Inverse first-order reliability method for probabilistic fatigue life prediction. of composite laminates under multiaxial loading

Transcription

1 Inverse irst-order reliabilit method or probabilistic atiue lie prediction o composite laminates under multiaial loadin Yibin Xian and Yonmin Liu* Department o Civil and Environmental Enineerin Clarson Universit Potsdam Y.3699 USA Abstract: Multiaial atiue reliabilit is a challenin problem despite etensive proress made durin the past ew decades. Anisotropic materials such as composite laminates are under eneral multiaial stress state even i the applied loadin is uniaial. A eneral methodolo or multiaial atiue reliabilit analsis o composite laminates is proposed in this paper. The proposed methodolo is based on a uniied multiaial atiue model or both isotropic and anisotropic materials and the inverse irst-order reliabilit method Inverse FORM or probabilistic lie prediction. The current atiue model is a critical plane-based model. The critical plane orientation is theoreticall determined b minimizin the damae introduced b the hdrostatic stress amplitude. One o the advantaes o the multiaial atiue model is that it has almost no applicabilit limitations with respect to dierent materials. A time dependent limit state unction o material ailure is developed based on the proposed mechanism model or probabilistic lie prediction. Inverse FORM method is proposed to calculate the atiue lie under a speciied ailure probabilit. Various uncertainties rom materials properties pl coniurations and volume ractions are included in the proposed methodolo. A wide rane eperimental atiue data o composite laminates is used to validate the proposed methodolo. It is observed that the proposed methodolo ives a satisactor * Correspondin author Tel.: ; Fa: ; liu@clarson.edu

2 prediction or both median lie and its conidence bounds. Kewords: multiaial atiue composite laminate reliabilit inverse FORM Introduction Composite materials are widel used or man dierent industries such as aerospace and automobile because o their hih strenth and stiness. The lon term durabilit o composite materials is critical or the saet and interit o structural and mechanical sstems. Composite materials are inhomoeneous and anisotropic which maes the atiue problems o composite materials more complicated than that o homoenous and isotropic materials e.. metallic materials. The atiue o composite laminates is multiaial and a special analsis approach is required or an accurate lie prediction. In eneral the multiaial problem can be divided into two cases: one is caused b the anisotrop o composite materials and the other is caused b the eternal multiaial loadin. Man enineerin materials ehibit some deree o anisotrop in mechanical properties such as unidirectional and multidirectional composite. Unlie the etensive proress in multiaial atiue analsis o isotropic materials much urther eort is needed to include the anisotrop o the material Miller and Brown985. Several investiations have been reported or anisotropic composite laminates. Deriec and Van Paepeem 00 classiied eistin atiue models into three cateories: atiue lie models S- curves phenomenoloical models or residual strenth or residual stiness and proressive damae models. Hasin and Rotem 973 proposed a ailure criterion which mimic the orm o static strenth criterion based on two major ailure modes iber ailure and matri ailure. Sims and Brodon 977 etended the static ailure

3 principle proposed b Tsai-Hill to atiue problem. Aboul Waa et al. 997 investiated the application o some polnomial ailure criteria or atiue analsis. Reisnider and Gao 99 proposed a micromechanics-based model which can tae into account the interacial bond. Wu 993 proposed dierent ailure criteria based on the Tsai-Hill criterion. Philippidis and Vassilopoulos Sep 999 proposed a ailure criterion based on the Tsai-Wu criterion. Petermanna and Plumtree 00 proposed a critical plane model or unidirectional laminates under o-ais tension-tension atiue loadin. Kawai 00 Kawai et al 00 proposed an eective stress model or the unidirectional laminates under o-ais loadin which is based on the Tsai-Hill static strenth theor. Liu and Mahadevan 005 proposed a multiaial damae accumulation model or multidirectional laminates under tension-tension atiue loadin. Sinle applied o-ais loadin causes proportional multiaial stress state within the laminates. Most o the atiue models or anisotropic composite laminates are or proportional multiaial stress state; however ver ew theoretical and eperimental studies are ound in the literature or the non-proportional multiaial atiue analsis. o universall accepted multiaial atiue damae model eists or dierent materials and dierent loadin conditions Liu and Mahadevan007. In addition hue uncertainties are associated with the atiue damae process o composite materials which is usuall larer than that o metallic materials Liu and Mahadevan007. The uncertainties associated with the atiue damae accumulation can be caused b material properties structural coniurations and the manuacturin processes. A probabilistic approach is more suitable or atiue analsis o composite materials. 3

