1. Composite Structural Analysis and Design Issues ME 7502 – Lecture 14 Dr. B.J. Sullivan 1

2. Finite Element Analysis of Composite Structures Lamina stress analysis in FEA of composites Considerations in selection of element types Modeling individual layers with orthotropic elements FEA model construction Boundary conditions in FEA of composite plates and shells Thermal and Thermo-Structural Analysis Methods Assessment of Calculated Stresses and Strains Determination of Allowable Stresses and Strains Calculation of Margins and Safety FEA Assessment of Delamination Growth in Composite Structures 2

3. 3 Lamina stress analysis in FEA of composites

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5. 5 Some public domain FEA codes have at least the capabilities a), b) and c) above; Capability d) can be user-dependent; i.e., the user may wish to be very specific about how lamina level stresses are combined within a failure criterion to make judgments regarding failure

6. 6 Lamina stress analysis in FEA of composites ANSYS, ABAQUS, NATRAN public domain FEA codes all have laminated composite plate and shell elements Required input includes: a.Lamina (ply) properties in local material directions b.Orientation of ply relative to laminate (global) coordinate direction c.Lamina (ply) thickness d.Lamina failure algorithm (e.g., Hashin, Puck, etc.) and associated parameters Output features: a.Stresses in each ply in local material axes b.Stress contour plots within plies across continuous elements c.Margin of safety contour plots within plies across continuous elements, based on failure criterion and its parameters

7. Considerations in Selection of Element Types Three basic questions in element selection:  Should elements be represented by plate elements or solid elements? Plate elements do a good job of capturing bending behavior and will require far fewer elements to simulate response Solid elements provide a much better assessment of the interlaminar stresses  If plate elements are selected, should homogenous orthotropic properties be used, or should individual ply properties be used? Homogenous orthotropic properties require single plate/shell elements through the thickness Use of individual ply properties in FE analysis requires use of layered plate/shell elements through the thickness  Should elements with transverse shear deformation be employed? 7

8. Considerations in selection of element types 8 Laminate approximated by homogeneous orthotropic plate properties (average properties throughout)

9. Considerations in selection of element types 9 Laminate approximated by homogeneous orthotropic plate properties (average properties throughout)

10. Modeling Individual Layers with Orthotropic Elements 10

11. Modeling Individual Layers with Orthotropic Elements 11

12. FEA Modeling of Laminated Plates 12

13. 13 FEA Modeling of Laminated Plates

14. Use of Elements with Transverse Shear Deformation Most general purpose FEA codes provide plate and shell elements with transverse shear capability As we have seen, this can be an important aspect of composite structural analysis in two cases:  The ratio of in-plane Young’s modulus to through thickness shear modulus is relatively large (in some cases as low as 5; for metals, E/G < 3 is typically)  The span-to-thickness ratio is small (~25 or less) If you are not sure if shear deformation is important, try to perform identical analyses with and without this effect  Near identical results will indicate shear deformation is not important  Different results will indicate the importance of this feature in your analysis 14

15. FEA Model Construction Typical FEA model construction practice is not substantially different from metallic parts  Shell / flat plate elements for thin gauge members, e.g. facesheets  3D solid elements for thicker parts, e.g. solid leading edge materials and core material of sandwich structure components  3D solids or 3D beam elements for longerons and ribs 15

16. FEA Model Construction One area of difference between FEA analysis of metallic parts and FEA analysis of composite parts is that more submodels are needed to accurately assess responses of composite parts, since interlaminar strengths of these materials are typically low  This is particularly true for refractory (e.g., Carbon-Carbon and Ceramic Matrix Composites) Submodeling effort is accomplished by creating a smaller model of the components where overstress in the full scale model is calculated  More elements through the thickness and better aspect ratio elements used 16

17. Cut Boundary Interpolation is performed by taking the resulting displacements from the full scale model that occur at the cut boundary of the sub model and applying them to the sub model (shown with bright blue arrows in picture on right) Submodel temperatures are applied using a Body Force interpolation, where temperatures are taken from the full scale model and applied to the submodel Submodel is composed of volumes outlined in orange. Submodel shown with applied temperatures and displacements from full scale model Example of FEA Submodel 17

