puck s ailure theory is based on a physical background. The model can be used to determine the exertion E (in earlier papers called stress exposure actor ) o a lamina, separately or ibre ailure (FF) and inter ibre ailure (IFF) and thus the corresponding reserve actor or the margin o saety. Moreover, it supplies a lot o inormation on the mode o ailure, which is a very important prerequisite or a realistic post ailure analysis o a laminate as e.g. the degradation o the transverse modulus E and the inplane shear modulus G in a layer ater IFF o Mode A, that allows calculating the stresses and the deormations under post ailure conditions [, 8]. This is described briely in [9] and in ull detail by Knops [0]. Puck s Action Plane Criteria or IFF In his well known 980-paper [] Hashin developed separate invariant-based interpolation criteria or ibre ailure FF and inter ibre ailure IFF (the so called Hashin-Criteria ). In that paper he also mentioned, that more realistic IFF-results could be expected, i the IFF-criteria would be based on O. Mohr s [] racture hypothesis or brittle racture. This states that only the normal and shear stress on the racture plane are responsible or racture. The problem is, that the racture plane, on which IFF occurs under a certain stress combination ( n, τ nt, τ n see Fig. ) is not known in advance. It must be detected by solving an extremum problem. Hashin in 980 was araid that the amount o numerical calculations necessary or detecting the racture plane which is the plane with maximum IFF-exertion could not be tolerated in engineering practice. At the beginning o the 990ies the computing capacity available to design engineers had increased enormously. Thereore, in 99 Puck started his development work on IFF-criteria [3] based on O. Mohr s hypothesis and a modiication by B. Paul (96) [4]. Hence, Puck s Criteria or IFF are based on the assumption that FRP actually do behave like intrinsically brittle material. The necessary veriication or the brittle ailure was given by Huybrechts [5] and Kopp [6, 7], who perormed tests on prismatic UD specimens under uniaxial transverse compression to determine whether the racture would occur under an inclined racture plane, as predicted by Mohr s hypothesis or not. In the experiments the racture angle θ p (see Fig. ) was ound to be θ p =53 ±3 (just the same as or grey cast iron). Based on these results one can state that the basis o Puck s model has been completely veriied. Fig.. Stresses acting on planes perpendicular to the natural axes x, x, x3 o the UD-element and on an inclined ibre parallel plane As can be ound in literature, or instance [8, 9, 8, 9], Puck s ormulation allows to predict IFF even or most general 3D states o stress. In this paper the D state o stress is ocussed which is mostly used when dimensioning FRP components. For (,, τ )-combinations 4

5 Puck has ound an analytical solution or the extremum problem o detecting the inclination angle θ p o the IFF-racture plane: with A R τ cosθ = + p ( + (8) p ) R R R = + p p R R A and p = p R R A (9) - (0) The stress parallel to the ibres has no inluence on θ p [8, 9, 8-0]. Using a modiied equation or θ p Puck could ormulate the IFF-ailure conditions or (, τ )-combinations as polynomials in and τ [8, 9, 9]: ( ) Mode A ( 0): R ( + ) + = + + = 0 ( ) E p τ p = + () θ p R Mode B ( <0); A R [ ( ) ] E = τ + p 0 + p = () θ p = τ τ c Mode C ( <0) ; τ τ c A = τ R ( ) + = θ rom Eq. 8 A ( ) (3) 4 ( R + p R ) ( ) R E p with c R + τ = p (see Fig. 3) (4) I the numerical value o the IFF-exertion E is set to as it has been done in Eqns. () to (3) one gets the racture conditions rom which the racture curve in Fig. 3 is calculated. I or a certain state o stress the numerical value o the polynomial given in Eqns. () to (3) is not equal to, this numerical value is the IFF-exertion E or the given state o stress. With Eqns. () to (3) Puck accounts or three dierent modes o IFF: A, B and C (see Fig. 3). Mode A signiies ailure in the positive range, and Mode B a longitudinal shear ailure in the presence o high τ and moderate compressive stresses. For both Mode A and Mode B the racture plane is the common action plane o and τ (that means θ p = 0, see Fig. ). Mode C prevails, i under suiciently high compressive stress