4 The e objective o this stud is to develop a probabilistic lie prediction ramewor or composite laminates. Man atiue reliabilit analsis methodoloies used the simulation-based approach such as direct Monte Carlo simulation to calculate the probabilistic lie prediction Liu and Mahadevan 007; Liu and Mahadevan009. This approach is time consumin or sstem level applications. Other approaches solved the problem in a similar wa with that o the time-dependent reliabilit i.e. calculatin the reliabilit level at a taret lie Liu and Mahadevan007 Liu and Mahadevan009. This approach cannot ive direct probabilistic lie prediction. A novel inverse irst-order reliabilit method FORM is proposed in this stud to calculate the residual lie directl. The inverse FORM method is oriinall proposed or reliabilit-based optimal desin RBDO problem Der Kiurehian Zhan and Li994. Several studies or static ailure usin the inverse FORM method have been reported in the literature. Sanranasoontorn et al. 004 developed an inverse reliabilit procedure or wind turbine components. Chen et al. 006 applied the inverse FORM method to estimate the cable saet o lon-span bride. Ver ew studies have been ound on the investiation o the inverse FORM method to time dependent atiue problem probabl due to the diiculties with the implicit response unction Chen Zhan Cai and Xiao007. In this paper a eneral methodolo or multiaial atiue reliabilit analsis o composite laminates is proposed. The proposed methodolo is based on a uniied multiaial atiue model or both isotropic and anisotropic materials Liu and Mahadevan 007 and the inverse irst-order reliabilit method FORM or time dependent atiue reliabilit analsis. The current atiue model is a critical plane-based model. Most o the earlier models based on the critical plane approach assume that the critical plane onl 4

5 depends on the stress state. In the current model the critical plane not onl depends on the stress state but also on the material properties. The critical plane is theoreticall determined b minimizin the damae introduced b the hdrostatic stress amplitude which maes the proposed model have almost no applicabilit limitations with respect to dierent materials. Various uncertainties rom materials properties pl coniurations and volume ractions are included in the proposed methodolo. Probabilistic lie prediction usin the inverse FORM method is compared with direct Monte Carlo simulation or model veriication. A wide rane o eperimental atiue data o composite laminates are used to validate the proposed methodolo. Generall the predictions based on the proposed model aree with the eperimental observations ver well. Multiaial atiue model Multiaial atiue model or isotropic materials A multiaial atiue damae criterion Liu and Mahadevan005 was developed based on the nonlinear combination o the normal stress amplitude shear stress amplitude and hdrostatic stress amplitude actin on the critical plane as σ aα τ t aα σ H aα β where σ aα a α t and H σ aα are the normal stress amplitude shear stress amplitude and hdrostatic stress amplitude actin on the critical plane respectivel; α is the anle between the critical plane and the maimum normal stress plane; and t are atiue limits in pure uniaial and pure shear tests respectivel; and and β are material parameters which can be determined b uniaial and pure shear atiue limits. Detailed 5

6 derivation and validation o the used multiaial atiue model can be ound in the reerred article Liu and Mahadevan005. Onl the results o the model parameters are reported here in Table. In Table s t is the atiue strenth ratio under the pure shear loadin and the pure uniaial loadin. For an arbitrar multiaial loadin histor the maimum stress amplitude plane is identiied irst. This is achieved b enumeration b chanin the anle b deree increments. Then the anle α and material parameters are determined or dierent materials accordin to Table. The critical plane is the plane which has an anle α with the maimum normal stress amplitude plane. Finall the stress components on the critical plane are calculated and the atiue damae is evaluated usin Eq.. ote that the critical plane in the proposed model depends not onl on the stress state maimum normal stress amplitude plane but also on the material propert anle α. Multiaial Fatiue Model or anisotropic material Man enineerin materials ehibit mechanical anisotrop such as wood rolled metals iber reinorced composite laminates etc. The uniaial and torsional atiue strenths also depend on the orientations o the aes at the critical point within the material. In the proposed multiaial atiue criterion Eq. atiue limits and t become unctions o the orientation θ sa θ and t θ. In order to etend the atiue model Eq. to anisotropic materials we need to speci a reerence plane on which the atiue strenth under uniaial and pure shear loadin can be evaluated. In the current model the e point is to calculate the anle between the maimum normal stress 6

