S. Srinivasan, Technip Offshore, Inc., Houston, TX

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1 9 th ASCE Specialty Conerence on Probabilistic Mechanics and Structural Reliability PROBABILISTIC FAILURE PREDICTION OF FILAMENT-WOUND GLASS-FIBER Abstract REINFORCED COMPOSITE TUBES UNDER BIAXIAL LOADING S. Srinivasan, Technip Oshore, Inc., Houston, TX B. Bhattacharya, M. ASCE, University o Delaware, Newark, DE Filament-wound, iber-reinorced polymer-matrix composite laminate tubes have been used or a wide range o engineering applications, owing to their high speciic stiness, strength and superior corrosion resistance. Leakage ailure o composite tubes under multiaxial loading has been an important design concern. Extensive analytical and experimental investigations have recently been reported on leakage ailure o composite tubes. These investigations developed deterministic estimates o leakage ailure or dierent combinations o axial load and internal pressure. Eects o uncertainties in material properties and laminate construction on leakage ailure need to be studied in detail. In this study; probabilistic analyses are perormed to predict irst-ply ailure o angle-ply composite laminate tubes under combined internal pressure and axial loading. Randomness is considered in (1) elastic moduli along principal lamina directions; (2) uniaxial matrix-dominated strengths o each lamina; and (3) local thickness variation o individual plies. The probabilistic inite element analysis is perormed using the ANSYS TM program and Latin hypercube sampling. Inluence o each basic variable on irst-ply ailure is evaluated. These are compared with experimental results available in the literature. Introduction Fiber-reinorced composite laminate tubes have been extensively used in the aerospace, automotive, and petroleum industries. One o the concerns in the use o these tubes is leakage o contents when the tubes are subjected to various combinations o internal pressure and axial tension. Leakage ailure is primarily attributed to matrix-dominated cracking in individual plies o the composite laminate tubes. A complex process o initiation and growth o these cracks through the tube wall provides a passage or luid to leak through the wall. Stresses in individual plies o the composite laminate tube are a unction o the stiness o the ibers and matrix, volume raction o ibers, ply angles and ply thicknesses, laminate lay-up, and applied loads among others. Analytical and experimental studies on leakage ailure o composite laminate tubes have been perormed by Soden et al. (1993), Wang et al. (1997) etc. These studies ocused primarily on deterministic mechanics and on phenomenological development o composite tube leakage ailure methods. Accurate prediction o leakage ailure o composite laminate tubes, however, should involve rigorous analytical models coupled with a detailed appreciation o the random aspects o the structure. The stochastic nature o response o composite laminates due to inherent uncertainties in mechanical properties at the iber/matrix scale, ply scale, and the structural scale is well Srinivasan, S., and Bhattacharya, B. 1

2 recognized (e.g., Chamis and Shiao, 1993). Probabilistic estimates o initial and system ailure o composite laminate plates and tubes (Mahadevan et al., 1997, Yushanov and Bogdanovich, 1998, and Frangopol and Recek, 2003) have been obtained. Probabilistic analyses o iber-reinorced composite laminates using the stochastic inite element method (SFEM) have been used to obtain reliability estimates o structural response, system ailure (Hong et al., 1995) and buckling and irst ply ailure (Lin et al., 1998). Comparisons o reliability estimates and computed ailure probabilities with experiments on composite laminates are limited in the literature. Furthermore, the studies do not consider local uncertainties in ply thickness in the plane o the composite laminate. While these may not be as prevalent in laminates constructed using prepegs, ilament wet winding could potentially induce such uncertainties in the ply and laminate thickness. In this study the eect o uncertainties in geometric and material parameters in the manuacture o ilament-wound composite laminate tubes on matrix-dominated irst ply ailure is examined. The basic variables include uncertainties in local variations o individual ply thicknesses, elastic moduli o a unidirectional ply, and ailure strains. The basic variables are chosen rom observations reported in the work by Srinivasan (1996). Uncertainties in ply iber angles were not measured in that work and not considered in the study. Furthermore, material nonlinearity and damage mechanics are beyond the scope o the paper. Linear inite element analysis and Latin hypercube sampling (LHS) are used or the probabilistic calculations. Method o Analysis Composite tube lamination, geometry and loading The composite laminate tube is made by ilament wet winding o E-glass ibers and epoxy resin. The lay-up is [±55 o ]2. The iber angle is measured rom the longitudinal axis o the tube. The nominal values o inner diameter and wall thickness o the tube are 29 mm and 2.03 mm, respectively. The tube is subjected to two combinations o internal pressure and axial tension. They are hoop to axial stress ratios o 0:1 (denoted as 0h:1a) and 1.86:1 (denoted as 1.86h:1a). One end o the tube is restrained and a uniorm axial load is applied at the other end. Finite element modeling The probabilistic analysis o the iber-composite laminate tube is perormed using linear analysis and the ANSYS TM inite element program (ANSYS, 2003). Eight-node isoparametric shell elements (SHELL99) with ive degrees o reedom per node, i.e., three displacements and two rotations, are used or structural discretization. The elements are C o continuous, with Mindlin-type displacement ield assumptions which include transverse shear deormations. The element is deined using corner layer thicknesses, layer material direction angles, and orthotropic material properties. The inite element model, load and constraints are shown in Figs. 1 and 2. Regions with local thickness variations are depicted with dierent colors in Fig. 1. Srinivasan, S., and Bhattacharya, B. 2