18. Boundary Conditions in FEA of Composites For isotropic plates and shells, we frequently use symmetry B.C.’s to avoid having to analyze the entire body  For example, a plate under a load symmetric about one or two axes parallel to the edges can frequently be analyzed with a reduced model by employing symmetry B.C.’s: 18 Here, the pressure load p is symmetric about both X and Y axes

19. For loads and edge B.C.’s symmetric about both X and Y axes, an isotropic plate can be analyzed with a quarter segment and the following B.C.’s: where Edge #1 could be simply-supported (u Z =  X =  Z = 0) or clamped (u Z =  X =  Y =  Z = 0) and Edge #2 could be simply-supported (u Z =  Y =  Z = 0) or clamped (u Z =  X =  Y =  Z = 0) 19 Boundary Conditions in FEA of Composites

20. For composite plates, the elements of the A, B and D matrices must be examined before deciding if symmetry B.C.’s such as those used above can be employed to make the model smaller  For example, even if the load and edge B.C.’s are symmetric about the X and Y axes, if the laminate has shear- extensional coupling (i.e., if A 16 and A 26 are not zero), then symmetry B.C.’s are the type shown above cannot be used, since u Y is not zero across the X-axis cut, and u X is not zero across the Y-axis cut  If the laminate has no shear extensional coupling (i.e., if A 16 = A 26 =0) but does have bending twisting coupling (i.e., if D 16 and D 26 are not zero), then  Z is not zero across either axis cut, so that here again symmetry B.C.’s could not be used  In either case, the entire plate would have to be analyzed. 20 Boundary Conditions in FEA of Composites

21. 21 Boundary Conditions in FEA of Composites

22. The same conclusion can be drawn for plates with bending- extensional coupling (i.e., non-zero B ij coefficients) since any loads causing bending would mean  u Y would not be zero across the X-axis cut, and  u X would not be zero across the Y axis cut Suppose we have a balanced and symmetric laminate (A 16 = A 26 = B ij = 0) making up a plate which has loads and edge boundary conditions symmetric about one or both axes  Typically, we will have bending-twisting coupling (i.e., D 16 and D 26 will be non-zero), so that, strictly speaking, the model cannot be made smaller by employing symmetry B.C.’s across the X- axis or Y-axis cuts Practically speaking, however, if there are many plies and the plies are well dispersed within the laminate, the magnitude of D 16 and D 26 relative to the other D ij will be small  In these cases we treat the laminate as if it were specially orthotropic and employ symmetry B.C.’s 22 Boundary Conditions in FEA of Composites

23. 23 Boundary Conditions in FEA of Composites Examples of laminate stack-ups and associated bending-twisting coupling coefficients: The greater the number of plies and the more dispersed the plies within the laminate, the smaller the D 16 and D 26 coefficients relative to the others, the better the representation of the plate as Specially Orthotropic, and the more appropriate the use of symmetry B.C.’s if loads and edge B.C.’s are themselves symmetric

24. Boundary Conditions in FEA of Composites The same concepts apply to the analysis of composite shells For example, the half-symmetry model of the cylindrical shell shown on the next page would not be appropriate if  The laminate contained shear-extensional coupling, or  If loads causing bending were present and there was substantial bending-twisting coupling 24

25. 25 Boundary Conditions in FEA of Composites

26. Thermal and Thermo-Structural Analysis Methodology In assessment of composite components on vehicles subjected to time-varying thermal loads, both transient heat transfer and thermal stress analyses are performed Stress analyses typically require greater discreteness in the FE grid than what is required for thermal analyses  Displacement, strain and stress spatial gradients are typically much greater than spatial temperature gradients Nevertheless, same FE model is used for both transient heat transfer and thermal stress analyses  Elements selected for the FE analyses must be capable of switching from thermal to stress types 26

27. Transient heat transfer analyses must usually be performed for some period of time beyond the cruise/re- entry period Thermal soak must be permitted to occur  Highest temperatures in thermal protection system (TPS) components often do not occur until after “wheel stop” Wheel Stop Period of thermal soak may actually be 2-3 times as long as period of aerothermal heating Thermal and Thermo-Structural Analysis Methodology 27

28. “Snap shot” thermal stress analyses are performed for a few discrete times of flight  Times corresponding to peak thermal gradients These times will lag time of peak gradient(s) in aero- thermal heating  Times corresponding to peak temperatures of different materials in CMC components These times will lag time of peak aero-thermal heating Candidate times for thermal stress analyses Thermal and Thermo-Structural Analysis Methodology 28