7 amplitude plane and the critical plane. The reerence plane is irst deined or the anisotropic material as the plane that eperiences the maimum normal stress amplitude. Thus Eq is rewritten as a uniied multiaial atiue criterion: H σ a α τ a α σ a α β θ t θ θ ma ma where θ ma indicates the direction o maimum stress amplitude. For isotropic materials Eq. reduces to Eq. since the unctions θ and t θ become constants. The atiue lie model or anisotropic materials can be epressed as: ma β σ ac θ ma H τ ac σ ac θ ma 3 t θ ma Eq. 3 can be rewritten as: p θ ma β σ ac τ ac σ s θ ma H ac 0 4 where s t θ ma θ ma is the strenth ratio o under pure shear loadin and the θ ma uniaial loadin alon the direction o θ ma. p θ ma θ ma is the ratio o 0 uniaial strenth alon the directions o θ and θ 0. The let side o Eq. 4 can θ ma be treated as an equivalent stress amplitude. It can be used to correlate with the atiue lie usin the uniaial S- curve alon the direction o zero deree. Detailed derivation and concept can be ound in Liu and Mahadevan007 The procedure or the atiue analsis o anisotropic materials is almost identical with that o isotropic material. For an arbitrar loadin histor the maimum stress 7

8 amplitude plane is identiied irst. The uniaial and pure shear atiue strenth alon this direction is also evaluated rom eperimental data. Then the anle α and the material parameters are determined or dierent materials accordin to Table. otice that the quantit s in Table is now redeined as t θ ma s s θ ma. Finall the θ ma equivalent stress amplitude and the atiue lie are calculated usin Eq. 4. For an arbitrar anisotropic material the variation o the uniaial and pure shear atiue strenths correspondin to the orientation o the aes is quite comple and requires etensive eperimental wor to quanti. However or some special anisotropic materials this can be simpliied usin one o the strenth theories available in the literature. In this paper an eample o orthotropic composite laminate is used or illustration. Consider a iber reinorced composite laminate. Several static strenth theories have been proposed or orthotropic laminates such as Tsai-Hill and Tsai-Wu theor Daniel and Ishai006. In this stud the Tsai-Wu theor is used. For the case o plane stress the Tsai-Wu theor is epressed as: F 66 6 σ F σ F σ F σ F σ F σ σ 5 where σ and σ are the stresses alon the iber direction and transverse to the iber direction respectivel and σ 6 is the in-plane shear stress. F F F 66 F F and F are strenth parameters and can be calibrated usin eperiments. F F 66 F F F s s s s s s s s L s LT L F L F L F 0.5 T T T T 6 8

9 where s ± L s ± T are the strenths alon the iber direction and transverse to the iber direction respectivel. The plus smbol indicates tension strenth and the minus smbol indicates compression strenth. s LT is the in-plane shear strenth. For the atiue problem the stress terms in Eq. 5 reer to the stress amplitudes alon dierent directions. I the strenths are deined usin stress amplitude values the plus and minus smbols in the above strenth notation disappear since the stress amplitude is alwas positivel deined. Thus Eq. 5 and Eq. 6 are rewritten or the atiue problem as: F 66 6 σ F σ F σ F σ σ 7 F F F F F66 F 8 sl st slt Usin the Tsai-Wu strenth theor the uniaial strenth and shear strenth alon an arbitrar direction θ can be easil obtained as θ / t θ / F F 4 cos θ F F 8F 4 sin θ F 66 F sin θ cos θ F 66 sin θ cos θ cos θ sin θ 9 For the atiue lie model the atiue strenth coeicients are also unctions o the atiue lie which can be evaluated rom the eperimental S- curves. Eq. 9 is rewritten as: t θ / θ / F F 4 cos θ F F 8F 4 sin θ F 66 F sin θ cos θ F 66 sin θ cos θ cos θ sin θ 0 Substitutin Eq. 0 into Eq. 4 we can solve or the atiue lie. Similar to isotropic materials Eq. 4 usuall has no closed orm solution. In practical calculation a trial and error method can be used to ind. For an orthotropic composite laminate the 9

