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# Table of Contents

  1. Cover
  2. Title
  3. Copyright
  4. Foreword
  5. Preface
  6. Introduction
  7. 1 Presentation of an Aeronautical Unidirectional Composite
    1. 1.1. Introduction
    2. 1.2. Carbon/epoxy composite T300/914
    3. 1.3. Polymers
  8. 2 Characteristics of UD Ply
    1. 2.1. State of stress of UD ply
    2. 2.2. Tensile test in the l-direction
    3. 2.3. Tensile test along the t-direction
    4. 2.4. Shear test
    5. 2.5. General case
  9. 3 Characteristics of a UD Ply in a Given Direction
    1. 3.1. Off-axis tensile test
  10. 4 Fracture of a Laminated Composite
    1. 4.1. Fracture of a UD ply
    2. 4.2. Fracture of a laminate
  11. 5 Fracture Criteria of a UD Ply
    1. 5.1. Maximum stress fracture criterion
    2. 5.2. Maximum strain fracture criterion
    3. 5.3. Hill's criterion
    4. 5.4. Tsai–Wu criterion
    5. 5.5. Yamada–Sun criterion
    6. 5.6. Conclusion
  12. 6 Membrane Behavior of a Laminated Composite Plate
    1. 6.1. Generalities and notations
    2. 6.2. Membrane behavior, bending behavior and mirror symmetry
    3. 6.3. Resultant forces
    4. 6.4. Displacement field, stress field and strain field
    5. 6.5. Tension / shear coupling
  13. 7 Bending Behavior of a Laminated Composite Plate
    1. 7.1. Notations
    2. 7.2. Resultant moments
    3. 7.3. Displacement field, stress field and strain field
    4. 7.4. Bending/twisting coupling
  14. 8 The Fracture Criterion of a Laminate
    1. 8.1. The sizing criterion
    2. 8.2. Test on a composite structure
    3. 8.3. Sizing principle
    4. 8.4. Sizing a given structure for a given loading
    5. 8.5. Optimal structure for a given load
  15. 9 Damage Tolerance
    1. 9.1. The principle of damage tolerance
    2. 9.2. Damage during impact and compression after impact
    3. 9.3. Sizing for impact damage tolerance
  16. 10 Interlaminar and Out-of-Plane Shear Stress
    1. 10.1. Tension of a cross-ply laminate [0,90]S
    2. 10.2. Tension of a cross-ply laminate [45,–45]S
    3. 10.3. Out-of-plane shear stress
  17. 11 Holed and Bolted Plates
    1. 11.1. Calculating holed composite plates
    2. 11.2. Calculating the multi-bolt composite joints
  18. 12 Buckling
    1. 12.1. Reminder surrounding beam buckling
    2. 12.2. Buckling of plates under compression
    3. 12.3. Plate buckling under shear loading
  19. 13 Miscellaneous Rules for Stacking
  20. 14 Exercises
    1. 14.1. Experimental determination of the characteristics of a UD material
    2. 14.2. Fracture of a laminate
    3. 14.3. Shear modulus
    4. 14.4. Optimization of stacking sequence
    5. 14.5. Composite tube
    6. 14.6. Laminate calculation without calculation.
    7. 14.7. Sandwich beam under bending
    8. 14.8. Laminate plate under compression
    9. 14.9. Tube under torsion/internal pressure
    10. 14.10. Optimization of a fabric with a strain fracture criterion
    11. 14.11. Open hole tensile test
    12. 14.12. Multi-bolt composite joint
  21. 15 Solutions to the Exercises
    1. 15.1. Experimental determination of the characteristics of a UD material
    2. 15.2. Fracture of a laminate
    3. 15.3. Shear modulus
    4. 15.4. Optimization of stacking sequence
    5. 15.5. Composite tube
    6. 15.6. Laminate calculation without calculation
    7. 15.7. Sandwich beam under bending
    8. 15.8. Laminate plate under compression
    9. 15.9. Tube under torsion/internal pressure
    10. 15.10. Optimization of a fabric with a strain fracture criterion
    11. 15.11. Open hole tensile test
    12. 15.12. Multi-bolt composite joint
  22. Bibliography
  23. Index
  24. End User License Agreement

## List of Tables

  1. 1 Presentation of an Aeronautical Unidirectional Composite
    1. Table 1.1. _Comparison of the three primary resins used in aircraft structures_
  2. 14 Exercises
    1. Table 14.1. _Stacking sequence for the four studied plates_
    2. Table 14.2. _Calculation of buckling resistance, bending stiffnesses and RF for eight stacking sequences (the two pure UD stacking sequences are entirely textbook cases only included for the purpose of the study, but are not real-world stacking sequences!)_
  3. 15 Solutions to the Exercises
    1. Table 15.1. _Stacking sequence for the four studied plates_
    2. Table 15.2. _Calculation of buckling resistance, bending stiffnesses and RF for eight stacking sequences (the two pure UD stacking sequences are entirely textbook cases only included for the purpose of the study, but are not real-world stacking sequences!)_
    3. Table 15.3. _Calculation of a stacking sequence for a tube under torsion/internal pressure_

## List of Illustrations

  1. Introduction
    1. Figure I.1. _Weight ratio of composite material in aircraft structures from the Airbus group and a few others (http://www.airbus.com/)_
  2. 1 Presentation of an Aeronautical Unidirectional Composite
    1. Figure 1.1. _Material breakdown of the Boeing 787 (according to http://www.boeing.fr). For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    2. Figure 1.2. _Unidirectional carbon/epoxy laminate_
    3. Figure 1.3. _Unidirectional and quasi-isotropic laminate_
    4. Figure 1.4. _Tension along the longitudinal direction of a composite: behavior of fiber, resin and composite. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    5. Figure 1.5. _Tension along the longitudinal direction: damaged area_
    6. Figure 1.6. _Young's modulus according to density [ASH 00a] CFRP: carbon fiber reinforced plastic; GFRP: glass fiber reinforced plastic. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    7. Figure 1.7. _Strength according to density [ASH 00a] CFRP: carbon fiber reinforced plastic; GFRP: glass fiber reinforced plastic. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    8. Figure 1.8. _Structure of polyethylene_
    9. Figure 1.9. _Structure of polymer in monomer chains_
    10. Figure 1.10. _Rigidity of a polymer depending on temperature. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    11. Figure 1.11. _Secondary links between monomer chains: cross-linking. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    12. Figure 1.12. _Rigidity of a polymer depending on temperature and cross-linking. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    13. Figure 1.13. _Autoclave for curing thermosetting compounds_
    14. Figure 1.14. _Photo of a roll of pre-preg (photo Hexcel ®)_
    15. Figure 1.15. _Thermoforming (SMC) for curing and shaping a thermoplastic_
    16. Figure 1.16. _Tensile test of a polymer_
    17. Figure 1.17. _Crystalline phases and amorphous areas of a thermoplastic polymer_
    18. Figure 1.18. _Rigidity of a thermoplastic polymer depending on temperature and degree of crystallinity. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
  3. 2 Characteristics of UD Ply
    1. Figure 2.1. _Defining the coordinate system of a UD ply. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    2. Figure 2.2. _Tensile test of a UD ply along the longitudinal direction. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    3. Figure 2.3. _Stress / strain curves of a tensile test of a UD ply in the longitudinal direction. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    4. Figure 2.4. _Tensile test of a UD ply in the transverse direction. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    5. Figure 2.5. _Stress / strain curves of a tensile test of a UD ply in the transverse direction. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    6. Figure 2.6. _In-plane shear test. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    7. Figure 2.7. _Stress / strain curve of an in-plane shear test of a UD ply. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    8. Figure 2.8. _General case: tension along l-direction, along t-direction and in-plane shear. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    9. Figure 2.9. _The six elementary stress states illustrated in the case of a UD ply. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
  4. 3 Characteristics of a UD Ply in a Given Direction
    1. Figure 3.1. _Off-axis tensile test of a UD ply. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    2. Figure 3.2. _Principle for calculating the stiffness matrix in (x, y). For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    3. Figure 3.3. _Elasticity characteristics depending on the angle of tensile loading. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    4. Figure 3.4. _The four typical directions of UD plies_
    5. Figure 3.5. _The three basic stress states applied to a UD ply at 45°. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    6. Figure 3.6. _The three tensile tests used to characterize a UD ply. For_
  5. 4 Fracture of a Laminated Composite
    1. Figure 4.1. _Longitudinal tension and schematic curves of stress/ strain in the fiber and resin of a composite. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    2. Figure 4.2. _Fracture of a composite with a high a) and low b) interface resistance. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    3. Figure 4.3. _Micrograph of a fracture facies after longitudinal tensile test [PET 05]_
    4. Figure 4.4. _Tensile test specimen, with tabs and strain gauges, of a quasi-UD composite [ABI 08]. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    5. Figure 4.5. _Tensile test at 0° and 90° of a quasi-UD composite [ABI 08]. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    6. Figure 4.6. _Longitudinal compression fracture: diagram a) and micrographic cut b) [PIN 06] of a kink band. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    7. Figure 4.7. _Longitudinal compression a) and geometry of the specimen b) [ABI 08]. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    8. Figure 4.8. _Longitudinal compression test of a UD composite [ABI 08]. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    9. Figure 4.9. _Transverse tension a) and associated damage scenario b). For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    10. Figure 4.10. _Transverse compression a) and associated damage scenario b). For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    11. Figure 4.11. _Shear fracture without a) and with b) compression. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    12. Figure 4.12. _In-plane shear loading a) and appearance of shear cracks along 45°-direction b). For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    13. Figure 4.13. _In-plane fracture: creation of cusps. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    14. Figure 4.14. _Shear fracture: creation of cusps in a carbon/epoxy UD [ROG 08]. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    15. Figure 4.15. _Different fracture modes of composites [EVE 99]_
    16. Figure 4.16. _Matrix cracking in transverse tension a) and shearing b). For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    17. Figure 4.17. _Characterization test of composite under out-of-plane shearing. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    18. Figure 4.18. _Damage scenario of a quasi-isotropic laminate_
    19. Figure 4.19. _Damage scenario of a laminate [45,–45] S_
  6. 5 Fracture Criteria of a UD Ply
    1. Figure 5.1. _The five elementary stress states of a UD ply. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    2. Figure 5.2. _Maximum stress criterion. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    3. Figure 5.3. _Off-axis tensile test of a UD ply. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    4. Figure 5.4. _Tensile stress limit of an off-axis UD ply. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    5. Figure 5.5. _Shear test of an off-axis UD ply. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    6. Figure 5.6. _Shear limit stress of an off-axis UD ply. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    7. Figure 5.7. _Effect of the shear sign on the fracture of a UD ply at 45°. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    8. Figure 5.8. _Maximum stress and strain fracture criteria. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    9. Figure 5.9. _Tensile limit stress of an off-axis UD ply. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    10. Figure 5.10. _Fracture criteria for a UD ply. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    11. Figure 5.11. _Tension of an off-axis UD ply. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    12. Figure 5.12. _Tensile limit stress of an off-axis UD ply. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    13. Figure 5.13. _Comparison of the four terms of the Hill's criterion. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    14. Figure 5.14. _Study of a tube under torsion. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    15. Figure 5.15. _Fracture of a UD tube under torsion: shear stress limit a) and influence of the different terms of the Hill's criterion b). For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    16. Figure 5.16. _Shear fracture of a ply at 45°. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    17. Figure 5.17. _Shear fracture of a ply at_ − _45°. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    18. Figure 5.18. _Limit stress of a UD ply. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    19. Figure 5.19. _Tensile limit stress of an off-axis UD ply. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    20. Figure 5.20. _Comparison of different fracture criteria during the "World Wide Failure Exercise" in the plane (σl, σt) [SOD 04] a) and comparison with the criteria presented here b). For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    21. Figure 5.21. _Comparison of different fracture criteria during the "World Wide Failure Exercise" in the plane (σt, τlt) [SOD 04] (a) and comparison with the criteria presented here (b). For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
  7. 6 Membrane Behavior of a Laminated Composite Plate
    1. Figure 6.1. _Notation of the plies within a laminate. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    2. Figure 6.2. _Diagram of membrane and bending behavior. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    3. Figure 6.3. _Effect of mirror symmetry during cooldown after manufacturing of the laminate_
    4. Figure 6.4. _Diagram and draping sequence of a rear helicopter rotor blade [BIZ 09]. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    5. Figure 6.5. _Displacements during membrane and bending loadings. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    6. Figure 6.6. _Stresses on a face of normal vector y. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    7. Figure 6.7. _Resultant forces for membrane behavior. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    8. Figure 6.8. _Forces on a square with sides dx and dy under membrane loading. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    9. Figure 6.9. _Displacement of a plate under membrane loading. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    10. Figure 6.10. _Strains and stresses in a laminate subject to membrane loading. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    11. Figure 6.11. _Laminate [0,90] S. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    12. Figure 6.12. _Tension of a laminate [0,90] S. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    13. Figure 6.13. _Strains and stresses in the plies of a laminate [0,90] S under tensile loading. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    14. Figure 6.14. _Off-axis tension of a laminate [0,90] S. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    15. Figure 6.15. _E lasticity characteristics of an off-axis laminate [0,90]S. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    16. Figure 6.16. _Laminate [45,–45] S_
    17. Figure 6.17. _Shear stress of a 45° ply. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    18. Figure 6.18. _Quasi-isotropic laminate [0,45,90,–45] S. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    19. Figure 6.19. _Quasi-isotropic laminate [0,45,90,–45] S under tension. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    20. Figure 6.20. _Quasi-isotropic laminate [0,45,90,–45] S under off-axis tensile loading. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    21. Figure 6.21. _E lasticity characteristics of a quasi-isotropic laminate [0,45,90,–45]S under off-axis traction. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
  8. 7 Bending Behavior of a Laminated Composite Plate
    1. Figure 7.1. _Notation of the plies within a laminate. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    2. Figure 7.2. _Laminate under membrane/bending loading (3D diagram). For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    3. Figure 7.3. _Laminate under membrane and bending loading (in-plane diagram). For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    4. Figure 7.4. _The three bending/twisting resultant moments. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    5. Figure 7.5. _Displacement field of a plate under bending loading. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    6. Figure 7.6. _Displacement field of a plate under bending loading. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    7. Figure 7.7. _Membrane and bending strains within a plate. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    8. Figure 7.8. _Bending strain and stress within a laminate. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    9. Figure 7.9. _Illustration of membrane/bending coupling within a plate. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    10. Figure 7.10. _Laminate [0,90] S. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    11. Figure 7.11. _Stress field within the plies of a laminate [0,90] S under bending moment Mx. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    12. Figure 7.12. _Stress field within the plies of a laminate [0,90] S under bending moment My. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    13. Figure 7.13. _Stress field within the plies of a laminate [0,90] S under twisting moment Mxy. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    14. Figure 7.14. _Laminate [0,90] S. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    15. Figure 7.15. _Strain field within the plies of a laminate [0,90] S under bending moment Mx. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    16. Figure 7.16. _Laminate [45,-45] S. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    17. Figure 7.17. _Quasi-isotropic laminate [0,45,90,-45] S. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    18. Figure 7.18. _Quasi-isotropic laminate [0,45,90,-45] S under off-axis loading. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    19. Figure 7.19. _E lastic bending characteristic D11 of a quasi-isotropic laminate [0,45,90,-45]S under off-axis loading_
  9. 8 The Fracture Criterion of a Laminate
    1. Figure 8.1. _Diagram curve of a fracture test on a composite structure. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    2. Figure 8.2. _Sizing a composite laminate under membrane loading. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    3. Figure 8.3. _Sizing of a composite laminate under bending loading. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    4. Figure 8.4. _Tension of a laminate [0,90] S. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    5. Figure 8.5. _Quasi-isotropic laminate [0,45,90,–45] S under tensile loading. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    6. Figure 8.6. _Quasi-isotropic laminate [0,45,90,-45] S under off-axis tension. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    7. Figure 8.7. _RF under off-axis tension and compression of a quasi-isotropic laminate [0,45,90,–45] S. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    8. Figure 8.8. _Laminate [0,90] S. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    9. Figure 8.9. _Strains within the plies of a laminate [0,90] S under bending resultant moment Mx. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    10. Figure 8.10. _Optimization of a laminate for a given load under membrane loading. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    11. Figure 8.11. _Equivalence between shearing and tension / compression at 45°. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    12. Figure 8.12. _Optimization of a laminate for a given bending loading. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
  10. 9 Damage Tolerance
    1. Figure 9.1. _General principle of damage detection and repair in metallic a) and composite materials b). For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    2. Figure 9.2. _Diagram of fatigue curves in a UD laminate composite composed of carbon and glass fibers_
    3. Figure 9.3. _Residual strength after impact and detectability of impact. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    4. Figure 9.4. _Size of detectable damage depending on the type of inspection. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    5. Figure 9.5. _The two types of matrix cracks: transverse cracks due to transverse tension σt a) and at 45° due to out-of-plane shear stress τtz b). For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    6. Figure 9.6. _Damage scenario during impact of a UD laminate [0,90] S [CHA 87]_
    7. Figure 9.7. _Damages in a carbon / epoxy UD laminate [02,452,902,–452] S after a 25 J impact. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    8. Figure 9.8. _Typical scenario of damage and fracture during CAI. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
  11. 10 Interlaminar and Out-of-Plane Shear Stress
    1. Figure 10.1. _Diagram of interlaminar stresses during the tensile test of a laminate [0,90] S. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    2. Figure 10.2. _Diagram curve of the interlaminar stress during the tensile test of a laminate [0,90] S_
    3. Figure 10.3. _Diagram of interlaminar stresses during the tensile test of a laminate [45,–45] S. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    4. Figure 10.4. _Diagram curves of interlaminar stresses during the tensile test of a laminate [45,–45] S. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    5. Figure 10.5. _Diagram curve of out-of-plane shear stresses of a laminate. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    6. Figure 10.6. _Diagram a) [CHO92] and micrograph b) [PET05] of the interaction of matrix cracks (due to out-of-plane shear stress) with delamination_
  12. 11 Holed and Bolted Plates
    1. Figure 11.1. _Holed plate under tension. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    2. Figure 11.2. _Stress concentration at hole edge for an isotropic material_
    3. Figure 11.3. _Stress concentration ratio at the hole edge depending on the stacking sequence_
    4. Figure 11.4. _Splitting at the hole edge_
    5. Figure 11.5. _Holed specimen after fracture [XU 14]_
    6. Figure 11.6. _Principle of net stress in a holed plate under tension_
    7. Figure 11.7. _Comparison between net stress and shape factor coefficient. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    8. Figure 11.8. _Principle of the "point stress" theory. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    9. Figure 11.9. _Geometry of a holed plate under tension_
    10. Figure 11.10. _Illustration of the hole size effect_
    11. Figure 11.11. _Hole size effect under tensile loading_
    12. Figure 11.12. _Influence of d0 in calculating the hole size effect under tensile loading. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    13. Figure 11.13. _Point stress method for different orientations of the plies. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    14. Figure 11.14. _Blind fastener Fybrflush ® (www.ahg.fr). For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    15. Figure 11.15. _Application of a Fybrflush ® blind fastener (www.ahg.fr). For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    16. Figure 11.16. _Principle of equivalent stress for the calculation of a bolt composite joint_
    17. Figure 11.17. _The four primary fracture modes in a bolt composite joint. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    18. Figure 11.18. _Filled hole tension with non-loaded bolt a) and loaded bolt b). For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    19. Figure 11.19. _Open hole tension a), filled hole with non-loaded bolt b) and loaded bolt c). For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    20. Figure 11.20. _Stress at the hole edge of an open-holed plate, filled hole with non-loaded bolt, and with loaded bolt. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    21. Figure 11.21. _Principle of equivalent stress for calculating a bolt composite joint. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    22. Figure 11.22. _Bearing stress. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    23. Figure 11.23. _Sizing of a multi-bolt composite joint depending on the number of bolts. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    24. Figure 11.24. _Diagram of a multi-bolt composite joint with the bypass loads. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
  13. 12 Buckling
    1. Figure 12.1. _Beam buckling. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    2. Figure 12.2. _Plate under compression simply supported at two ends. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    3. Figure 12.3. _Calculation of a plate under compression [ESD 95]. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    4. Figure 12.4. _Laminate [0n,90 n] S. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    5. Figure 12.5. _Laminate plate [0n,90n] S under compression. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    6. Figure 12.6. _Buckling of a simply supported plate under shear loading. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    7. Figure 12.7. _Calculation of a plate under shear loading [ESD 95]. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
  14. 14 Exercises
    1. Figure 14.1. _Tensile tests for the characterization of a UD ply. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    2. Figure 14.2. _Geometry of the tensile specimen_
    3. Figure 14.3. _UD ply at 45°. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    4. Figure 14.4. _Composite tube under tension/compression/torsion/bending. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    5. Figure 14.5. _Studied laminates. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    6. Figure 14.6. _Sandwich beam under bending. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    7. Figure 14.7. _Nomex ® honeycomb_
    8. Figure 14.8. _Simply supported panel under compression. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    9. Figure 14.9. _Stiffened panel under buckling. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    10. Figure 14.10. _Stiffened panel using T-shaped stiffeners. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    11. Figure 14.11. _Composite tube under torsion/internal pressure. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    12. Figure 14.12. _Ply of 8 harness satin fabric. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    13. Figure 14.13. _Holed plate under tension_
    14. Figure 14.14. _Stress concentration ratio at the hole edge_
    15. Figure 14.15. _Principle of the point stress method_
    16. Figure 14.16. _Multi-bolt composite joint. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
  15. 15 Solutions to the Exercises
    1. Figure 15.1. _Tensile tests for the characterization of a UD ply. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    2. Figure 15.2. _Position of the principal strain frame. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    3. Figure 15.3. _Compression test [ABI 08]. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    4. Figure 15.4. _Example of a fabric ply (satin 8HS). For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    5. Figure 15.5. _Composite tube under tension/compression/torsion/bending. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    6. Figure 15.6. _Composite tube under tension. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    7. Figure 15.7. _Composite tube under torsion. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    8. Figure 15.8. _Shear induced by shear force: if rigidity xy were the same everywhere a), and in reality b). For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    9. Figure 15.9. _Composite tube under bending: stress resultant due to the bending moment a) and shear force b). For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    10. Figure 15.10. _Sandwich beam under bending. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    11. Figure 15.11. _Nomex ® honeycomb_
    12. Figure 15.12. _Sandwich beam under bending. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    13. Figure 15.13. _Internal forces in a clamped beam under bending. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    14. Figure 15.14. _Stresses in a clamped solid cross-section beam under bending. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    15. Figure 15.15. _Stresses in a clamped sandwich cross-section beam under bending. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    16. Figure 15.16. _Plate hypothesis or beam hypothesis_
    17. Figure 15.17. _Simply supported panel under compression. For_
    18. Figure 15.18. _Stiffened panel under buckling. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    19. Figure 15.19. _T-shaped stiffener and omega-shaped stiffener_
    20. Figure 15.20. _Fuselage with omega-shaped stiffeners (San Diego Composites ®) and a low-cost composite fuselage with T-shaped stiffeners (Muratec®)_
    21. Figure 15.21. _Stiffened panel with T-shaped stiffeners_
    22. Figure 15.22. _First buckling modes of a flat panel [45, −45] 8S and a panel with T-shaped stiffeners [(45, −45)5, 45]S (the perspective of the stiffened panel can be deceiving; the panel is seen from below and the blister is facing away from the stiffeners). For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    23. Figure 15.23. _Buckling of a stiffened panel with longitudinal and transverse stiffeners. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    24. Figure 15.24. _Composite tube under torsion/internal pressure. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    25. Figure 15.25. _Resultant force under torsion a) and internal pressure b). For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    26. Figure 15.26. _Ply of 8 harness satin fabric. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    27. Figure 15.27. _Strains of the laminate under bending Mx. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    28. Figure 15.28. _Holed plate under tension_
    29. Figure 15.29. _Principle of the "point stress" theory. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    30. Figure 15.30. _Hole size effect of holed plate under tension and accounting for this using the "point stress" theory. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    31. Figure 15.31. _Multi-bolt composite joint_
    32. Figure 15.32. _Diagram of the multi-bolt composite joint with varying thickness. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    33. Figure 15.33. _Ply drop-offs at the edge of the multi-bolt joint. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    34. Figure 15.34. _Real and theoretical ply drop-offs [ABD 15]. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    35. Figure 15.35. _Multi-bolt joint with alignment of the midplanes. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_
    36. Figure 15.36. _Multi-bolt joint with stiffener. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_

## Landmarks

  1. Cover
  2. Table of Contents
  3. Begin Reading

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  307. e1

# Mechanics of Aeronautical Composite Materials

Christophe Bouvet

First published 2017 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.

Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address:

ISTE Ltd

27-37 St George's Road

London SW19 4EU

UK

www.iste.co.uk

John Wiley & Sons, Inc.

111 River Street

Hoboken, NJ 07030

USA

www.wiley.com

© ISTE Ltd 2017

The rights of Christophe Bouvet to be identified as the author of this work have been asserted by him in accordance with the Copyright, Designs and Patents Act 1988.

Library of Congress Control Number: 2017941155

British Library Cataloguing-in-Publication Data
A CIP record for this book is available from the British Library
ISBN 978-1-78630-114-7

# Foreword

When my young colleague gave me his manuscript to proofread and asked me to write a few words for a foreword, it is with great pleasure that I accepted, even though the preamble and title were quite sufficient to present this book.

Many books have been written on composite materials. The originality of this one comes in part from the objective, which is targeted towards aircraft structures made of composite materials. To this end, the calculation of holed structures, and their subsequent multi-bolt joints, and the theme of damage tolerance are presented alongside the problems that they carry with them, as well as the calculation methods that are currently in use. These two axes are rarely presented in the literature, yet they are essential to perform an accurate and realistic sizing since they often govern the resistance of a given structure and thus its sizing.

This book covers all classic notions surrounding composite structures. Theoretical formulae are presented concisely and without unnecessary developments allowing the focus to remain on the essentials. When studying criteria, another crucial point, the differences between first damage and final fracture are made abundantly clear, something that is essential for achieving optimal structures and performing realistic calculations. This is another essential aspect of composite calculations that is nonetheless quite absent from the literature.

Lastly, a number of exercises surrounding all studied themes are proposed with complete and detailed corrections. They will allow the readers to assimilate the presented themes more easily. This is without doubt one of this book's strongest assets as there are few other titles that carry quite this many examples.

I recommend reading this book as it uses concision and precision to provide essential bases for designing and sizing composite structures.

Jean-Jacques BARRAU
Former Professor at the University Paul Sabatier
Toulouse, France

# Preface

The objective of this lesson on composite structure sizing is to present the principles that allow the sizing of composite laminates widely used in composite structures. After a brief presentation of the primary material used in aircraft structures, the basic theory of laminated plates under membrane and bending loading as well as their associated fracture criteria is touched on. The fracture of a UD ply is then explained in detail, in order to demonstrate its inherent complexity and the limits of the criteria in use. Next, these criteria of the base ply are used to size a complete composite laminate. Lastly, two fundamental cases of structure calculations are presented: sizing holes and multi-bolt joints, as well as the study of buckling.

The criteria that are specific to aviation, in particular the notion of limit loads and ultimate loads, are addressed. The notion of damage tolerance specific to aviation is then presented, and in particular the notion of impact damage tolerance. These notions are fundamental to understanding the specificities of sizing aircraft composite structures.

Corrected exercises then allow curious readers to test their understanding of the different subjects. These corrected exercises are typical for sizing aircraft composite structures. Engineers will also find exercises that resemble their case studies. Lastly, an Excel spreadsheet allows the presentation of the calculations in the book in detail for review (available to download at www.iste.co.uk/bouvet/aeronautical2.zip).

The originality of this particular title is that it places itself very clearly in the field of aviation, where the sizing criteria are very specific. Take the calculation of holed plates and multi-bolt joints that are typical for composite aircraft structures and yet are not often touched on in the literature. Nonetheless, the notions in this book remain valid for most industrial purposes.

Another originality of this book is that it shows a number of typical calculations for aeronautics. These examples illustrate the many complexities of sizing for composite laminates. Readers can also easily perform these calculations using the Excel spreadsheet provided.

Lastly, this book groups many of the notions required to understand sizing for aircraft composite structures, and it should interest engineers who work in that field.

We will also note that the field of aeronautics is not the only one interested in using composite structures, and other areas, such as automotive, railway or civil engineering (bridges, etc.) increasingly use composite structures. While the field of aviation has precedence when it comes to composite structure sizing, it should be noted that these methods will be widely used in other fields in the years to come.

Christophe BOUVET
June 2017

# Introduction

Composite materials are increasingly used within industry, thanks to their high performance to mass ratio. This is, of course, particularly true in aviation and airspace due to the crucial importance of the mass of such structures (Figure I.1). This high performance to mass ratio is due to the use of materials with specific mechanical characteristics such as carbon, glass or Kevlar. This type of material nonetheless presents the major drawback of being brittle, and therefore needs to be used in combination with a less brittle material such as resin. This is the basic concept of composites, which join a brittle resistant material (typically fibers of varying length depending on the application) with a less effective but more resistant matrix (typically resin). Don't forget that there is then an interface that appears between the two materials which also plays an important role in the behavior of the composite.

**Figure I.1.** _Weight ratio of composite material in aircraft structures from the Airbus group and a few others (<http://www.airbus.com/>)_

The structure of the composite is therefore more complex than a standard homogeneous material such as metal, and requires an entirely new way of designing parts. Composite structure design means designing the structure at the same time as the material; this is the fundamental difference between designing a metallic structure and a composite structure. This composite design also requires classic design iterations on the geometry as well as design iterations on the material itself; these two types of iterations are, of course, inherently linked. In practice, on top of deciding geometry for the structure, the choice of stacking sequence and manufacturing process will have to be taken into account.

The majority of composite materials also present an important anisotropy (meaning its characteristics depend on the direction considered); they are presented in the form of unidirectional fibers, i.e. all facing the same direction. The performance of the composite is, of course, much better in the direction of the fibers, and since in reality a structure is generally under different loadings depending on the direction, a good choice of fiber orientation will make for a bespoke material adapted to real situations. This optimization of the direction of the fibers then allows a mass benefit and therefore a high performance to mass ratio. This benefit will nonetheless require an optimization process in the direction of the fibers to external loadings, which will also depend on the geometry of the structure, which is why the material and structure must be designed together.

Another important point for composite structures is the possibility of obtaining complex forms in one shot, for example, using layer-by-layer production processes along with molds and counter-molds or dry pre-forms. The advantage is that it decreases the mass and complexity of the assembly of the structure by reducing the number of screws or rivets. We can, for instance, cite the tail fin of the Tristar plane (Lockheed-USA) that was composed of 175 elements and 40,000 rivets with a classic construction, and 18 elements and 5,000 rivets with a composite construction [GAY 97]. Once again, this allows mass reductions while decreasing the number of parts and assembly elements, but requires a more complex design process and forces us to integrate the design of the structure and the material.

However, despite the advantages of composites, one of their main drawbacks is the price of the manufacturing process. Examples include the shelf life of the epoxy resins, the prices of laminate ovens (autoclaves), the resin injection devices, shaping using molds and counter-molds, and even the necessity for non-destructive control and assessment to guarantee that the material is healthy. All of these things make the production process more complex and thus increase the price.

Their brittleness under impact is also a negative factor as it leads to structures being oversized thus counteracting the potential mass benefit in order to guarantee their residual strength after impact. This brittleness is also associated with complexity in repairing these impact damages. These repairs can be complex and repair methods are still ill-suited and need to be refined for large-scale structures subject to impact, such as the fuselages of planes like the Boeing 787 or the Airbus A350.