and τ combinations ailure occurs on an inclined racture plane. This racture is brought about by the transverse shear stress τ nt in cooperation with the longitudinal shear stress τ n (see Fig. ). While IFF developing under Mode A and Mode B may be tolerable, Mode C ailures can not be tolerated because o the wedge-eect o the inclined racture planes, that can initiate delamination in adjacent layers, urthers the risk o ibre local-buckling o adjacent layers causing ibre racture and can lead up to a catastrophic ailure o the whole laminate by massive delamination [8, 8, 9]. The three dierent sections o the racture curve in Fig. 3 resulting rom Eqns. (-3) are (+) ( ) excellently in line with the test results in Fig.. In addition to the basic strengths, ( + ) ( ) (+) and R Puck uses our inclination parameters,,, (the latter one just needed or 3D-states o stress). It could be shown in the past [] that these coeicients can be taken as constants or one type o ibre-matrix combination and vary only very little or dierent materials. Table lists recommend values. Thus, or the complete Puck theory the 5 p p p p 5

6 ( + ) (+) ( ) basic material strengths ( R, R,, R, R ) as or the Tsai-Wu Criterion and most other ailure criteria are suicient. θ p τ θ p τ ±36 ±4 0 ±4 ±47 Mode B τ (-) τ ±50 Mode C τ c p (+) p Mode A τ ±54 - (-) A (+) Fig. 3. Predicted racture curve rom Puck s IFF criterion, set up with reasonable material constants Table. Recommended values or the inclination parameters []. p p ( + ) ( ) p = (+) p GFRP 0,5 0,30 0,0 0,5 CFRP 0,30 0,35 0,5 0,30 Important: Using Eqns. () to (4) normally is chosen rom Table. By doing this is always ixed by the parameter coupling given in Eq. (0) which is mandatory in this case. p Puck s Criteria or FF For the well understood ibre ailure Puck recommends or (,, τ )-combinations to use the maximum stress ormulation Eq. (5) he proposed already in 969 [, ]. It expresses the physical idea that ibre ailure under multiaxial stresses in a UD-lamina occurs when the ibre parallel stress is equal to or exceeds the stress necessary or a ailure under uniaxial stress. From this hypothesis ollows the simple FF-condition ( ) p E( FF ) = = ( ± ) R R with R ( + ) or > 0 or < 0 (5) Puck [8, 8, 9] and Fischer [3] worked on a urther improvement o the ibre ailure criterion. This work is based on a more sophisticated ailure hypothesis: FF o a UD-lamina under combined stresses will occur when in the ibres the same stress is reached which is acting in the ibres at an FF o the lamina caused by uniaxial tensile stress or a uniaxial compressive stress respectively. Based on this ailure hypothesis o maximum normal stress o the ibres the FF-condition or the UD-lamina is: ( R ( + ) E( FF ) = ) ν 3 = ± ν m E E with ( R ( R + ) ) or or [...] [...] 0 < 0 (6) 6

7 E(FF) is the FF-exertion, ν is the major Poisson s ratios o the UD-lamina, and ν the major Poisson s ratio o the ibre. E is the longitudinal modulus o the lamina parallel to the ibres and E the longitudinal modulus o the ibres. m is a magniication actor or the transverse stress in the ibres (GFRP m,3 and or CFRP m,). Puck has ound that in case o plane (,, τ )-stress, the results o Eq. (5) and Eq. (6) dier only by a ew percent. (However, the inluence o transverse stresses on FF can become important in the region o combined <0 and 3 <0 o similar magnitude, where and 3 can exceed the transverse ( ) compressive strength by a actor o up to 3 or 4 [7]). Inluence o Stresses parallel to the ibres on IFF-Strength Following strictly O. Mohr s hypothesis, stating that only the stresses acting on a common racture plane are producing the racture, in case o an IFF only the stresses n, τ nt, τ n (see Fig. ) would be responsible or IFF and not the ibre parallel stress, which acts on a plane transverse to the ibres. But as already mentioned in his early work o 969 [, ] Puck expects an inluence o high stresses on IFF-strength. There is no interaction o and the stresses n, τ nt, τ n in the usual sense o an cooperation in bringing about the ailure like the interaction shown in Eq. (6). The stress does certainly not cooperate directly with n, τ nt, τ n in producing the brittle racture because is not acting on the racture plane. Instead ( + ) ( ) ( ) reduces the IFF-strengths