29. Measured stress-strain curves of composite material used in structural components will dictate type of analysis performed  Non-linear response requires nonlinear material analysis and strain allowable assessments  Linear response permits linear material analysis and allowable stress assessments Thermal and Thermo-Structural Analysis Methodology 29

30. Assessment of Calculated Stresses and/or Strains Peak stresses or strains appearing on FEA contour plots are not appropriate for realistic assessment of component performance  When failure is initiated within a composite test specimen, the initial failure occurs over a region containing several (3-5 at a minimum) textile unit cells  A textile unit cell is defined as the “smallest volume of the fiber reinforcement containing all unique fiber orientations in the preform” (i.e., longitudinal, lateral, and through thickness) In-plane stress in torque tube of flaperon EDU resulting from mechanical loading In Plane Stress σ x (6.3ksi) In Plane Stress σ y (6.0ksi) ILT and ILS are localized stresses σ z (6.0ksi) Stress Contours in access panel of flaperon EDU resulting from mechanical loading 30

31. Accordingly, in the comparison of FEA calculated strains or stresses, an average of any given strain or stress component over a volume of at least three (3) unit cells should be compared to the measured failure strain or stress of the material This approach definitely makes the structural analysis more time consuming One textile unit cell Assessment of Calculated Stresses and/or Strains 31

32. Photomicrographs of Composites and Definition of Unit Textile Cells Assessment of Calculated Stresses and/or Strains One textile unit cell 32

33. Generation of design properties through material property testing How are the thermo-elastic properties used in the material models of the composite FE analyses determined? How are the composite strength or strain-to-failure values measured? How are the measured strengths used to define allowable values for the comparison with calculated stresses/strains? 33

34. Design Properties Most basic element of design properties database is material thermo-elastic properties themselves  Young’s moduli (tension and compression)  Axial shear modulus  Poisson’s ratios  Coefficients of thermal expansion  Thermal conductivities Thermo-elastic moduli and thermal conductivities are typically temperature-dependent 34

35. For full three-dimensional orthotropic materials, all properties (e.g., through thickness Young’s and shear moduli) cannot be measured Micromechanics models are correlated using measurable data and then used to predict properties that cannot be measured Temperature-dependent strengths are frequently measured at same time as moduli  Axial tensile and compressive  In-plane shear strengths Through thickness strengths (tensile, compressive, shear) are measured alone Design Properties 35

36. Tension Specimen This specimen is used to measure the composite in-plane axial tensile modulus and strength 36

37. Compression Specimen This specimen is used to measure the composite in-plane axial compressive modulus and strength 37

38. Rumanian Shear Specimen This specimen is used to measure the composite in-plane axial shear modulus and strength 38

39. Double Notch Shear Specimen This specimen is used to measure the composite through thickness or interlaminar shear strength 39

40. Curved Beam Specimen This specimen can be used to measure the composite through thickness tensile strength 40

41. Thermal Expansion Specimen This specimen can be used to measure the composite in-plane coefficients of thermal expansion (CTE) 41

42. Design Properties Definition of A-basis allowable property:  Design property for which there is a 95 percent confidence that 99 percent of the tested material samples will exceed this value B-basis allowable property defined as property for which there is a 95 percent confidence that 90 percent of the tested material samples will exceed this value Guidelines for composite material test programs to achieve A-basis or B-basis allowable properties are provided in MIL-HDBK-17 42

43. A-basis Allowable Test Matrix from MIL-HDBK-17 43

44. Allowable Material Properties A statistical software package known as STAT17, a by product of the MIL-HDBK-17 Working Group, is available for calculating A- basis and B-basis allowable properties from measured material property test data  A-basis and B-basis properties for polymer matrix composites used in military aircraft exist  Reinforced Carbon-Carbon used on Space Shuttle is the only refractory composite material for which A-basis properties exist  B-basis properties for specific material properties exist for ACC-6 and CVI C/SiC 44

45. Other Critical Design Properties Tensile strength of specimens with butt-joint plies Strength of notched specimens (i.e., specimens containing open holes) Strength of specimens containing loaded holes  Bearing strength  Net tension and/or compressive strength  Shear tear-out strength 45