The objective of this lesson is to present the primary sizing principles of composite structures, in particular the ones used in the field of aviation. The objective is to show the sizing of simple composite structures such as plates, and also to provide engineers with the elements required to perform and interpret composite structure calculations made using a finite element code. The complexity of real structures makes it impossible to size these structures by hand, as is presented in this book, but understanding case studies is necessary for the interpretation of results as complex as those obtained by finite elements or even simply from experimental results.

Before beginning this book, it is recommended that readers have prior knowledge of different existing composite materials, of their reinforcements and resins, and of the semi-products used such as pre-pregs and their primary advantages and drawbacks. Indeed, only a few unidirectional composites will be presented in the first chapter, but there are plenty of books out there that have more in-depth presentations in this area [GAY 97, BAT 13, BER 99, CAS 13, DAN 94, DEC 00, SES 04].

# 1
Presentation of an Aeronautical Unidirectional Composite

## **1.1. Introduction**

As it stands, the two main materials in use in aircraft structures are aluminum and a carbon fiber-based material. These two materials make up approximately 70% of the mass of the structure of a typical commercial plane such as the Boeing 787 (Figure 1.1).

**Figure 1.1.** _Material breakdown of the Boeing 787 (according to<http://www.boeing.fr>). For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_

The remaining structural materials are glass fiber-based composites, sandwich structures (a honeycomb core covered with two composite sheets), titanium, steel, etc.

Keep in mind that the weight ratio of composite materials is of over 50% of the overall mass for the Boeing 787 or the Airbus A350, but more standard commercial aircrafts such as the Airbus A320 or A380 are primarily composed of aluminum alloy, which makes up over 60% of its mass.

## **1.2. Carbon/epoxy composite T300/914**

We will now look at a carbon/epoxy composite that is widely used in aircraft structures, called T300/914. The T300 portion of the name refers to a carbon fiber produced by Toray® TOR 16], while 914 is a reference to an epoxy resin produced by Hexcel® [HEX 16]. T300/914 is a first generation carbon/epoxy composite that is 50% (in volume) carbon fiber and 50% epoxy resin. It takes the form of a thin fabric (less than a millimeter thick) that can subsequently be cut and draped to obtain a desired thickness ([Figures 1.2 and 1.3).

**Figure 1.2.** _Unidirectional carbon/epoxy laminate_

**Figure 1.3.** _Unidirectional and quasi-isotropic laminate_

Performing a test along the direction of the fibers, also called the longitudinal direction, we can observe a brittle elastic behavior comparable to the fibers. The elastic limit is obviously lower than that of the fibers, since approximately 50% of the resin has been added, which has a relatively low elastic limit (Figure 1.4). This resin is necessary in order to obtain a less brittle material and to shape it. The carbon fibers are indeed thoroughly interesting elements from the aspect of their mechanical characteristics but cannot be used to adopt a desired geometry. Furthermore, when a crack appears in the material and propagates perpendicularly to the fibers, it will cause a lot of fiber failures and fiber debonding, thus requiring an elevated dissipation of energy; the material will therefore be less brittle (Figure 1.5). In practice, a crack would travel parallel to the fibers if it could, which is why there are plies in other directions, so as to reinforce the material in different orientations of loading (in practice, we can demonstrate that four directions 0°, +45°, −45° and 90° are enough). It is then a case of a composite laminate, as opposed to a composite with fibers facing only one direction, which is called a unidirectional composite (Figure 1.3).

**Figure 1.4.** _Tension along the longitudinal direction of a composite: behavior of fiber, resin and composite. For a color version of this figure, seewww.iste.co.uk/bouvet/aeronautical2.zip_

**Figure 1.5.** _Tension along the longitudinal direction: damaged area_

Comparing Young's modulus and the strength of the main structural materials according to density, we note that composite materials are very well positioned compared to the metals (Figures 1.6 and 1.7). Ceramic materials are also very interesting but often too brittle for any structural use.

**Figure 1.6.** _Young's modulus according to density ASH 00a] CFRP: carbon fiber reinforced plastic; GFRP: glass fiber reinforced plastic. For a color version of this figure, see[www.iste.co.uk/bouvet/aeronautical2.zip_

**Figure 1.7.** _Strength according to density ASH 00a] CFRP: carbon fiber reinforced plastic; GFRP: glass fiber reinforced plastic. For a color version of this figure, see[www.iste.co.uk/bouvet/aeronautical2.zip_

The way to better understand a composite material is to take a close look at its composition and microstructure, in particular for epoxy resin.

## **1.3. Polymers**

The epoxy matrix is part of the polymer family commonly referred to as plastics. The word plastic comes from the mechanical behavior of polymers that present plastic strains, i.e. the deformation does not return to its original point when loading is released.

Polymers are composed, as indicated by the name, of chains of monomers linked together with covalent bonds. In this particular work, we will limit ourselves to organic polymers. Keep in mind that organic matter is created by living creatures (plants, mushrooms, animals, microorganisms), particularly by their decomposition. In contrast, inorganic or mineral matter is composed of metals, glass, ceramics, rocks, etc.

Organic polymers are therefore based on chains of monomers linked together by carbon atoms. The carbon–carbon covalent bond is strong and will provide elevated mechanical properties. These carbon–carbon bonds serve as a basis for macromolecules that make up the skeleton of the polymer material. On top of these strong bonds, these macromolecules are linked together via the intermediary of weak bonds (hydrogen bonds, Van der Waals bonds, etc.). It is the deformation of these weak bonds which will induce important plastic strain on behalf of the polymers.

Take for instance, the case of polyethylene, one of the simplest and cheapest polymers. It is made up of the polymerization of ethylene monomers (CH2=CH2) which leads to the creation of long chains. These chains are simply linked together via weak bonds. The mechanical characteristics obtained are therefore relatively weak and depend on temperature.

**Figure 1.8.** _Structure of polyethylene_

**Figure 1.9.** _Structure of polymer in monomer chains_

If we trace, for example, the Young's modulus according to temperature, we then obtain a characteristic three-part curve that demonstrates the existence of two distinct temperatures points within the studied material; its glass transition temperature _T g_ ( _g_ for glass; this glass appellation will be refined further on) and its melting temperature _T m_. Below _T g_, the behavior of a material is typical of a solid material; beyond _T m_, it takes on a practically fluid (more or less viscous) behavior and in between the two, we observe a rubbery behavior characterized by very low rigidity and a high capacity for deformation. Typically, polymers cannot be used as structure materials, in particular as a resin for a composite material, for a temperature beyond _T g_ (there are, nonetheless, exceptions which we will see further along).

**Figure 1.10.** _Rigidity of a polymer depending on temperature. For a color version of this figure, seewww.iste.co.uk/bouvet/aeronautical2.zip_

This rubber-like behavior is made possible because the molecular chains present only weak bonds (hydrogen bonds, Van der Waals bonds, etc.) linking them. They can therefore reorient and align when loaded. This type of polymer is called thermoplastic, because its plastic behavior depends on temperature.

If we want to increase the mechanical characteristics of a polymer, we need to lock the relative movement of molecules by creating covalent bonds between molecule chains; this phenomenon is called cross-linking. We can then describe the polymer as thermoset if the level of cross-linking is significant, while it is referred to as thermoplastic in the absence of these cross-links. We note that the term thermoset stems from the fact that the chemical reaction that allows this cross-linking is activated by temperature; in other words, the material hardens with temperature. This is true during manufacturing, but is no longer the case once the polymer is already cross-linked.

**Figure 1.11.** _Secondary links between monomer chains: cross-linking. For a color version of this figure, seewww.iste.co.uk/bouvet/aeronautical2.zip_

There is a third class of polymers, elastomers, that present an intermediary behavior somewhere between thermoset and thermoplastic. Elastomers generally present a low level of cross-linking and these cross-links are realized from sulfur-based covalent bonds (a process referred to as vulcanization). These sulfur-based secondary bonds grant the material a high level of elasticity (nonlinear in general).

Going back to Young's modulus depending on temperature, we distinguish a thermosetting polymer from a thermoplastic polymer via a less pronounced – if not non-existent – glass transition, and the absence of a melting temperature. The glass transition is a result of the dissociation of weak bonds by thermal agitation. Nonetheless, because thermosetting materials present high levels of cross-linking, this network of cross-links will last beyond _T g_ and grant the thermosetting material a decent mechanical behavior after _T g_. Increasing the temperature further, we reach the material's decomposition pyrolysis, i.e. decomposition into different forms of gas and residue (CO, CO2, H2, etc.).

**Figure 1.12.** _Rigidity of a polymer depending on temperature and cross-linking. For a color version of this figure, seewww.iste.co.uk/bouvet/aeronautical2.zip_

From a chemical standpoint, this thermoset/thermoplastic difference is due (upon first approximation) to the type of chemical reaction used to synthesize the polymer. Thermoplastics are obtained using a simple reaction of addition or polyaddition (making it difficult to create cross-links), while thermosetting polymers are obtained using a condensation or polycondensation reaction (that induces many cross-links). We can also talk about polymerization reactions, which cover both of these types of reactions.

The presence (or lack thereof) of cross-links in the network of a polymer modifies its mechanical behavior entirely, and also changes the processes for manufacturing and recycling it. Thermosetting materials are obtained once and only once during the polycondensation reaction. It therefore needs to be shaped before this reaction and cannot be reshaped afterwards. Going back to the carbon/epoxy T300/914 composite discussed previously (the epoxy is a thermosetting polymer), it is currently sold in the form of a thin film, 0.125 or 0.25 mm thick, made up of approximately 50% epoxy matrix and 50% carbon fiber (with diameter measuring approximately 7 μm), called pre-impregnated or pre-preg. This designation comes from the fact that the fibers are impregnated with resin beforehand. This resin is composed of monomers and a hardener, which improves the polymerization reaction. In order to avoid polymerizing the pre-preg, it is conserved at a low temperature (typically in the freezer at −20 °C) and it has a short shelf life (generally no more than 2 years). It is then a matter of assembling the pre-pregs into the desired shape for the part required, and curing the whole thing. This curing will serve, first of all, to melt the resin (or more exactly reduce its viscosity) and shape it. This phase is particularly critical as it evacuates any air bubbles trapped in the pre-preg, in particular between the layers of pre-preg which avoids porosity (which can cause cracks, making the material more brittle) within the final material. In order to improve this evacuation of the air bubbles, the thermosetting compound is generally place in a vacuum sealed bag, which is sealed tight to begin with, and then placed under pressure (between 5 and 12 bars) and heated. To achieve this, an autoclave is used (essentially a big pressure cooker), which must obviously be larger than the part that is being treated; something that may be an issue when constructing a wing for a plane.

**Figure 1.13.** _Autoclave for curing thermosetting compounds_

The temperature will then secondly activate the polymerization reaction (polycondensation in this instance) and consolidate the material. If the composite is heated a second time, nothing else will happen, unless pyrolysis temperature is achieved. It is therefore impossible to shape the material a second time or recycle it. This inability to recycle it and the obligation to keep it at a low temperature over a short period of time are the major drawbacks of thermosetting composites.

**Figure 1.14.** _Photo of a roll of pre-preg (photo Hexcel ®)_

Unlike thermosetting materials, thermoplastics present no cross-linking and can therefore be easily reshaped by heating past the melting point. They can therefore be recycled with ease and maintained at ambient temperature with no sell-by date.  Their major disadvantage is that they present inferior mechanic characteristics to thermosetting materials. Despite this, they are currently being tested to hopefully replace thermosetting materials in aeronautics. They are also being studied for the automotive industry. They can be used for thermoforming or SMC (sheet molding compound), allowing high volume production at a fast rate. For comparison, the typical curing cycle for a thermosetting compound such as T300/914 is approximately 2 h as opposed to a few minutes for thermoformed thermoplastic. Only the use of thermoplastics is possible for the automotive industry (other than a few high-end vehicles which use thermosetting materials).

**Figure 1.15.** _Thermoforming (SMC) for curing and shaping a thermoplastic_

It is nonetheless possible to increase the mechanical characteristics of a thermoplastic by playing with the morphology of the molecular chains. Simply aligning the molecular chains increases rigidity in the direction of the elongation. Consider, for instance, a tensile test for a thermoplastic polymer (Figure 1.16), we then observe a typical three-phase behavior:

  * – Phase 1 corresponds to the linear elasticity of the material.
  * – If we continue to stretch, we observe a necking (localized narrowing of the cross-section) and a plastic behavior (if we release the stress, the strain does not return to its original position). We then observe a stress plateau corresponding to the material stretching and the development of the necking zone to the whole specimen. We typically observe maximum strain in the order of 100–300%!
  * – Finally, once the whole specimen has been stretched, we observe a strong stiffening of mechanical behavior. If we continue to stretch, we see that the molecular chains begin to break. This stiffening phase is very interesting, because it allows us to obtain a much more rigid material than the initial one.

**Figure 1.16.** _Tensile test of a polymer_

In practice, this stretching can provide fibers (nylon, for example, is obtained by stretching under liquid form) or thin films that can be stretched in one or two directions. Obviously, this technique does not result in a composite material matrix for which we need rigidity in all directions.

This alignment of molecular chains can also be naturally obtained during cooldown; this is called crystallization. We can demonstrate that to minimize their energy, the molecules need to align in certain preferred directions. This alignment is only possible in the absence of cross-linking, and this is only possible for thermoplastics.

**Figure 1.17.** _Crystalline phases and amorphous areas of a thermoplastic polymer_

In practice, the degree of crystallization (or degree of crystallinity), i.e. the proportion of crystalline phase in the material, can reach 80–90% in thermoplastics and depends on the cooldown rate. This level of crystallization is even more important when there is a low cooldown rate; in other words, the macromolecules need time to reorganize if we wish to obtain crystals. If cooldown is too fast, the material will not present any particular order, and the result is an amorphous material: this state is also referred to as glassy. This is obviously where the term "glass transition" comes from, since, as it passes through the glass transition temperature _T g_, these crystals disappear and the material return to an amorphous state. It is also in part due to this disappearance of the crystalline phase that the mechanical characteristics of thermoplastics reduce when passing through _T g_.

REMARK.– The glass that is commonly used in our windows is a glassy material (but is in no way a polymer, it is primarily composed of silica which is evidently inorganic). It is thanks to that amorphous state that it is transparent since light rays can pass through it without interacting with the crystalline network (which is absent).

To produce a thermoplastic matrix with interesting mechanical properties, an interesting molecule must be selected, and the production process to favor its crystallization established. If we now go back to the evolution curve of Young's modulus depending on temperature, we note that a highly crystalline thermoplastic presents similar behaviors to a thermosetting material while having a lower _T g_. Obviously, this thermoplastic cannot be used too close to its melting temperature so as to avoid reducing its degree of crystallinity and cause it to return to its amorphous state.

**Figure 1.18.** _Rigidity of a thermoplastic polymer depending on temperature and degree of crystallinity. For a color version of this figure, seewww.iste.co.uk/bouvet/aeronautical2.zip_

Currently, composite materials in use for structural applications in the aeronautical field are for the most part epoxy matrix-based composites, i.e. thermosets. Two thermoplastic matrices are currently under consideration to replace the PPS matrix (polyphenylene sulfide) and the PEEK (polyether ether ketone) matrix. The advantage of thermoplastics is, as mentioned previously, their ease of recyclability and preservability at ambient temperature without expiring. Another advantage is their improved ductility. Thermoplastics do not have cross-linking and can therefore deform more easily before breaking and will therefore present better ductility. We restate that one of the major inconveniences of composite materials is their brittleness, in particular to impact, and that better ductility of the matrix can reduce that brittleness. Something else to keep in mind is that the ductility of a material can be characterized by fracture toughness, i.e. the energy necessary to create a crack (thus, the use of J/m2 units). Obviously, the higher the fracture toughness, the more energy the cracks need to propagate and thus the less brittle the material.

In order to compare these three materials, here is a comparison table. The values should be used cautiously, in particular the cost that depends on the supplier and the amounts produced, and the level of the mechanical properties that depend both on crystallinity and temperature. The values given here correspond to measures at ambient temperature for high levels of crystallinity and under control during the production process (essentially applying to a material created in good conditions).

**Table 1.1.** _Comparison of the three primary resins used in aircraft structures_

| **Epoxy matrix (thermoset)** | **PPS matrix (thermoplastic)** | **PEEK matrix (thermoplastic)**
---|---|---|---
**Density**
 **(kg/dm 3)** | 1.29 | 1.35 | 1.32
**_T g_ (°C)** | 190 | 90 | 143
**Maximum useable temperature (°C)** | 110 | 100 | 260
**_T m_ (°C)** | – | 285 | 380
**Young's Modulus (GPa)** | 4 | 3.3 | 3.3
**Tensile stress at rupture**
 **(MPa)** | 100 | 50 | 100
**Fracture toughness (J/m 2)** | 100–500 | 700 | 4,000
**Price (€/kg)** | 10 | 10 | > 100

PPS has the advantage, over epoxy, of having a comparable cost, but presents inferior mechanical characteristics, except for fracture toughness that is much better. Its temperature for use is also much lower than epoxy.

PEEK is also an interesting candidate for replacing epoxy, even though its price remains prohibitive. It can be used up to 260 °C, presents similar mechanical characteristics to epoxy, except its fracture toughness which is much better. It is, however, much harder to use, as it must be heated past its Tm in order to be shaped; in practice, at around 500 °C, while epoxy resin requires a heat of around 190 °C. It should, therefore, in all logic see an increase in use in the field of aeronautics over the next few years, once its fabrication process and costs are better controlled.

The presentation of polymers in this section has been largely simplified and the reader is invited to refer to [ASH 00a, ASH 00b, BER 99, DEQ 12, DOR 86, GAY 97, SES 04, etc.].

# 2
Characteristics of UD Ply

## **2.1. State of stress of UD ply**

A UD ply (common abbreviation for unidirectional, which we will use throughout this book) is composed of fibers, all facing the same way, and resin. We call _l_ the longitudinal direction, _t_ the transverse direction and _z_ the out-of-plane direction.

**Figure 2.1.** _Defining the coordinate system of a UD ply. For a color version of this figure, seewww.iste.co.uk/bouvet/aeronautical2.zip_

REMARK.– In this book the matrices will be double underlined and vectors will have a single line under them, but only where confusion is possible (which will rarely be the case) in order to alleviate notations.

Obviously, the behavior of such a material is not isotropic but rather orthotropic: it presents three planes of symmetry _(l, t)_ , _(t, z)_ and _(l, z)_ , and _(l, t, z)_ which define the orthotropic coordinate system. As a reminder, an isotropic material is one that presents the same behavior in all directions in space and a material that presents no symmetry is considered anisotropic.

In practice, a UD ply being thin (a few tenths of a millimeter in thickness), we can suppose that stress is planar:

[2.1]

And the stress matrix is:

[2.2]

We will begin by studying its behavior in a plane.

## **2.2. Tensile test in the _l_ -direction**

If we perform a tensile test along the direction of fibers _l_ , we typically observe an elongation along the _l_ -direction, shrinking along the _t_ -direction and no shearing, meaning that the specimen maintains its initial rectangular shape. The stress and strain during the test are therefore:

[2.3]

**Figure 2.2.** _Tensile test of a UD ply along the longitudinal direction. For a color version of this figure, seewww.iste.co.uk/bouvet/aeronautical2.zip_

Furthermore, we observe that the strain is globally linear depending on the stress (keeping in mind that these behavior laws mainly serve as theoretical models and that when we take a closer look, reality is always more complicated):

[2.4]

With _E l_ being Young's modulus (in either MPa or GPa) in the _l_ -direction and _ν lt_ Poisson's ratio (no unit) along the _l_ -direction.

If we continue to increase the load, we observe a sudden rupture with a practically linear behavior all the way through to the end.

**Figure 2.3.** _Stress / strain curves of a tensile test of a UD ply in the longitudinal direction. For a color version of this figure, seewww.iste.co.uk/bouvet/aeronautical2.zip_

## **2.3. Tensile test along the _t_ -direction**

If we now perform a tensile test in transversal direction _t_ , we similarly observe an elongation proportional to _σ t_ along the _t_ -direction, a shrinking proportional to _σ t_ along the _l_ -direction, and no shearing.

[2.5]

**Figure 2.4.** _Tensile test of a UD ply in the transverse direction. For a color version of this figure, seewww.iste.co.uk/bouvet/aeronautical2.zip_

With:

[2.6]

With _E t_ being Young's modulus in the _t_ -direction and _ν tl_ Poisson's ratio along the _t_ -direction. _E l_ and _E t_ are generally different and so are _ν lt_ and _ν tl_. We can also demonstrate, thanks to energetic considerations, that if the relations (equations [2.4] and [2.6]) are linear, then we necessarily get:

[2.7]

And if we continue to increase the load, we observe a sudden rupture with a practically linear behavior all the way through to the end.

**Figure 2.5.** _Stress / strain curves of a tensile test of a UD ply in the transverse direction. For a color version of this figure, seewww.iste.co.uk/bouvet/aeronautical2.zip_

## **2.4. Shear test**

Performing a pure shear test, we get a shear strain _γ lt_ proportional to _τ lt_, and elongations _ε l_ and _ε t_ are null:

[2.8]

**Figure 2.6.** _In-plane shear test. For a color version of this figure, seewww.iste.co.uk/bouvet/aeronautical2.zip_

With:

[2.9]

With _G lt_ as the shear modulus (in MPa or GPa). We note in passing that performing a shear test is far more complicated than performing a tensile test: for example, we need to perform a torsion test of a thin tube. In practice, we can also find shear characteristics using an off-axis tensile test (see Chapter 3).

Continuing to increase the load, we observe nonlinear behavior with an irreversible strain and a sudden rupture.

**Figure 2.7.** _Stress / strain curve of an in-plane shear test of a UD ply. For a color version of this figure, seewww.iste.co.uk/bouvet/aeronautical2.zip_

## **2.5. General case**

If we now add these three elementary stresses, we then obtain:

[2.10]

**Figure 2.8.** _General case: tension along l-direction, along t-direction and in-plane shear. For a color version of this figure, seewww.iste.co.uk/bouvet/aeronautical2.zip_

That can be put in the following form:

[2.11]

However, this relation is only true in the orthotropic coordinate system. This 3×3 matrix is called a compliance matrix; the matrix is called __S__ and is symmetrical:

[2.12]

With:

[2.13]

We can reverse this compliance matrix to obtain stiffness matrix _ _:

[2.14]

For example, T300/914 – which is a composite formed of approximately 50% in volume of 7 μm-diameter carbon fibers and 50% in volume of epoxy resin and is presented under the form of 0.125 mm-thick pre-preg – has the following characteristics:

[2.15]

As a reminder, if the material were isotropic, i.e. had the same behavior in all the directions, we would then get:

[2.16]

And the behavior is then simply described using two material coefficients, whereas in the case of an orthotropic material four coefficients are required. For aluminum for example, another widely used material in aircraft structures:

[2.17]

Looking at the issue within a 3D space, two coefficients obviously remain necessary to describe the behavior of an isotropic material but now a total of nine coefficients are required for the description of an orthotropic material:

[2.18]

In practice, UD composite materials are almost transversally isotropic, i.e. all the directions of the _(t, z)_ plane present the same behavior, and only five constants are required to describe their 3D behavior:

[2.19]

As a reminder, the six elementary stress states are:

**Figure 2.9.** _The six elementary stress states illustrated in the case of a UD ply. For a color version of this figure, seewww.iste.co.uk/bouvet/aeronautical2.zip_

Bear in mind that a stress _τ lt_ means a force along the _l_ -direction on a face of normal vector _t_ or a force along the _t_ -direction on a face of normal vector _l_ (which is the same due to the symmetry of the stress tensor). And this also works for _σ t_, which strictly speaking should be noted _σ tt_ and means a force along the _t_ -direction for a face of normal vector _t_.

REMARK CONCERNING NOTATIONS.– The notation of the stress and the strain in the form of vector is obviously inappropriate; they are of course symmetrical tensors and therefore present the mathematical properties of symmetrical tensors. In the example in the following chapter, we will see that the rotation rule applies for a tensor and not a vector.

The order of indices can also appear surprising but comes from the direction of the path of the matrix:

[2.20]

We index the stress or strain from 1 to 6 which gives us, for the stiffness matrix:

[2.21]

With:

[2.22]

Which in 2D then gives us:

[2.23]

With:

[2.24]

# 3
Characteristics of a UD Ply in a Given Direction

## **3.1. Off-axis tensile test**

When performing a tensile test of a UD composite off of its orthotropic coordinate system, we observe that this tension along the _x_ -direction creates an elongation along the _x_ -direction, shrinking along the _y_ -direction and also shearing. This is described as tension–shear coupling. In physics terms, this coupling is explained by a lower stiffness in the transverse direction, meaning that the material has a tendency to stretch in that direction rather than in the longitudinal direction.

**Figure 3.1.** _Off-axis tensile test of a UD ply. For a color version of this figure, seewww.iste.co.uk/bouvet/aeronautical2.zip_

This coupling is also visible on the stiffness matrix that presents non-zero components _Q 16_ and _Q 26_ in the frame _(x, y)_. In order to determine the stiffness matrix in this frame, we use the following procedure: we begin by inserting the stress in _(x, y)_. Obviously, we know what they are since they are imposed by the test (even if this is not initially evident, we can demonstrate that the stress shown in the previous figure is in fact a state of pure tension despite the overall deformation in the shape of a parallelogram when dealing with small displacement theory). Next, we determine this stress within _(l, t)_ via a rotation then we determine the strain within _(l, t)_ using the reference stiffness matrix and, finally, we determine the strain within _(x, y)_ via rotation. We then obtain the stiffness matrix in _(x, y)_ , i.e. the relation that binds stress and strain in _(x, y)_.

**Figure 3.2.** _Principle for calculating the stiffness matrix in (x, y). For a color version of this figure, seewww.iste.co.uk/bouvet/aeronautical2.zip_

We therefore begin by determining the stress in _(l, t)_ from that in _(x, y)_.

[3.1]

Where   is the rotation matrix from _(x, y)_ to _(l, t)_ :

[3.2]

This can also be written as follows:

[3.3]

An equivalent procedure can be performed with the strain by changing _θ_ to _–θ_ and thus _s_ to _–s_ without modifying _c_ :

[3.4]

Taking care not to forget that in the strain matrix the non-diagonal component is _ε xy_ and not _γ xy_, with:

[3.5]

Thus:

[3.6]

When we substitute this:

[3.7]

This therefore gives us the compliance matrix:

[3.8]

With:

[3.9]

And the two coupling terms _η x/Ex_ and _η y/Ey_, which in general are non-zero save for certain specific values of _θ_ , then cause shear strain for tensile loading and tensile strain for shear loading. If we trace these different coefficients with the values of aforementioned T300/914, we get:

**Figure 3.3.** _Elasticity characteristics depending on the angle of tensile loading. For a color version of this figure, seewww.iste.co.uk/bouvet/aeronautical2.zip_

We see that _E x_ is maximum with a value of _E l_ for _θ_ _=_ 0 _°_ , and is minimum with a value of _E t_ for _θ_ _=_ 90°. The evolution of _E y_ is obviously opposite to that of _E x_. Poisson's ratio _ν xy_ has a value of _ν lt_ for _θ_ _=_ 0° and _ν tl_ for _θ_ _=_ 90°, and is maximum with a value of 0.48 for _θ_ ≈ 24°. Variations in coupling coefficients are harder to interpret. Coefficient _η x_ is maximum, in absolute value, when _θ_ _=_ 10° and both coefficients _η x_ and _η y_ are evidently null when _θ_ _=_ 0° and _θ_ _=_ 90°. We can nonetheless show that depending on _E l_ and _E t_ values, they can present surprising variations. The shear modulus varies relatively less (compared to what it is possible to attain with two plies at ±45°) and is maximum when _θ_ _=_ 45°. This result is significant as it demonstrates that if we wish to obtain an important shear modulus, it isn't enough to place fibers at _θ_ _=_ 45°, and that there also need to be some at –45°. This result will be demonstrated in the following chapter.

Similarly, we can obtain the stiffness matrix in _(x, y)_ :

[3.10]

With:

[3.11]

**Figure 3.4.** _The four typical directions of UD plies_

In practice, we mainly use plies at 0°, +45°, –45° and 90° and get:

[3.12]

And:

[3.13]

With:

[3.14]

EXAMPLE (Ply at +45°).– For example, in the case of a ply at +45°, supposing that _E l >> Et_ and _E l >> Glt_, we get:

[3.15]

We see that the behavior is piloted by the resin, which is logical as no matter what the external stress ( _σ x_, _σ y_ or _τ xy_) may be, the resin causes an important strain. In  particular, if you wish to create a material with a high shear modulus using a UD ply at +45° is insufficient; you will also need a ply at –45° (see Chapter 6).

And for the three basic stress states we obtain:

**Figure 3.5.** _The three basic stress states applied to a UD ply at 45°. For a color version of this figure, seewww.iste.co.uk/bouvet/aeronautical2.zip_

EXAMPLE (Experimental determination of stiffness).– In order to determine the in-plane elastic characteristics of a UD composite, we typically perform three tensile tests (see exercise 1):

**Figure 3.6.** _The three tensile tests used to characterize a UD ply. For a color version of this figure, seewww.iste.co.uk/bouvet/aeronautical2.zip_

The first test consists of a tensile test in the direction of the fibers with two strain gauges bonded along the _l-_ and _t_ -directions. We can therefore determine _E l_ with the use of the first gauge and _ν lt_ using the second gauge:

[3.16]

A second tensile test in the transverse direction with a strain gauge in that direction gives us _E t_:

[3.17]

Then, the final test at 45° with a gauge bonded along _x_ -direction gives us _G lt_ knowing _E l_, _E t_ and _ν lt_. We can demonstrate that:

[3.18]

Where the only unknown is _G lt_.

# 4
Fracture of a Laminated Composite

The fracture mechanisms of UD plies vary strongly depending on the nature of the fiber, matrix, interfaces, stress state and fiber ratio. In this chapter, we will limit ourselves to describing the primary modes of fracture observed on a laminated composite formed of UD plies of resistant and brittle fibers such as glass, carbon or Kevlar and an organic resin that is more ductile than the fiber.

## **4.1. Fracture of a UD ply**

The fracture of a UD ply reveals a fracture scenario that a combination of elementary fractures, namely fiber fracture, matrix cracking and matrix/fiber debonding. These scenarios will be detailed mainly for plane stress.

### **4.1.1. _Longitudinal tension_**

On this type of UD composite, where the matrix is generally more ductile than the fiber, fracture from longitudinal tension is governed by the fiber fracture. Indeed, the longitudinal strain experienced by the fiber and the matrix being the same, and the strain limit of the fiber being generally lower than the resin, the fiber will break first.

Nonetheless, the fiber being quite brittle, there is a high dispersion of the tensile strength and it will begin to break in a scattered pattern within the test-piece during the second part of the loading, with an increase in the number of broken fibers at the end of the test.

**Figure 4.1.** _Longitudinal tension and schematic curves of stress / strain in the fiber and resin of a composite. For a color version of this figure, seewww.iste.co.uk/bouvet/aeronautical2.zip_

**Figure 4.2.** _Fracture of a composite with a high a) and low b) interface resistance. For a color version of this figure, seewww.iste.co.uk/bouvet/aeronautical2.zip_

Once an isolated fiber has broken, this fracture will either cause a transverse crack of the matrix (in the case of a brittle matrix) and high interface resistance (Figure 4.2(a)) which will diffuse stress concentration onto the neighboring fibers, or a fracture of the interface in the case of low interface resistance or of high fiber tensile strength (Figure 4.2(b)). The types of materials currently in use in the aeronautical field present relatively low interface resistance which therefore fracture in this way (Figure 4.2(b)). This also means that the fracture of an isolated fiber is not a problem, as the load is passed on to neighboring fibers (as long as they are not fractured). These local fractures will then multiply and eventually coalesce leading to a catastrophic failure of the UD composite.