R, R, ( is proportional to the action plane racture A resistance ) by micro-damage caused by ailures o single ibres or small ilaments, which ( + ) occur progressively, when comes close to R or R respectively. Based on this consideration Puck ormulated a degradation actor η w (w = weakening). All (+) A three action plane racture resistances,, R governing IFF are multiplied with the same weakening actor η w <. Decreased IFF-strength means increased IFF-exertion. Thereore, the IFF-exertion E respecting a weakening due to ollows rom E according to Eqns. () to (3) by setting E E = η w = E c ( a c² ( a² s² ) + + s) ( ca) + with c = ( E / E(FF) ), valid or /s c m. The parameter s marks the starting point o the weakening eect and a is the axis o an assumed elliptical racture curve given by (7) s a = (8) m² where m is the assumed minimum value o η w. The eects o s and m can be seen in Fig. 4. Eq. (7) is valid or /s c m. c > /s means no weakening and c < m means no IFF beore FF. For the time being no test results are available rom which igures or s and m could be taken. It is recommended to use preliminary medium igures s = 0.5 and m = 0.5 [9]. 7

8 IFF (+) m = s = 0.5 m = 0.5 m = 0 FF (-) (-) (-) (+) (+) -R -s R - s R R Fig. 4. Possible shapes o the IFF curve depending on the parameters s and m The treatment o the weakening eect in IFF-analysis is described briely in [9] and in more detail in [0]. The parameters s and m can be chosen between 0 and. This presents an excellent adoptability o the racture envelope to test results. But, the severe problem at the moment is, that no reliable test results are available or (, )-compression as such tests are extremely diicult. 5. TEST-SETUP FOR BIAXIAL INPLANE COMPRESSION Tubular specimens loaded by axial compression F X and an external pressure p can be used to obtain the desired state o biaxial inplane (, )-compression in an inner test layer (see Fig. 5 and 6). The design o adequate tubular specimen is a complex task. An optimization process resulted in a lay-up [89 ;(+ ;- ) 4 ;89 ]. The circumerentially wound 89 -layers are made o two plies and have a nominal thickness o 0.3 mm and are made rom T300 Carbon ibre (00 tex). The supporting -layers are manuactured as single plies rom T700 Carbon ibre (300 tex), their total nominal thickness is mm. The used epoxy resin is Araldite LY556/HY97/D070. Fibre volume content is 60%. The supporting layers were used to reach the necessary saety against buckling which is extremely critical or shells under combined external pressure and axial compression which. These layers in combination with the inside and outside circumerential layers produce a sort o sandwich eect that results in a rather higher circumerential bending stiness. Angles larger than would result in the need o too high external pressure, smaller angles would lead to an undesirably high necessary axial compression orce to obtain the desired stresses in the test layer. pressure vessel membrane specimen F X Ø 45. mm Ø 40 mm F X p 80 mm 00 mm Fig. 5. Test setup principle or biaxial compression loading and photo o test specimen. 8

9 Fig. 6. Test setup or biaxial compression loading. In spite the act that the circumerential 89 test layer was placed on the inside o the tube, due to the external pressure p a certain radial compressive stress 3 is acting on the outside surace o the test layer. 3 does however not exceed 0 N/mm which appears acceptable. 6. RESULTS & DISCUSSION So ar the specimens were tested under our load cases (TS-TS4) leading to dierent (, )- compression stresses combinations. The stresses in the test layer were determined or each o the load cases by an FE-analysis. For these FE-analyses a model o the specimen meshed with volume elements (Solid 46 in ANSYS) was used and the averaged external loads at ailure o the specimens tested or one load case was applied. The stresses resulting or the dierent load cases are given together with average load at ailure and standard deviation in Table. Inspection o the ailed specimen showed that ailure is most likely not caused by the desired pure ibre racture in the test layer. Instead, it is suspected that two dierent undesired kinds o ailure occur. Some evidence o buckling which leads to the destruction o the shell could