10 eperimental S- curves alon the iber direction transverse to the iber direction and in-plane shear stress are required in the proposed model. Then the atiue lie under arbitrar multiaial loadin can be predicted. The atiue model or the isotropic material is consistent with the atiue model or the anisotropic material derived in this section. I F F F66 the atiue model or the 3 orthotropic material is identical with the atiue model or the isotropic material with s 3 in which the Tsai-Wu criterion reduces to the von Mises criterion. The above discussion can be easil applied to a laminate with multiple plies ollowin the steps described in Liu and Mahadevan005. First divide the total atiue lie into several blocs. In each bloc chec the ailure o each pl usin the above model. I no ailure occurs accumulate the atiue damae or each pl. I ailure occurs assume that the pl strenth and stiness decrease to zero. Then update the lobal stiness matri and proceed to the net step. The computation continues till the entire laminate ails. The number o the loadin ccles to ailure is the atiue lie o the composite laminate. Inverse FORM method The above discussion is or deterministic analsis and is not suicient to capture the stochastic behavior o atiue damae o composite materials. A eneral inverse reliabilit methodolo is proposed in this stud to include various uncertainties rom materials eometries and manuacturin or probabilistic atiue lie prediction o unidirectional and multidirectional composite laminates. Details are shown below. Inverse FORM method The irst-order reliabilit method is a widel used numerical technique to calculate the reliabilit or ailure probabilit o various enineerin problems. Man studies have been 0

11 reported on static ailure problems usin the FORM method Thorndahl and Willems008 Sas and Barr996 Cizelj Mavo and Riesch-Oppermann994. It has been applied to atiue problems to calculate the time dependent reliabilit. Unlie the FORM method Liu and Mahadevan009 Haldar and Mahadevan000 the inverse FORM method tries to solve the unnown parameters under a speciied reliabilit or ailure probabilit level which is more suitable or probabilistic lie prediction i.e. remainin lie estimation correspondin to a taret reliabilit level. In the inverse FORM method a limit state unction needs to be developed irst such as the eneric epression o Eq.-a. is the vector o random variables and is the vector o indein variables. For eample could be the random material properties loadins and environmental actors and could be the time and spatial coordinates. The limit state unction need be transormed to the standard normal space or the calculation which is similar to the classical FORM method Haldar and Mahadevan000. The numerical search or the unnown parameters needs to satis the reliabilit constraints which are described in Eqs.b-c. β is the reliabilit inde which is deined as the distance rom oriin to the most probable point MPP in the standard normal space. The ailure probabilit P can be calculated usin the cumulative distribution unction CDF Φ o the standard Gaussian distribution. umerical search is required to ind the optimum solution which satisies the limit state unction Eq. d. Details o the eneral inverse FORM method and concept can be ound in Der Kiurehian Zhan and Li994.

12 Φ 0 : : : 0 : d p c b a β β The overall objective o the inverse FORM method is to ind a non-neative unction satisin all constraint conditions speciied in Eq.. Then the numerical search alorithm can be used to ind the solutions o the unnown parameters. Followin the eneral concept o the irst-order reliabilit method the limit state unction is approimated usin the irst-order Talor s series epansion to acilitate the calculation. First the limit state unction Eq. a is epanded around random variable vector and the indein variable vector is ied i n i i O Eq. can be rewritten as [ ] u u u u u 3 where n u The increments o and can be epressed as [ ] 0 d d 4 A non-neative merit unction considerin the constraints o Eq. a and Eq. d can be written as