**Figure 4.3.** _Micrograph of a fracture facies after longitudinal tensile test [PET 05]_

In order to perform these tests, we take care to place tabs on the ends of the tensile test specimen in order to avoid stress concentration in the clamps and ensure that the fracture takes place at the center of the sample.

**Figure 4.4.** _Tensile test specimen, with tabs and strain gauges, of a quasi-UD composite ABI 08]. For a color version of this figure, see[www.iste.co.uk/bouvet/aeronautical2.zip_

We obtain the stress from the force sensor of the tensile machine and the strain from the strain gauges placed in the central part of the specimen. In the following figure, we observe the difference in behavior between the fiber direction, noted 0°, and the transversal direction, noted 90°.

**Figure 4.5.** _Tensile test at 0° and 90° of a quasi-UD composite ABI 08]. For a color version of this figure, see[www.iste.co.uk/bouvet/aeronautical2.zip_

### **4.1.2. _Longitudinal compression_**

During a longitudinal compression test, the fracture is associated with the phenomenon of micro-buckling or kink bands. During the test, we observe buckling of the fibers. This buckling is then stabilized thanks to the presence of the resin (which is loaded in shear). Once the resin reaches its shear stress limit, it will break and cause shear bands or kink bands (Figure 4.6).

In order to perform these tests, we use specimens with specific geometries (with a short useful length in order to avoid premature fracture from buckling) (Figure 4.7). Furthermore, in order to verify that the real stress state of the specimen is compression, we generally use two gauges on both the faces of the specimen in order to isolate the compressive stress by calculating their half-sum, and the bending component by calculating their half-difference. The heavy increase in this bending strain at the end of the test is a result of damage caused to the specimen which causes it to open (Figure 4.8).

**Figure 4.6.** _Longitudinal compression fracture: diagram a) and micrographic cut b) PIN 06] of a kink band. For a color version of this figure, see[www.iste.co.uk/bouvet/aeronautical2.zip_

**Figure 4.7.** _Longitudinal compression a) and geometry of the specimen b) ABI 08]. For a color version of this figure, see[www.iste.co.uk/bouvet/aeronautical2.zip_

**Figure 4.8.** _Longitudinal compression test of a UD composite ABI 08]. For a color version of this figure, see[www.iste.co.uk/bouvet/aeronautical2.zip_

The compression test is one of the more delicate tests to perform, and we observe a high scattering in the results depending on the geometry of the sample used. The problem arises from the fact that to have an even compression stress state within the specimen, we would need to be able to increase its length (to avoid the boundary effect due to the clamps on the machine); however, that is impossible as we want to avoid the overall buckling of the specimen!

### **4.1.3. _Transverse tension_**

During the tensile test along the transversal direction of a UD composite, a high stress concentration coefficient within the fiber/matrix interface in the loading direction will lead to the debonding of these interfaces at multiple points of the specimen and, eventually, the coalescence of these cracks will lead to the fracture of the specimen. This type of fracture is probably the most critical fracture mode for a UD, along with shearing. Keep in mind though that UD is designed to work along the fiber direction and that the very design of the structure should avoid this type of loading. In practice, other plies of the laminate should withstand this load. In fact, it's to avoid this type of fracture that there are at least 10% of fibers in every direction (0°, 45°, −45° and 90°), even non-stressed ones. Furthermore, a moderate fracture is generally non-critical for the structure and can often be tolerated. Obviously, the limit strength in transverse tension is lower to that in longitudinal tension (

Figure 4.8). **Figure 4.9.** _Transverse tension a) and associated damage scenario b). For a color version of this figure, seewww.iste.co.uk/bouvet/aeronautical2.zip_

### **4.1.4. _Transverse compression_**

During compression in the transversal direction of the UD, a high stress concentration coefficient within the fiber/matrix interface will lead to the debonding of these interfaces at multiple spots within the test specimen and the coalescence of these cracks will lead to the fracture of the specimen. This fracture scenario resembles, upon first glance, that of transverse tension. However, upon closer inspection compression is slightly more complex. We observe a crack at angles between approximately 45° and 60° compared to the load direction.

**Figure 4.10.** _Transverse compression a) and associated damage scenario b). For a color version of this figure, seewww.iste.co.uk/bouvet/aeronautical2.zip_

We could have thought, upon first glance, that without the presence of a positive principal stress the fracture would come as a result of shearing. If that had been the case, we then would have observed a crack at 45° (the direction of maximum shear stress is at 45° from the compression direction).

Mohr–Coulomb emitted the hypothesis that this was indeed a shear fracture, only impeded by friction due to the negative normal stress; in other words, the more negative the normal stress, the more the material can endure shearing before breaking.

**Figure 4.11.** _Shear fracture without a) and with b) compression. For a color version of this figure, seewww.iste.co.uk/bouvet/aeronautical2.zip_

This crack will then cause a shear fracture of the fiber/matrix interface or the matrix. This eventually leads to a final break of the sample by shearing.

### **4.1.5. _In-plane shear_**

During an in-plane shear test of a UD, we observe the appearance of cracks at 45° (the direction of maximum principal stress is at 45° of the shear direction) (Figure 4.12(b)).

These cracks will then develop and coalesce and then create a fracture facies composed of cusps which are typical of shear fracture (Figures 4.13 and 4.14).

**Figure 4.12.** _In-plane shear loading a) and appearance of shear cracks along 45°-direction b). For a color version of this figure, seewww.iste.co.uk/bouvet/aeronautical2.zip_

**Figure 4.13.** _In-plane fracture: creation of cusps. For a color version of this figure, seewww.iste.co.uk/bouvet/aeronautical2.zip_

This type of fracture, like transverse tensile fracture, is highly critical for a UD but must obviously be avoided at the design stage of the structure. In practice, fibers at ±45° should be used to withstand these types of stress. In fact, it's to avoid these types of fractures that there is at least 10% of fiber at ±45° even in the absence of shear stress. In other words, if this type of fracture occurs in a laminate it means that it was badly designed. We should always aim for the fibers to withstand the main loads and they will be the ones to govern the final fracture.

**Figure 4.14.** _Shear fracture: creation of cusps in a carbon/epoxy UD ROG 08]. For a color version of this figure, see[www.iste.co.uk/bouvet/aeronautical2.zip_

In practice, the loading of a UD composite is complicated and reveals a number of stress states; its fracture is therefore a combination of these different basic fracture modes.

## **4.2. Fracture of a laminate**

A laminate being an assembly of UD plies, its fracture mechanisms will, for each ply, regroup the elementary fracture mechanisms previously mentioned. However, we will also see the appearance of a potential splitting of the interface between layers, called delamination.

**Figure 4.15.** _Different fracture modes of composites [EVE 99]_

Another mode of fracture of the UD ply that has not been previously mentioned can also appear: out-of-plane fracture. For a laminate, especially a thick one, out-of-plane stress, in particular that resulting from out-of-plane transverse shear _τ tz_, can become very intense and cause the fracture of plies of the laminate. Furthermore, we note that there is a high interaction between out-of-plane matrix cracking and delamination: when a matrix crack reaches an interface, it initiates delamination.

**Figure 4.16.** _Matrix cracking in transverse tension a) and shearing b). For a color version of this figure, seewww.iste.co.uk/bouvet/aeronautical2.zip_

The characteristics of out-of-plane shearing are more difficult to obtain and we generally use three point bending tests, even if the results should be taken with a grain of salt.

**Figure 4.17.** _Characterization test of composite under out-of-plane shearing. For a color version of this figure, seewww.iste.co.uk/bouvet/aeronautical2.zip_

EXAMPLE (Fracture of a quasi-isotropic laminate).– If we perform a tensile test of a quasi-isotropic laminate _0,45,90,–45] S_ (see [Chapter 8), the first plies to break are along 90°-direction due to transverse cracking (Figure 4.18). This fracture is obviously not critical as most of the load is withstood by the plies along 0 °-direction. At the same time, a few isolated fibers at 0° will break, mainly the ones that are badly oriented and/or more brittle than the average. Secondly, the plies at ±45° will break from transverse cracking and in-plane shearing. Once again, this fracture is not critical. At the same time, isolated fibers will continue to fracture. Finally, the plies at 0° will break, leading to the ultimate fracture.

**Figure 4.18.** _Damage scenario of a quasi-isotropic laminate_

EXAMPLE (Fracture of a laminate 45,–45]S).– If we perform a tensile test of a laminate _[45,–45] S_ (see [Chapter 8), we first observe the matrix cracking of ±45° plies, primarily due to in-plane shear stress _τ lt_, followed by delamination between layers that leads to ultimate fracture.

**Figure 4.19.** _Damage scenario of a laminate [45,–45] S_

# 5
Fracture Criteria of a UD Ply

There are a great deal of criteria for fracture of a UD ply under planar stress in the existing literature but they are all based on the five elementary stress states: tension/compression along the longitudinal direction, tension/compression along the transversal direction, and in-plane shear.

**Figure 5.1.** _The five elementary stress states of a UD ply. For a color version of this figure, seewww.iste.co.uk/bouvet/aeronautical2.zip_

As mentioned in the previous paragraph, we note a considerable difference between the fracture behavior under tension and compression, unlike the cases of elastic stiffness where this difference is less notable, even though it can reach 20% in certain cases.

## **5.1. Maximum stress fracture criterion**

The maximum stress criterion consists of considering that the UD ply is fractured when one stress reaches a stress limit under tension, compression or shear:

[5.1]

Where _σ lt_ and _σ lc_ are the tensile and compressive fracture stresses in the longitudinal direction, _σ tt_ and _σ tc_ are the tensile and compressive fracture stresses in the transversal direction, and _τ ltf_ is the shear fracture stress. For example, for T300/914 that we touched on previously, we get:

[5.2]

For a biaxial loading with no shear, we get a fracture envelope in the form of a rectangle.

**Figure 5.2.** _Maximum stress criterion. For a color version of this figure, seewww.iste.co.uk/bouvet/aeronautical2.zip_

We can, for example, use this criterion to determine the fracture point of a ply under tension/compression off of its orthotropic axes.

**Figure 5.3.** _Off-axis tensile test of a UD ply. For a color version of this figure, seewww.iste.co.uk/bouvet/aeronautical2.zip_

We then get:

[5.3]

And the limit value of _σ x_ is obtained when one of the five inequalities is no longer verified:

[5.4]

And the value of _σ x_ must then be located within the six curves in order to avoid fracture.

On this graph we can observe the different fracture modes. We also note that a small error on the fiber orientation near 0° can cause an important fall in the fracture characteristics, passing from one mode of fiber fracture to a mode of resin shearing. The orientation of fibers will therefore have to be controlled with utmost precision during manufacturing.

**Figure 5.4.** _Tensile stress limit of an off-axis UD ply. For a color version of this figure, seewww.iste.co.uk/bouvet/aeronautical2.zip_

The benefit of this criterion is that it is particularly simple and only requires the user to identify five material constants. It also gives an indication as to the encountered mode of fracture, which can give an indication of the modification in the stacking sequence that should be used, or which directions we have to add plies to avoid fracture. This can also nuance the result, as a small fracture in the resin can be accepted as long as it is not critical to the final fracture of the structure.

It nonetheless presents important discrepancies with experimental results that are due to the fact that interaction phenomena between different damage modes are not being taken into account. This is particularly significant for interaction between stress along the longitudinal direction ( _σ l_) and the in-plane shear stress ( _τ lt_) that is not accounted for, even though it is observed in practice (see upcoming sections).

EXAMPLE (Shear fracture of an off-axis UD ply).– We can also use the maximum stress criterion in order to determine the shear limit stress of an off-axis UD ply:

**Figure 5.5.** _Shear test of an off-axis UD ply. For a color version of this figure, seewww.iste.co.uk/bouvet/aeronautical2.zip_

We obtain the following stresses:

[5.5]

Thus:

[5.6]

**Figure 5.6.** _Shear limit stress of an off-axis UD ply. For a color version of this figure, seewww.iste.co.uk/bouvet/aeronautical2.zip_

We note that the tensile fracture in the longitudinal direction never actually occurs.

The effect of the shear sign, which can be surprising, is explained by the fact that a positive shear stress with fibers at 45° will put transverse compressive stress in the matrix, while a negative shear stress with fibers at 45° will put transverse tensile stress in the matrix.

**Figure 5.7.** _Effect of the shear sign on the fracture of a UD ply at 45°. For a color version of this figure, seewww.iste.co.uk/bouvet/aeronautical2.zip_

## **5.2. Maximum strain fracture criterion**

The maximum strain fracture criterion involves considering the UD ply as broken once one of the strains reaches a strain limit under tension, compression or shear:

[5.7]

Where _ε lt_ and _ε lc_ are the tensile and compressive fracture strains along the longitudinal direction, _ε tt_ and _ε tc_ are the tensile and compressive fracture strains in the transversal direction, and _γ ltf_ is the shear fracture strain. And we can link the different maximum strains with the maximum stresses by:

[5.8]

This criterion thus gives us the same values as a maximum stress criterion under pure tension/compression but induces an interaction between the two stresses via Poisson's ratio. It can appear as follows:

[5.9]

We will note, however, that there is a large difference between these two interactions. The first one (1st line of the previous equation) characterizes the effect of the stress _σ t_ on the fiber fracture, whereas the second (2nd line of the previous equation) characterizes the effect of the stress _σ l_ on the matrix fracture. And while the 1st criterion makes sense and can be observed in practice, the second one does not. As the stress _σ l_ passes primarily through the fibers, it will have very little effect on the matrix fracture.

In reality, this criterion is practically never used to characterize fracture in the transversal direction or under shear loading. However, it does make sense for predicting fiber fracture; thus, in the cases of a UD composite, it is common to just use the following:

[5.10]

Obviously, the written criterion does not take into account matrix cracking, which is not necessarily an issue in the case of a real-world stacking sequence (with fibers in all the four directions 0°, ±45° and 90°). The stress in the transverse direction of the studied ply will not be taken on by the latter but rather by its closest perpendicular neighbor (the transverse stress of plies at 0° is supported by plies at 90°, and _vice-versa_ , and those at +45° by plies at −45°, and _vice-versa_ ). This notion is fundamental as it allows us to gain some perspective over all the criteria integrating matrix cracking! In evidence, it is often not necessary to study in too much detail the effect of matrix cracking in fracture criteria because in the case of real stacking sequences, surrounding perpendicular fibers will support the loading. Obviously, this is only the case if there are fibers in all the directions (0°, ±45° and 90°), which is the case most of the time; and indeed, that is partly the reason why industrial materials use fibers in all the directions! Having said that, note that studying a fracture criterion on a UD composite (with all the fibers in the same direction) is very practical because it simplifies the problem, though it is generally not an accurate representation of real-world situations. In evidence, the behavior of a ply observed in isolation is very different to its behavior within a laminate. This is something we will touch on again further along in the upcoming sections.

The strain fracture criterion is also used in the case of woven plies, i.e. with fibers along the _l-_ and _t_ -directions woven together (satin, serge, taffetas, etc.). In this case, it makes sense to write a strain criterion in these two directions:

[5.11]

The strain criterion in its general form (equation [5.7]) translates, for a biaxial loading and no shearing, to a fracture envelope in the shape of a quadrangle. Keep in mind that in this figure, the incline of both the lines of the strain criterion characterizing the matrix fracture (the two lines passing through _σ tt_ and _σ tc_) makes no sense because stress _σ l_ will, in reality, have practically no influence on the matrix fracture (since _σ l_ passes through the fibers and not the resin).

**Figure 5.8.** _Maximum stress and strain fracture criteria. For a color version of this figure, seewww.iste.co.uk/bouvet/aeronautical2.zip_

We can also use this criterion to determine the fracture of a ply under tensile/compressive stress off of its orthotropic axes.

We then get:

[5.12]

And the limit value of _σ x_ is obtained when one of the five inequalities is not verified:

[5.13]

If we compare this criterion to the one with maximum stress, we note that these two criteria are practically equal.

**Figure 5.9.** _Tensile limit stress of an off-axis UD ply. For a color version of this figure, seewww.iste.co.uk/bouvet/aeronautical2.zip_

The strain criterion presents similar advantages to the previous one, i.e. that it requires five material constants, that it gives an indication as to the fracture mode and that it presents a weak interaction between stresses.

We nonetheless favor criteria such as Hill's, Tsai–Wu's or Yamada–Sun's over this type of criterion, as they better account for the different components of the stress tensor.

## **5.3. Hill's criterion**

This criterion can be seen as a generalization of the Von Mises criterion. The Von Mises criterion is based on a calculation of the maximum strain energy and appears in 2D:

[5.14]

Where _σ e_ is the elasticity limit in tension. As a reminder, this criterion is used to determine the elastic limit of a ductile material by comparing an invariant of the stress tensor, the Von Mises stress, to the tensile stress limit. It is established for an isotropic material and without distinguishing between tension and compression. The Hill's criterion can be seen as a generalization of this criterion:

[5.15]

And we can easily demonstrate that under tension along the longitudinal direction, we get:

[5.16]

Under tension in the transversal direction:

[5.17]

And under pure shearing:

[5.18]

The last parameter _C_ is determined by considering that under biaxial tension ( _σ l = σt_ and _τ lt = 0_) the fracture will be reached when _σ t_ reaches _σ tt_ and we can therefore show that:

[5.19]

In practice, this hypothesis is very difficult to verify because biaxial tests on this type of orthotropic material, with a large difference between behaviors in the _l-_ and _t_ -directions, are hard to interpret. Nonetheless, the influence of this parameter _C_ remains overall quite weak in reality and compared to the other parameters as its value has little impact on the criterion. Furthermore, we can demonstrate that this coefficient can even take on absurd values in certain cases and is often removed from the criterion.

The Hill's criterion is thus written in its general form as:

[5.20]

With:

[5.21]

This criterion gives the same values as the maximum stress and strain criteria under pure tension/compression but induces coupling between the stress components that accounts for part of the interaction between different damage types.

**Figure 5.10.** _Fracture criteria for a UD ply. For a color version of this figure, seewww.iste.co.uk/bouvet/aeronautical2.zip_

It accounts, in particular, for the effect of shearing on the biaxial stress criterion. For instance, if the shear stress is equal to three quarters of the shear limit stress:

[5.22]

The criterion is thus reduced by as much (Figure 5.10).

We can then use this criterion to determine the stress limit under tension / compression of a UD ply:

[5.23]

We can easily demonstrate that the cross-product term _σ l.σt_ will be negligible compared to the shear term in most cases (Figure 5.13).

**Figure 5.11.** _Tension of an off-axis UD ply. For a color version of this figure, seewww.iste.co.uk/bouvet/aeronautical2.zip_

We can then trace the evolution of the maximum stress depending on the angle _θ_ :

We note that the results are close to the strain or stress criteria (which are practically the same) and, in reality, we can demonstrate that this criterion is slightly closer to real-world applications. We can also trace the evolution of the four terms of the criterion to determine the predominant term for each angle.

**Figure 5.12.** _Tensile limit stress of an off-axis UD ply. For a color version of this figure, seewww.iste.co.uk/bouvet/aeronautical2.zip_

**Figure 5.13.** _Comparison of the four terms of the Hill's criterion. For a color version of this figure, seewww.iste.co.uk/bouvet/aeronautical2.zip_

Once again, for low angles, we see that fiber tension/compression is predominant, while for high angles, it's the matrix tension/compression that is predominant and between the two, the predominant stress is the matrix shearing. As for the cross-term, it is always negligible compared to the others.

The main inconvenience of the Hill's criterion is that it exaggerates the effect of matrix cracking. For example, in the case of a tensile test on a laminate with a ply at 0° and one at 90°, the criterion gives an early fracture of the 90° ply under matrix cracking (see section 6.4 in Chaper 6). Nonetheless, this premature fracture under transverse cracking of the 90° ply is not too detrimental for the structure, because the stress in that direction is supported by the fibers of the 0° ply (direction _t_ of the 90° ply is also the direction _l_ of the 0° ply). Moreover, it is generally pointless to take matrix cracking into account if there are fibers in all the four directions (0°, ±45° and 90°) to support the loading; this is practically always the case in the industry (this is partly the reason why there are plies in all the directions). Once again, keep in mind that studying a fracture criterion on a UD composite (and with all the fibers in the same direction) is very useful as it helps simplify the problem, but it is often not accurate to reality because the behavior of a ply observed independently is very different to its behavior within a laminate.

To avoid granting too much importance to the effects of matrix cracking in the Hill's fracture criterion, we can, for example, remove the terms at   and at _σ l.σt_, which then gives us the Yamada–Sun's criterion, which will be presented in section 5.5 in Chapter 5.

EXAMPLE (Tube under torsion with the Hill's criterion).–

**Figure 5.14.** _Study of a tube under torsion. For a color version of this figure, seewww.iste.co.uk/bouvet/aeronautical2.zip_

A thin tube, with a thickness of _e_ and a radius of _R_ , loaded under torsion, is performed from a UD ply wrapped with an angle _α_. Determine the fracture torque _C r_ with the Hill's criterion.

Supposing the tube is thin, we can show that upon first glance only one stress _τ zθ_ is generated by torsion, and:

[5.24]

Therefore, in the frame of the UD, we have:

[5.25]

Then the Hill's criterion:

[5.26]

We can therefore trace the shear limit stress and the respective influence of each fracture mode:

[5.27]

**Figure 5.15.** _Fracture of a UD tube under torsion: shear stress limit a) and influence of the different terms of the Hill's criterion b). For a color version of this figure, seewww.iste.co.uk/bouvet/aeronautical2.zip_

We then observe that the tube resists better for positive angle _α_. For this type of orientation, the torsion loading will cause transverse compression, whereas for negative _α_ , it will cause transverse tension. For example, for _α_ _=_ 45°:

**Figure 5.16.** _Shear fracture of a ply at 45°. For a color version of this figure, seewww.iste.co.uk/bouvet/aeronautical2.zip_

And for _α_ _=_ −45°:

**Figure 5.17.** _Shear fracture of a ply at_ − _45°. For a color version of this figure, seewww.iste.co.uk/bouvet/aeronautical2.zip_

## **5.4. Tsai–Wu criterion**

Tsai–Wu proposed a fracture criterion that presents the advantage of passing continuously from tension to compression without having to verify the sign of stress. To achieve this, they added a linear stress term to the quadratic term, proposed by Hill, to add information on the sign of the stresses:

[5.28]

Where notation _s_ comes from shearing. As, furthermore, this criterion must not depend on the sign of _τ lt_, then:

[5.29]

If we write this tension/compression criterion along the longitudinal and transversal directions, then under shearing, we get:

[5.30]

And the last coefficient _f lt_ could be reached thanks to an additional test. For example, if we performed an off-axis tensile test at 45°, we would then get an expression of this coefficient depending on the stress limit under tension at 45°, _σ_ 45 _t_ :

[5.31]

Nonetheless, in most cases, this coefficient is of lesser importance and can be approximated in a satisfying manner by (it is even sometimes null):

[5.32]

Thus, the Tsai–Wu criterion:

[5.33]

This criterion presents a high difference with the Hill's criterion under compression:

**Figure 5.18.** _Limit stress of a UD ply. For a color version of this figure, seewww.iste.co.uk/bouvet/aeronautical2.zip_

This criterion accounts for the beneficial effect of a transverse compression on the resistance to longitudinal compression. In practice, the fracture under longitudinal compression is due to micro-buckling of the fibers; a transverse compression will indeed stabilize this buckling and delay the compressive fracture.

In practice, it is difficult to conclude on the best of the both criteria under compression/compression. For example, we can show that the stabilizing effect of transverse compression over the longitudinal compression is rather true for a confined ply (surrounded by other plies in different directions) but it is less verified for a thick or isolated UD ply. And, more generally, the sizing of composite structures under compression remains a largely open question and is the object of numerous researches.

Moreover, the Tsai–Wu criterion, just like Hill's, accounts for the effect of shearing on the biaxial stress criterion. For example, if the shear stress is equal to three quarters of the shear limit stress:

[5.34]

The criterion is then reduced by as much (Figure 5.18).

We can then use this criterion in order to determine the stress limit of an off-axis UD ply.

**Figure 5.19.** _Tensile limit stress of an off-axis UD ply. For a color version of this figure, seewww.iste.co.uk/bouvet/aeronautical2.zip_

Unlike the case of biaxial stresses, the Tsai–Wu criterion is very close to that of Hill's. This is due to the fact that the shearing part of the criterion quickly overpowers the compression part of the criterion. In practice, this criterion gives relatively good results and is, like the Hill's criterion, widely utilized.

In 2003, an international exercise (the "World Wide Failure Exercise" [SOD 04]) was carried out to test different fracture models for UD composite. The material used was a UD with fiberglass and epoxy resin. The first comparison was performed with biaxial stress in the plane ( _σ l, σt_) (called _σ x_ and _σ y_ in this publication); unfortunately, there are no experimental results under tension/compression. Nonetheless, we see that the Hill and Tsai–Wu criteria (referred to as the Tsai criterion in this publication) translate the interaction between tension/compression, unlike maximum stress and strain criteria.

**Figure 5.20.** _Comparison of different fracture criteria during the "World Wide Failure Exercise" in the plane (σ l, σt) SOD 04] a) and comparison with the criteria presented here b). For a color version of this figure, see [www.iste.co.uk/bouvet/aeronautical2.zip_

The second comparison was carried out with transverse tension/compression _σ t_ and in-plane shear _τ lt_. We then observe an increase in shear resistance of the UD composite in the presence of transverse compression. This is due to the fact that compression will reduce the tension component induced by shearing (which causes two principal stresses equal to ± _τ_ at ±45° under pure shearing):

[5.35]

**Figure 5.21.** _Comparison of different fracture criteria during the "World Wide Failure Exercise" in the plane (σ t, τlt) SOD 04] (a) and comparison with the criteria presented here (b). For a color version of this figure, see [www.iste.co.uk/bouvet/aeronautical2.zip_

Thus:

[5.36]

Yet, the material being more resistant under compression than under tension, the effect of a low transverse compressive stress will be beneficial. Obviously, if we continue to increase the compression (in absolute value), the fracture will appear under compression. This effect is relatively well rendered by the Tsai–Wu criterion (referred to as the Tsai criterion in this publication), even if we can show that this effect is totally fortuitous in this criterion (which is a convenient coincidence).

## **5.5. Yamada–Sun criterion**

To avoid over-accounting for the matrix cracking in the Hill's criterion, we can, for example, remove the terms at   and at _σ l.σt_, which then gives the Yamada–Sun criterion:

[5.37]

With:

[5.38]

Keep in mind that this criterion mainly focuses on the fiber fracture (even if we can show that the shear term also contains matrix cracking). It must therefore not be used for the fracture of a UD ply on its own but to predict fractures of a laminate. In fact, that is why tracing this criterion in plane ( _σ l, σt_) makes no sense (and will therefore not be traced). It would just give us two vertical straight lines:

[5.39]

This doesn't mean that there is no fracture in transverse tension along the _t_ \- direction but that this fracture is not accounted for in this criterion because it is secondary in the case of a laminate. In other words, this criterion can only be used if  there exists at least one other perpendicular ply to the load (in addition to the plies in the loading direction).

In the same way, this criterion cannot be used to predict the fracture of an off-axis UD ply. However, it can be used to trace the fracture criterion in plane ( _σ l, τlt_), and you will easily be able to demonstrate that we find the same result using the Hill's criterion (Figure 5.21).

Moreover, it is generally pointless to account for matrix cracking if there are fibers in all the four directions (0°, ±45° and 90°) to support the loading, which is practically always the case in industrial uses (indeed, this is part of the reason why there are always plies in all directions).

Therefore, the Yamada–Sun criterion is the most common in the industry and will be widely used in this book.

## **5.6. Conclusion**

Lastly, to conclude this chapter, never forget that these criteria are never absolute truths: they are models that attempt to translate reality, one which almost always remains far more complex the closer you look at it. In evidence, a fracture criterion is considered correct until proven false or, more precisely, until experimental tests have disproven it (which will happen sooner or later, because a criterion remains true only in a given domain)!

# 6
Membrane Behavior of a Laminated Composite Plate

## **6.1. Generalities and notations**

We call laminate the superposition of a number of UD plies linked together by interfaces.

**Figure 6.1.** _Notation of the plies within a laminate. For a color version of this figure, seewww.iste.co.uk/bouvet/aeronautical2.zip_

The plies are numbered from the bottom up. We note _e k_ as the thickness of the ply _k_ , and _z k-1_ and _z k_ as the _z_ coordinates of its lower and upper interfaces. The  midplane is taken at midthickness of the laminate, and the thickness is noted as _h_. And for example, the draping sequence:

will be written in the following form:

Where the subscript number expresses the number of plies and _S_ the eventual symmetry.

We will suppose that the laminate is thin, and therefore that the stress state is plane; thus, for example, the stress and strain in ply _k_ :

[6.1]

which in most cases will depend on _z_.

Before entering into the detail of calculating a laminated plate in membrane and bending behavior, it is useful to restate what a plate model is and why is it useful. A plate model serves to transform the real 3D problem to a 2D problem (and thus makes it simpler than the initial problem).

_The objective of a plate model is to regroup all the necessary information in the midplane in order to perform the required calculations using nothing but this information. Once the calculation has been done in the midplane, we can use these data to evaluate the stress and strain fields everywhere within the plate (in sum, depending on the thickness)._

In practice, we will see that, knowing resultant forces, resultant moments, membrane strains and plate curvatures within the midplane, we can evaluate the stress and strain fields throughout the thickness of the plate (and, for example, apply fracture criteria). The entire constitution of the plate will then be grouped into a stiffness matrix which will link the forces (in the sense of generalized forces, in this case, resultant forces and moments) to strains (in the sense of generalized strains, in this case, membrane strains and plate curvatures). Thus, we can perform all the calculations simply using these three values (forces, strains and stiffness matrix), and once these three values have been established, we can return to the distribution of stress and strain throughout the thickness of the plate. This is evidently a similar  procedure to the beam theory, where we simplify a 3D problem into a 1D problem along the neutral fiber of the beam.

## **6.2. Membrane behavior, bending behavior and mirror symmetry**

We consider that a plate works as a membrane when all the external loadings are forces within its midplane, and its midplane does not suffer out-of-plane deformation.

**Figure 6.2.** _Diagram of membrane and bending behavior. For a color version of this figure, seewww.iste.co.uk/bouvet/aeronautical2.zip_

And conversely, a plate works under bending if the external loadings deform its midplane with out-of-plane displacements, for example, through the intermediary of moments or out-of-plane forces in this midplane.

For a laminate plate, we can demonstrate that the plate will deform with membrane displacements if the external loadings are forces (and not moments) in its midplane, and that it presents mirror symmetry.

A laminate plate presents mirror symmetry if its midplane is the symmetry plane of the draping sequence, for example:

In practice, mirror symmetry is respected for most laminates because it avoids twisting of the plate during cooldown. Indeed, laminates tend to be polymerized at high temperature, typically 120 °C for an epoxy resin, and therefore the cooldown will induce twisting if the plate does not present mirror symmetry.

**Figure 6.3.** _Effect of mirror symmetry during cooldown after manufacturing of the laminate_

There are, nonetheless, cases where this mirror symmetry is not respected, for example, if the plate is curved (and we will then call it a shell rather than a plate), because the very shape of the structure avoids twisting. For example, for a rear propeller blade on a helicopter, the draping sequence is typically composed of two fabric plies _[0/90°, 45/–45°]_ , and the shape of the blade box and presence of foam avoids twisting during cooldown.