be observed in strain measurements on a circumerence which show characteristically diverging strains in some tests. Characteristics o IFF in Mode C could be seen on some other specimens. This is in contrast to the results with the Puck action plane criterion, which did not predict an IFF-exertion E in any layer higher than 50% o the exertion or FF in the test layer. Failure in the outer 89 layer should not occur according to dierent diameters o the inner 89 layer (40.3 mm) and the outer layer (44.9 mm). The exertion or FF due to external pressure is % higher in the inner layer (44.9/40.3=.) than in the outer layer. Microscopic investigations showed that the layers have very high void content which could explain inaccurate predictions as the void content strongly degrades critical strength or Mode C IFF. Table. Average external loads at ailure and calculated stresses, in the inner test layer. Load at ailure Stresses at ailure in the inner 89 layer valid specimens F X [N] p [MPa] [MPa] [MPa] TS 0 8,5-8 -6, 7 STAND [%] - 3,3 TS , ,5 5 STAND [%] 5,5 6,9 TS , ,9 6 STAND [%] 5,9 5,8 TS , ,3 5 STAND [%] 6,4 7, 9

10 To solve the current problem o specimens ailing much earlier than expected several approaches o improvement are under investigation: The lay-up shall be modiied in order to obtain a test tube which withstands the external pressure even ater the ailure o the inner test layer. This would allow proving deinitely ibre racture as the cause o ailure in the test layer. The ± supporting layers shall be substituted by pure resin. The strain to a Mode C ailure in the pure resin is expected to be much higher than that o the unidirectional layer. Voids appeared in the ± layers can be avoided by a high quality resin casting process. 7. TEST RESULTS FROM LITERATURE FOR BIAXIAL COMPRESSION In the literature the highest reported strengths under biaxial (, )-compression are rom Swanson et al. [4, Fig. 0]. They were measured on tubular specimens with a quasi isotropic 0 /90 /±45 -lay-up. Unortunately, each o these test points belongs to a single tested specimen and Swanson also suspects a circumerential buckling to cause the ailure. Statistically more secured results are reported rom Welsh and Adams [5]. They perormed tests on cruciorm specimens made rom plates with a [0 /90 ] 0 lay-up. It is not sure whether the stress distribution in their test region is homogenous enough as it should be or tests used to actually benchmark ailure criteria. To the knowledge o the authors the results which Welsh and Adams report allow to draw the most complete racture curve measured so ar. But because o the 0 /90 -ibre orientation o the test specimens it is very questionable i the ailure curves are applicable to unidirectional layers. Following the exertion analysis o these results it is estimated that buckling was involved in the tests as well. Swanson et al. and Welsh and Adams report their racture stresses in global stresses ( x, y ) on the tested laminate and thus their results can not directly be compared to the predictions o existing ailure criteria. For this reason the results are transormed into stresses or the UDlamina using inormation given on the specimen s geometry and the lamina properties or the ibre matrix combination tested (AS4/350-6) as given in Part A o the WWFE [6]. transverse stress [N/mm ] TS TS FF according to Puck Eq. (5) IFF ac. to Puck, Eqns (),(3) and Eqns (7),(8) with s = m = 0.5 Tsai-Wu Criterion Eqns ()-(7) F *= -0.5 Results o the Test Series in this study O. Fischer [3] Swanson et al. [4] Welsh & Adams [5] TS3 TS4 ibre parallel stress [N/mm ] Fig. 7. Stresses at ailure in one layer, calculated rom external loads or stresses. From the racture stresses (, ) in the layers with dierent ibre direction the highest stresses are taken as strength values and plotted in Fig. 7. In this igure the results o Swanson 0