13 3 [ ] 5 In Eq.5 both and are constants. et the reliabilit constraint Eq. b needs to be included. Substitute Eq.-b into Eq.-d one can obtain ar et t β 6 Usin irst order Talor s series epression the limit state unction can be epanded around and as m i i m m i i n i i O 7 Substitute Eq.7 into Eq.8 the indein variables can be epressed as [ ] et t ar β α 8 The increments o and can be epressed as [ ] d d et t et t ar ar β β 9 A merit unction considerin the reliabilit constraints can be written as ar 3 et t β α 0 Combine the two merit unction Eq. 5 and Eq. 0 a eneral unction is obtained as

14 4 [ ] 3 β umerical search alorithm is developed to iterativel solve the Eq.. The search alorithm is epressed as Eq. ater iterations. d a d a X d X X where d and d are the search directions correspondin to dierent merit unctions and can be calculated usin Eq. 4 and Eq. 9 respectivel. a and a are the weiht actors and can be calculated as a a 3 The converence criterion or the numerical search alorithm is ε 4 where ε is a small value and indicates that the relative dierence between two numerical solutions is small enouh to ensure the converence. Usin the proposed methodolo the comple probabilistic atiue lie prediction problem can be solved eicientl compared to the direct Monte Carlo simulation method. It is noted that the above derivation assumes the random variables are standard Gaussian variables. In practical enineerin application non-gaussian variables are commonl used or some nonneative phsical quantiies such as strenth and Youn s modulus. The proposed

15 inverse FORM method can be etended to non-gaussain variables with proper random variable transormation. This paper uses the transormation method proposed b Racwitz and Fiessler June 976 to transorm the non-gaussian variables to their equivalent standard normal space. Ater that the proposed inverse method can be used. Once the solutions are obtained in the standard normal space the inverse transormation can be used to transorm the solution to its oriinal space. The random variable transormation can be epressed as * X Φ F X σ X * X φ σ X σ X * * * X Φ [ FX ] σ X * * φ { Φ [ FX ]} X σ X * X 5 where Φ and * F X are the cumulative distribution unctions CDF o the standard normal random variable and the non-normal random variable respectivel. φ and * X are the probabilit densit unction PDF o the standard normal random variable and the non-normal random variable respectivel. This transormation wors well or the atiue problem o composite laminates since the distributions o random variables are not hihl sewed. For hihl sewed distribution the transormation proposed b Racwitz and Fiessler 978 can be used instead. umerical eample and model veriication The above discussed inverse FORM method is applied to probabilistic atiue lie prediction o composite laminates interatin the mechanism model. The limit state unction is shown as X X... X

16 X -3 are the random variables. is the indein actor and represents the ailure time. represents the proposed model o atiue lie prediction or composite materials. It is noted that the is a eneric implicit unction and no analtical solution or the derivatives is available. The perturbation-based inite dierence method Sauer006 is used to calculate the irst-order derivatives in the proposed inverse FORM ramewor. Thirteen random variables are included in the calculation. The includes the elastic modulus E E Poisson s ratio υ shear modulus G volume raction o ibers V pl thicness t and pl orientations θ. These random variables represent the basic material properties eometric coniurations and manuacturin actors. Material random atiue properties are also included. The ittin parameters o the material S- curves are assumed to be random variables. The classical power unction is used to describe the atiue S- curves under lonitudinal transverse and pure shear loadins i.e. S B A 7 where S is the stress amplitude level and is the atiue lie. A and B are material properties and are assumed to be random variables. Since three independent atiue S- curves are required in the proposed mechanism model si random variables are included. A D55 balanced laminate is selected or numerical eample which consists o three pairs o pl with identical thicness and elastic properties but with ±0 deree orientations. The mean value o the above mentioned random variables can be ound in Mandell J.F.Feb For demonstration purpose the coeicient o variation or all random variables is assumed to be All random variables are assumed to be 6