Mirror symmetry also allows us to avoid the coupling of membrane and bending behaviors, keeping in mind that:

  * – external force loadings in the midplane only cause in-plane displacements of the midplane;
  * – external moment loadings in the midplane only cause out-of-plane displacements of the midplane (in first approximation).

**Figure 6.4.** _Diagram and draping sequence of a rear helicopter rotor blade BIZ 09]. For a color version of this figure, see[www.iste.co.uk/bouvet/aeronautical2.zip_

**Figure 6.5.** _Displacements during membrane and bending loadings. For a color version of this figure, seewww.iste.co.uk/bouvet/aeronautical2.zip_

In the following section, we will limit ourselves to studying membrane behavior of plates with mirror symmetry, and then in the next chapter, we will look at the most common cases.

## **6.3. Resultant forces**

Let us take a look at the stresses on a face of normal vector _y_ :

**Figure 6.6.** _Stresses on a face of normal vector y. For a color version of this figure, seewww.iste.co.uk/bouvet/aeronautical2.zip_

This face is subject to stresses _σ y_ along _y_ -direction and _τ xy_ along _x_ -direction. These stresses will, in general, depend on _z_ , and we can sum them along _z_ -direction in order to define a generalized force characteristic of stresses throughout the thickness of the plate:

[6.2]

Similarly, for a face of normal vector _x_ subject to stresses _σ x_ along _x_ -direction and _τ xy_ along _y_ -direction, we define:

[6.3]

We can therefore define the normal resultant force along _x_ -direction, _N x_, along _y_ -direction, _N y_, and the shear resultant force, _T xy_:

[6.4]

**Figure 6.7.** _Resultant forces for membrane behavior. For a color version of this figure, seewww.iste.co.uk/bouvet/aeronautical2.zip_

It is important to pay attention to the units; these resultant forces are expressed in N/mm and represent the force supported by a square plate with a one unit side. In particular, a plate with sides _dx_ and _dy_ is subject to the following forces:

**Figure 6.8.** _Forces on a square with sides dx and dy under membrane loading. For a color version of this figure, seewww.iste.co.uk/bouvet/aeronautical2.zip_

We can also demonstrate that this plate is in equilibrium no matter what the values of _dx_ and _dy_ are.

## **6.4. Displacement field, stress field and strain field**

If a laminate plate presenting mirror symmetry is subject to external loading of resultant forces ( _N x_, _N y_ and _T xy_) in its midplane, then the displacements are planar and homogeneous along the thickness.

[6.5]

**Figure 6.9.** _Displacement of a plate under membrane loading. For a color version of this figure, seewww.iste.co.uk/bouvet/aeronautical2.zip_

Where we note _M(x, y, z)_ a random point of the plate; _u(M)_ and _v(M)_ its in-plane displacements; _M 0(x, y, 0)_ its projection in the midplane; and _u 0(M0)_ and _v 0(M0)_ its displacements.

We can then determine the strains, which are homogeneous through the thickness:

[6.6]

The strains being homogeneous throughout the thickness and the behavior being elastic and homogeneous within a single ply but different from one ply to the next, depending, among other things, on the orientation of the ply, the stresses will therefore be homogeneous within one ply, but different from one ply to the next (potentially with discontinuity among the interfaces). For instance, for ply _k_ :

[6.7]

**Figure 6.10.** _Strains and stresses in a laminate subject to membrane loading. For a color version of this figure, seewww.iste.co.uk/bouvet/aeronautical2.zip_

This will simplify the expression of resultant forces as follows:

[6.8]

With an elastic behavior law for ply _k_ :

[6.9]

And therefore a relation between the resultant forces and the strains for the laminate:

[6.10]

With:

[6.11]

Again, note the units; the strains are unitless, the resultant forces are in N/mm and stiffness _A ij_ is in N/mm.

We also sometimes define the average stresses within the thickness:

[6.12]

Note that these stresses are just an average and do not apply to the material. Applying a fracture criterion with these average stresses would be particularly pointless.

We also define the average stiffness matrix of the laminate by:

[6.13]

where this time the component _A ij/h_ represents the average stiffness in MPa.

## **6.5. Tension / shear coupling**

Because terms _A 16_ and _A 26_ are generally not 0, there is a coupling between the tension / compression behavior and the shearing behavior in the same way as for a UD subjected to stress off its orthotropic axes. However, these terms are null if the number of plies at _+θ_ is equal to the number of plies at _–θ_. In practice, we often see plies at 0°, 90°, +45° and –45° and as many plies at +45° as at –45°; thus:

[6.14]

EXAMPLE (Laminate [0,90]S).– Consider a plate with a thickness of _h = 1 mm_ , with a _[0,90] S_ draping sequence (i.e. four plies of 0.25 mm thickness) made from T300/914:

[6.15]

**Figure 6.11.** _Laminate 0,90] S. For a color version of this figure, see [www.iste.co.uk/bouvet/aeronautical2.zip_

We then have:

[6.16]

And:

[6.17]

In the case where _E l >> Et_ and _E l >> Glt_:

[6.18]

We obtain a material where the average modulus along _x_ ( _y_ )-direction is due to the fibers along _x_ ( _y_ )-direction and is equal to half the UD modulus. We also obtain a shear modulus equal to the original, which makes sense as the shear moduli of different plies are equal in _(x, y)_. And we obtain a low Poisson's ratio since, during tension along _x_ ( _y_ )-direction, any shrinking along _y_ ( _x_ )-direction will be impeded by the fibers along _y_ ( _x_ )-direction.

Let us now study this laminate in the case of a simple tension:

[6.19]

then an average stress of:

[6.20]

**Figure 6.12.** _Tension of a laminate 0,90] S. For a color version of this figure, see [www.iste.co.uk/bouvet/aeronautical2.zip_

In order to determine the different stresses and strains in each ply, we begin by inverting the stiffness matrix  , and then multiplying it by the resultant forces to obtain the membrane strain (with _N y = Txy = 0_ in this instance):

[6.21]

We can then determine the strains in the frame of the ply:

  * – For plies 1 and 4 at 0°:

[6.22]

  * – For plies 2 and 3 at 90°:

[6.23]

_Use caution when using these notations; the frame (l, t) depends on the selected ply!_

Then, by multiplying these strains with the stiffness matrix _ _expressed in _(l, t)_

(meaning it will be the same for all plies):

  * – For plies 1 and 4 at 0°:

[6.24]

  * – For plies 2 and 3 at 90°:

[6.25]

And, in all logic, the tension stress along _x_ -direction is almost entirely supported by the fibers at 0°:

**Figure 6.13.** _Strains and stresses in the plies of a laminate 0,90] S under tensile loading. For a color version of this figure, see [www.iste.co.uk/bouvet/aeronautical2.zip_

Note in the previous figure that we took care to present each notation in full form, specifying each time whether _l_ and _t_ is applied to ply at 0° or 90° (which is generally not the case, as it creates too many notations). We therefore observe that:

  * – Strain _ε_ x is homogeneous throughout the thickness, as it is a membrane loading and is equal to the longitudinal strain of plies at 0°, _ε_ l0°, and to the transverse strain of plies at 90°, _ε_ t90°.
  * – Strain _ε_ y is homogeneous throughout the thickness, as it is a membrane loading and is equal to the longitudinal strain of plies at 90°, _ε_ l90°, and to the transverse strain of plies at 0°, _ε_ t0°.
  * – Stress _σ_ x is homogeneous for each ply (but varies from one ply to the next), as it is a membrane loading, and is equal, in plies at 0°, to longitudinal stress, _σ_ l0°, and in plies at 90°, to transverse stress _σ_ t90°. Also note that the average of _σ_ x is indeed equal to the average stress _σ 0x = 100 MPa_, but that plies at 0° support almost twice as much loading than the average stress.
  * – Stress _σ_ y is homogeneous for each ply (but varies from one ply to the next), as it is a membrane loading, and is equal, in plies at 90°, to longitudinal stress, _σ_ l90°, and in plies at 0°, to transverse stress, _σ_ t0°. Also note that the average of _σ y_ is null because the average stress along _y_ -direction is null ( _σ 0y = 0 MPa_).

We can then trace the evolution of the moduli depending on the orientation of the laminate (Figure 6.14).

**Figure 6.14.** _Off-axis tension of a laminate 0,90] S. For a color version of this figure, see [www.iste.co.uk/bouvet/aeronautical2.zip_

Where _E 0 = 70.7 GPa_ (Figure 6.15) is Young's modulus along _x_ -direction when _θ_ _= 0°_. We obtain the maximum Young's modulus at 0° and at 90°, and equal moduli _E x_ and _E y_. The shear modulus has a value equal to _G lt_ at 0° and 90° and reaches its maximum at 45°: indeed shearing induces tension / compression at ±45°. This phenomenon will be specified in the following example. The Poisson's ratio is practically null at 0° and 90° as previously mentioned, and we can demonstrate that it is maximum and has a value of approximately 0.8 at 45°. As for the coupling coefficients, they are null at 0°, 45° and 90° thanks to the symmetries. In these three cases, the material is orthotropic (at least for its membrane behavior) and loaded along its orthotropic axes.

**Figure 6.15.** _E lasticity characteristics of an off-axis laminate 0,90]S. For a color version of this figure, see [www.iste.co.uk/bouvet/aeronautical2.zip_

EXAMPLE (Laminate [45,–45]S).– Consider a plate with a thickness _h_ and a _[45,–45] S_ draping sequence.

**Figure 6.16.** _Laminate [45,–45] S_

We then obtain:

[6.26]

And:

[6.27]

Or in the case where _E l >> Et_ and _E l >> Glt_:

[6.28]

We therefore obtain a low average modulus along _x_ ( _y_ )-direction facing the fiber modulus because loads along _x_ ( _y_ )-direction mainly generate shear strain _γ lt._

Let us now study this laminate for a simple tensile loading:

[6.29]

Then an average stress of:

[6.30]

By then inverting the stiffness matrix _ _, we get:

[6.31]

Or in the coordinate systems of the plies:

  * – For plies 1 and 4 at 45°:

[6.32]

  * – For plies 2 and 3 at –45°:

[6.33]

Tension along _x_ -direction will therefore be supported by tension in the fibers and shearing. Evidently, the shear stiffness is lower than the fiber stiffness, and this low stiffness will govern the stiffness of the laminate under tensile loading; stiffness that will therefore also be low and be governed by the shear modulus. This explains that Poisson's ratio is in the order of 1, because this tension will mainly cause shear strain.

Finally, we obtain a shear modulus governed by the stiffness of the fibers and therefore far higher than the initial shear modulus. This is due to the fact that a shear stress in _(x, y)_ will be decomposed into a tension at 45° and a compression at –45° and will therefore load the fibers at 45° and –45°.

**Figure 6.17.** _Shear stress of a 45° ply. For a color version of this figure, seewww.iste.co.uk/bouvet/aeronautical2.zip_

This fundamental point can be summed up as:

_In order to obtain high shear stiffness, the fibers need to be placed at +45° and at –45°._

Finally, we can study the stiffness moduli of this laminate depending on the orientation of the laminate; this obviously gives us the same curves as for laminate _[0,90] S_ offset by 45°. And we observe once again that the maximum shear modulus is at 45°.

EXAMPLE (Quasi-isotropic laminate).–

If we wish to realize a planar isotropic laminate, meaning that it presents the same behavior in all orientations (and since for manufacturing reasons, we only use plies at 0°, 90°, 45° and –45°) while respecting mirror symmetry, then we generally use the draping sequence:

We restate that using composite to realize an isotropic plate generally does not make sense as the point of composite is to adapt to the loading of the structure; this reflex to use an isotropic material is a habit that arises from metal design where plates are generally isotropic.

**Figure 6.18.** _Quasi-isotropic laminate 0,45,90,–45] S. For a color version of this figure, see [www.iste.co.uk/bouvet/aeronautical2.zip_

We then obtain:

[6.34]

Then in the case where _E l >> Et_ and _E l >> Glt_:

[6.35]

We will therefore have a modulus along _x_ ( _y_ )-direction linked to the stiffness of the fibers, mainly to fibers at 0° (90°) and in part to fibers at ±45°. As for the shear modulus, it is entirely due to the fibers at ±45°. Note also that we observe the relation linking the shear modulus to Young's modulus and Poisson's ratio of an isotropic material:

[6.36]

In reality, _E t_ and _G lt_ are not really negligible compared to _E l_, and these conclusions need to be nuanced.

For example, if we consider a T300/914 subject to a tensile resultant force:

[6.37]

**Figure 6.19.** _Quasi-isotropic laminate 0,45,90,–45] S under tension. For a color version of this figure, see [www.iste.co.uk/bouvet/aeronautical2.zip_

We then obtain:

  * – For plies 1 and 8 at 0°:

[6.38]

  * – For plies 2 and 7 at 45°:

[6.39]

  * – For plies 3 and 6 at 90°:

[6.40]

  * – For plies 4 and 5 at –45°:

[6.41]

In this way, we observe that, when a quasi-isotropic laminate is loaded by an average tensile stress, the plies at 0° experience almost three times more stress (in the fiber direction), the plies at ±45° experience more or less the same stress (in the fiber direction), and the plies at 90° experience a compression stress almost equal (in absolute value). This illustrates in particular that we must always watch the notation of average stress since no plies feel it (some experience more and others less), and that it should never be used to apply a fracture criterion!

We can then trace the evolution of the moduli of a quasi-isotropic laminate depending on the orientation of the laminate:

**Figure 6.20.** _Quasi-isotropic laminate 0,45,90,–45] S under off-axis tensile loading. For a color version of this figure, see [www.iste.co.uk/bouvet/aeronautical2.zip_

**Figure 6.21.** _E lasticity characteristics of a quasi-isotropic laminate 0,45,90,–45]S under off-axis traction. For a color version of this figure, see [www.iste.co.uk/bouvet/aeronautical2.zip_

And we then get a lovely Lapalissade, i.e. that a quasi-isotropic laminate is actually isotropic. We can then wonder why we use the term quasi-isotropic and not isotropic: in the upcoming chapter, we will demonstrate that if the membrane stiffness is planar isotropic, this is not the case for the bending stiffness, and thus the use of "quasi"!

# 7
Bending Behavior of a Laminated Composite Plate

## **7.1. Notations**

The notations here will be the same as in the previous chapter:

**Figure 7.1.** _Notation of the plies within a laminate. For a color version of this figure, seewww.iste.co.uk/bouvet/aeronautical2.zip_

## **7.2. Resultant moments**

On top of resultant forces raised in the previous chapter:

[7.1]

Stresses _σ x_, _σ y_ and _τ xy_ will cause moments in the mid-plane:

[7.2]

**Figure 7.2.** _Laminate under membrane/bending loading (3D diagram). For a color version of this figure, seewww.iste.co.uk/bouvet/aeronautical2.zip_

**Figure 7.3.** _Laminate under membrane and bending loading (in-plane diagram). For a color version of this figure, seewww.iste.co.uk/bouvet/aeronautical2.zip_

Note the units: these resultant moments are in N.mm/mm or in N and represent the moment supported by a plate with a single unit side.

Due to their definitions, _M x_ ( _M y_) represents the moment resulting from stress _σ x_ ( _σ y_) on a face of normal vector _x_ ( _y_ ) and is therefore directed along _y_ (- _x_ )-direction; _M xy_ is the resultant moment from stress _τ xy_ on a face of normal vector _x_ and is therefore directed along _-x_ -direction or by symmetry due to stress _τ xy_ on a face of normal vector _y_ and therefore directed along _y_ -direction.

_M x_ ( _M y_) represents the resultant moment of bending around _y_ ( _x_ ) and _T xy_ the resultant moment of twisting :

**Figure 7.4.** _The three bending/twisting resultant moments. For a color version of this figure, seewww.iste.co.uk/bouvet/aeronautical2.zip_

## **7.3. Displacement field, stress field and strain field**

Consider a laminate plate presenting mirror symmetry and under external resultant moments ( _M x_, _M y_ and _M xy_) in its mid-plane. We apply the Love–Kirschoff hypothesis: a normal vector in the mid-plane before deformation will remain a line segment perpendicular to the mid-plane after deformation.

**Figure 7.5.** _Displacement field of a plate under bending loading. For a color version of this figure, seewww.iste.co.uk/bouvet/aeronautical2.zip_

**Figure 7.6.** _Displacement field of a plate under bending loading. For a color version of this figure, seewww.iste.co.uk/bouvet/aeronautical2.zip_

And the displacement of any given point _M(x,y,z)_ of the laminate can then be presented as follows:

[7.3]

Where displacements _u 0(x,y)_ and _v 0(x,y)_ are (upon first glance) the results of the membrane loading as discussed in the previous chapter, displacement _w 0(x,y)_ is the displacement along _z_ -direction of the mid-plane that results from bending and _∂w 0/∂x (∂w0/∂_ _y_ ) represents the angle of rotation of the normal vector of the mid-plane along _-y_ __( _x_ )-direction. We can then deduce the strains, which are not constant throughout the thickness as in the case of membrane loading but rather linear throughout the thickness:

**Figure 7.7.** _Membrane and bending strains within a plate. For a color version of this figure, seewww.iste.co.uk/bouvet/aeronautical2.zip_

[7.4]

Or even:

[7.5]

With:

[7.6]

Where  _0 (M0(x,y))_ represents the curvature of the plate defined in the mid-plane. Subsequently the information of the mid-plane, i.e. its in-plane strain and curvature, allows us to determine the strains of any given point of the plate (meaning we have successfully built a plate model).

As the strains are linear throughout the thickness and the behavior of the plies is evenly elastic within a single ply but different from one to the next, the strains are therefore linear throughout each ply with potential discontinuity in the interfaces:

**Figure 7.8.** _Bending strain and stress within a laminate. For a color version of this figure, seewww.iste.co.uk/bouvet/aeronautical2.zip_

As in the case of the membrane, once we know the behavior law of a ply, we can determine the stress from the resulting strain:

[7.7]

Which can be presented as follows:

[7.8]

After integration according to the thickness, this allows us to determine the resultant forces and moments depending on the strain and curvature of the mid-plane:

[7.9]

With:

[7.10]

Where   is the 3×3 membrane stiffness matrix that we discussed previously,   is the 3×3 bending stiffness matrix and   is the membrane/bending coupling matrix between the membrane behavior and the bending behavior.

Once again, note the units: the strain is unit-less, the curvature is in mm–1, the resultant force is in N/mm, the resultant moment is in N, meaning stiffness _A ij_ is in N/mm, stiffness _B ij_ is in N and stiffness _D ij_ is in N.mm.

And of course:

[7.11]

The (membrane/bending) coupling matrix   in particular implies that, when subject to loading within its mid-plane, the laminate will deform outside of its mid-plane or, subject to bending moment, it will be deformed as a membrane:

**Figure 7.9.** _Illustration of membrane/bending coupling within a plate. For a color version of this figure, seewww.iste.co.uk/bouvet/aeronautical2.zip_

Coupling matrix   in particular will be null for a laminate presenting mirror symmetry. Since this mirror symmetry avoids the twisting of the plate during the cooldown of the polymerization, a significant amount of laminates used in the industry present mirror symmetry.

We can also note that due to the expression of   and  ,   does not depend on the position of the plies, while   depends on their respective positions. This is due to the fact that   represents the in-plane stiffness of the plate and basically depends on the number of fibers in each direction, while   represents the bending stiffness of the plate and depends both on the number of fibers in each direction and their position compared to the mid-plane (the further away they are, the higher the bending stiffness).

## **7.4. Bending/twisting coupling**

As in general the terms _D 16_ and _D 26_ of the bending stiffness matrix are non-zero, there is a coupling between the behavior of pure bending and that of twisting. Unlike the case of the membrane, these terms are non-zero even when the number of plies at _+θ_ is the same as the number of plies at _–θ_. Nonetheless, they will be null if there are only plies at 0° and at 90°, and will remain relatively low (but not null except in very specific cases) in the case where there are as many plies at _+θ_ as there are at _–θ_ and if these plies are bunched together in sets of two (1 at _+θ_ and 1 at _–θ_ ).

EXAMPLE (Laminate [0,90]S).– Consider a plate with a thickness of _h_ and a draping sequence of _[0,90] S_.

**Figure 7.10.** _Laminate 0,90] S. For a color version of this figure, see [www.iste.co.uk/bouvet/aeronautical2.zip_

We then have:

[7.12]

And in the case where _E l>>Et_ and _E l>>Glt_:

[7.13]

We obtain a far higher value for _D 11_ than we do for _D 22_, which is due to the fact that a bending moment _M x_ will load the 0° plies in the fiber direction, while _M y_ will load these plies in the transverse direction. The 0° plies are the furthest away from the mid-plane, meaning they are the most capable of providing bending stiffness _D 11_. We obviously reach the same conclusion as for membrane behavior which is that to achieve a high level of stiffness, the fibers must be aligned with the loading.

**Figure 7.11.** _Stress field within the plies of a laminate 0,90] S under bending moment Mx. For a color version of this figure, see [www.iste.co.uk/bouvet/aeronautical2.zip_

**Figure 7.12.** _Stress field within the plies of a laminate 0,90] S under bending moment My. For a color version of this figure, see [www.iste.co.uk/bouvet/aeronautical2.zip_

**Figure 7.13.** _Stress field within the plies of a laminate 0,90] S under twisting moment Mxy. For a color version of this figure, see [www.iste.co.uk/bouvet/aeronautical2.zip_

You should note that in the previous figure, the direction of stresses _τ xy_ should be along _x_ ( _y_ )-direction for a face of normal vector _y_ ( _x_ ), which is not the case in the diagram as that would render it unreadable (thus the dotted lines).

The low value of _D 12_ translates the coupling between directions _x_ and _y_ that exists thanks to the effect of the resin. Similarly, _D 66_ is low and is only due to the shear stiffness and therefore the resin. Lastly, _D 16_ and _D 26_ are null as there are no plies at ±45°.

For example, if we consider a T300/914 subjected to:

[7.14]

**Figure 7.14.** _Laminate 0,90] S. For a color version of this figure, see [www.iste.co.uk/bouvet/aeronautical2.zip_

We then have:

[7.15]

And we then calculate the strain field from:

[7.16]

**Figure 7.15.** _Strain field within the plies of a laminate 0,90] S under bending moment Mx. For a color version of this figure, see [www.iste.co.uk/bouvet/aeronautical2.zip_

Thus:

  * – For ply 1 at 0°, the value is maximum (in absolute value) in _–h/2_ and is:

[7.17]

  * – For ply 2 at 90°, the value is maximum (in absolute value) in _–h/4_ and is:

[7.18]

  * – For ply 3 at 90°, the value is maximum (in absolute value) in _–h/4_ and is:

[7.19]

  * – For ply 4 at 0°, the value is maximum (in absolute value) in _–h/2_ and is:

[7.20]

EXAMPLE (Laminate [45,–45]S).– Consider a plate with a thickness of _h_ and a draping sequence _[45,–45] S_.

**Figure 7.16.** _Laminate 45,-45] S. For a color version of this figure, see [www.iste.co.uk/bouvet/aeronautical2.zip_

We then have:

[7.21]

In the case where _E l>>Et_ and _E l>>Glt_:

[7.22]

Due to symmetry, _D 11_ and _D 22_ are equal and have relatively high values, as they are mainly due to the loading applied to 45° fibers. The values, however, remain below that of _D 11_ for _[0,90] S_ as the fibers are not aligned with the stress induced by bending. The twisting stiffness _D 66_ is also relatively high since this loading ( _M xy_) will also load the fibers. Lastly, the coupling term between bending along _x_ -direction and the one along _y_ -direction, _D 12_, as well as the terms for the coupling between bending and twisting will be relatively high for the same reasons.

EXAMPLE (Quasi-isotropic laminate).– Consider a plate with a thickness of _h_ and a draping sequence _[0,45,90,-45] S_

**Figure 7.17.** _Quasi-isotropic laminate 0,45,90,-45] S. For a color version of this figure, see [www.iste.co.uk/bouvet/aeronautical2.zip_

We then have:

[7.23]

In the case where _E l>>Et_ and _E l>>Glt_:

[7.24]

We then have a high value for _D 11_ thanks to the stiffness of the 0° fibers loaded along their direction by a moment _M x_, a lower value _D 22_ because the 90° plies are closer to the mid-plane, and a low value _D 12_ as it is a result of the resin.

We can also question the isotropy of this laminate, indeed, in the previous chapter, we saw that the membrane behavior of this laminate was isotropic in the plane. If we trace, for instance, _D 11/h3_ according to the _θ_ -orientation by taking the characteristics of the aforementioned T300/914, we observe a considerable variation of _D 11_:

**Figure 7.18.** _Quasi-isotropic laminate 0,45,90,-45] S under off-axis loading. For a color version of this figure, see [www.iste.co.uk/bouvet/aeronautical2.zip_

**Figure 7.19.** _E lastic bending characteristic D11 of a quasi-isotropic laminate [0,45,90,-45]S under off-axis loading_

The maximum value of _D 11_ is reached for a value close to 0° (or 180°, which is the same thing), meaning that the fibers of the furthest plies from the mid-plane, i.e. the ones at 0°, are directly loaded in their direction. Nonetheless, here the stiffness of the other plies, and in particular that of those at 45°, is obviously non-negligible. Similarly, the minimum value is obtained for a value close to 90° but not for 90°.

We can therefore once again conclude that in order to obtain a high bending stiffness, the fibers that are furthest from the mid-plane must be loaded along their direction.

Lastly, we note that the quasi-isotropic laminate is not completely isotropic, unlike the case of an aluminum plate:

[7.25]

The most common case in the aeronautical field, where a high bending stiffness is desired, is the case of buckling. Indeed, we can show (see Chapter 12) that the resistance to buckling of a plate is directly linked to its bending stiffness. It is important not to forget that the resistance to buckling of a laminate plate said to be "quasi-isotropic" is not isotropic! This isotropic designation comes from the membrane stiffness.

# 8
The Fracture Criterion of a Laminate

## **8.1. The sizing criterion**

Chapter 5 showed how fracture within a UD ply can be estimated. In the case of a laminate, the problem is slightly more complex because fracture must be evaluated for each ply of the laminate. Further more, in reality, fracture within the first ply rarely leads to the total failure of the laminate which can generally absorb a lot more loading before reaching its limit. We must also keep in mind that a structure will be subjected to different loads throughout its life cycle (and not just one isolated load, which is what we see in academic testing situations) known as "load cases". An engineer's duty is then to ensure that the structure will support all of these load cases without breaking and / or straining too much.

In the aeronautical field, the matter is slightly more complicated as we must distinguish between limit loads (LL) and ultimate loads (UL). The sizing of aeronautical structures is indeed highly complex but can be concisely summarized by the statement that the structure must statically support the LL without sustaining damage (and / or plasticity for metallic materials) and the UL without experiencing catastrophic failure [BOU 16, FED 88, JOI 78]. More specifically, the limit load (LL) is defined as the load that the structure will experience on average once in its lifetime, which for the aeronautical industry gives it a probability of occurrence of approximately 10–5 per flight hour (an aircraft flies about 105 hours). As for ultimate load (UL), it is defined as a highly improbable load, which for the aeronautical industry gives it a probability of occurrence of approximately 10–9 per flight hour (ACJ 25.1309 [FED 88]). In practice, there is a coefficient _k_ generally sitting between 1.1 and 1.5 from one to the other:

[8.1]

The sizing philosophy of an aircraft structure can thus be summarized as:

  * – No damage or permanent deformation is permitted during service, i.e. under realistic loads or even loads less than or equal to the LL.
  * – The structure must remain whole under test loads, i.e. with highly improbable loads or even loads less than or equal to the UL.

We must also add to that any damage problems. The structure, on top of resisting LL and UL, must be able to sustain damages (falling objects, stone impacts, hitting birds, lightning, etc.) without experiencing catastrophic failure. This is the notion of damage tolerance (which we will come back to in the next chapter).

However, no matter the complexity of the structure sizing, it always comes down to demonstrating that the structure (with any potential sustained damage, cracks or missing portions) can withstand such a load without fracturing or overly straining. It therefore needs sizing criteria based on the stress and / or strain fields (or on the strain energy release rate) to demonstrate that the structure can withstand the load. These criteria will obviously need to be verified everywhere in the structure and for all the foreseeable load cases. The engineer's task will then consist of modifying the design of the structure if that is not the case or if these criteria are not practical (in sum, if the structure is too oversized and thus too heavy).

While the list will never truly be exhaustive, we can attempt to review the different sizing cases of a composite aircraft as follows:

  * – Buckling: the laminate will then primarily be sized in terms of bending stiffness. It is very common in the aeronautical field for structures to be made from thin plates designed to withstand compressive loading in particular.
  * – Stiffness: the laminate will then be sized in terms of membrane or bending stiffness. We can use the example of space where the primary sizing criterion is often the 1st natural frequency of the system.
  * – Damage resistance: the damaged structure must be able to withstand sufficient loads. The most critical damage for composite structure generally tends to be impact and the residual strength after impact criterion is often the most restrictive of all for aircraft structures (see next chapter).
  * – Ageing: the resins used in aviation, epoxy for example, degrade over time with exposure to humidity, UV, etc. The structure must therefore be able to withstand loads even after ageing. In practice, this isn't really a criterion, in the sense that the sizing criterion of the composite remains the same as previously but with the characteristics of the pristine material replacing those of the aged one.

  * – Interface fracture: all the calculations performed in this lesson are based on the hypothesis that the laminate remains laminate, in other words that all interfaces are healthy. In reality, there are cases where these interfaces can be damaged, for example on the edges of the plate or when the laminate sustains a small impact, or even from a manufacturing defect such as during a drilling operation. In practice, these cases are complicated to size numerically and global criteria are put in place. These criteria nonetheless remain unfavorable and lead to the composite being used beyond its real capabilities.

## **8.2. Test on a composite structure**

During the certification process of a composite structure in the aeronautical field, a number of experimental tests are required in order to ensure its resistance. If we perform, for example, an experimental test until final fracture, we generally obtain a curve with an initial linear part representing the stiffness of the pristine structure, followed by a nonlinear segment due to various structural damages.

**Figure 8.1.** _Diagram curve of a fracture test on a composite structure. For a color version of this figure, seewww.iste.co.uk/bouvet/aeronautical2.zip_

Even in the linear portion we can see that there are numerous damages within the structure. Matrix cracks appear in the most loaded areas, but because the stiffness reductions caused by these cracks remain low (most of the stiffness is linked to the fibers) and localized, their effect on the overall stiffness is, generally speaking, practically null.

If we continue to load we then observe the first fiber fractures in certain plies. When these fiber fractures remain localized, their effect on the overall stiffness remains quite low. It is therefore often quite difficult to locate these initial fiber fractures from the force / displacement curve (however they can be heard during a test as fiber fractures are highly energy-concentrated phenomena and can be very loud). In order to characterize these initial fiber fractures, a criterion of overall stiffness reduction is often used. Consider, for example, a stiffness reduction of 5% in the structure (taking the initial stiffness of the structure then multiplying it by 0.95 so that the point of intersection between this line and the real curve will give the desired point; see previous figure). This is then called a "structural fracture".

If we continue to increase the load, fiber fractures will increase and their effect on the overall stiffness of the structure will become greater and greater. In this portion, we often observe quite violent and noisy fractures (which are quite impressive during the test) that are easily visible on the global force / displacement curve. Lastly, if we increase the load any more, we then obtain the final fracture of the structure (which is very impressive!).