11 et al (squares), Welsh and Adams (circles), Fischer (rhombi) [3] and the test results o this study (triangles, calculated by FEA) are compared with the predicted ailure envelopes o the Tsai-Wu Criterion and Puck s Theory. As one can see rom Fig. 7, in none o the quoted works the measured strengths under combined biaxial (, )-compression stresses gets close to the uniaxial compressive strength R or high transverse compression as given by [5]. Compared to these stresses, the stresses achieved in the presented test campaign represent already a remarkable improvement. The stresses come rom at least 5 valid tests per load ratio. The ability to urther develop the present specimen in order to avoid buckling and other undesired ailures beore reaching the ibre compressive strength at presence o a high transverse compression stress is absolutely given. O course, none o the results in Fig. 7 can be used to adapt ailure criteria or evaluate the predictions made by them. Another question arises rom these results: as up to now it has not been possible to reach the real ibre strength under combined compressive loading, not even with the most sophisticated test set-ups, it is diicult to imagine in which real structure stresses much higher than those achieved in the tests could be reached without a prior ailure o the structure because each specimen is nothing else than an simple FRP structure thoroughly designed to withstand very high (, )-compressive stresses. However, o course this consideration does not relieve us rom the task o inding the true strengths under combined (, )-compressive stresses, but it still shows that all criteria would lead to unconservative predictions as even the uniaxial compression strength R predicted as the strength by all maximum stress criteria has not been reached so ar in any tests. 8. CONCLUSION As the results rom the literature and the presented new experimental results have shown that the determination o the strength or biaxial (, )-compression remains a very diicult task. However, it can be stated that the large ellipse in the Tsai-Wu ailure envelope predicting strengths 4 times higher than any test result (Fig. 7) is unrealistic. In spite o the encountered diiculties, one shall continue to elaborate the test set-up and do other veriication tests because only in this way a major problem in the dimensioning o FRP can be solved. There is not a lack o theories but a lack o reliable test results to benchmark the available theories. Thus, it is diicult or the design engineer to know which criteria to trust on the basis o the summary [7] o the WWFE. Also Part C o the WWFE recently published and taken into account the contribution o the so-called post-runner o the exercise [8, 9] has not helped to clariy the situation. Thereore, urther testing is necessary as it is in no ones interest to ollow one o the advices [30] given in connection with the WWFE which was simply to use several criteria simultaneously and then trust the highest calculated exertion. This would mean that the calculating eort reaches a maximum, and the design will most o the time be very conservative. A conservative design, without knowing how conservative it is, will always represent a risk or the designing engineers and maybe even more important, the weight and costs o the designed parts will in most cases be too high, because the light weight potential is not ully exploited. ACKNOWLEDGEMENTS The presented work was unded by the European Space Agency (ESA) under the contract no. 66/0/NL/CP.

12 Reerences. Kaiser, C., Weihs, H., Wittke, H. and Obst, A., Failure Criteria or Non-Metallic Materials Part II: Ceramic Matrix Composites, Proceedings o the ECCM, 3. May June 004, Rhodes, Greece. Puck, A., Festigkeitsberechnung an Glasaser/Kunststo-Laminaten bei zusammengesetzter Beanspruchung; Bruchhypothesen und schichtweise Bruchanalyse ( Strength analysis on GRPlaminates under combined stresses; racture hypotheses and layer-by-layer ailure analysis ), Kunststoe, German Plastics 59 (bilingual edition English and German), (969), 8-9, German text Hinton, M.J. and Soden, P.D., Predicting Failure in Composite Laminates: The Background to the Exercise Comp. Sci. and Techn. 58 (998) Hinton, M.J., Kaddour, A.S and Soden, P.D., Evaluation o the Failure Prediction in Composite Laminates: Background to Part B o the Exercise Comp. Sci. and Techn. 6 (00) Cuntze, R. et al., Neue Bruchkriterien und Festigkeitsnachweise ür unidirektionalen Faserkunststoverbund unter mehrachsiger Beanspruchung Modellbildung und Experimente (New Failure Criteria and Strength Proos or unidirectional Fiber Reinorced Plastics under multiaxial load models and experiments), VDI-Fortschrittberichte Reihe 5 Band 506, VDI-Verlag, Düsseldor, Liu, K. and Tsai, S., A Progrssive Quadratic Failure Criterion or a Laminate, Comp. Sci. Techn. 