46. Calculation of Margins of Safety Margins of safety (MOS) are used to quantify the state of stress or strain relative to design allowable values Margins of safety must be calculated using either calculated stresses or strains In expression below, substitute  Allowable and  Actual for corresponding stress quantities, if strain allowable design approach is used Typically for composite stresses calculated via FEA, the MOS for one component at a time is calculated 46

47. Example of a typical MOS table: This approach ignores potential adverse affects of stress or strain interaction Calculation of Margins of Safety 47

48. Interlaminar Strain Interaction Material characterization testing to define interaction curves for all relevant components and at all temperatures of interest can be costly 48

49. Comments on Load Factors and Factors of Safety Load factors and factors of safety are used to amplify loads and calculated stresses, respectively, prior to determining MOS values Load factors are a reflection of the degree of uncertainty in the applied loads Factors of safety are intended to reflect the degree of uncertainty in the calculated stresses due to the complexity of the structure and/or the uncertainty in the math model representation of physical structure Typical factors of safety for composite structures (from NASA-STD-5001: Structural Design and Test Factors of Safety for Spaceflight Hardware) 49

50. Hot Structure / TPS Components Load Factors and Factors of Safety 50 Vehicle Limit Load FactorsFactor of Safety Margin of Safety Calculation MechanicalThermalMechanicalThermal AHW~1.21.25 X-331.251.01.25 Estimated B-Basis Allowables X-371.5 1.0 Estimated B-Basis Allowables X-431.01.21.51.0 Estimated B-Basis Allowables X-512.01.21.5 / 2.0*1.0 Estimated B-Basis Allowables Space Shuttle 1.4 1.0 True A-Basis Allowables NASA Depends on uncertainty (NASA-STD-5002) 1.5 A-Basis allowables if man-rated; B-basis allowable otherwise (NASA-STD-5001A) * 1.5 if tested; 2.0 if not tested

51. FEA Assessment of Delamination Growth Delaminations can occur in composite structures from  Residual stresses from processing  Overstress conditions from operation of composite structure Computational, finite element based methods are available to make an assessment of the likelihood of the delaminations to grow in an unstable fashion The strain energy release rate (SERR) is calculated at the “front” of the delamination or crack  SERR is then compared to the critical strain energy release rate (i.e., the Fracture Toughness) within a specific fracture mode (G Xc ) of the material If SEER > G Xc, delaminations will grow If SEER < G Xc, delaminations are stable and will not continue to grow under the applied loads 51

52. FEA Delamination Growth Analysis Methods 52 Initially developed FEA method was Finite Crack Extension Technique for calculating dU/da (or SERR ).  This method requires two FE analyses: one using the original flaw size, and another using a marginally increased flaw size (+1%).  The strain energy U is calculated for both cases  dU = U (2) – U (1). U is a solution quantity calculated by the FE code.  The change in the flaw size is known  da = a (2) – a (1). Subsequently, theVirtual Crack Closure Technique** (VCCT) was developed, which is more robust, requires only one FE analysis, does not require user-defined quarter-point elements, and has been shown to yield good agreement with the closed form solution for a plate with a central crack. VCCT essentially replaced Finite Crack Extension Technique R. Krueger, “The Virtual Crack Closure Technique: History, Approach and Applications,” ICASE Final Report No. 2002-10, NASA/CR-2002-211628. **

53. Delamination Growth Analysis Method 53 Delaminations in composite structures frequently result from excessive ILS stresses. Accordingly, fracture propagation (if it occurs) would most likely occur in Mode II, but could also occur in Modes I or III. Delamination grows if SERR X ≥ G X,c where SERR X = strain energy release rate in Mode X and G X,c = Mode X critical strain energy release rate. While Mode II ILS may be the primary mode of concern, VCCT allows SERR to be easily calculated for all three modes to obtain a more complete assessment of the likelihood of propagation.