We can now question what LL and UL this structure can support or, in other words, where we can place the LL and UL on this curve (in practice, determining the maximum LL and UL of a structure is almost equivalent to the real problem, which is obviously to design a structure with an awareness of the LL and UL; for example, all it takes is to change the thickness of the plate to lock the maximum LL and UL with the desired ones):

  * – The LL are generally placed just before the structural fracture, i.e. just before getting too many fiber fractures. Also keep in mind that no damage or permanent deformation is permitted for loads less than or equal to the LL. We will note, however, that minor damages (in this case, more or less localized matrix cracking) are tolerated, on the condition that they do not reduce the structure's stiffness too much. In other words, even after minor damages, if you perform another test it should not be noticeable!
  * – The UL are located between the LL and the final fracture. It is important not to locate them too close to the final fracture in order to avoid unwanted fracture. It is worth keeping in mind that there is a relatively high variability in composite fracture and variations in the order of 20% of the fracture limit are often encountered.

In the context of this book, we will limit ourselves to predicting the fracture of the first ply and thus establish the LL. If we wish to size a structure to the UL then we will have to take into account ply fracture; we can, for instance, determine the fracture of each ply one after the other, remove their stiffness, and evaluate the stiffness of the structure without them until there are none left. However, sizing to the UL is far more complicated and far surpasses the scope of this book. It has, however, been the subject of many other studies and research [WIS 12, ABD 15, BOU 09, CHA 87, US 97].

## **8.3. Sizing principle**

There are two main cases when sizing a structure:

  * – The structure and the loads are known and the objective of the sizing is to demonstrate that it withstands the loads without fracturing and then determine the "reserve factors" (RF). This leads, for example, to applying a Yamada–Sun type fracture criterion to all the plies of the laminate throughout the structure, not forgetting that the laminate plate can hold a large number of plies, that the stacking sequence is not the same throughout the structure, and that the resultant forces and moments vary from one point to the next in the structure.
  * – The structure is unknown, and the loads are known. This is the classic case in the aeronautical field where we know the resultant forces that a structure will have to withstand and we are looking to design the lightest possible structure. It is then a matter of determining which stacking sequence to use, which can evidently vary from one point to the next in the structure, in order to withstand the load cases. We can note that not only will the loads depend on the point that is being studied but also on the characteristics of the structure, in other words, of the stacking sequence. We can therefore demonstrate that for a hyperstatic structure (which tends to be the case quite often in reality), the stress distribution depends on the stiffness of a structure. An iterative design process is then necessary in order to optimize the structure.

## **8.4. Sizing a given structure for a given loading**

Consider a composite structure with known geometrical characteristics, in particular its stacking sequence and load cases, which will, in principle, change depending on the given point. The problem can then be reduced to determining the RF, i.e. the safety coefficient, at any given point and in each ply for the loading at that given point.

For a membrane sizing with mirror symmetry, this can be presented as follows:

**Figure 8.2.** _Sizing a composite laminate under membrane loading. For a color version of this figure, seewww.iste.co.uk/bouvet/aeronautical2.zip_

Based on the resultant forces we determine the membrane strain of the midplane and then the strains in each ply (which are obviously homogeneous throughout the thickness since we are studying a membrane loading and plate with mirror symmetry). In practice, these strains will vary in _x_ and _y_ because the stiffnesses vary in the plane of the laminate (if, for example, the number of plies changes) and because the loads vary depending on the position in the structure. We can then determine, for each ply and each point in plane _(x, y)_ , the stresses and therefore the RF. We then obtain non-fracture of the first ply if the minimum RF of all the other plies and all the points in the plane _(x, y)_ remain above 1.

For a sizing in bending, the problem is slightly more complicated because we have to add the variation of these strains depending on the thickness:

**Figure 8.3.** _Sizing of a composite laminate under bending loading. For a color version of this figure, seewww.iste.co.uk/bouvet/aeronautical2.zip_

Based on the resultant forces and moments, we determine the membrane strains and curvatures in the midplane and then the strains in the plies that are obviously not homogeneous throughout the thickness. In practice, these strains will vary depending on _x_ and _y_ because the stiffnesses vary in the plane of the laminate (if, for example, the number of plies changes) and because the stresses depend on the position in the structure. We can then determine, in the thickness of each ply and for each point in plane _(x, y)_ , the stresses and therefore the RF. We then obtain non-fracture of the first ply if the minimum RF throughout the thickness of all plies and for all points in plane _(x, y)_ remains above 1. In all evidence, for a real problem this work is performed automatically on calculation results by the finite element method.

EXAMPLE (Laminate [0, 90]S).– Consider a plate with a thickness _h_ and a stacking sequence _[0,90] S_ composed of UD plies of T300/914, 0.25 mm thick (both 0.125 mm and 0.25 mm standard thicknesses exist) subjected to a resultant force _N x = 400 N/mm_. We restate the characteristics of T300/914:

[8.2]

**Figure 8.4.** _Tension of a laminate 0,90] S. For a color version of this figure, see [www.iste.co.uk/bouvet/aeronautical2.zip_

We begin by calculating the membrane stiffness of the laminate:

[8.3]

We can then determine the strains, which here are homogeneous throughout the thickness of the laminate:

[8.4]

Where we restate that the strains are unit-less and that microstrain (μ _ε_ ) is simply a coefficient of 10–6. Thus, for each ply in its axes:

  * – For plies 1 and 4 at 0°:

[8.5]

  * – For plies 2 and 3 at 90°:

[8.6]

Keeping in mind that frame _(l, t)_ is not the same depending on the direction of the ply.

We can then deduce the stresses which here are homogeneous within each ply:

  * – For plies 1 and 4 at 0°:

[8.7]

  * – For plies 2 and 3 at 90°:

[8.8]

We then apply a fracture criterion in each ply. We will begin by using the Hill criterion:

  * – For plies 1 and 4 at 0°:

[8.9]

  * – For plies 2 and 3 at 90°:

[8.10]

Where the RF corresponds to the coefficient by which the load must be multiplied in order to obtain the fracture of the ply:

  * – For plies 1 and 4 at 0°: _RF = 1.87_
  * – For plies 2 and 3 at 90°: _RF = 1.27_

We see that the 1st ply to break will be the one at 90° (for a load equal to 1.27 the set load) and if we trace the respective contribution of each term of the Hill criterion, we note that it breaks under transverse tension. Nonetheless, this premature fracture (compared to the real fracture) under transverse cracking of the 90° ply is not too penalizing for the structure, as long as the plies at 0° are safe. In practice, in order to avoid penalization and not grant too much importance to these secondary matrix cracks, we artificially increase the limit strain of the matrix in the criteria. To do this, we just artificially reduce its Young's modulus while maintaining the same transverse fracture stress. As an example, by dividing the transverse Young's modulus by two, we obtain:

[8.11]

And with the Hill criterion:

  * – For plies 1 and 4 at 0°: _RF = 1.90_
  * – For plies 2 and 3 at 90°: _RF = 2.46_

This time, we see that the minimum RF is that of the plies at 0° and the RF for the primary ply, i.e. the one at 0°, has suffered little variation. We can show that the results obtained by reducing _E t_ are then closer to the experimental results in terms of the fracture of the 1st ply. In sum, this method gives us good results by taking into account the influence of the fracture of the plies due to matrix cracking in a more realistic way.

To distance ourselves from this matrix cracking, we can also use the Yamada–Sun criterion. Keeping in mind that we can use this criterion here because there are plies perpendicular to the studied ones (plies at 90° if studying plies at 0°, and vice-versa), we then get:

  * – For plies 1 and 4 at 0°: _RF = 1.97_
  * – For plies 2 and 3 at 90°: _RF = 156_

The RF of plies at 90° is obviously very far from the calculation with the Hill criterion, as the fracture of these plies is primarily due to matrix cracking which is not accounted for in the Yamada–Sun criterion. Nonetheless, this fracture is secondary and is often accepted in real-world applications. Furthermore, it is evidently the weakest RF that will size the structure and therefore, in this case, the one for the plies at 0°, which is close for both the criteria. We can show that this result is representative of the real fracture of the structure.

Lastly, in order to better account for the potential matrix cracks in a real structure, we generally take a modulus _E t_ that is lower than the real modulus of the pristine material. To do this we can, for example, perform a tensile test on a laminate, then identify _E t_ retroactively on the mean modulus _E x_ of the laminate (the modulus _E l_ is then identified directly in a pure UD tensile test). Performing a tensile test with a sufficiently high load will allow the matrix cracking to develop and let us identify the real laminate (with comparable damage in reality). In the case of T300/914 studied here, the value of 7 GPa for _E t_ accounts for these minor damages.

EXAMPLE (Quasi-isotropic laminate [0,45,90,–45]S).– Consider a laminate with a stacking sequence of _[0,45,90,–45] S_ composed of UD plies of 0.125-mm-thick T300/914, subjected to a resultant force _N x = 400 N/mm_. Note here that the plies are thinner than in the previous example. This will allow us to achieve the same overall thickness for the laminate, making both the examples easier to compare.

**Figure 8.5.** _Quasi-isotropic laminate 0,45,90,–45] S under tensile loading. For a color version of this figure, see [www.iste.co.uk/bouvet/aeronautical2.zip_

We begin by calculating the membrane stiffness of the laminate:

[8.12]

We can then determine the strains that, in this case, are evenly distributed in the thickness of the laminate:

[8.13]

We can then deduce the strains in the axes of each ply and the stresses that, in this case, are homogeneous throughout each ply:

  * – For plies 1 and 8 at 0°:

[8.14]

  * – For plies 2 and 7 at 45°:

[8.15]

  * – For plies 3 and 6 at 90°:

[8.16]

  * – For plies 4 and 5 at –45°:

[8.17]

Then apply the Yamada–Sun fracture criterion to each ply:

  * – For plies 1 and 8 at 0°: _RF = 1.41_
  * – For plies 2 and 7 at 45°: _RF = 1.59_
  * – For plies 3 and 6 at 90°: _RF = 4.51_
  * – For plies 4 and 5 at –45°: _RF = 1.59_

The RF are higher than 1, meaning that the laminate will withstand the load in question. Of course, since laminates are unsymmetrical in tension / compression, the RF will be different under compression. If, for example, _N x = –400 N/mm_, we then get:

  * – For plies 1 and 8 at 0°: _RF = 1.31_
  * – For plies 2 and 7 at 45°: _RF = 1.57_
  * – For plies 3 and 6 at 90°: _RF = 4.84_
  * – For plies 4 and 5 at –45°: _RF = 1.57_

We can then wonder about the maximum load that will cause the final fracture of the laminate (under tension). If we increase _N x_, we obtain an RF of 1 in the plies at 0° for _Nx = 564 N/mm_. The result is obviously equal to the product of the previous RF ( _RF0° = 1.41_ ) by the corresponding resultant force ( _Nx = 400 N/mm_ ); this is in fact the very definition of RF (the coefficient by which the load must be multiplied in order to achieve the fracture)!

Similarly, we can trace the RF when the direction of the resultant force _Nx = 400 N/mm_ varies:

**Figure 8.6.** _Quasi-isotropic laminate 0,45,90,-45] S under off-axis tension. For a color version of this figure, see [www.iste.co.uk/bouvet/aeronautical2.zip_

**Figure 8.7.** _RF under off-axis tension and compression of a quasi-isotropic laminate 0,45,90,–45] S. For a color version of this figure, see [www.iste.co.uk/bouvet/aeronautical2.zip_

And we then observe that the minimum RF (which is obviously the size-determining factor) is practically isotropic both under tension and under compression. In practice, it varies between 1.3 and 1.4 depending on the orientation.

This stacking sequence, known as quasi-isotropic, is therefore practically isotropic (as its name would indicate) in the characteristics of the fracture of its 1st ply.

In sum, the quasi-isotropic stacking sequence is isotropic in its membrane stiffness, quasi-isotropic as far as its resistance to fracture under membrane loading, but non-isotropic in both its bending stiffness and its characteristics of fracture under bending (I'll leave the demonstration to you).

EXAMPLE (Laminate [0, 90]S).– Consider a laminate with a stacking sequence _[0,90] S_ composed of 0.25-mm-thick UD plies of T300/914 subjected to a resultant moment _M x = 100 N_.

**Figure 8.8.** _Laminate 0,90] S. For a color version of this figure, see [www.iste.co.uk/bouvet/aeronautical2.zip_

We begin by calculating the bending stiffness of the laminate:

[8.18]

We can then determine the strains and curvatures of the midplane (the strains are obviously not homogeneous throughout the laminate's thickness):

[8.19]

Then:

**Figure 8.9.** _Strains within the plies of a laminate 0,90] S under bending resultant moment Mx. For a color version of this figure, see [www.iste.co.uk/bouvet/aeronautical2.zip_

Thus, the maximum strains and stresses in each ply:

  * – For ply 1 at 0°: maximum (in absolute value) at _z = –h/2_

[8.20]

  * – For ply 2 at 90°: maximum (in absolute value) at _z = –h/4_

[8.21]

  * – For ply 3 at 90°: maximum (in absolute value) at _z = h/4_

[8.22]

  * – For ply 4 at 0°: maximum (in absolute value) at _z = h/2_

[8.23]

We then apply the Yamada–Sun fracture criterion to each ply:

  * – For ply 1 at 0°: _RF = 2.06_
  * – For ply 2 at 90°: _RF = 69.4_
  * – For ply 3 at 90°: _RF = 64.8_
  * – For ply 4 at 0°: _RF = 2.20_

We see that evidently the most loaded plies are the ones at 0° since they are the ones that provide bending stiffness and will support most of the loading. We also note that ply 1 is more loaded than ply 4, as its stress mainly comes from compression along the _l_ -direction and that the stress limit under compression is lower than under tension.

The method presented in this section therefore allows us to relatively easily estimate the RF of a laminate for a given load. Obviously, the calculations quickly become tiresome when done by hand and are generally performed with the use of numerical tools. For example, a calculation code using finite element method will provide us with the RF for every ply within a structure, allowing us to redesign the said structure so as to optimize the RF closest to and above 1, while minimizing its mass. Obviously, each modification to the structure will require new calculations, which can be costly in terms of calculation and analysis time. And some analytical optimization methods for the stacking sequence, such as the one we will see in the next section, can help save time by leading us to an initial relevant solution, which can then be optimized using finite element tools.

## **8.5. Optimal structure for a given load**

In many cases, the first stages of design will determine the shape of the structure and the loads it will have to support. The problem then becomes deciding on a material and a stacking sequence in order to support the load while having a minimal mass. The choice of material is generally intricately tied to the cost and its specific characteristics, which is something we will not delve into here. So the last thing to do is to determine the stacking sequence everywhere in the structure as it must vary gradually from one area to the next so as to avoid overstressing due to sudden changes in the stiffness and the complex manufacturing. Over a given area, the stacking sequence will therefore be constant and the problem is then determining the stacking sequence for the point of highest stress while minimizing its mass.

For a membrane optimization with mirror symmetry, this can be summed up as:

**Figure 8.10.** _Optimization of a laminate for a given load under membrane loading. For a color version of this figure, seewww.iste.co.uk/bouvet/aeronautical2.zip_

From the resultant force, we determine the ratio of the ply in each direction from the ratio of the stress in that direction. If we name:

  * – _α_ the fiber ratio at 0°;
  * – _β_ the fiber ratio at 90°;
  * – _γ_ the fiber ratio at 45°;
  * – _γ_ the fiber ratio at –45°;

supposing, of course, that the fiber ratio at 45° is equal to that at –45°. The fibers at ±45° are used to support shear stress which then decomposes into tensile stress at +45° and compressive stress at –45°; the fiber ratio in these two directions must in principle be the same. We will note, however, that composite materials tend to be more resistant to tension than they are to compression. We can afford to place fewer fibers at +45° than at –45° if shear loading remains constantly positive.

**Figure 8.11.** _Equivalence between shearing and tension / compression at 45°. For a color version of this figure, seewww.iste.co.uk/bouvet/aeronautical2.zip_

We obviously get the sum of the fiber ratios equal to 1:

[8.24]

Then:

[8.25]

Where _N x_, _N y_ and _T xy_ are the resultant forces that must be supported. We can then determine the fiber ratio in each direction.

Lastly, in reality, loads will tend to vary throughout the life cycle of a structure depending on its complexity and, in order to circumvent certain unexpected loads, it is often expected to place at least 10% of the plies in each direction, 0°, 45°, 90° and –45°. Having a minimum number of plies in each direction also allows them to support loads near holes or rivets. In these areas that receive a lot of stress, even if the average loading is simple, the presence of holes creates complex localized stress fields with all the components of the resultant forces. Setting a minimum of 10% of plies in every direction allows these secondary loads to not be treated and instead bear a couple of maximum loads. If we didn't have plies in one direction, even a case involving a load causing low levels of stress could prove to be critical.

We can then determine the stiffness matrix of the laminate, which will be linear in _h_ , then the strain, which will vary in _1/h_. Obviously, the strain can vary along _x-_ and _y_ -directions, because the stress varies according to the location in the structure. We can then determine, in each ply and for each point in the plane _(x, y)_ , the stress and the RF that will vary in _1/h_. We then obtain non-fracture of the 1st ply by dictating that the minimum RF of all the plies and for all the points in the plane _(x, y)_ be equal to 1. We then get the value of _h_ and thus the stacking sequence (Figure 8.10).

For an optimization under bending loading, the problem is more complicated because the variations in strain depending on the thickness must be included. Subsequently, there isn't really any usable technique other than experience and the previous solutions:

**Figure 8.12.** _Optimization of a laminate for a given bending loading. For a color version of this figure, seewww.iste.co.uk/bouvet/aeronautical2.zip_

In evidence, the trick is to iterate by starting with a stacking sequence, checking that all the RF remain higher than 1 and then, depending on the result, increasing or reducing the number of plies.

EXAMPLE (Let us determine the optimal stacking sequence for) .–

[8.26]

According to equations [8.24] and [8.25], we get:

[8.27]

We maintain the minimum of 10% in each direction. We then determine the membrane stiffness of the laminate, which is linear in _h_ :

[8.28]

Thus, for the strains that here are homogeneous throughout the thickness of the laminate:

[8.29]

Thus:

  * – For plies at 0°:

[8.30]

  * – For plies at 90°:

[8.31]

  * – For plies at 45°:

[8.32]

  * – For plies at –45°:

[8.33]

We then apply the Yamada–Sun fracture criterion to each ply:

  * – For plies at 0°: 1.52 _/h ≤ 1_
  * – For plies at 90°: 1.18 _/h ≤ 1_
  * – For plies at 45°: 1.59 _/h ≤ 1_
  * – For plies at –45°: 0.61 _/h ≤ 1_

We take the maximum value of _h_ for all the four directions, i.e.:

[8.34]

Taking thirteen 0.125 mm-thick plies:

  * – 7 plies at 0°, i.e.: _α_ _= 55%_
  * – 2 plies at 90°, i.e.: _β_ _= 15%_
  * – 2 plies at 45°, i.e.: _γ_ _= 15%_
  * – 2 plies at –45°, i.e.: _γ_ _= 15%_

Note that to respect mirror symmetry, you necessarily need even numbers of plies at ±45° and one of the two numbers of plies at 0° and 90° to be even. Evidentally, only the number of plies at 0° or (mutually exclusively) at 90° can be uneven. In fact, it is this uneven ply that will be at the center of the stacking sequence and the midplane of the laminate will be the center of this ply.

We then get the following RF:

  * – For plies at 0°: _RF = 1.01_
  * – For plies at 90°: _RF = 1.29_
  * – For plies at 45°: _RF = 0.97_
  * – For plies at –45°: _RF = 2.46_

We then obtain a RF below 1 for plies at 45°: this is due to not respecting the exact ratios of the plies in each direction. The number of plies being integer, the calculated ratios cannot exactly be respected. And indeed, if we redo the previous test, we get a Yamada–Sun criterion for the most critical ply, the one at 45°, of:

[8.35]

It is then just a matter of adding, for example, a ply at 0°. We could also have added a ply at 45° but that would have forced us to add two plies at 45° and two plies at –45° to respect the mirror symmetry and have a balanced stacking sequence (as many plies at +45° and at –45°):

  * – 8 plies at 0°: _RF = 1.07_
  * – 2 plies at 90°: _RF = 1.32_
  * – 2 plies at 45°: _RF = 1.03_
  * – 2 plies at –45°: _RF = 2.67_

And this time the RF are all higher than 1. We can, for instance, use a stacking sequence:

Which will mean that for a T300/914 with a density of 1,600 kg/m3, we obtain an area density of 2.8 kg/m2. We can then compare this mass to that of a standard aeronautical aluminum alloy (type 2024):

[8.36]

This material being isotropic, we get:

[8.37]

Then writing the Von Mises criterion for in-plane stress state:

[8.38]

Thus:

[8.39]

And:

[8.40]

Meaning, for a density of aluminum alloy of 2,700 kg/m3, an area density of 7.4 kg/m2.

We then get a structure 2.6 times as light in composite. This result should nonetheless be nuanced with other criteria such as buckling resistance, hole and bolt sizing, tolerance to impact damage, etc.

# 9
Damage Tolerance

## **9.1. The principle of damage tolerance**

Damage tolerance was defined in the 1970s for civil aircraft structures. These requirements are expressed in the context of European certification JAR 25.571 [JOI 78]: "the damage tolerance evaluation of the structure is intended to ensure that should serious fatigue, corrosion or accidental damage occur within the operational life of the airplane, the remaining structure can withstand reasonable loads without failure or excessive structural deformation until the damage is detected" and by the American certification FAR 25.571 [FED 78]: "an evaluation of the strength, detail design, and fabrication must show that catastrophic failure due to fatigue, corrosion, manufacturing defects, or accidental damage, will be avoided throughout the operational life of the airplane". As a reminder, the JAR (Joint Aviation Requirement) concerns certification regulations of different classes of aircraft, and the JAR 25 [JOI 78] regulates aircrafts of over 10 passengers or more than 6 tons at take-off. The FAR (Federal Aviation Regulations) 25 [FED 78] is the American equivalent regulation.

The study of damage tolerance can therefore be defined as the behavior of a structure damaged from fatigue loading, corrosion or accidental damage ROU 95]. The idea is to make sure that the structure is capable of withstanding acceptable loads without fracturing or straining beyond a certain point until damage detection. This concept is the basis for damage tolerance in metallic aircraft structures ([Figure 9.1(a)). Indeed, damage propagation throughout metallic structures is relatively slow and generally well controlled. It is then possible to define the intervals between inspections in order to guarantee that the damage will not propagate too much before being detected and will not lead to an interval under UL that is too long. It must also be demonstrated that the structure's residual strength will always be higher than the LL.

This case is illustrated in the following figure, which offers a diagram presenting the residual strength through time. At first, this residual strength is constant (and higher than the UL) until the appearance of damage (for example, a crack starting at the edge of a hole). This damage will then propagate and the residual strength will then decrease until reaching, if not detected, the LL. We then talk about critical damage. In practice, this damage is detectable starting at a certain length, or rather the airline company must implement the procedure required to detect it before it becomes critical! The minimum interval between the inspections is defined by the amount of time between the moment when the damage is detectable and the moment it becomes critical; the point is to reduce the time spent under the UL. Then, once detected, this damage must be repaired and restored to a residual strength, higher than the UL.

**Figure 9.1.** _General principle of damage detection and repair in metallic a) and composite materials b). For a color version of this figure, seewww.iste.co.uk/bouvet/aeronautical2.zip_

Nonetheless, it is generally not possible to apply this concept of damage tolerance to composite structures, because composite materials are almost insensitive to fatigue ROU 95]. This result is (mostly) true for carbon-fiber-based composites, and slightly less for fiber-glass-based composites ([Figure 9.2) [ALI 97, BIZ 09]. However, at the same time, the majority of primary aircraft structures (the ones supporting the main loads in the plane) are made from carbon fibers, and we observe that aircraft structures are practically insensitive to fatigue.

**Figure 9.2.** _Diagram of fatigue curves in a UD laminate composite composed of carbon and glass fibers_

This insensitivity to fatigue that is present in composites generally prevents damage from propagating in service. This makes it impossible to define maintenance intervals based on the slow crack growth concept (Figure 9.1), the way we would for a metallic structure. Furthermore, because composites are sensitive to impact, impact damage can rapidly decrease the structure's residual strength below the UL. Thus, the standard (AC 20-107B FED 78]) requires the composite be at least as secure as the metallic ones. The objective is therefore to demonstrate that the time spent under UL of a composite structure remains lower than a metallic structure. On top of considering the amount of time spent under UL, another thing to account for is the difference between the residual strength and the UL. In sum, any damage resulting in a residual strength just below the UL (damage 2 [Figure 9.1(b)) will take longer to detect resulting in a residual strength that is just higher than the LL (damage 1 Figure 9.1(b)).

Impact damage tolerance in composite structures depends on two main phenomena:

  * – The loss of strength caused by the impact; this can reach 50 to 75% of strength without impact ABR 98, BOU 16]. Obviously, the higher the energy of the impact, the more damage there is and the lower the residual strength ([Figure 9.3).
  * – The detectability of the impact: the higher the energy of the impact, the more damage there is and the easier it is to detect it. Furthermore, for aircraft laminate composites of standard thickness (a few millimeters), the damage is generally visible from the non-impacted side before it is visible from the impacted side. However, since there is often no way of accessing the non-impacted side (interior fuselage, wings, wing box, etc.), it is only once the impact is visible from the impacted side that it is considered (Figure 9.3).

**Figure 9.3.** _Residual strength after impact and detectability of impact. For a color version of this figure, seewww.iste.co.uk/bouvet/aeronautical2.zip_

This curve (Figure 9.3) of residual compressive strength depending on the permanent indentation after impact (i.e. the dent left by the impact on the composite) is particularly important in the context of impact damage tolerance. First of all, note that the compression is most often taken into account to justify a composite structure to impact damage tolerance, as it is the characteristic that is most affected by the impact damage [ABR 98, BOU 16, AST 05, BER 99]. Indeed, compression fracture is often due to buckling (for standard laminates of a few millimeters in thickness). During impact, numerous delaminations are created throughout the laminate's thickness that begin to "separate" the laminate into thinner sub-laminates. These sub-laminates obviously present bending stiffnesses that are far lower than the full laminate (the bending stiffness varies depending on the cube of the thickness); this results in local premature buckling under compression, while tension will not be affected by these delaminations.

This curve (Figure 9.3) of residual compressive strength depending on the permanent indentation after impact helps define the load that a structure needs to withstand, and in particular, the three sizing areas of a composite structure to impact damage tolerance:

#### _The undetectable damage area_

The structure must then withstand the UL. This area is also defined as the area of static requirements, because the structure's sizing is governed by static load cases.

#### _The detectable damage area_

The structure must then "only" withstand the LL. Obviously, an inspection procedure must detect these damages as quickly as possible. Once they are detected, these damages must be repaired so that the structure can once again withstand the UL.

We also define the BVID (barely visible impact damage) as the smallest detectable damage using a detailed visual inspection. For example, it has been demonstrated that a permanent indentation of 0.5 mm deep was detectable during a detailed visual inspection with a probability of more than 90% TRO 94] ([Figure 9.4); we therefore retain this value 0.5 mm as a BVID.

**Figure 9.4.** _Size of detectable damage depending on the type of inspection. For a color version of this figure, seewww.iste.co.uk/bouvet/aeronautical2.zip_

This area is also defined as the area of damage tolerance, because a structure's sizing is governed by its damage tolerance. In sum, the damage is there, but we just have to live with it and guarantee that it won't become more critical before being detected.

Lastly, it is often this area, and in particular the BVID, which is the most critical for sizing a composite aircraft structure to damage tolerances (Figure 9.3), indeed:

_A structure presenting impact damage resulting in a permanent indentation equal to the BVID must withstand the UL, and this is throughout the life span of the aircraft_.

Imagine, for example, an impact (a tool being dropped) occurring during the manufacturing of the structure causing a permanent indentation just under the BVID. There is no guarantee that this damage will be detected, and it needs to be demonstrated to the authorities that the structure can withstand the UL, and this is for the entire life span of the aircraft. This is, in fact, how a structure is certified for damage tolerance. You start by impacting the structure all over (in particular at critical points!) with impacts that provoke permanent indentations lower than the BVID, then demonstrate that regardless of these damages, the structure withstands the UL.

#### _The obviously detectable damage area_

The structure must then withstand the loads required for flight and function as normally as possible (obviously in accordance with the size of the damage!). Once detected, they must be repaired (or the damaged area be changed) in order to withstand the UL once more...

We also define the VID (visible impact damage) as the smallest damage immediately detectable in standard inspection conditions. It has, for example, been demonstrated that a permanent indentation that is 2 mm deep (or a perforation that is 20 mm in diameter) was detectable during a general visual inspection (inspection that is shorter and less in-depth than a detailed visual inspection) with a probability of more than 90% ROU 95, TRO 94] ([Figure 9.4); this value of a 2 mm indentation is therefore generally used to define the VID.

This area is also defined as the flight loads area, as these flight loads govern the sizing of the structure. In sum, the impact was sufficiently high that it was easily detected, and the flight must be completed despite that, in the best possible conditions. In reality, this area is divided into smaller subsections with serious damage, very serious damage and critical damage TRO 94, BOU 16] (for example, the minor and large VID defined in [Figure 9.4).

## **9.2. Damage during impact and compression after impact**

Before focusing on the criteria used to size a composite in the context of impact damage tolerance, we should start by discussing what impact damage looks like and how this damage develops to lead to the final fracture during compression after impact (CAI). Take the simple case of a laminate _0,90] S_. During impact, we observe the appearance of two types of matrix cracking ([Figures 9.5 and 9.6):

  * – Vertical matrix cracks in the lower part of the laminate due to high transverse stress _σ t_.
  * – Matrix cracks at 45° at the center of the laminate under the impactor due to high out-of-plane shear stress _τ tz_. We restate that when there is a bending applied to a plate, the out-of-plane shear stresses _τ xz_ and _τ yz_ are null on and under the plate, and reach their maximum at the center.

**Figure 9.5.** _The two types of matrix cracks: transverse cracks due to transverse tension σ t a) and at 45° due to out-of-plane shear stress τtz b). For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_

**Figure 9.6.** _Damage scenario during impact of a UD laminate [0,90] S [CHA 87]_

As the density of these matrix cracks increases, they create delaminations between the plies, delaminations that will then propagate more or less through the impact area (Figure 9.6). We then see the plies fracturing, beginning with the lower plies (and upper plies) due to longitudinal tension (and compression) _σ l_, then the plies located at mid-thickness due to longitudinal tensile stress _σ l_ and out-of-plane shear stress _τ lz_.

Now take the realistic example of a laminate _0 2,452,902,–452]S_ ([Figure 9.7). In this micrographic cut, we observe the different damages we discussed in the case of laminate _0,90] S_: matrix cracks, delaminations and fiber fractures. We note in this figure that the delaminations are present until the junction between two groups of plies in different orientations (and not between two plies directly in the same direction). This is due to the high interlaminar shear stresses _τ xz_ and _τ yz_ in these areas. Matrix cracks are very pronounced in the area just below the impactor, mainly at 45° ([Figure 9.7), due to high out-of-plane shear stresses _τ tz_ in this area. In the ply located on the non-impacted face at the impactor location, a wide open vertical matrix crack is present, resulting from high transverse tensile stress _σ t_ in this area.

The micrographic cut shows a high delamination of the first interface (noted 1 in Figure 9.7) on the non-impacted face between the two plies at 0° and the two plies at 45°. This delamination is still wide open, and this is despite a null impact force (the cut is obviously taken after impact). We also observe a number of junctions between matrix cracks and delaminations showing the high interaction between these two types of damage. We also note from this figure the permanent indentation under the impactor (which here was approximately 0.5 mm) due to various damages in the impacted area.