17 lonormal variables ecept or the power coeicient B in Eq. 4 which can tae the neative value and is assumed to ollow normal distribution. Fi. Comparison o the direct MC method with the inverse FORM method Fi. shows the probabilistic lie prediction usin both inverse FORM method and the direct Monte Carlo method. The solid line is the result o Monte Carlo MC Simulation with one million samples at a certain stress level at MPa. The inverse FORM results are shown as the trianular points which aree well with the MC simulation. The proposed inverse FORM can ive an accurate result and siniicantl reduce the computational time. It taes 9044 seconds usin the MC simulation. The inverse FORM method taes 4 seconds. All computations are perormed usin Matlab 007 on a dualcore PC.66 GHz with 3 Gb memor. The operatin sstem is Windows XP Proessional. Validation o the proposed method A wide rane o eperimental data on unidirectional and multidirectional composite laminates is used to demonstrate and validate the proposed probabilistic lie prediction methodolo. Eperimental data and statistics o input random variables Seven sets o atiue eperimental data or unidirectional composite laminate under o-ais loadin are emploed in this section and are listed in Table. The eperimental S- curves alon the iber direction transverse to the iber direction and pure in-plane shear stress are required in the proposed atiue model. The curves alon and transverse to the iber direction are usuall reported. However most o the atiue eperimental data do not include the pure shear test results. It is possibl due to 7

18 the diicult o applin the pure shear loadin to the composite laminate. In the proposed stud the S- curve under pure in-plane shear stress is calibrated use one additional o-ais atiue test data b a trial and error method Liu and Mahadevan005. For eample the S- curves or a D55 balanced laminate alon the iber and transverse to the iber are reported in Mandell J.F.Feb The eperimental data is shown in Fi. a and b or 0 o and 90 o respectivel. Statistical analsis can be done and the distribution o Alon Blon Atran Btran can be obtained. However no eperimental data were reported under pure shear loadin to obtain Ashear or Bshear directl. The pure shear S- curve is calibrated usin the balanced laminate [±45 o ] 3 and Ashear and Bshear can be obtained. Once the S- curves are obtained the atiue lie o composite laminates can be predicted or arbitrar orientations. The mean values o the strenth coeicients are shown in Table 3. All the si strenth coeicients are the input random variables or both unidirectional materials. All the input random variables are assumed to ollow lo-normal distribution ecept Blon Btran and Bshear which ollow normal distribution. Fi. Eperimental data: a [±0] 3 b [±90] 3 For multidirectional material another seven input random variables the elastic modulus E E Poisson s ratio υ shear modulus G volume raction o ibers V pl thicness t and pl orientations θ are included in the current model or calculation. The eometr properties and volume raction o the balanced laminate D55 are reported in Mandell J.F.Feb The coeicient o variation o all the other ive random variables are assumed to be 0.05 Liu and Mahadevan005. Validation or unidirectional composite materials 8

19 In Fi. 3 the model prediction o the median lie and its 90% conidence bounds are plotted toether with the eperimental data. The -ais is the atiue lie and the -ais is the stress amplitude. A semi-lo scale plot is used i.e. onl the -ais is in lo scale. The solid lines are the median prediction results and dashed lines are the 90% conidence bounds. All points are the eperimental observations under o-ais loadin at dierent anles. The anles o the o-ais loadin are shown in the leends. As shown in Fi. 3 the median prediction results aree ver well with the eperimental results. In addition dierent uncertainties o eperimental data can be quantitativel predicted usin the proposed probabilistic methods. Almost all the eperimental data lies in the 90% conidence bounds. Fi.3 Comparison o lie prediction with eperimental data or unidirectional composite laminates Validation or multidirectional composite laminates Fatiue test data o lass-iber-based multidirectional composite laminates Mandell J.F.Feb 003. are used to validate the proposed atiue model. The material chosen D55 is a balanced laminate which consists o pairs o laers with identical thicnesses and elastic properties but with ±0 o ±30 o ±40 o ±50 o ±60 o ±70 o ±80 o. Aain the atiue S- curve or pure shear test is not available. In the current stud the balanced laminate [±45] 3 is used to calibrate the shear S- curve. The prediction results and the eperimental observations are plotted in Fi. 4. The - ais is the atiue lie and the -ais is the applied stress amplitude. The solid lines are the median prediction results and dashed lines are the 90% conidence bounds. From Fi. 4 enerall satisactor results can be observed with a ew eceptions. In all cases the 9