3 Fig. 1. Finite element model Constitutive equations and ailure criteria Fig. 2. Loads and constraints The stress-strain expression or a iber composite ply, in material coordinates (1,2) can be written as σ' i = ' ij 1 Q ε j, τ = Q2 γ tj (1) ti ' ij where ' Q 1ij is the in-plane stiness matrix o the ply, ' Q 2ij is the transverse shear stiness matrix o the ply and {σ'} = {σ 1, σ 2, τ 12 } T, {τ t '} = {τ 13, τ 23 } T, {ε'} = {ε 1, ε 2, γ 12 } T, and {γ t '} = { γ 13, γ 23 } T are the in-plane and out-o-plane stresses and strains in the material coordinate system. Here 1 and 2 indicate iber direction and ply transverse directions, respectively. The inite element laminate constitutive ormulation ollows the well-known Classical Lamination Theory (Jones, 1975) (CLT) with appropriate modiications or ply thickness variations. Further details are discussed in the ANSYS Theoretical Manual (ANSYS, 2003). In this study Hashin s (Hashin, 1980) criterion or matrix-dominated ailure o unidirectional iber-composites expressed in terms o strains is used. The perormance index or matrix-dominated ailure o a unidirectional iber-composite due to a tensile strain transverse to the iber is summarized as 2 2 ε2 γ F = + 12 ε2 γ12, (2) where 2 ε is the uniaxial ailure strain transverse to the iber and 12 γ is the uniaxial inplane shear ailure strain. Failure in any ply o a composite laminate is said to occur when the perormance index F 1 in Eq. (2). Srinivasan, S., and Bhattacharya, B. 3

4 Probabilistic modeling and simulation procedure Four sets o basic variables were chosen or evaluating irst ply ailure o the composite laminate tubes. They are individual ply and total tube wall thickness, axial and circumerential locations o local thickness variations in the tube, and elastic moduli and ailure strains o a unidirectional E-glass/epoxy ply. As noted beore, these were chosen based on ad-hoc observations in the work o Srinivasan (1996). The statistics o ply thicknesses, moduli and ailure strain are shown in Table 1. The elastic moduli and ailure strains basic variables do not vary spatially. Note that the plies are numbered sequentially rom the inner most ply at the internal diameter (Ply 1) towards the outside o the tube (Ply 4). The total area exhibiting local thickness variations is assumed to be a constant 5% o the total surace area o the tube and always constitutes six regions o equal area. The centroids o these regions ollow a 2D conditional Poisson process thinned so that no two regions overlap. Random Var. Distribution Mean Stdev Ply 1 thickness Trunc Gauss 0.51 mm mm Ply 2 thickness Trunc Gauss 0.51 mm 0.1 mm Ply 3 thickness Trunc Gauss 0.51 mm 0.1 mm Wall thickness Trunc. Gauss 2.03 mm 0.20 mm E 11 Gauss 45.0 GPa 2.25 GPa E 22 Gauss Gpa 1.3 Gpa G 12 Gauss 5.85 Gpa 0.59 Gpa Weibull Exponent 25 Char value ε 2 Min. value γ 12 Weibull Exponent 25 Char value 0.05 Min. value Table 1. Probability distribution unctions o input variables Latin hypercube sampling implemented in ANSYS (ANSYS, 2003) is used to perorm the probabilistic calculations. This is a cost eective procedure or computing probabilistic estimates o irst ply ailure o the tubes. Four dierent basic variable cases were considered: (BV1) The ply thickness variation occurs over the whole domain o the tube, (BV2) local ply thickness variation; all basic variables uncorrelated (BV3) local ply thickness variation; ailure strains are 50% correlated, and (BV4) local ply thickness variation; thickness o plies in six regions are ully correlated, ailure strains are 50% correlated. Results and Discussion Linear elastic inite element analyses were perormed using ANSYS (ANSYS, 2003) or the our basic variable cases and two load conditions described in the previous sections. The number o simulations range rom 40 or BV1 to a maximum number o 150 or BV2 and BV3. For each simulation, irst ply ailure is identiied as a irst passage o the perormance index. Perormance indices are computed at each element and the maximum perormance index in each ply over the whole domain o the tube was Srinivasan, S., and Bhattacharya, B. 4