**Figure 9.7.** _Damages in a carbon / epoxy UD laminate 02,452,902,–452] S after a 25 J impact. For a color version of this figure, see [www.iste.co.uk/bouvet/aeronautical2.zip_

The delaminations propagate during the impact in the direction of the lower ply (for example, along 0°-direction for the delamination of interface 1 in the previous figure) because of the high interlaminar stress in this direction. We also observe that the delaminated areas are larger when we go from the impacted face to the non-impacted face (Figures 9.7 and 9.8).

These different impact damages are likely to propagate during the CAI if the compressive stress is too high. During a CAI test, we mainly observe two damage phenomena leading to the fracture:

  * – The first damage phenomenon is the buckling of the sub-laminates delaminated during impact (Figure 9.8). These sub-laminates are likely to buckle under compression and then cause the delaminations to propagate, which can amplify the effects of the buckling. This propagation of impact delaminations often takes place in the perpendicular direction to the load. This phenomenon is often catastrophic and quickly leads to the final fracture of the structure. This buckling will be all the more premature during CAI if the plate is deformed, meaning the permanent indentation will be deeper.
  * – The second phenomenon is the propagation of compressive fiber fractures of plies oriented along the loading direction. The laminates loaded by compression were designed for this, and obviously presented more plies at 0° than in other directions. A compression crack is likely to appear at the edge of the area damaged by the impactor and propagates during the CAI. This compression crack is due to the local buckling in the fibers; this is the kinking band phenomenon raised at §4.1.2 (Figure 9.8). Furthermore, this crack is generally initiated during impact (because of the high compressive and shear stresses just below the impactor). This phenomenon is generally invisible from the outside, because the surface plies are generally at ±45° (to increase the buckling resistance).

**Figure 9.8.** _Typical scenario of damage and fracture during CAI. For a color version of this figure, seewww.iste.co.uk/bouvet/aeronautical2.zip_

## **9.3. Sizing for impact damage tolerance**

The composites are sensitive to impact, which means the impact damage tolerance is a case of critical (and complex) sizing for composite aircraft structures. The complexity of these sizing cases arises from the fact that both the impact damage, or more specifically the structure's loss of residual strength due to the impact and its detectability, or more exactly the permanent indentation left by the impact, must be considered simultaneously.

Thus, in order to justify a composite aircraft structure to impact damage tolerance, we begin by determining its residual strength curve depending on the permanent indentation, and we then show that the conditions of impact damage tolerance are met. It is then a matter of showing that for undetectable damage, the structure is able to withstand the UL, and that for detectable damage, the structure can withstand the LL (Figure 9.3).

Obviously, in reality, the problem is not 1D, and there can be additional stresses on top of the primary compressive loading, in particular shearing. The shear stresses can also have an impact on the post-impact residual strength of a structure. Impact damage (delaminations in particular) is very sensitive to buckling, and since a shear stress is composed of tension at +45° and compression at –45°, this shear stress can induce buckling and therefore cause a significant loss of characteristics after impact. Ironically, there are very little data on the shear strength of a plate after impact; this arises from the fact that shear tests are difficult to implement and utilize.

To characterize the residual strength of a composite structure after impact, we often use the following compression/shear strain fracture criterion:

[9.1]

Where _ε lcomp_ is the compressive strain (the tensile strength is not much affected by impact) and _ε l_res and _γ lt_res are the longitudinal and shear residual strains respectively. The most complicated thing is obviously defining the residual strains after impact, which depend on the energy of said impact (and therefore the permanent indentation), the stacking sequence, the material, etc. because that requires a large number of experimental tests, which can be long and costly.

One solution consists of determining the data using a numerical model. Let us imagine, for instance, a model that would allow us to determine numerically the residual strength and permanent indentation for all types of structures (built, for example, from a chart depending on the thickness, stacking sequence, material, etc.). Then it can be used directly to justify or design a composite structure with the desired impact damage tolerance. The drawback with this type of approach lies with the reliability of the charts, which are only valid for certain types of impacts and structures (i.e. the ones that were used to create the charts in question). This type of approach is generally used in the aviation industry.

We can also imagine a numerical model that could simulate the impact damage completely, including the permanent indentation and residual strength after the impact. The drawback this time lies with the reliability and complexity of the model. The complete numerical simulation of the impact damage and the CAI of a composite structure is currently a real challenge and is the object of various studies [WIS 12, GON 12, TAN 15, BOU 09].

The problem is all the more complex if we wish to optimize a composite structure to damage tolerance. In this case, only the load is known and the objective is to determine the lightest and/or the cheapest structure possible, satisfying the conditions of impact damage tolerance (as well as static requirements). The problem is complicated by the contradictory nature of both the impact damage tolerance criteria, which are the residual strength and permanent indentation. It is generally required to increase the thickness of a composite laminate in order to increase its residual strength, but its thickness also needs to be decreased in order to improve the detectability of the impact. We can therefore encounter cases where increasing the thickness of a structure will paradoxically reduce its strength to impact damage tolerance by, for example, making the impact undetectable.

# 10
Interlaminar and Out-of-Plane Shear Stress

## **10.1. Tension of a cross-ply laminate _[0,90] S_**

If we perform the tension of two laminates _[0°] S_ and _[90°] S_ independently with an imposed strain _ε x_ equal for both laminates, we then obtain shrinking along the _y_ -direction:

For laminate _[0°] S_:

[10.1]

For laminate _[90°] S_:

[10.2]

The laminate at 0° will therefore shorten a lot more than the laminate at 90°. For example, for the aforementioned T300/914:

[10.3]

**Figure 10.1.** _Diagram of interlaminar stresses during the tensile test of a laminate 0,90] S. For a color version of this figure, see [www.iste.co.uk/bouvet/aeronautical2.zip_

If we now perform a tensile test on the laminate _[0,90] S_ with the interface imposing the same strains on all the plies under in-plane loading, shear stress _τ yz_ will appear at the level of the interfaces. Since the lateral faces are free, we will necessarily have _τ yz_ null on these two faces (of normal vector _y_ ). We can also show that this shear stress is practically null at the center and maximum near the edges (while being null at the actual edges):

**Figure 10.2.** _Diagram curve of the interlaminar stress during the tensile test of a laminate [0,90] S_

We will therefore see that interlaminar stresses appear on a composite structure, especially near the free edges, and therefore the potential appearance of delaminations.

## **10.2. Tension of a cross-ply laminate _[45,–45] S_**

If we perform the tension of two laminates _[45°] S_ and _[–45°] S_ independently with an imposed strain _ε x_ equal for both laminates, we then obtain opposing shear strains on both the plies:

**Figure 10.3.** _Diagram of interlaminar stresses during the tensile test of a laminate 45,–45] S. For a color version of this figure, see [www.iste.co.uk/bouvet/aeronautical2.zip_

For laminate _[45°] S_:

[10.4]

For laminate _[–45°] S_:

[10.5]

If we now perform a tensile test on the laminate _[45,–45] S_, the interface will impose the same strain on both the layers with a null shear strain. There is then creation of a shear stress field _τ xy_ in each ply. Furthermore, the edges being free of any external stress, the stress must be null on all four faces of the plies (faces of normal vectors _x_ and _y_ ). This stress _τ xy_ is itself transmitted from one ply to another via the intermediary of shear stress _τ xz_ throughout the interface. For example, at the center of the specimen (in _x = 0_ ), we get:

**Figure 10.4.** _Diagram curves of interlaminar stresses during the tensile test of a laminate 45,–45] S. For a color version of this figure, see [www.iste.co.uk/bouvet/aeronautical2.zip_

These phenomena of interlaminar stress are high on the free edges of the composite structures, and more particularly at the level of the holes where, in addition to the free edge phenomena, there exist phenomena of stress concentration due to the presence of the hole, which will amplify the phenomena.

## **10.3. Out-of-plane shear stress**

So far, we have supposed that the laminate was sufficiently thin to support in-plane stresses. The reality is obviously more complex, for instance, for thick laminates, the out-of-plane shear stresses, and in particular the out-of-plane shear stresses _τ xz_ and _τ yz_, are non-negligible. We can also show that they are tied to the variations in the plane, and obviously eventual out-of-plane loads. If we write the equilibrium equation in the absence of volume forces:

[10.6]

We obtain along _x-_ and _y_ -directions:

[10.7]

And, having seen the hypotheses performed on the displacement field, we get:

[10.8]

We can then obtain the shear stress variations of _τ xz_ and _τ yz_, and we can then demonstrate that they are parabolic for each ply. Furthermore, they must be null for upper and lower faces of the plate and continuous at the interfaces; they will appear as follows:

**Figure 10.5.** _Diagram curve of out-of-plane shear stresses of a laminate. For a color version of this figure, seewww.iste.co.uk/bouvet/aeronautical2.zip_

They are generally maximum at the center of the plate and can cause two fracture mechanisms:

  * – The fracture of the plies located at the center of the plate under out-of-plane shear stress. These fractures are characteristic because they are globally oriented at 45°.
  * – Interface fractures. Indeed, these stresses will have to cross through the interfaces in order to pass from one ply to the other and will therefore overload there.

In practice, these two phenomena are intricately linked, and in particular the stress concentration present at the end of a matrix crack under out-of-plane shear stress will tend to initiate delamination.

**Figure 10.6.** _Diagram a) [CHO92] and micrograph b) [PET05] of the interaction of matrix cracks (due to out-of-plane shear stress) with delamination_

# 11
Holed and Bolted Plates

## **11.1. Calculating holed composite plates**

So far, we have seen how it is possible to determine the fracture, and moreover the RF, of a composite laminate subject to a given load. In reality, before applying these criteria, we must determine the stress field, which will obviously vary depending on the location of the structure. To determine this stress field, i.e. here the resultant force and moment, we generally use the finite element method. However, the finite element method presents various drawbacks.

One such disadvantage is the time it takes to calculate. The most obvious case is that of sizing bolt or rivet holes in aircraft structures. Keeping in mind that a standard passenger aircraft counts upward of a million rivets, it is obviously out of the question to mesh every single hole to determine the stress field at the hole edge. We therefore use either an analytical method to determine the stress field at the hole edge, knowing the far-field stress, or a hole ratio using a test or a finite element model surrounding a reference hole.

Another disadvantage is the representativeness of the finite element models in the areas of stress concentration. It can be demonstrated that in these areas, the behavior of the composite is more complex than the simple elastic behavior law and plate model raised in this lesson, and that it is then necessary to use complex 3D models to account for composite damage and achieve better realism. We note that using only an elastic behavior law and a plate model to simulate the behavior of the composite in the areas of stress concentration results in efforts far greater than in reality (which are smoothed by damage), which leads to the structure being oversized. This phenomenon can be compared to the plasticity phenomenon encountered in metallic structures; the plasticity reduces overstressing at singular areas (hole, crack, notch, etc.). These damage or plasticity effects must therefore be seen as beneficial (in general), which help smooth the stress in critical areas.

Let us take the case of a holed laminate composite plate subject to a tensile loading.

**Figure 11.1.** _Holed plate under tension. For a color version of this figure, seewww.iste.co.uk/bouvet/aeronautical2.zip_

We obviously observe a stress concentration at the hole edge due to the presence of the hole. We note that reasoning tends to be with average stress in the thickness of the laminate _(σ x0_), but keep in mind that this stress does not apply to the material: certain plies support more stress than others, and it is also forbidden to apply a fracture criterion directly to this average stress, without first determining the stress in each ply. Furthermore, index _0_ is generally omitted in order to simplify notations (this will be the case in the present work), but should not be forgotten all the same!

We can determine analytically the stress field at the hole edge for an infinite plate:

[11.1]

Where _R_ is the radius of the hole,   is the stress away from the hole, _y_ is the coordinate of the considered point (varying between _R_ and _+∞_ at the hole edge) and   is the stress concentration ratio at the hole edge, expressed as:

[11.2]

**Figure 11.2.** _Stress concentration at hole edge for an isotropic material_

This stress concentration ratio represents the overstress at the hole edge, in relation to the stress away from the hole. Coefficients _E x_, _E y_, _ν xy_ and _G xy_ obviously represent the equivalent moduli of the laminate. It should be noted that this calculation is only valid if the orthotropic axes of the laminate coincide with the stress coordinate system _(x, y)_.

This stress concentration ratio is equal to 3 in the case of an isotropic material; in other words, the stress at the hole edge is three times higher than the stress in the absence of a hole. In the case of an isotropic material ( _E x = Ey = E_ and _G = E/2/(1+v)_ ), we then get:

[11.3]

Nonetheless, we note that in the case of an orthotropic material, the stress concentration is higher if the ratio of 0° plies is high. Take, for example, T300/914 that we mentioned previously:

[11.4]

If we use a stacking sequence with 100% of plies at 0°,   reaches 7.5, whereas it is only at 2.2 with 100% of plies at 90°!

**Figure 11.3.** _Stress concentration ratio at the hole edge depending on the stacking sequence_

In practice, this coefficient of 7.5 in the case of a stacking sequence with 100% of 0° ply is to be taken with a pinch of salt, because the hole edge will experience a type of damage called splitting, which reduces this stress concentration. Furthermore, a laminate with 100% of its fibers at 0° doesn't really make sense; this type of stacking is too sensitive to matrix cracking and is always supplemented with at least a few plies at 90° and at ±45° which will reduce the stress concentration ratio.

**Figure 11.4.** _Splitting at the hole edge_

**Figure 11.5.** _Holed specimen after fracture [XU 14]_

Another important effect is that of the width of the specimen noted _W_. We thus define the net stress   as the average stress through the net section:

[11.5]

Where _D_ is the diameter of the hole and _W_ the width of the plate. We verify that these two stresses are equal for _W >>_ _D_.

**Figure 11.6.** _Principle of net stress in a holed plate under tension_

Nonetheless, we can demonstrate that to account for the width of the specimen, it is best to do it via the intermediary of a coefficient _β_ , known as the shape factor:

[11.6]

With:

[11.7]

Once again, we verify that _β_ tends towards 1 when _W >> D_. This coefficient is slightly weaker than the net stress coefficient. Also, the tendency is to trace the inverse of these coefficients so as to evidence the linear character of the section reduction _((W–D)/W_ _)_ , rather than the inverse of the shape factor coefficient ( _1/_ _β_ ).

**Figure 11.7.** _Comparison between net stress and shape factor coefficient. For a color version of this figure, seewww.iste.co.uk/bouvet/aeronautical2.zip_

We note that in this case, the stress concentration ratio at the hole edge becomes _β. _ rather than  !

Considering that the concentration ratio for   is 3 (in the quasi-isotropic case), one could think that a holed plate would support three times less stress than a plate with no hole. However, the calculation that determines the stress at the hole edge is only true if the material is linear elastic, and if the plate remains a plate. And, at the hole edge, the material suffers damage. Therefore, not only does the material's  behavior stop being linear elastic, but the plate theory we just presented is no longer valid (in particular, the damage at the hole edge will cause small delaminations and the laminate plies will therefore no longer be perfectly joined, and the homogeneous strain model in the thickness for a membrane loading will no longer be valid).

To take the effect of damage into account, we use the theory of "point stress". This theory consists of considering that the stresses calculated are only valid at a distance _d 0_ of the hole.

**Figure 11.8.** _Principle of the "point stress" theory. For a color version of this figure, seewww.iste.co.uk/bouvet/aeronautical2.zip_

We thus define a stress concentration ratio  , lower than the previously used  , and much closer to reality! All that remains is to determine the _d 0_ in order to account for the experimental observations, and this is where the real problem with this method lies: how does the _d 0_ vary with experimental conditions? In practice, this _d 0_ will indeed vary with the type of stress (tension, compression, shear, etc.), the stacking sequence, thickness, material, etc.

Values in the order of 1 to 2 mm are often used for standard aircraft cases (carbon/epoxy type material T300/914 with a thickness of a few millimeters). This _d 0_ can be seen, upon first glance, as the size of the damaged area at the hole edge on which the plate theory used for the calculation of composite laminates is not valid. In reality, though, the problem is more complicated, as d0 is simply repositioned to identify the experimental fracture. In practice, this calculation method works quite  well and is widely used in the aviation industry. Nonetheless, the value of _d 0_ remains to be determined in the most varied cases possible, depending on the thickness, the material, the stacking sequence, the diameter of the hole, the type of drilling used, etc.

Take, for instance, the following example:

**Figure 11.9.** _Geometry of a holed plate under tension_

We then obtain _β_ _= 1.15_ and:

[11.8]

when considering an isotropic laminate. However, if we take _d 0 = 1.5 mm_, we then obtain:

[11.9]

Which means that this laminate will break for a stress 2.26 times lower than in the absence of a hole (and not 3.52 times as   would have led us to believe). This result complies better with the experimental results.

Another important effect for composite laminates is the "hole size effect". It is characterized by the fact that the composite is less sensitive to a small hole than a large one. Note, however, that to compare different hole sizes, one must either consider an infinite holed plate, or use a constant _W/D_ ratio (or we'll just be observing that a small hole removes less matter than a large one!):

**Figure 11.10.** _Illustration of the hole size effect_

**Figure 11.11.** _Hole size effect under tensile loading_

This phenomenon is evidenced by the point stress. The stress distribution depends only on _y/R_ , and will therefore be the same regardless of the size of the specimen if traced according to _y/R_ , but will be much more sudden if traced according to its true size (according to _y_ ) for a small hole:

**Figure 11.12.** _Influence of d 0 in calculating the hole size effect under tensile loading. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_

Thus, taking the stress at a constant distance _d 0_ from the hole edge, the stress will be lower for a small hole. This type of behavior is very different for metallic materials for which the size has no (or little) effect on the resistance of a hole (obviously if the size is considered large compared to the size of the grain, typically a couple tenths of a millimeter, which is widely acceptable for structures that are more than one millimeter thick).

The point stress method can also be generalized whatever the load, on the condition that it is applied to every direction of the fibers:

**Figure 11.13.** _Point stress method for different orientations of the plies. For a color version of this figure, seewww.iste.co.uk/bouvet/aeronautical2.zip_

We must then verify the point stress criterion for two points located at a distance _d 0_ of the hole:

  * – in the direction _y_ for the plies at 0°;
  * – in the direction _x_ for the plies at 90°;
  * – in the direction at –45° of _x_ for the plies at 45°;
  * – in the direction at 45° of _x_ for the plies at –45°.

## **11.2. Calculating the multi-bolt composite joints**

The aforementioned point stress methods are widely used for empty holes, but less so for bolt or rivet holes. These holes are obviously filled by the bolt or rivet, and then referred to as filled holes. The presence of tightening and potential washers will obviously modify the way the stress travels through the composite, and it can be delicate to take into account using the point stress method. In the case of a bolt composite joint, the point stress method is therefore often substituted for an equivalent stress method, which we will be describing here. Keep in mind, however, that the choice between these two methods often comes down to the habits of engineering firms rather than any actual advantages/disadvantages of either method.

**Figure 11.14.** _Blind fastener Fybrflush ® (www.ahg.fr). For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_

**Figure 11.15.** _Application of a Fybrflush ® blind fastener (www.ahg.fr). For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_

One common approach often used to size filled holed plates is based on the hole ratio. This hole ratio is defined as the relation between the stress limit with the hole   and the stress limit with no hole  :

[11.10]

This hole ratio is therefore far lower than 1 (unlike the stress concentration ratio). It can be compared, upon first glance, to the inverse of the aforementioned stress concentration ratio, specifically the inverse of the stress concentration ratio at a distance _d 0_ from hole  .

We can thus define an equivalent stress,  , defined as the stress that will lead to the same amount of damage (or the same RF) as with no hole:

[11.11]

Furthermore, in order to take into account the effect of the width of the specimen, we use net stress:

[11.12]

**Figure 11.16.** _Principle of equivalent stress for the calculation of a bolt composite joint_

In practice, the hole ratio is determined by an experimental test, then used to define an equivalent stress. This ratio is approximately _α t = 0.6_ under tensile loading, _α c = 0.85_ under compressive loading and _α s = 0.6_ under shear loading. The presence of the hole is less disturbing under compression as the bolt supports part of the load.

This is the same as saying that a filled holed plate will break _1/0.6 = 1.67_ times earlier under tension (or shear) and _1/0.85 = 1.18_ times earlier under compression. As with the _d 0_, all that remains is to determine these ratios _α t_, _α c_ and _α s_ which can depend on the thickness, the stacking sequence, the material (of the laminate, bolt or rivet), the type of bolt or rivet, the tightening, etc.

This type of approach characterizes the net section fracture of the structure. In the case where the bolt is loaded, we would need to make sure the bolt composite joint will not break under different fracture modes. There are four primary fracture modes for a bolt composite joint:

  * – Net section fracture that we have just mentioned. It primarily concerns plies at 0° that support most of the stress along _x_ -direction (this is only true for standard stacking sequences containing a minimum of plies at 0°).

  * – Bearing fracture. This one is due to high compressive stress existing at the contact between the bolt and the composite laminate. It primarily affects plies at 0° and at ±45° which support most of this stress.
  * – Shear out fracture. This is due to high shear stress _τ lt_ generated in the plies at ±45°. It is observed when the distance L between the hole and the side of the plate is low.
  * – Cleavage fracture. This is due to the stress along the _y_ -direction generated by the load on the bolt. These forces cause stress that is mainly supported under the form of longitudinal stresses _σ l_ __by the plies at 90°. This fracture is observed when the distance L between the hole and the side of the plate is low.

**Figure 11.17.** _The four primary fracture modes in a bolt composite joint. For a color version of this figure, seewww.iste.co.uk/bouvet/aeronautical2.zip_

Let us compare the filled hole tension loaded with far-field tensile stress with that of a filled hole tension loaded via the intermediary of the bolt. We must obviously have _F = .W.e_ (where _e_ is the thickness of the plate) to verify the equilibrium equation.

**Figure 11.18.** _Filled hole tension with non-loaded bolt a) and loaded bolt b). For a color version of this figure, seewww.iste.co.uk/bouvet/aeronautical2.zip_

If we model this problem with the finite element method, using a quasi-isotropic stacking sequence ( _E x = Ey = 50 GPa_) and a far-field tensile stress _σ x∞ = 100 MPa_, we then obtain the following mean stress fields (once again, this is the average stress throughout the thickness of the laminate):

**Figure 11.19.** _Open hole tension a), filled hole with non-loaded bolt b) and loaded bolt c). For a color version of this figure, seewww.iste.co.uk/bouvet/aeronautical2.zip_

If we now perform a cut at _x = 0_ (at the hole edge) of the stress field, we get:

**Figure 11.20.** _Stress at the hole edge of an open-holed plate, filled hole with non-loaded bolt, and with loaded bolt. For a color version of this figure, seewww.iste.co.uk/bouvet/aeronautical2.zip_

We observe that the maximum stress is far higher when the stress is introduced into the bolt (550 MPa) compared to the case where the load is introduced by a far-field tensile stress (270 MPa). We also observe that the stress, when there is no bolt (300 MPa) is slightly higher than when there is a bolt (270 MPa). Lastly, we also note in this graph that stress _σ x_ tends towards _σ x∞_, far from the hole for the case of far-field tensile loading, and towards a much lower value for the case where the tensile loading is imposed by the bolt. We can also demonstrate that the average stress on the net section is equal for all three cases with net stress, or 125 MPa.

To account for this overstress of the case where the load is introduced by the bolt, compared to the case of far-field tensile loading, we define an equivalent stress as:

[11.13]

Where _K b_ is the bearing ratio and _σ b_ the bearing stress defined as:

[11.14]

**Figure 11.21.** _Principle of equivalent stress for calculating a bolt composite joint. For a color version of this figure, seewww.iste.co.uk/bouvet/aeronautical2.zip_

This bearing stress is the average contact pressure experienced by the bolt. We restate that if the contact pressure was constant over a half cylinder and always normal on the surface, then this formula would be correct (you can in fact obtain it by integrating a constant and a normal pressure to the surface). In practice, the pressure distribution is more sinusoidal, and therefore this formula gives us (as its name would indicate) the average of this pressure.

**Figure 11.22.** _Bearing stress. For a color version of this figure, seewww.iste.co.uk/bouvet/aeronautical2.zip_

The next step is to experimentally determine the bearing ratio _K b_ in order to define the equivalent stress. In practice, this ratio is in the order of 0.2, but can obviously vary depending on the material, the thickness, the stacking sequence, etc.

If we return now to our example, that would give us, for the following equivalent stresses:

  * –   _= 208 MPa_ in the case of the far-field tensile loading;
  * –   _= 375 MPa_ in the case of the loaded bolt.

Once again, the stress is higher in the case of the loaded bolt, as observed by the finite element method. These stresses can be compared to one of the point stress methods by realizing that they correspond to stress obtained at approximately 0.5 mm from the hole; this would be the same as using a point stress of _d 0 = 0.5 mm_. This remark should nonetheless be used with caution, as it is the tests that dictate the results. Do not forget that a method remains valid until proven false! In sum, comparing these different sizing criteria with experimental results will allow us to evaluate the parameters ( _α t_, _α s_, _α c_, _K b_ and the stress limits) and assess their relevance.

This equivalent stress method helps size a bolt composite joint to the net section; it now remains to verify that other fracture modes are not met. The bearing fracture model is simply verified by comparing the average bearing pressure to an acceptable pressure:

[11.15]

Where   is the bearing limit stress obtained experimentally. This calculation can appear surprising since we know that the distribution of bearing stress is more sinusoidal than constant. However, at the same time, the bearing limit stress is obtained experimentally with the same hypotheses and during a similar test, meaning the procedure is consistent. The objective here is to demonstrate that this bearing limit stress does not vary too much depending on the thickness, the stacking sequence, or the type of bolt or rivet. In practice, the order of magnitude for this stress is near 600 MPa (or –600 MPa to be precise, since it is a compression stress).

We note that it corresponds to an intermediary value between the compression stress limit of the fibers and the compressive limit stress of the resin. For example, for T300/914 and using a quasi-isotropic stacking sequence, if we consider that only the fibers that are perpendicular to the load will break under fiber fracture, and that the other directions will break under resin compressive fracture, we then get:

[11.16]

In other words, regardless of the direction of the bearing, and for a quasi-isotropic stacking, 25% of fibers will work in the direction of the fibers and 75% in the direction of the resin. This reasoning is coherent and gives us the first estimate of the bearing limit stress. We also note that following this reasoning, the bearing limit stress will vary depending on the stacking sequence and the direction of the load, if the stacking sequence is non-isotropic.

Lastly, the bearing limit stress varies depending on the tightening of the bolt: the tighter the bolt (until a certain point), the higher the bearing limit stress. The value of –600 MPa corresponds to a standard aircraft tightening, and can be twice as low in the absence of tightening (in other words, it becomes closer to the compressive limit stress of the matrix).

Having demonstrated that the bolt composite joint resisted bearing, it now remains necessary to demonstrate that it withstands the last two load cases, which are the shear out fracture and the cleavage fracture. In practice, these two cases are covered by net section and bearing sizing. But this result is true only if the edge distance is not too small and if there is a minimum number of plies at ±45°. In conclusion, the result is true if the aviation guidelines are respected. In practice, the edge distance must be at least equal to approximately three or four times the diameter of the hole in every direction.

If we now trace the limit stress of a multi-bolt composite joint (sized by the limit stress of the material) according to the ratio of the diameter to the width, we obtain the following curves:

**Figure 11.23.** _Sizing of a multi-bolt composite joint depending on the number of bolts. For a color version of this figure, seewww.iste.co.uk/bouvet/aeronautical2.zip_

These curves help define the efficiency of a multi-bolt composite joint depending on the _D/W_ ratio. We note that for high _D/W_ values, the net section criterion (i.e. in  equivalent stress) is the one that limits the bolt joint's resistance, while as for lower values of _D/W_ , it is the bearing that limits the bolt joint's resistance. To increase the bolt joint's resistance, the solution is to increase the number of bolts (in the direction of the load) in order to limit the amount of stress being transmitted by each bolt and tinker with the _D/W_ ratio. We note that even with four bolts, the stress does not distribute evenly; the first bolts (in this example, bolts 1 and 4 of Figure 11.24) transmit more load than the others. In practice, this ratio depends on the stiffness of the plates, the bolts (or rivets), the tightening, etc., and a value of approximately 30% for the first bolts and of 20% for the others is a good order of magnitude (Figure 11.24).

Considering the previous figure, we understand why a _D/W_ ratio in the order of 0.2–0.25 is currently used; it is the value that maximizes the resistance of a multi-bolt composite joint. We also note that in order to avoid having an overly sudden fracture, we will favor bearing a fracture that is more progressive than a net section fracture. Lastly, we see that again increasing the number of bolts does not have much impact on the overall resistance of the bolt joint.

The _D/W_ ratio of the bold joint can be seen as the distance (in the perpendicular direction to the load) between two consecutive bolts. In practice, a ratio of 4 or 5 is commonly used; meaning that the pitch distance between two consecutive bolts is in the order of four or five times the diameter ( _D/W ≈ 0.2–0.25_ ). This order of magnitude of the pitch distance between two consecutive bolts is also a good order of magnitude for the pitch distance used in the direction of the load.

**Figure 11.24.** _Diagram of a multi-bolt composite joint with the bypass loads. For a color version of this figure, seewww.iste.co.uk/bouvet/aeronautical2.zip_

# 12
Buckling

## **12.1. Reminder surrounding beam buckling**

When performing compression on a beam, we observe a compressive loading at the beginning of the test! By increasing compressive force _F_ , we can achieve buckling, depending on the slenderness of the beam. Buckling is an elastic instability, and we can demonstrate that the problem presents an eigenmode when the compressive load is equal to the critical buckling load:

[12.1]

In this case, the solution to the problem is then undetermined (the solution displacement field depends on an undetermined multiplicative coefficient), and in particular any value of deflection _δ_ is a solution to the problem.

**Figure 12.1.** _Beam buckling. For a color version of this figure, seewww.iste.co.uk/bouvet/aeronautical2.zip_

_F cr_ is called Euler's critical load and has the following value:

[12.2]

where _E_ is the Young's modulus of the beam, _I_ is its minimum bending inertia, _L_ its length and _α_ is a ratio that depends on the boundary conditions, and is equal to 1 in the example of the beam simply supported at both the ends presented here.

The solution ( _δ_ , _F_ ) to the problem therefore lies in the horizontal blue line of the previous figure. In reality, the problem is more complicated, as the initial beam is never perfectly linear and will always present a slight initial deflection _δ_ 0 and the real curve tends towards Euler's critical load without ever actually reaching it, because the structure breaks under bending first (Figure 12.1).

## **12.2. Buckling of plates under compression**

Just as with the beams, a compression stress can cause buckling in a plate.

**Figure 12.2.** _Plate under compression simply supported at two ends. For a color version of this figure, seewww.iste.co.uk/bouvet/aeronautical2.zip_

We can demonstrate that the critical buckling load can be presented as follows:

[12.3]

where _C_ and _K_ are the ratios that depend on boundary conditions and can be determined using a chart according to the characteristics of the plate.

**Figure 12.3.** _Calculation of a plate under compression ESD 95]. For a color version of this figure, see[www.iste.co.uk/bouvet/aeronautical2.zip_

In aviation, the case of both ends simply supported is often used (this corresponds to ends and sides simply supported in the chart above). This comes from the fact that the studied plate often corresponds to a panel skin, i.e. a plate limited by four stiffeners. Yet, the torsional stiffness of stiffeners being relatively low, the rotation of the sides of the plate is not impeded, and the boundary conditions cannot be considered clamped. We can nonetheless show that reality is often somewhere between simply supported and clamped, but at the same time the resistance to buckling is dependent on boundary conditions (which are often not well known), and so we prefer to remain conservative and consider the case of simply supported ends.