20 median predictions capture the major trends in the eperimental observations. The 90% conidence bounds covers majorit o the eperimental data. Fi. 4 Comparison o lie prediction with eperimental data or multidirectional composite laminates Conclusion A eneral probabilistic lie prediction methodolo is proposed in this paper combinin a critical plane-based multiaial model and the inverse irst-order reliabilit method. The multiaial atiue model can be applied to both isotropic and anisotropic materials. The proposed inverse FORM method can eicientl calculate the atiue lie prediction correspondin to dierent taret reliabilit level compared to the direct Monte Carlo simulation method. Several conclusions can be drawn based on the current investiations. - Overall satisactor results are observed between model predictions and eperimental results or both unidirectional and multidirectional composite laminates. - The proposed inverse FORM method has been veriied with direct Monte Carlo simulation results and validated with etensive eperimental data. - The scatter o eperimental data can be predicted usin the quantiied uncertainties o material properties eometric coniurations and manuacturin processes and the proposed probabilistic ramewor. - It is observed that the predictions results have a better areement or unidirectional composite laminates which suests that a more comprehensive mechanism model or multidirectional composite laminates is required to include 0

21 other actors such as delaminatin between plies. Current investiation ocuses on the constant proportional multiaial loadin. Further model development and validation are needed or eneral nonproportional random loadin. Geometric eects such as holes and notches need urther stud or structural level applications. Acnowledement The research reported in this paper was supported b unds rom SF Award o. CMMI-0900 Project Manaer: Dr. Mahendra Sinh and b unds rom ational Aeronautics and Space Administration ASA Contract o. X09AY54A Project Manaer: Dr. Kai Goebel. The support is rateull acnowleded.

22 Reerences: Aboul Waa M.. Hamd A. and El-Midan A.A. Combined bendin torsional atiue o woven rowin GRP. J En Mater Technol Trans ASME Chen J. Zhan J. Cai C.S. and Xiao R.-C. A new approach or solvin inverse reliabilit problems with implicit response unctions. Enineerin Structures Cizelj L. Mavo B. and Riesch-Oppermann H. Application o irst and second order reliabilit methods in the saet assessment o craced steam enerator tubin. uclear Enineerin and Desin Daniel I.M. and Ishai O. Enineerin mechanics o composite materials. Oord Universit Press ew Yor 006. Deriec J. and Paepeem W.V. Fatiue damae modelin o ibre-reinorced composite materials: Review. Applied Mechanics Reviews Der Kiurehian A. Zhan Y. and Li C.-C. Inverse reliabilit problem. J. Enr. Mech.ASCE El Kadi H. and Ellin F. Eect o stress ratio on the atiue o unidirectional lass ibre/epo composite laminae. Composites Haldar A. and Mahadevan S. Probabilit reliabilit and statistical methods in enineerin desin. John Wile ew Yor ; Chichester [Enland] 000. Hasin Z. and Rotem A. A atiue criterion or iber reinorced composite material. Journal o Composite Material Jin Chen R.-C.X. Application o inverse reliabilit method to estimation o cable saet actors o lon span suspension brides. Structural Enineerin and Mechanics 006 3o Kawai M. Hachinohe A. Taumida K. and Kawase Y. O-ais atiue behaviour and its damae mechanics modellin or unidirectional ibre-metal hbrid composite: GLARE. Composites Part A: Applied Science and Manuacturin Kawai M. Suda H. Eects o on-eative Mean Stress on the O-Ais Fatiue Behavior o Unidirectional Carbon/Epo Composites at Room Temperature. Journal o Composite Materials Ma Liu Y. and Mahadevan S. Multiaial hih-ccle atiue criterion and lie prediction or metals. International Journal o Fatiue Liu Y. and Mahadevan S. Probabilistic atiue lie prediction o multidirectional composite laminates. Composite Structures Liu Y. and Mahadevan S. A uniied multiaial atiue damae model or isotropic and anisotropic materials. International Journal o Fatiue Liu Y. and Mahadevan S. Stochastic atiue damae modelin under variable amplitude loadin. International Journal o Fatiue Liu Y. and Mahadevan S. Eicient methods or time-dependent atiue reliabilit analsis. AIAA Journal Mandell J.F. a.s.d.d. DOE/MSU Composite Materials Fatiue Database: Test Methods Materials and Analsis. Sandia ational Laboratories Albuquerque M Feb 003.