58 (998) Puck, A., Festigkeitsanalyse von Faser-Matrix-Laminaten Modelle ür die Praxis (Strength analysis o ibre-matrix laminates models or practice), in German, Carl Hanser Verlag, München, Wien, N.N., VDI-Guideline 04, Development o FRP components, Part 3 Analysis (bilingual, German and English), Beuth-Verlag, Berlin, to be published in Knops, M., Sukzessives Bruchgeschehen in Faserverbundlaminaten (Gradual ailure process in ibre/polymer laminates), Technisch-wissenschatlicher Bericht, IKV Aachen, Band 40, Verlag Mainz, Aachen, ISBN Hashin, Z., Failure Criteria or Unidirectional Fiber Composites, J. Appl. Mech. 47 (980), Mohr, O., Welche Umstände bedingen die Elastizitätsgrenze und den Bruch eines Materials? (Which circumstances are causing yield limit and racture o a material?), in German, Z.d.VDI 4 (900) 44,S u.46, Puck, A., Ein Bruchkriterium gibt die Richtung an (A ailure criterion shows the direction), Kunststoe, German Plastics (bilingual edition English and German) 8 (99) 7, 9-3, German text Paul, B., A Modiication o the Coulomb-Mohr Theory o Fracture, J. App. Mech. June (96) Huybrechts, D., Ein erster Beitrag zur Veriikation des wirkebenebezogenen Zwischenaserbruchkriteriums nach Puck (a irst contribution in veriying the action-plane-related inter-iber-ailure criterion ater Puck) (996), Technisch-wissenschatlicher Bericht, IKV Bd. 44, Verlag Mainz, Aachen, ISBN Kopp, J. and Michaeli, W., More eicient composite component design using action plane related IFF strength criteria, Proceedings 43 rd International SAMPE Conerence, Anaheim, June Kopp, J., Zur Spannungs- und Festigkeitsanalyse von unidirektionalen Faserverbundkunststoen (Contribution to the Stress and strength analysis o unidirectional ibre reinorced plastics) (000), Technisch-wissenschatlicher Bericht, IKV Bd. 99, Verlag Mainz, Aachen, ISBN Puck, A. and Schürmann, H.; Failure analysis o FRP Laminates by means o physically based phenomenological models, Comp. Sci. and Techn. 58 (998) Puck, A. and Schürmann, H., Failure analysis o FRP Laminates by means o physically based phenomenological models, Comp. Sci. and Techn. 6 (00) Puck, A. and Knops, M., State o the art o action plane racture criteria or ibre/polymer laminates, Comp. Sci. and Techn., to be published in 004. Puck, A., Kopp, J. and Knops, M., Guidelines or the determination o the parameters in Puck s action plane strength criterion, Comp. Sci. Techn. 6 (00), 3, and 9, 75. Puck, A. and Schneider, W., On ailure mechanisms and ailure criteria o ilament-wound glass-ibre / resin composites Plastics & Polymers, The Plastics Institute Transactions and Journal, (969), 33-4, Pergamont Press, Oxord (UK) 3. Fischer, O., Faserbruchgeschehen in kohlenstoaserverstärkten Kunststoen (Fibre racture behaviour in carbon ibre reinorced plastics) (003), Technisch-wissenschatlicher Bericht, IKV Bd. 43, Verlag Mainz, ISBN X

13 4. Soden, P.D. and Hinton, M.J. and Kaddour, A.S., biaxial test results or strength and deormation o a range o E-glass and carbon ibre reinorced composite laminates: ailure exercise benchmark data Comp. Sci. and Techn. 6 (00) Welsh, J. and Adams, D., An experimental investigation o the biaxial strength o IM6/350-6 carbon/epoxy cross-ply laminates using cruciorm specimens, Composites: Part A 33 (00) Soden, P.D., Hinton, M.J. and Kaddour, A.S., Lamina properties, lay-up conigurations and loading conditions or a range o ibre-reinorced composite laminates, Comp. Sci. and Techn. 58 (998) Hinton, M.J., Kaddour, A.S. and Soden, P.D., A comparison o the predictive capabilities o current ailure theories or composite laminates, judged against experimental evidence, Comp. Sci. Techn. 6 (00), Hinton, M.J., Kaddour, A.S and Soden, P.D., Evaluation o the Failure Prediction in Composite Laminates: Background to Part C o the Exercise Comp. Sci. and Techn. 64 (004) Soden, P.D., Kaddour, A.S and Hinton, M.J., Recommendations or Designers and Researchers resulting rom the World Wide Failure Exercise Comp. Sci. and Techn. 64 (004) Hinton, M.J., personal note at the workshop on ailure analysis o FRP (9. Oct. 00), Seville, Spain 3