Note that the previous formula (equation [12.3]) is only valid for a plate with an orthotropic behavior in ( _x, y_ ), so if the mirror symmetry is respected (  = 0) and if:

[12.4]

We also note that in this formula the critical load is dependent on the characteristics along the _y_ -direction, _D_ 22 and _b_ , and on the coupling characteristics _D_ 12 and _D_ 66. It also obviously depends on the characteristics along the _x_ -direction, _D_ 11 and _a_ (via coefficient _K_ ). This dependence on direction _y_ can be explained by the deformation which will cause bending around both _x_ and _y_. As for the dependence on the coupling characteristics, this can be explained by the shape of the blisters. We can demonstrate that the coupling coefficients will upset the blisters and thus delay buckling.

We also note that, depending on the size and bending stiffness of the plate, buckling can occur with different numbers of blisters. For example, for low values of _λ_ :

[12.5]

The minimum value of _K_ is obtained for a single blister, while for higher values of _λ_ , it is obtained with more than one blister. However, the amount of blisters is influenced by the boundary conditions, and we often use the asymptotic value of the simply supported case, i.e. _K_ = 19.7 to obtain a conservative result. Indeed, buckling is very sensitive to boundary conditions, while the boundary conditions are often not well known.

EXAMPLE (Laminate [0n,90n]S).– Consider a laminate with a stacking sequence [0 _n ,_90 _n_ ] _S_ composed of UD 0.25 mm thick plies of T300/914 subject to a compressive resultant force _N x_ = –1,000 N/mm, with its side _a_ = 300 mm and _b_ = 200 mm simply supported on all four sides (essentially, a sheet of A4 subject to 20 tons of compression).

Determine _n_ , so that it will support the load.

**Figure 12.4.** _Laminate 0n,90 n] S. For a color version of this figure, see [www.iste.co.uk/bouvet/aeronautical2.zip_

**Figure 12.5.** _Laminate plate 0 n,90n]S under compression. For a color version of this figure, see [www.iste.co.uk/bouvet/aeronautical2.zip_

We begin, for example, by taking _n_ = 2 or a plate with a thickness of _h_ = 2 mm. We then calculate the bending stiffness of the laminate:

[12.6]

We then obtain:

[12.7]

Thus:

[12.8]

And:

[12.9]

This is evidently insufficient to support the presented load. And you will note that the negative sign for the compression is generally not used in buckling tests, but it should be remembered that it is indeed a compression.

By gradually increasing _n_ , we obtain the lowest value of _n_ that will support the load:

[12.10]

which gives us:

[12.11]

We note that to determine _n_ , we can use the fact that the buckling resistance evolves with the cube of the thickness, the same way the bending stiffness does. In sum, if we multiply by 2 the thickness of a plate (with the same stacking sequence but plies that are twice as thick), then we are multiplying by 23 = 8 the bending stiffness and buckling.

We can then verify that under _N x_ = −1,000 N/mm, the RF (with a Yamada–Sun criterion, for instance) remain over 1:

  * – For plies at 0°: _RF_ = 5 _._ 89
  * – For plies at 90°: _RF_ = 563

This is widely verified here.

We can, nonetheless, easily improve this stacking sequence by adding a ply at 45° and one at −45° at the surface:

We then obtain for:

[12.12]

a buckling resistance of:

[12.13]

And the RF are still well over 1:

  * – For plies at 0°: _RF_ = 4 _._ 81
  * – For plies at 90°: _RF_ = 65 _._ 5
  * – For plies at 45°: _RF_ = 6 _._ 11
  * – For plies at −45°: _RF_ = 6 _._ 11

This increase remains small but non-negligible. We can therefore demonstrate that to increase the resistance to buckling of a plate, it would be best to place the plies at ±45° outside it. In aviation, it is in fact common to place two plies at ±45° at the surface. This then presents the dual advantage of increasing the buckling resistance and protecting the plies at 0° (which are often the most loaded).

We can then further improve the buckling resistance of this stacking sequence by using only plies at ±45° (even though this type of stacking may result in issues due to matrix cracking):

We then obtain for:

[12.14]

a buckling resistance of:

[12.15]

But this time the RF are lower than 1, since there are no plies at 0° to support the load:

  * – For plies at 45°: _RF_ = 0 _._ 89
  * – For plies at −45°: _RF_ = 0 _._ 89

Furthermore, in practice, there is always a minimum of 10% of fibers in every direction! We can therefore correct these RF by adding plies at 0° and 90°:

or:

[12.16]

A buckling resistance of:

[12.17]

And RF above 1:

  * – For plies at 0°: _RF_ = 1 _._ 85
  * – For plies at 90°: _RF_ = 3 _._ 40
  * – For plies at 45°: _RF_ = 1 _._ 93
  * – For plies at −45°: _RF_ = 1 _._ 93

Keep in mind that, compared with a metallic structure, determining an optimal stacking sequence for a composite structure remains a complicated process even when using only two criteria, here the fracture of the first ply and the resistance to buckling.

## **12.3. Plate buckling under shear loading**

As for a plate under compressive loading, the shear loading can induce buckling. Indeed, the shear stress being tension at 45° and compression at −45°, the compressive component can induce buckling.

**Figure 12.6.** _Buckling of a simply supported plate under shear loading. For a color version of this figure, seewww.iste.co.uk/bouvet/aeronautical2.zip_

The critical buckling load under shear loading _T xycr_ is obtained using a chart that gives us the evolution of:

[12.18]

According to _λ_ :

[12.19]

Note that this chart is only valid for a simply supported plate on all four sides, but since, at the same time, it is the most commonly used boundary condition in aviation, this chart covers a wide number of cases.

**Figure 12.7.** _Calculation of a plate under shear loading ESD 95]. For a color version of this figure, see[www.iste.co.uk/bouvet/aeronautical2.zip_

Once again, this formula is only valid for an orthotropic plate in ( _x, y_ ), therefore if the mirror symmetry is respected (  = 0) and if:

[12.20]

In the case of a multiaxial load, with _N x_, _N y_ and _T xy_, the calculation of buckling is more complicated, and even if there exist other charts [ESD 95, MIL 97], we soon find ourselves using finite element solutions. For more details surrounding buckling calculations, interested readers can consult [ESD 95, MIL 97, BER 99, GAY 97], and refer in particular to exercise 8 for the calculation of a stiffened panel.

# 13
Miscellaneous Rules for Stacking

To conclude this lesson, here are the primary rules for stacking composite laminates of aircraft structures, along with their explanations. Use caution, however, as these rules are constantly evolving, and are set to evolve during the coming years (composite materials are still in their youth and evolve every day).

  1. 1) Mirror symmetry:

This avoids twisting during cooldown (which would be very inconvenient for building flat pieces and sizing tolerances) and the membrane/bending coupling (the second reason is less important). This rule is very often respected.

  2. 2) As many plies at +45° as at −45°:

We restate that a laminate with as many plies at +45° as at −45° is called a balanced laminate. This rule makes sense due to shear resistance (shear stress produces the same amount of tension at +45° and compression at −45°); it also avoids tension/shear coupling and shearing during cooldown (the two last reasons are less important compared to the 1st one). However, as composite materials are generally more resistant to tension than to compression, engineers will sometimes opt to have less fibers in the direction receiving tensile stress (+45° if shear is positive). This rule remains, nonetheless, widely respected.

  3. 3) At least 10% in each direction:

Even though this rule is starting to be put into question in order to achieve lighter structures, it is crucial and has different explanations (do note, however, that at the time of writing this, this 10% is tending towards 7–8%). The first reason concerns matrix cracking. When loading is applied in one direction, cracks appear in the matrix in the perpendicular direction to it, and a few plies in that perpendicular direction can protect the laminate. In fact, this is one of the necessary conditions to use a criterion such as the Yamada–Sun which does not account for matrix cracking (or at least only the part that is due to shear). The second reason is not having to account for secondary loads and simply consider maximum loads. If there are no plies in one direction, even a case presenting a low loading in that direction could end up being critical. Another reason is the unexpected variation of external loads. Loads will sometimes vary during the life cycle of a structure in more or less complex ways, and as a way to avoid unexpected load, a minimum of plies are required in each direction to support these unexpected loads. Lastly, having a minimum of plies in each direction provides better support for loads around the rivets and holes. In these areas under high stress level, even if far-field loading is simple, the presence of holes will produce complex local stress fields with all of the components of the resultant force.

  4. 4) Dispersed stacking sequence:

Avoid placing too many consecutive plies in the same direction, and try to distribute them throughout the thickness. This limits the interlaminar stress and thus delamination. This rule is generally properly respected.

  5. 5) Plies at ±45° at the surface:

This increases buckling resistance and protects primary plies that support primary load (often plies at 0°). This rule is generally respected, leading to a great deal of industrial stacking sequences in [+45, −45,...]S.

  6. 6) Avoid 90° between two consecutive plies:

This limits interlaminar stresses and thus delamination. This rule can be hard to respect and is therefore often overlooked.

# 14
Exercises

## **14.1. Experimental determination of the characteristics of a UD material**

We wish to assess the elastic characteristics of a glass/epoxy UD with 50% of fiber volume content and a thickness of 0.25 mm per ply. For this, we use three specimens at 0°, 90° and 20°.

**Figure 14.1.** _Tensile tests for the characterization of a UD ply. For a color version of this figure, seewww.iste.co.uk/bouvet/aeronautical2.zip_

We glue two strain gauges along _x-_ __and _y_ -directions on the specimen at 0°, and one gauge along _x_ -direction on the two other specimens. Norm (ASTM D3039-76) proposes a geometry for this type of specimen:

**Figure 14.2.** _Geometry of the tensile specimen_

We measure for a stress _σ x_ = 20 MPa the following strains:

[14.1]

_Question 1_

Explain why the tabs are necessary.

_Question 2_

Determine the elastic characteristics of the ply.

_Question 3_

Determine the principal stresses, strains and directions.

_Question 4_

We wish to determine the tensile fracture limit of this UD. We perform these three tests until fracture and obtain the following:

[14.2]

Use these results to deduce the tensile fracture characteristics using the Hill criterion.

_Question 5_

We wish to determine all compressive fracture characteristics of this UD to apply a Hill criterion. What further tests would be needed? What shape would you recommend for the specimen?

## **14.2. Fracture of a laminate**

Consider a carbon/epoxy T700/M21 UD with a ply thickness of 0.25 mm that has the following characteristics:

[14.3]

_Question 1_

Determine whether the laminate [45, −45, 02, 90, 0] _S_ breaks when subject to a tensile resultant force _N x_ = 2,500 N/mm using the Hill criterion.

_Question 2_

Determine the tensile fracture resultant force  .

_Question 3_

In reality, this laminate breaks for a resultant force  . How do you explain this difference? Propose a solution so that the Hill criterion calculates the real fracture of the laminate.

_Question 4_

Determine the compressive resultant force  .

## **14.3. Shear modulus**

Consider a carbon/epoxy T700/M21 UD oriented at 45° with the following elastic characteristics:

[14.4]

**Figure 14.3.** _UD ply at 45°. For a color version of this figure, seewww.iste.co.uk/bouvet/aeronautical2.zip_

_Question 1_

Determine the shear modulus _G xy_.

_Question 2_

We now use, instead of this UD, a balanced carbon/epoxy fabric with 50% of carbon fiber volume content (25% in the warp direction and 25% in the weft direction) oriented at 45° that presents the following elastic characteristics:

[14.5]

Determine the shear modulus _G xy_.

## **14.4. Optimization of stacking sequence**

We wish to create a laminate plate with a UD carbon/epoxy T300/914 composite of 0.125 mm thickness with the following characteristics:

[14.6]

It is subjected to the following resultant forces:

[14.7]

_Question 1_

Determine the ply ratio in each direction.

_Question 2_

Determine the number of plies in each direction and the stress in each of the plies.

_Question 3_

Determine its buckling limit under pure compression along _x_ -direction for a square panel of 200 mm per side simply supported on all four sides.

## **14.5. Composite tube**

Consider a composite tube with a thin outer wall compared to its radius, clamped at one end and free at the other, made out of a carbon/epoxy UD composite.

**Figure 14.4.** _Composite tube under tension/compression/torsion/bending. For a color version of this figure, seewww.iste.co.uk/bouvet/aeronautical2.zip_

Determine the fiber ratio in each direction for the following cases:

  * – Tension: __F_ = F. _x__ in _A_ with _F_ > 0
  * – Compression: __F_ = F. _x__ in _A_ with _F_ < 0
  * – Torsion: __M_ = M. _x__ in _A_
  * – Bending: __F_ = F. _y__ in _A_
  * – Bending and torsion: __F_ = F. _y__ in _A_ and __M_ = M. _x__ in _A_

## **14.6. Laminate calculation without calculation**

Consider the following four plates:

**Table 14.1.** _Stacking sequence for the four studied plates_

| **Plate 1** | **Plate 2** | **Plate 3** | **Plate 4**
---|---|---|---|---
**Ply 1** | 0° | 0° | 45° | 0°
**Ply 2** | 90° | 90° | −45° | 0°
**Ply 3** | Honeycomb | 90° | −45° | 0°
**Ply 4** | 90° | 0° | 45° | 0°
**Ply 5** | 0° | – | – | –

where the plies have the following characteristics:

  * – 0°, 90°, ±45°: carbon/epoxy UD plies of 0.25 mm thickness (we will name the longitudinal and transverse moduli _E l_ and _E t_, respectively)
  * – Core: aluminum honeycomb of 10 mm thickness

**Figure 14.5.** _Studied laminates. For a color version of this figure, seewww.iste.co.uk/bouvet/aeronautical2.zip_

Without using calculations, determine:

  * – the plate with the highest bending stiffness;
  * – the plate with the highest tensile stiffness along _x_ -direction;
  * – the plate with the highest shear stiffness in the plane ( _x, y_ );
  * – the ratio between the tensile stiffnesses along _x-_ and _y_ -directions;
  * – the plate with the highest fracture limit along _x_ -direction;
  * – the plate with the highest buckling resistance along _x_ -direction.

## **14.7. Sandwich beam under bending**

Consider a sandwich beam under bending clamped at one end and free at the other, subjected to a bending force:

**Figure 14.6.** _Sandwich beam under bending. For a color version of this figure, seewww.iste.co.uk/bouvet/aeronautical2.zip_

The Nomex® honeycomb core of this beam has the following characteristics:

[14.8]

Where the direction _L_ (for "longitudinal") of the honeycomb corresponds to the direction _x_ of the beam, _W_ (for "width") to the direction _y_ of the beam, and _T_ (for "thickness") to the out-of-plane direction _z_ :

**Figure 14.7.** _Nomex ® honeycomb_

The top and bottom skins are made out of carbon/epoxy UD that are 0.125 mm thick, laminated in a quasi-isotropic manner [0, 45, 90, −45] _S_ with the following characteristics:

[14.9]

_Question 1_

Comment on the characteristics of the honeycomb, and in particular, on why it is missing certain modulus values and fracture limits.

_Question 2_

Trace the bending moment and shear force along the beam.

_Question 3_

Qualitatively explain what the bending moment and shear force will be supported by. From this, deduce the induced stresses in the top and bottom skins and the core for the most loaded cross-section.

_Question 4_

Determine the deflection due to the bending moment.

As a reminder, for a beam under bending with a homogeneous cross-section, we obtain:

[14.10]

The negative sign, which is not present in the classic relation when _x_ is the direction of the beam and _y_ the direction of the deflection ( _M_ _z_ = _E_. _I z_ . ∂2 _v_ /∂ _x_ 2) , comes from the definition of the frame. You may also note that the relation is similar to that between the resultant moment _M x_ and curvature _k 0x_ (see Chapter 7).

_Question 5_

Determine the deflection due to the shear force.

As a reminder, for a beam subject to a shear force with a homogenous cross-section, we get:

[14.11]

This relation simply comes from _τ xz = G xz.γ_ _xz_ with a homogeneous shear stress _τ xz = Tz_/ _S_ and a shear strain equal to the derivative of the deflection.

_Question 6_

Determine the total deflection and compare this to the case with no honeycomb. Show that without honeycomb, the beam cannot support a load of 2.5 kN.

_Question 7_

Determine if this beam is fractured or not.

## **14.8. Laminate plate under compression**

We wish to make a panel with an area of one square meter out of a carbon/ epoxy T300/914 UD (plies 0.25 mm thick) subject to a compressive resultant force of −100 N/mm and simply supported on all four sides.

**Figure 14.8.** _Simply supported panel under compression. For a color version of this figure, seewww.iste.co.uk/bouvet/aeronautical2.zip_

_Question 1_

Suggest a stacking sequence and determine the thickness to allow it to resist fracturing.

_Question 2_

Demonstrate that with this type of stacking sequence, the plate buckles.

_Question 3_

What stacking sequence modification would you suggest to improve buckling resistance?

To simplify the problem, we will set the total thickness to 8 mm. Using the method given by the ESDU 80023, we find the following critical buckling resultant forces:

**Table 14.2.** _Calculation of buckling resistance, bending stiffnesses and RF for eight stacking sequences (the two pure UD stacking sequences are entirely textbook cases only included for the purpose of the study, but are not real-world stacking sequences!)_

|
(N/mm) | _D 11_
(N.mm) | _D 22_
(N.mm) | RF (0°) | RF (90°) | RF (±45°)
---|---|---|---|---|---|---
_[0 32] (textbook case)_ | 68 | 5.7 106 | 3.0 105 | 112 | – | –
_[90 32] (textbook case)_ | 35 | 3.0 105 | 5.7 106 | – | +∞ | –
_[0, 90] 8S_ | 68 | 4.0 106 | 2.0 106 | 59 | 5,364 | –
_[0, 45, 90,_ − _45] 4S_ | 84 | 3.9 106 | 1.3 106 | 42 | 155 | 50
_[45,_ − _45, 0, 90] 4S_ | 113 | 2.1 106 | 1.6 106 | 42 | 155 | 50
_[45,_ − _45] 8S_ | 119 | 1.7 106 | 1.7 106 | – | – | 12

_Question 4_

Why are plies at 0° better than plies at 90°?

How do you explain the influence of plies at 45°?

_Question 5_

What solutions do you suggest for making this panel lighter?

_Question 6_

We wish to make a stiffened panel with T-shaped stiffeners:

**Figure 14.9.** _Stiffened panel under buckling. For a color version of this figure, seewww.iste.co.uk/bouvet/aeronautical2.zip_

What shape would you propose for these stiffeners? How would you stack the stiffeners?

_Question 7_

To simplify the problem, we will limit ourselves to T-shaped stiffeners of 40 mm in height, sitting at 200 mm from one another, and we will consider the same stacking sequence for the skin and the stiffeners:

**Figure 14.10.** _Stiffened panel using T-shaped stiffeners. For a color version of this figure, seewww.iste.co.uk/bouvet/aeronautical2.zip_

Determine the necessary stacking sequence to resist buckling. To answer this, you will need to perform a few calculations using the finite element method.

## **14.9. Tube under torsion/internal pressure**

Consider a thin tube made out of a UD carbon/epoxy T300/914 composite (ply 0.125 mm thick) subject to a combination of torsion and internal pressure:

[14.12]

**Figure 14.11.** _Composite tube under torsion/internal pressure. For a color version of this figure, seewww.iste.co.uk/bouvet/aeronautical2.zip_

_Question 1_

Determine the resultant force the laminate is subjected to.

_Question 2_

We make this tube using plies at 0°, 90° and ±45°. Determine the stacking sequence (in this instance, we will not insist on the minimum 10% of fibers in each direction in order to simplify the problem).

_Question 3_

We manufacture this tube in filament winding with a stacking sequence such as _[+α,_ − _α] n_. Determine _α_ and _n_ so that there are as few plies as possible.

## **14.10. Optimization of a fabric with a strain fracture criterion**

Consider a fabric ply (satin 8HS) carbon/epoxy G803/914 that is 0.5 mm thick and that presents the following characteristics:

[14.13]

**Figure 14.12.** _Ply of 8 harness satin fabric. For a color version of this figure, seewww.iste.co.uk/bouvet/aeronautical2.zip_

We are only interested in the final fracture, and we will suppose that the material obeys a strain fracture criterion:

[14.14]

### **14.10.1. _Part 1: preamble_**

_Question 1_

Determine the compliance matrix of this ply at 0° (depending on _E_ , _ν_ and _G_ ).

_Question 2_

Determine the stiffness matrix of this ply at 0° (depending on _E_ , _ν_ and _G_ ).

_Question 3_

Determine the compliance matrix of this ply at 45° (depending on _E_ , _ν_ and _G_ ).

Explain why _S 16_ and _S 26_ (or _Q 16_ and _Q 26_) are null.

_Question 4_

Determine the stiffness matrix of this ply at 45° (depending on _E_ , _ν_ and _G_ ).

What do you think of the term _Q 66_ compared to the case of the ply at 0°?

### **14.10.2. _Part 2: quasi-isotropic stacking sequence_**

Consider a laminate composed of carbon/epoxy G803/914 fabric with two plies at 0° and two plies at 45°.

_Question 5_

Determine its Young's modulus along _x_ -direction.

_Question 6_

Determine its Young's modulus along _u_ -direction, where _u_ is vector in the plane ( _x_ , _y_ ) that forms an angle of _θ_ with _x_.

_Question 7_

Determine the shear stress limit _τ xy0_. What do you think of the RF of the plies at 0°?

_Question 8_

Determine the RF in each ply under the following load:

[14.15]

_Question 9_

From the previous result and using as few calculations as you can, propose a stacking sequence that would resist this loading.

### **14.10.3. _Part 3: stacking sequence optimization_**

_Question 10_

Adapt the stacking sequence optimization method presented in Chapter 8 in the fabric case study and determine the optimal stacking sequence to resist this loading:

[14.16]

Why does the solution here end up being heavier than in the previous one?

### **14.10.4. _Part 4: stacking sequence optimization under bending_**

Consider a stacking sequence using two plies at 0° and six plies at 45°.

_Question 11_

Determine an optimal stacking sequence to resist bending resultant moment _M x_. Then determine the fracture resultant moment

## **14.11. Open hole tensile test**

We study a holed composite plate made out of a carbon/epoxy IM7/M21 fabric subject to a far-field tensile resultant force (meaning far away from the hole)  . The laminate presents a quasi-isotropic stacking sequence [0,45] _S_ with 0.5 mm plies, for a total thickness of _e_ = 2 mm. We will suppose that the radius of the hole is small compared to the dimensions of the plate.

**Figure 14.13.** _Holed plate under tension_

The elastic and fracture characteristics of fabric IM7/M21 are the following:

[14.17]

We are only interested in the final fracture, and we will suppose that the material obeys a strain fracture criterion:

[14.18]

_Question 1_

Determine the fracture resultant force   without hole.

_Question 2_

Show that the equivalent moduli _E x_, _E y_ , _ν xy_ and _G xy_ are equal to:

[14.19]

_Question 3_

We can show that the resultant force concentration ratio at the hole edge (meaning the ratio between the maximum resultant force obtained at the hole edge and the far-field resultant force) is worth:

[14.20]

and that the resultant force at a distance ( _y–R_ ) from the hole has a value of:

[14.21]

With _α_ = ( _y_ − _R_ )/ _R_ (we will note that _α_ has a value of 0 at the hole edge). The trace of the relation   gives us:

**Figure 14.14.** _Stress concentration ratio at the hole edge_

Deduce the fracture resultant force   in the presence of a hole. Show that the resultant force does not depend on the size of the hole. What do you think?

_Question 4_

In practice, we have experimentally obtained:

  * – _N x10_ = 910 N/mm for a hole with a radius _R_ 10 = 10 mm
  * – _N_ _x_ 6 = 1,050 N/mm for a hole with a radius _R_ 6 = 6 mm
  * – _N_ _x_ 1 = 1,760 N/mm for a hole with a radius _R_ 1 = 1 mm

What do you think of this hole size effect? Why is a smaller hole less dangerous?

In practice, to account for this hole size effect, we use a point stress criterion. This is the same as considering the stress at a certain distance _d 0_ from the side of the hole where the fracture criterion is reached. What is the physical meaning of the point stress method? Determine the distance of the point stress, _d 0_, for _R 1_, _R 6_ and _R 10_.

**Figure 14.15.** _Principle of the point stress method_

## **14.12. Multi-bolt composite joint**

We wish to size a multi-bolt composite joint of two composite plates made of T300/914 (ply thickness of 0.25 mm) subject to a resultant force _N x_ = 1,500 N/mm in the direction of the joint and of which one of the two plates is subject to a resultant force _N y_ = 1,000 N/mm, using 5 mm-diameter rivets.

**Figure 14.16.** _Multi-bolt composite joint. For a color version of this figure, seewww.iste.co.uk/bouvet/aeronautical2.zip_

_Question 1_

Determine the thickness of the plates, their stacking sequences, the size of the joint _L_ as well as the number and position of the rivets.

_Question 2_

What is the point of making changes to the stacking and the thickness of the plates in this case? Suggest new corresponding stacking sequences as well as ply drop-offs.

# 15
Solutions to the Exercises

## **15.1. Experimental determination of the characteristics of a UD material**

**Figure 15.1.** _Tensile tests for the characterization of a UD ply. For a color version of this figure, seewww.iste.co.uk/bouvet/aeronautical2.zip_

##### _Question 1_

The tabs are necessary in order to minimize the stress concentration in the grips of the testing machine. They are particularly important when we are looking for the fracture limits. In sum, if we simply tighten the specimen alone, we would quickly obtain a fracture at the grip edge, and the maximum stress would therefore be far lower than the real fracture limit!

##### _Question 2_

The elastic behavior law of a UD ply is characterized by the two Young's moduli _E l_ and _E t_, by the Poisson coefficient _ν lt_ and the shear modulus _G lt_; the relation between the stress and strain is:

[15.1]

Evidently, this relation is only true in the frame ( _l_ , _t_ )!

In the case of the test at 0°, the frame ( _x_ , _y_ ) is also the frame ( _l_ , _t_ ); we therefore get:

[15.2]

Thus:

[15.3]

And:

[15.4]

The test at 90° then allows us to determine the Young's modulus _E t_ and:

[15.5]

Thus:

[15.6]

And:

[15.7]

The test at 20° allows us to determine the shear modulus. We must begin by determining the stress tensor in the frame _(l, t)_ by performing the rotation from _(x, y)_ to _(l, t)_ :

[15.8]

With _c_ = _cosθ_ and _s_ = _sinθ_. Thus:

[15.9]

And then all that is left to do is put the strain tensor back into the frame _(x, y)_ performing a rotation of − _θ_ (just change _s_ to – _s_ , and do not forget coefficients 2 as a result of the famous _ε xy = γxy/2_):

[15.10]

Knowing that only component _ε x_ is what we are interested in:

[15.11]

Thus:

[15.12]

##### _Question 3_

In the case of the tensile test at 0°, the frame _(x, y)_ is evidently the principal frame; thus:

[15.13]

As in the case of tension at 90°:

[15.14]

You can also choose __x_ II = − _x__ if you want; then _(x I, xII)_ is a direct orthonormal frame. In practice, only the directions of _x I_ and _x II_ are of any interest to us.

Obviously, the problem is slightly more complicated in the case of tension at 20°. We of course have:

[15.15]

However, this frame is not the principal frame for the strain; indeed, _γ xy_ is not null:

[15.16]

This is due to the fact that the material is orthotropic and is loaded outside of its orthotropic axes. The principal stress and strain frames are therefore different.

To determine the principal strains, it should be written:

[15.17]

Remembering to write the strains in the form of a matrix:

[15.18]

as the vector notation is practical, but is mathematically incorrect, and this pseudo-vector does not have the properties of a real vector. This equation (equation [15.17]) has two solutions, the principal strains:

[15.19]

All that remains is to determine the principal directions using:

[15.20]

Thus for example:

[15.21]

Noting at the same time that the two principal directions are of course perpendicular, and that these vectors are defined within a multiplication coefficient (in sum only the direction is important), and in particular that the vector units have no significance. Thus, the following illustration:

**Figure 15.2.** _Position of the principal strain frame. For a color version of this figure, seewww.iste.co.uk/bouvet/aeronautical2.zip_

##### _Question 4_

The Hill criterion is defined by:

[15.22]

with:

[15.23]

Keeping in mind that this relation is true only in frame ( _l_ , _t_ )!

In the case of tension at 0°, we obtain:

[15.24]

The Hill criterion can be summed up as:

[15.25]

Thus, the fracture:

[15.26]

In the case of tension at 90°, we have:

[15.27]

The Hill criterion can be summarized as:

[15.28]

Thus, the fracture:

[15.29]

And in the case of tension at 20°, we have:

[15.30]

The Hill criterion leads to:

[15.31]

Which allows us to determine _τ ltf:_

[15.32]

We note that in practice, we would avoid using an off-axis tensile test to determine the shear stress limit _τ ltf_, as this type of test has a tendency to underestimate _τ ltf_. Instead, a rail shear test would be more appropriate [BER 99, GAY 97], though this test is harder to perform.

##### _Question 5_

To completely determine the Hill criterion, we are missing the two compression characteristics, _σ lc_ and _σ tc_. To obtain these, the simplest approach is to perform a compression test at 0° and one at 90°. In order to correctly introduce the stress into the specimen, we will often use aluminum tabs onto which the composite specimen is glued:

**Figure 15.3.** _Compression test ABI 08]. For a color version of this figure, see[www.iste.co.uk/bouvet/aeronautical2.zip_

## **15.2. Fracture of a laminate**

##### _Question 1_

To determine if the laminate _45, −45, 0 2, 90, 0]S_ will break when subjected to a tensile resultant force of _N x = 2,500 N/mm_, we must first determine the stresses within each ply, then apply a fracture criterion such as the Hill criterion. To achieve this, we must begin by determining _A,_ the stiffness matrix of the laminate. Then, we can determine the membrane strains of the plate (these strains are homogeneous throughout the thickness, because the laminate presents mirror symmetry and is subject to membrane loading). The next step is to determine the strains within the frame _(l, t)_ of each ply, then the stresses within the frame _(l, t)_ of each ply, and then apply the fracture criterion (or determine the corresponding RF, which is essentially the same thing) ([Figure 8.2).

We therefore start by calculating the matrix _ _via the stiffness matrix _Q ijk_ of each ply (making sure to write these stiffness matrices in the frame _(x, y)_ , otherwise you cannot sum them up so simply):

[15.33]

Since we are dealing with membrane forces and the laminate presents a mirror symmetry, the order of the plies is unimportant, and we can regroup the plies in the same orientation:

[15.34]

After using the matrices _Q ijk_ in the frame _(x, y)_ :

[15.35]

We note that _A 16_ and _A 26_ are null since there are as many plies at +45° as there are at −45°, and that _A 11_ is higher than _A 22_ since there are more plies at 0° than there are at 90°.

We can then determine the strains:

[15.36]

The next step is to place all these strains in the frame _(l, t)_ of each ply, then determine the stresses (since we are now in the frame _(l, t)_ of each ply, the stiffness matrix _Q ijk_ is the same for all the plies):

  * – For plies at 0°:

[15.37]

  * – For plies at 90°:

[15.38]

  * – For plies at +45°:

[15.39]

  * – For plies at −45°:

[15.40]

We can then determine the RF for each ply, meaning the coefficient by which we must multiply the load to achieve fracture:

[15.41]

This gives us:

  * – For plies at 0°: _RF = 1.62_
  * – For plies at 90°: _RF = 0.82_
  * – For plies at +45°: _RF = 1.18_
  * – For plies at −45°: _RF = 1.18_

The minimum of the RF is therefore 0.82, and the fracture of the laminate is achieved for _N x = 2,500 N/mm_ in the plies at 90°. We will see further along that this fracture is largely exaggerated!