23 Miller K.J. Brown M.W. ASTM Committee E-9 on Fatiue. and ASTM Committee E-4 on Fracture Testin. Multiaial atiue : a smposium. American Societ or Testin and Materials Philadelphia Pa Petermann J. and Plumtree A. A uniied atiue ailure criterion or unidirectional laminates. Composites Part A: Applied Science and Manuacturin Philippidis T.P. Vassilopoulos A. P. Fatiue Strenth Prediction under Multiaial Stress. Journal o Composite Materials Sep Racwitz R.a.F. B. Structural Reliablit Under Combined Random Load Sequences. Computers & Structures Racwitz R.a.F. B. ote on Discrete Saet Checin When Usin on-ormal Stochastic Models or Basic Variables. Load Project Worin SessionMITCambrideMA June 976. Reisnider K.L. and Gao Z. A micromechanics model or composites under atiue loadin. International Journal o Fatiue Saranasoontorn K. and Manuel L. Eicient models or wind turbine etreme loads usin inverse reliabilit. Journal o Wind Enineerin and Industrial Aerodnamics Sauer T. umerical Analsis Sims D. and Brodon V. Fatiue behavior o composites under dierent loadin modes. Fatiue o ilamentar materials.astm STP Sas T.H. and Barr D.A. Assessin uncertaint in subsurace solute transport: eicient irst-order reliabilit methods. Environmental Sotware Thorndahl S. and Willems P. Probabilistic modellin o overlow surchare and loodin in urban drainae usin the irst-order reliabilit method and parameterization o local rain series. Water Research Wu L. Thermal and mechanical atiue analsis o CFRP laminates. Composite Structures

24 List o Fiures Fi. Comparison o the direct MC method with the inverse FORM method Fi. Eperimental data: a [±0] 3 ; b [±90] 3 Fi. 3 Comparison o lie prediction with eperimental data or unidirectional composite laminates a AS4/PEEK; b E-lass ibre/eop- with R 0; c E-lass ibre/eop- with R 0.5; d E-lass ibre/eop- with R -; e T800H/500 carbon/epo with R 0.; T800H/500 carbon/epo with R -0.3; T800H/500 carbon/epo with R 0.5; h E-lass/polester; i T800H/epo; j T800H/polmide; GLARE l GLARE Fi. 4 Comparison o lie prediction with eperimental data or multidirectional composite laminates a ±0 o ; b ±30 o ; c ±40 o ;d ±50 o ; e ±60 o ; ±70 o ; ±80 o List o Tables Table.Material parameters or atiue damae evaluation Table.Eperimental data or unidirectional materials Table 3. Eperimental data or uniaial materials and multidirectional material 4

25 Table.Material parameters or atiue damae evaluation Material Propert s t s t > α 4 4 / s 3 5 / s 4s cos α α 0 5 / s 4s 0 9 s β β [cos α s sin α ] β s Table.Eperimental data or unidirectional materials Material Reerences E-lass/polester Philippidis and Vassilopoulos Sep 999 E-lass ibre/epo- Kadi and Ellin 994 T800H/epo Kawai et al 00 T800H/polimide Kawai et al 00 AS4/PEEK Kawai et al 00 GLARE ibre metal laminates Kawai et al 00 T800H/500 carbon/epo Kawai and Suda Ma 004 5

26 Table 3. Eperimental data or uniaial materials and multidirectional material Alon Atran Ashear Materials \ Random variable MPa Blon MPa Btran MPa Bshear AS4/PEEK E-lass ibre/epo- R E-lass ibre/epo- R E-lass ibre/epo- R T800H/500 carbon/epo R T800H/500 carbon/epo R T800H/500 carbon/epo R E-lass/polester T800H/epo T800H/polimide GLARE ibre metal laminates D