54. Delamination Growth Analysis Method VCCT uses the forces at the crack front (red arrows) and relative crack opening displacements (green arrows) immediately behind the crack front to enable direct calculation of SERR for each mode, for each node along the crack front. Therefore, VCCT readily provides SERR for all three modes. If SERR X ≥ G X,c, delamination is expected to propagate. If SERR X < G X,c, delamination is stable. 54 Figure from Krueger

55. VCCT vs. Closed Form Solution 55 Simple Plate with Central Crack Plate Dimensions: 5” x 8” x 1” Crack Width (a): 0.50” Applied Load: 100 psi (on top face) Material: Aluminum (E = 10 Msi) Closed Form Solution: a = 0.50” Two Mesh Sizes Model 1: All elements are 0.1” x 0.1” x 0.1” 11 Nodes through-the-thickness Model 2: All elements are 0.25” x 0.25” x 0.25” 5 Nodes through-the-thickness Schematics of Model 1 100 psi Take Y-dirn displacements from these nodes… Take Y-dirn forces from these nodes… For each pair of nodes TTT, complete the SERR I calculation defined previously on slide 12.

56. VCCT vs. Closed Form Solution 56 VCCT results agree very well with closed form solution  Model 1 (10 elements TTT) yields a maximum error of 6.4%.  Model 2 (4 elements TTT) yields a maximum error of 12.7%.

57. Example: Finite Element Model Overview Original GTOR Plate Model Submodel of Hole Cross Section of Submodel Applied boundary conditions derived from full GTOR plate model. Coupled nodes representing the edge of the delamination (nodes inside this boundary at the midplane are unmerged). 57

58. Location and Size of Delamination in GTOR 58 Delamination is centered at the location of maximum ILS stress predicted in the global analysis.  Duplicate unmerged nodes are used at the mid-plane of the cross section over the appropriate region to physically represent the delamination.  Approximate size of the delamination is shown at right (previous work performed by S. Caperton of NASA JSC)  ~0.10” circumferentially x ~0.067” radially.

59. 59 RT Case (Hole 5) – GTOR Plate Results  From FE analysis of GTOR plate, peak ILS stress is found to occur at Hole 5  create submodel for this location.  First, identify the angle at which the peak ILS stress occurs  center the delamination at this location.  Next, use the GTOR plate solution to obtain the applied displacements for the boundary of the submodel. Peak ILS stress occurs at  = 325° Displaced shape

60. 60 RT Case (Hole 5) – Submodel Details Submodel details are shown below.  Image on left shows outline of delamination, centered about the location of the peak ILS stress.  Image on right shows correct application of boundary conditions as derived from complete GTOR plate model. Delamination centered at  = 325° Displaced shape (boundary conditions interpolated from GTOR plate model) 

61. 61 Calculation of SERR along the Crack Front As discussed previously, VCCT uses forces at the crack front and relative crack opening displacement behind the crack front to calculate SERR.  All calculations are made in the appropriate coordinate system (normal to the crack front) as discussed below. Delamination Nodes There is a pair of duplicate, unmerged nodes in each green circle. There is a pair of duplicate, coupled nodes in each red circle. Coordinate System  X  Black Line  Y  Yellow Line  Z  Out of the page For each green-red circle pair (boxed in black above), calculate the relative distance between the pair of nodes in the green circle, calculate the force between the two coupled nodes in the red circle, for each direction (X, Y, Z), then complete the SERR calculations discussed previously on slide 14.

62. 62 RT Case (Hole 5) – SERR Calculations As shown in the following tables, the SERR in all three directions along the entire crack front is at least five orders of magnitude lower than the lowest C/SiC fracture toughness at any temperature. Minimum C/SiC fracture toughness: 40.4 in-lbf/in 2. Maximum SERR: 1.785∙10 -4 in-lb/in 2. Radial (X-Direction) SERR Data – Mode II

63. 63 RT Case (Hole 5) – SERR Calculations Hoop (Y-Direction) SERR Data – Mode III Through-Thickness (Z-Direction) SERR Data – Mode I

64. Summary FEA modeling and analysis of composite structural components must be sensitive to relatively poor interlaminar properties, especially for refractory composites  Submodeling is frequently necessary Nonlinear material elastic analysis is more appropriate for certain types of composite materials, e.g. particular types of refractory composites, high strain-to-failure composites  Strain based design criteria applies in this case Post-processing of calculated stresses and strains must account for textile reinforcement architecture effects  3-5 unit textile cell volume averaging Interaction criteria are important in calculation of MOS values, primarily for interlaminar components Finite element analysis methods (VCCT) may be conveniently applied to the numerical assessment of delamination growth in composite materials and structures 64