5 identiied. Probability density unctions (PDF s) o irst ply ailure computed rom the maximum perormance index o all our plies or all cases are shown in Figs. 3 and 4. In Fig. 3, the dotted line indicates the ailure axial stress determined rom experiment (Wang et al., 1997). The experimental result alls in the tail o the predicted range o occurrence o irst ply ailure. For the load ratio o 1.86h:1a, tube ailure occurred at an axial stress o 24 ksi (Wang et al., 1997). First ply ailure was not identiied in the experiments. First ply ailure prediction (0h:1a) First ply ailure prediction (186h:1a) BV1 BV2 BV BV1 BV2 BV3 BV4 0.3 BV Axial stress (ksi) Axial stress (ksi) Fig. 3. PDF or 0h:1a load case Fig. 4. PDF or 1.86h:1a load case Load Case 0h 1a Load case 1.86h: 1 a Parameter BV 1 BV2 BV3 BV4 BV 1 BV2 BV3 BV4 E E G Tlay (max) Loc (max) ε γ Table 2 Correlation coeicients o basic variables with perormance index Sensitivities o the basic variables on irst ply ailure were determined in terms o correlation coeicients with the maximum perormance index in a ply (Table 2). The inluence o ply thickness is highest or case BV1. The quantities Tlay (max) and Loc (max) in the table indicate maximum o all correlation o layer thickness or location variables with the perormance index. Srinivasan, S., and Bhattacharya, B. 5

6 Conclusions Probabilistic inite element calculations o [±55 o ]2 E-glass/epoxy composite laminate tubes were perormed to determine uncertainties in predicting matrix-dominated ailure. The ailure predictions were compared with experiments. The basic variables considered were ply thickness, local ply thickness variations, elastic moduli, and ailure strains. The results indicate that (a) the perormance index or matrix-dominated ailure is sensitive to ply thickness variations and ply transverse and shear elastic moduli (b) the perormance index is not very sensitive to locations o ply thickness variations. More rigorous probabilistic analysis that includes spatial variation o ply thicknesses, material nonlinearity, and damage mechanics may be required or predicting leakage ailure o the tubes or load ratio 1.86h:1a. Reerences ANSYS TM Release 8.0, ANSYS Inc., Canonsburg, PA. Chamis, C.C. and M.C. Shiao (1993), Probabilistic Assessment o Composite and Smart Composite Structures, Probabilistic Structural Mechanics: Advances in Structural Reliability Methods, IUTAM Symposium, San Antonio, Texas, June 7-10, 1993, Proceedings, Frangopol, D.M. and S. Recek (2003), Reliability o iber-reinorced composite laminate plates, Probabilistic Engineering Mechanics, 18, Hashin, Z. (1980), Failure criteria or Unidirectional Fiber Composites, Journal o Applied Mechanics, 47, Hong, C. S., J.S. Park, and C.G. Kim (1995), Stochastic Finite Element Method and System Reliability Analysis or Laminated Composite Structures, Recent Advanced in Solids and Structures ASME Conerence, San Francisco, CA, Nov , 1995, Proceedings PVP, 321, Jones, R. M. (1975), Mechanics o Composite Materials, McGraw-Hill, New York. Lin S. C., T.Y. Kam, and K.H. Chu (1998), Evaluation o Buckling and First-Ply Failure Probabilities o Composite Laminates, International Journal o Solids and Structures, 35(13), Mahadevan, S. X. Liu, and Q. Xiao (1997), Probabilistic progressive ailure model o composite laminates, Journal o Reinorced Plastics and Composites, 16(11), Soden, P.D., R. Kitching, P.C. Tse, Y. Tsavalas, and M. J. Hinton (1993), Inluence o Winding Angle on the Strength and Deormation o Filament Wound Composite Tubes subjected to Uniaxial and Biaxial Loads, Composites Science and Technology, 46, Srinivasan, S. (1996), Long-Term Leakage Failure In Filament-Wound Glass-Fiber Reinorced Cylindrical Composite Laminate Shells Under Combined Internal Pressure and Axial Loading, Disseration, University o Houston. Wang, S.S., S. Srinivasan, and B.W.Cole (1997), Long-Term Leakage Failure and Lie Prediction o Fiber Composite Laminate Tubing under Combined Internal Pressure and Axial Loading: Theory and Experiments, Composite Materials or Oshore Operations, Houston, Texas, October 28-30, Proceedings, Yushanov, S.P. and A.E. Bogdanovich (1998), Analytical Probabilistic Modeling o Initial Failure and Reliability o Laminated Composite Structures, International Journal o Solids and Structures, 37(7-8), Srinivasan, S., and Bhattacharya, B. 6

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