##### _Question 2_

By definition of the RF, we obtain laminate fracture for:

[15.42]

And more specifically, we obtain the fracture of plies at 90° when _N x = Nxt_.

##### _Question 3_

In reality, this laminate breaks for a much higher resultant force than the one presented in the previous question. This is due to the fact that the fracture of the plies at 90° will not lead to the total failure of the laminate, but rather will induce matrix cracking in the plies at 90°. In order to avoid granting too much importance to these matrix cracks, we must modify the considered fracture criterion. For example, we must remove the part that is due to the transverse stress in the Hill criterion, which is essentially the same as using the Yamada–Sun criterion:

[15.43]

We then get:

  * – For plies at 0°: _RF = 1.62_
  * – For plies at 90°: _RF = 2.92_
  * – For plies at +45°: _RF = 1.61_
  * – For plies at −45°: _RF = 1.61_

And the fracture resultant force is therefore:

[15.44]

Which is relatively close to the experimental value.

The drawback with this type of criterion is that it slightly overvalues the stiffness of the material and does not account for matrix cracks. Once the plies at 90° break (and at ±45°, as we will see later on), the lowered level of modulus Et on the overall stiffness of the laminate should also be accounted for. In order to include this reduced stiffness, we can progressively remove transverse stiffness in the plies experiencing matrix cracking (keeping in mind that even with matrix cracks, a ply can continue to operate, on the condition that it is stabilized by a perpendicular ply).

We can then begin to use _E t = 0 MPa_ for the plies at 90°. We then obtain:

[15.45]

We will note at this point that matrix _ _has but slightly changed in comparison with the previous case; this makes sense since most of the stiffness is provided by the fibers.

We can then determine the membrane strains in the laminate:

[15.46]

Then the strains and stresses within each ply:

  * – For plies at 0°:

[15.47]

  * – For plies at 90°:

[15.48]

  * – For plies at +45°:

[15.49]

  * – For plies at −45°:

[15.50]

The primary difference is obviously the stress _σ t_ in the plies at 90° that is null (because _Et_ is null!).

Thus, the RF with the Hill criterion:

  * – For plies at 0°: _RF = 1.60_
  * – For plies at 90°: _RF = 2.82_
  * – For plies at +45°: _RF = 1.16_
  * – For plies at −45°: _RF = 1.16_

Which gives us a fracture resultant force:

[15.51]

which is still too low in comparison with the reality. The plies at ±45° are now the ones to break under matrix cracking, and if we decrease _E t_ in these plies, we then find the RF using the Hill criterion:

  * – For plies at 0°: _RF = 1.61_
  * – For plies at 90°: _RF = 1.61_
  * – For plies at +45°: _RF = 1.61_
  * – For plies at −45°: _RF = 2.84_

And we find the same fracture resultant force as with the Yamada–Sun criterion, which remains coherent with the experimental results:

[15.52]

We can then compare the equivalent moduli for the pristine laminate:

[15.53]

with those for the laminate where the _E t_ modulus of plies at 0° and ±45° have been decreased:

[15.54]

We note that the stiffnesses are close due to the fact that they are primarily due to the fibers.

In practice, you should avoid using a criterion that grants too much importance to matrix cracking, such as the Hill criterion, at the risk of oversizing your structure. We can also gradually take into account the matrix cracking, by gradually decreasing the transverse Young's modulus of the plies broken under matrix cracking. This approach is similar to the use of a fracture criterion that does not account for matrix cracking such as Yamada–Sun's, but can slightly overestimate the overall stiffness of the laminate (even if in reality, since the modulus _E t_ is assessed from tests that would induce matrix cracking, the experimental value already includes part of this damage).

##### _Question 4_

To determine the fracture resultant force under compression, simply use the approach we detailed above, making sure to use the correct stress limit value following the sign of _σ l_, which here will give you, for a resultant force _N x =_ − _2,500 N/mm_ (and a Yamada–Sun criterion):

  * – For plies at 0°: _RF = 0.84_
  * – For plies at 90°: _RF = 5.60_
  * – For plies at +45°: _RF = 1.39_
  * – For plies at −45°: _RF = 1.39_

Which gives us a fracture resultant force:

[15.55]

We note here that the 1st ply to break is the one at 0°, and it breaks for a value lower than under tension. This is due to the fact that the stress limit under longitudinal compression (−1,200 MPa) is far lower (in absolute value) than that under tension (2,300 MPa). The fiber fracture under compression is indeed guided by the buckling of fibers (and thus by the shear resistance of the resin; see Chapter 4) while the fiber fracture under tension is guided by the tensile fracture of fibers. This explains in particular that newer composite materials have better tensile stress limits than the 1st generation did, while the compressive stress limits remain similar.

## **15.3. Shear modulus**

##### _Question 1_

We must begin by remembering how the shear modulus is defined. Imagine holding this material in your hands without knowing that it is a UD ply at 45°. You would then have defined its stiffness matrix   or its compliance matrix   (the compliance matrix highlights the Young's and shear moduli more simply than the stiffness matrix) by:

[15.56]

You can then determine _G xy_ as follows:

  * – start by imposing a stress tensor in _(x, y)_ of which only _τ xy_ is non-zero;
  * – then determine the stress tensor in _(l, t)_ ;
  * – determine the strains in _(l, t)_ using the compliance matrix;
  * – perform the inverse rotation to determine the strains in _(x, y)_ ;
  * – you can then identify _G xy_ as the term linking _γ xy_ and _τ xy_.

However, you can also use the laminate membrane stiffness matrix  . I will let you verify that both the methods produce the same results. The membrane stiffness matrix  , or more specifically its inverse, compliance matrix  − _1_ , can be related to the previous compliance matrix. Indeed, we obtain:

[15.57]

The subscript 0 on the strains means that the strains are homogeneous throughout the thickness, which here is the case, because the laminate (in this case, a UD ply at 45°) presents mirror symmetry and is loaded only with membrane force.

We must now link the stresses and resultant forces. In this case, the stresses are homogeneous in the thickness; we just need to multiply it by the thickness. In the case of a real laminate (with multiple plies in different orientations), the resultant forces are tied to the average stress (thus the subscript 0 in the following equation) by:

[15.58]

Keeping in mind that the average stresses are simply the averages of the stresses and are therefore never supported by the material. In particular, applying a fracture criterion to these average stresses makes no sense.

Lastly, we can identify the compliance matrix of the laminate, multiplied by the thickness:

[15.59]

We should begin by determining the stiffness matrix of the laminate:

[15.60]

Where _h_ must be expressed in mm so that   is in N/mm. Thickness _h_ is unknown, and we can evidently demonstrate that the shear modulus _G xy_ does not depend on _h_. Indeed, if we invert matrix   and multiply the result by _h_ , the dependence on _h_ disappears:

[15.61]

This way, we can determine the elastic characteristics of the laminate (in this case, a UD ply at 45°):

[15.62]

Obviously, you will note that the coupling coefficients _η x_ and _η y_ are non-zero; meaning that there is coupling between tension and shear.

##### _Question 2_

We can use the same approach as previously with the fabric ply. We then obtain:

[15.63]

Then:

[15.64]

Then identifying the elastic characteristics of the laminate (in this case a ply of fabric at 45°):

[15.65]

We of course observe a shear modulus that is much higher than previously (25.7 GPa against 6.4 GPa for the UD)! A shear stress being equal to a tensile stress at +45° and a compressive stress at −45°; to properly support this stress, we need to place fibers at +45° and at −45° (which is not the case with a UD ply at +45°).

We also observe that the coupling coefficients are low. They would be null if there were as many fibers at +45° as there are at −45°. Here, this is practically the case; there are in fact as many fibers in both the directions of the fabric, but since the stiffness in the warp direction is slightly higher than in the weft direction, we create a small imbalance between both the orientations.

**Figure 15.4.** _Example of a fabric ply (satin 8HS). For a color version of this figure, seewww.iste.co.uk/bouvet/aeronautical2.zip_

## **15.4. Optimization of stacking sequence**

##### _Question 1_

To determine an optimal stacking sequence for a given load, we follow the procedure presented in Figure 8.10. We begin by determining the ply ratio in each direction ( _α_ at 0°, _β_ at 90 and _γ_ at ±45°) supposing, initially, that they are proportional to the amount of stress:

[15.66]

Furthermore, we note that as _T xy = 0_, we should find _γ_ _= 0_. However, we cannot guarantee that during the life cycle of the structure, there will never be a shear resultant force; therefore, we include a minimum of 10% of plies in each direction (see Chapter 13). Thus:

[15.67]

##### _Question 2_

Supposing that we know the thickness _h_ of the laminate, we can then determine the ply thickness in each direction:

[15.68]

We then determine the membrane stiffness of the laminate, which is linear in _h_ :

[15.69]

Thus the strains, which in this case are homogeneous in the thickness of the laminate:

[15.70]

Thus:

  * – For plies at 0°:

[15.71]

  * – For plies at 90°:

[15.72]

  * – For plies at +45°:

[15.73]

  * – For plies at −45°:

[15.74]

We then apply the Yamada–Sun fracture criterion to each ply:

  * – For plies at 0°: _1.67/h ≤ 1_
  * – For plies at 90°: _1.56/h ≤ 1_
  * – For plies at +45°: _1.97/h ≤ 1_
  * – For plies at −45°: _1.97/h ≤ 1_

We of course take the maximum values for _h_ for all four directions, i.e.:

[15.75]

Therefore, 16 plies of 0.125 mm thickness. Taking care to use an even number of plies at ±45° so as to maintain mirror symmetry, and an even number of plies at 0° or at 90° (one of the two can be odd, but not both), we obtain, for example:

  * – 6 plies at 0°, therefore: 38%
  * – 6 plies at 90°, therefore: 38%
  * – 2 plies at 45°, therefore: 12%
  * – 2 plies at −45°, therefore: 12%

We must then recheck the RF in each ply, indeed except for certain cases, we cannot respect the exact ply ratio in each direction (in the present case, we are very close to it), which will then give us:

  * – For plies at 0°: _RF = 1.14_
  * – For plies at 90°: _RF = 1.22_
  * – For plies at 45°: _RF = 0.96_
  * – For plies at −45°: _RF = 0.96_

We must therefore increase the number of plies, for example, by adding a ply at 0°:

  * – For plies at 0°: _RF = 1.29_
  * – For plies at 90°: _RF = 1.24_
  * – For plies at 45°: _RF = 1.03_
  * – For plies at −45°: _RF = 1.03_

We can for example take a stacking sequence of 17 plies _[45,_ − _45, 0, 90, 0, 90,_ _0, 90, 0 0.5]S_ (where the 0.5 factor means that with symmetry, we only have one ply at 0° in the center).

##### _Question 3_

To determine the buckling limit, we begin by determining the bending matrix by:

[15.76]

We then find for a stacking sequence _[45,_ − _45, 0, 90, 0, 90, 0, 90, 00.5] S_:

[15.77]

Thus:

[15.78]

And

[15.79]

This value is much lower to the resultant force −1,000 N/mm along _x_ -direction used to size this stacking sequence against failure. This result is typical; we indeed note that a panel under compression is generally more sensitive to buckling than to compression, and it is its buckling resistance that governs its stacking sequence and in particular its thickness. In the present case, we can make an initial estimate of the thickness required to hold under buckling along _x_ -direction, noting that the buckling resistance evolves with the cube of the thickness, thus the necessary thickness:

[15.80]

Upon 1st approximation, we would need to multiply the thickness by 2.8 to resist buckling along _x_ -direction. For more details, see exercise 8.

Also note at this point that the tension along _y_ -direction will limit buckling, but that the analytical calculation becomes more delicate and goes beyond the scope of the present book. Readers who are interested in finding out more can refer to [BER 99, US 97].

## **15.5. Composite tube**

  * – Tension: __F_ = F. _x__ in _A_ with _F > 0_

In the case where the tube is subject to tensile loading, we had obviously better place all the fibers at 0°, unless we are adhering to the 10% of fibers in each direction rule. In this case, in order to avoid unpredicted loading to other directions, we will use, for instance, 10% of fibers at 90°, 10% at +45°, 10% at −45° and 70% at 0°. In reality, we can still create this tube using only plies at 0°, but on the condition that it is protected. For example, we could place all the plies at 0°, then cover it with fabric to protect it and absorb loading other than the primary tension (for example, impacts resulting from falling objects during manufacturing, maintenance or in operation).

**Figure 15.5.** _Composite tube under tension/compression/torsion/bending. For a color version of this figure, seewww.iste.co.uk/bouvet/aeronautical2.zip_

In practice, the composite tubes are often made by the filament winding technique. We cover a mandrel with a composite wire (or with a strand of fibers that has been impregnated with resin, to be precise) or a small strip of prepreg. It is then just a matter of controlling the rotation speed of the tube and the movement of the wire to obtain different orientations. This allows us to achieve angles other than the traditional 0°, 90° and ±45° (see exercise 9). The drawback is that the 0° orientation is practically impossible to achieve, and we will use orientations of a few degrees minimum.

Once the stacking sequence has been determined, all that is left to do is use the Hill criterion to determine the thickness of the laminate, knowing the tensile resultant force _N x_ which is simply determined by (we divide the loading by the length of the plate in question):

[15.81]

**Figure 15.6.** _Composite tube under tension. For a color version of this figure, seewww.iste.co.uk/bouvet/aeronautical2.zip_

This result is obviously equivalent to the most classic calculation of tensile stress:

[15.82]

Note that in this formula, it is the average stress _σ 0x_ that we are determining, in other words, the average of the stresses in the various plies. This stress is therefore never supported by a single ply (unless the laminate is a pure UD or in the case of a material that is homogeneous through the thickness, for example, an aluminum plate), and it should not be used to calculate the fracture criterion.

Also note that the formula of the tensile resultant force does not depend on the stacking sequence of the laminate, and in particular, does not depend on its thickness; this is the point of working with resultant force rather than stress!

  * – Compression: __F_ = F. _x__ in _A_ with _F < 0_

In the case where the tube is under compressive loading, the problem is practically the same as with tensile loading; we will mainly use fibers at 0°. Of course the compressive stress limits must be used when determining the fracture criterion. The tube must also be sized for buckling. Global buckling of the tube can be achieved if it is slender, and it can also buckle locally if it is thin. To size it for global buckling, you can use Euler's critical buckling load:

[15.83]

Where _I z_ is the quadratic bending moment, _E_ is the mean Young's modulus in the direction of the compression, _L_ is the length of the tube and _α_ is a coefficient that depends on the boundary conditions and has a value of 1 in the case of a beam simply supported at both ends. For local buckling, the calculation is slightly more complicated [GER 57].

  * – Torsion: __M_ = M. _x__ in _A_

In the case where the tube is subject to torsion loading, the laminate will experience shearing _T xθ_. Indeed, if we cut the laminate by a face with normal vector _x_ , the resultant force is then directed along _θ_ -direction. To determine this shear resultant force, it is just a matter of adding the moment caused by all of these arrows (Figure 15.7), as:

[15.84]

Where _(2.π.R)_ corresponds to the circle's perimeter, and the second _R_ to the lever arm in order to calculate the moment.

**Figure 15.7.** _Composite tube under torsion. For a color version of this figure, seewww.iste.co.uk/bouvet/aeronautical2.zip_

The best stacking sequence to support this shear resultant force is therefore a stacking sequence at ±45°, with 50% of plies at 45° and 50% at −45°. We can also place 10% of plies at 0° and at 90° to support a potential unpredicted load.

To complete the sizing of the tube, we should also verify that there is no local buckling (the shear is a tension at +45° and a compression at −45° that can cause buckling). The calculation is slightly more complicated, and you can find charts in the literature [GER 57], or use finite element calculation methods.

  * – Bending: __F_ = F. _y__ in _A_

The case of bending is more complicated. We can show that the internal forces of the upper part of the beam on the lower part (by fictionally cutting the beam) decompose into a bending moment _M z_ and a shear force _T y_:

[15.85]

In other words, the forces are maximum at the clamp (in _x = 0_ ), and if the beam should break, it would be at the clamp.

The bending moment will induce a tensile/compressive resultant force _N x_ linear in _y_ (the resulting moment of which must be equal to _M z_; see Figure 15.9(a)):

[15.86]

This resultant force _N x_ will be highest at the clamp (in _x = 0_ ) and at the top and bottom (in _y = ±R_ ). You could demonstrate this formula by remembering your knowledge of mechanics of material [AGA 08] and the famous:

[15.87]

Where _I z_ is the quadratic bending moment, which in this case is worth _π.R 3.e_ and by multiplying by thickness _e_ to return to the resultant force.

As for the shear force _T y_, it will cause a shear resultant force. This shear force is often left out of calculations because its effect is negligible in the case of a solid beam, which will not be the case here. To support loading along _y_ -direction, the ideal solution would be to induce a shear stress _τ xy_ (we are located on a face of normal vector _x_ ), or a shear resultant force _T xy_ (which is essentially the same thing). This does not pose a problem for the tube skin located around _y = 0_ , as this shear is located in the plate plane, but it is impossible for the tube skin located around _z = 0_ , as this shear then becomes an out-of-plane shear stress (Figure 15.8).

As the plate is thin, its ability to resist out-of-plane shear is low, compared to its ability to resist in-plane shear. We will therefore see the appearance of a shear resultant force _T xθ_, which will be maximum near _y = 0_ (in _z = ±R_ ) and null near _z = 0_ (in _y = ±R_ ); in sum, the plate can only support loading in its plane (Figure 15.8(b)). To calculate the resultant force, we can use the Bredt formula [AGA 08] which gives us the shear force resultant at a point _A_ depending on its value in _B_ :

[15.88]

**Figure 15.8.** _Shear induced by shear force: if rigidity xy were the same everywhere a), and in reality b). For a color version of this figure, seewww.iste.co.uk/bouvet/aeronautical2.zip_

We can then say using symmetry, the shear force resultant must be null at _y = ±R_. We can then demonstrate that this shear force resultant is sinusoidal in _θ_ (Figure 15.9(b)):

[15.89]

**Figure 15.9.** _Composite tube under bending: stress resultant due to the bending moment a) and shear force b). For a color version of this figure, seewww.iste.co.uk/bouvet/aeronautical2.zip_

We can then determine the ply ratio in each direction ( _α_ , _β_ , _γ_ ):

[15.90]

Remember to consider the maximum for _N x_ and _T xθ_ throughout the structure. Once we know _L_ and _R_ , we can use these to determine the stacking sequence.

The drawback of this approach is that it only considers the maximum of the resultant forces. In reality, if we wished to optimize the weight of the tube, we would have to consider the resultant forces at each point and determine the stacking sequence in each point, which would obviously vary from one point to another. In the present case, the thickness of the laminate would be maximum at the level of the clamp and would be much lower at the end of the tube.

To complete the sizing of this tube, we would also have to verify that there is no local buckling (due to both the shear from the shear force and the compression from the bending moment). The calculation is even more complicated than previously, but you can refer to charts in the literature [GER 57] or perform a calculation using the finite element method.

  * – Bending and torsion: __F_ = F. _y__ in _A_ and __M_ = M. _x__ in _A_

This case is only the sum of the two previous cases:

[15.91]

The approach is therefore exactly the same, and if we consider the stacking sequence to be homogeneous throughout the structure, we obtain:

[15.92]

Even though, in reality, we would have to imagine a scalable stacking and remember to take buckling into account.

## **15.6. Laminate calculation without calculation**

Consider the following four plates:

**Table 15.1.** _Stacking sequence for the four studied plates_

| Plate 1 | Plate 2 | Plate 3 | Plate 4
---|---|---|---|---
Ply 1 | 0° | 0° | 45° | 0°
Ply 2 | 90° | 90° | −45° | 0°
Ply 3 | Honeycomb | 90° | −45° | 0°
Ply 4 | 90° | 0° | 45° | 0°
Ply 5 | 0° | – | – | –

Without using calculations, determine:

  * – The plate with the highest bending stiffness:

It is obviously the sandwich plate (i.e. with two skins and honeycomb in the center), and therefore plate no. 1, which presents maximum bending stiffness. We restate that the bending stiffness varies with the cube of the thickness. This variation following the cube of the thickness appears both in the formula for the bending stiffness of a composite laminate plate:

[15.93]

and also in the famous formula for the bending stiffness of a rectangular solid beam:

[15.94]

Where _b_ is the width of the beam and _h_ the height.

  * – The plate with the highest tensile stiffness along _x_ -direction:

The plate presenting the most plies at 0°; therefore, plate no. 4 will be the one that has the highest tensile stiffness along _x_ -direction.

  * – The plate with the highest shear stiffness in the plane _(x, y)_ :

The plate presenting the most plies at ±45°; therefore, plate no. 3 will have the highest shear stiffness in _(x, y)_. We restate that a shear stress is the sum of a tensile stress at +45° and a compressive stress at −45°, and that plies should be placed at +45° and at −45° to support this type of stress.

  * – The ratio between the tensile stiffnesses along _x-_ and _y-_ directions:

For plate no. 1, this ratio will be equal to 1; in other words, the tensile stiffness is the same along _x-_ and _y_ -directions. When performing a rotation of 90°, the roles of the plies at 0° and 90° are simply being inverted. As for the tensile stiffness of the honeycomb along _x-_ or _y_ -directions, it is very low, and its effect on the overall tensile stiffness can be neglected (in practice, it is so low that it isn't even provided by the supplier and is in the range of the kPa!). However, we will recall that the use of honeycomb is intended to separate the two skins and thus increase the bending stiffness. The only interesting honeycomb characteristics are its tensile/compressive characteristics along _z_ -direction and its out-of-plane shear characteristics in the plane _(x, z)_ and _(y, z)_ (see the following exercise).

For plates 2 and 3, this ratio will also be equal to 1. For plate 2, the reasoning is the same as for the previous one, and for plate 3, performing a rotation of 90° simply inverts plies at +45° and at –45°.

For plate 4, this ratio will be equal to _E l/Et_. Along _x_ -direction, the tensile stiffness is equal to _E l_, and along _y_ -direction, the tensile stiffness will be equal to _E t_.

  * – The plate with the highest fracture limit along _x_ -direction:

The plate with the most plies at 0°; therefore, plate no. 4 will be the one that has the highest fracture limit along _x_ -direction.

  * – The plate with the highest buckling resistance along _x_ -direction:

The plate with the highest bending stiffness; therefore, plate no. 1 will be the one that presents the highest resistance to buckling. We restate that the resistance to buckling along _x_ -direction of a plate is both linked to bending stiffness _D 11_ (which is along _y_ -direction, and is thus linked to stress along _x_ -direction, _σ x_), but also to the other bending stiffnesses _D 22_ (which is along _x_ -direction, and is therefore linked to stress along _y_ -direction, _σ y_), _D 22, D12, D16, D26_ and _D 66_, and the link between the bending stiffnesses and buckling resistance of a plate is not straightforward (see Chapter 12).

## **15.7. Sandwich beam under bending**

**Figure 15.10.** _Sandwich beam under bending. For a color version of this figure, seewww.iste.co.uk/bouvet/aeronautical2.zip_

##### _Question 1_

The honeycomb is a very light material (typically a couple of tenths of kg/m3, in comparison to 1,000 kg/m3 for water or 1,700 kg/m3 for a typical carbon/epoxy laminate), which serves as the core in sandwich structures (one honeycomb plus two upper and lower skins). The _honeycomb_ denomination comes from its evident resemblance to the honeycomb cells found in a beehive:

**Figure 15.11.** _Nomex ® honeycomb_

The three directions are generally called _L_ for longitudinal (the stretching direction), _W_ for the width and _T_ for thickness.

There are two main categories of honeycomb made from sheets of aluminum or from Nomex® with a thickness of approximately 0.1 mm. Nomex® was originally the brand name for a type of aramid fabric sold by Du Pont de Nemours made out of aramid paper impregnated in a phenolic resin.

The advantage of inserting a layer of honeycomb between two skins is obviously to increase the bending stiffness (which varies depending on the cube of the thickness) for a reduced increase in mass (see question 6). The main interesting characteristics of honeycomb are its stiffness and resistance to tension/compression in the out-of-plane direction z and out-of-plane shear in the planes _(x, z)_ and _(y, z)_. Its other characteristics, _E x, Ey, σxr, σyr, Gxy,_ and _τ xyr_, are very low and are not even provided by the supplier because they can be neglected when sizing.

##### _Question 2_

To determine the internal forces (force in the wider sense of the term, i.e. force in Newton, but also moment) from the right part of the beam to the left part, it is just a matter of isolating the right part of the beam and stating that the sum of the forces applying to this part is null.

**Figure 15.12.** _Sandwich beam under bending. For a color version of this figure, seewww.iste.co.uk/bouvet/aeronautical2.zip_

You will easily find that the internal forces from the right part of the beam to the left part of the beam (since of course the force from left to right is equal to the opposite from right to left), thus the bending moment and the shear force:

[15.95]

**Figure 15.13.** _Internal forces in a clamped beam under bending. For a color version of this figure, seewww.iste.co.uk/bouvet/aeronautical2.zip_

We note that the bending moment is maximum at the clamp, and that if there were to be a fracture, this is where the damage would initiate.

##### _Question 3_

The shear force will be supported by the intermediary of the shear stress _τ xz_ (cross-section of normal vector _x_ and stresses along _z_ -direction) and the bending moment by tensile/compressive stress _σ x_ (and we should verify that the sum of all these stresses is indeed worth zero in stress but not in moment; see Figure 15.14). In the case of a homogeneous solid cross-section, we can show that shear stress _τ xz_ is parabolic [AGA 08] (look closely at the following diagram; the blue dotted line indicates the distribution of stress _τ xz_, but not their direction, which is in fact along _z_ -direction!). It must, in particular, be null at the bottom and top due to the boundary conditions (null external forces above and below the beam), and the equilibrium equation written in the beam allows us to find the parabolic form from the linear form of _σ x_. As for stress _σ x_, we can show that it is linear in _z_. You will also note that _τ xz_ is constant throughout the length of the beam (along _x_ -direction), whereas _σ x_ is also linear in _x_ (and maximum at the clamp). If you do the math, you will find the classic result in the case of a homogeneous cross-section:

[15.96]

Where _I_ _y_ is the quadratic bending moment:

[15.97]

**Figure 15.14.** _Stresses in a clamped solid cross-section beam under bending. For a color version of this figure, seewww.iste.co.uk/bouvet/aeronautical2.zip_

Note, however, that these results are not valid in our case where the cross-section is not homogeneous. In reality, the stresses will have this form:

**Figure 15.15.** _Stresses in a clamped sandwich cross-section beam under bending. For a color version of this figure, seewww.iste.co.uk/bouvet/aeronautical2.zip_

The shear stress _τ xz_ is once again null at the bottom and the top, but is practically constant throughout the honeycomb (again, look closely at the diagram; the blue dotted line indicates the distribution of stress _τ_ xz, but not its direction, which is, in fact along _z_ -direction!). This is due to the fact that the tensile stiffness of the skins is much higher than that of the honeycomb (the Young's modulus for the honeycomb along _x_ -direction is practically 0). And we can demonstrate that the variation along _z_ -direction of _τ xz_ (or ∂ _τ xz_/∂ _z_ ) is equal (in absolute value) to the variation along _x_ -direction of _σ x_ (or ∂ _σ x_/∂ _x_ ). This comes from the equilibrium equation written for a planar stress in plane _(x, z)_ :

[15.98]

For more details surrounding these calculations, see [BAM 08, BER 99]. We can therefore initially suppose that the shear stress is null in the skins and constant in the honeycomb; thus:

[15.99]

Similarly, the tensile stiffness along _x_ -direction is so low in the honeycomb that the generated stress _σ x_ will be low in comparison with the one in the skins. And since on top of that the thickness of the skins is low compared with the honeycomb, we can initially suppose that _σ x_ is constant in the skins. It is then just a matter of stating that the moment caused by these stresses is equal to the bending moment:

[15.100]

Thus:

[15.101]

And _σ x_ will be negative in the upper part of the beam and positive in the lower part of the beam.

We note, nonetheless, that this result is only true for the average stress (or if the skins are homogeneous along the thickness). To generalize this result to typical laminate layers, we would have to reason directly in the resultant forces and then state that the moment caused by these two resultant forces _N x_ is equal to the bending moment:

[15.102]

In conclusion, the cross-section of the beam that is the most loaded will be at the clamp, and we then get:

[15.103]

Where _τ xz_ is the shear stress in the honeycomb and _N x_ is the resultant force in the skins.

##### _Question 4_

To determine the deflection at the end of the beam, we should integrate the beam constitutive equation:

[15.104]

Using boundary conditions that _w_ and its derivative according to _x_ are null at the clamp. We will then find the classic formula for the deflection _δ M_ due to the bending moment:

[15.105]

However, this result is only strictly true for a beam with a homogeneous solid cross-section. In practice, we can show that it is also true for beams with non-solid and non-homogeneous cross-sections, on the condition that the correct bending stiffness _E.I y_ be used. In our case, it is just a matter of asking where the bending stiffness is coming from (the honeycomb is only included to separate the skins and support the shear force). Modulus _E_ will therefore be the average modulus of the laminate along _x_ -direction, and _I y_ will be the quadratic bending moment of the skins. To be exact, we would have had to calculate the bending stiffness of each of the plies following their orientations, but since the thickness of the honeycomb is far greater than the skins, using an average modulus does not affect the results much. I leave you to demonstrate that with the proposed laminate, you will find for the skins _E x = 51.4 GPa_ and the following quadratic bending moment:

[15.106]

Thus a deflection of _δ M = 259 mm_.

We can note that the constitutive equation of a beam under bending is equivalent to that between a resultant moment _M x_ and the curvature _k 0x_:

[15.107]

It should nonetheless be noted that _M x_ is a resultant moment in N.mm/mm and _D 11_ is a plate bending stiffness in N.mm (whereas _E.I y_ is in N.mm2). Also note that the notation _M x_ means that this resultant moment is due to stresses along _x_ -direction, _σ x_, but that it is directed along _y_ -direction. If we multiply this relation by the plate width _b_ , we can then identify bending stiffness _b.D 11_ of the plate of length _b_ with bending stiffness _E.I y_ of the beam. Supposing that the modulus is constant throughout the thickness of the plate, we then obtain:

[15.108]

And you can show that in the case of a homogeneous isotropic material, we obtain:

[15.109]

The two formulae therefore differ simply from coefficient _β_ (which, in a classic case where _ν_ _= 0.3_ , is worth 1.1). This difference comes from the hypothesis used in both the calculations. In the case of a beam, the thickness along _y_ -direction is assumed to be small; we can therefore suppose that the stress is planar ( _σ y = 0_). In the case of a plate, the thickness along _y_ -direction is high, and we will suppose that the strain is planar ( _ε y = 0_). In practice, we have _b = h_ , and we are therefore closer to the beam hypothesis than the plate hypothesis.

**Figure 15.16.** _Plate hypothesis or beam hypothesis_

##### _Question 5_

In the case of a homogeneous solid cross-section beam, we can show that the deflection due to shear force is negligible compared with that of the bending moment and is therefore often forgotten. In the case where the cross-section is more complex (for example, in the case of a beam with thin walls), this is not the case.

If we suppose that the shear stress _τ xz_ is homogeneous throughout the cross-section _S_ of the beam, as is the case with honeycomb, we then have:

[15.110]

We can then determine the shear strain _γ xz_, also assumed homogeneous in the whole cross-section of the beam, by:

[15.111]

and by definition of strain:

[15.112]

Therefore, we can show that the derivative of the displacement along _x_ -direction, _w(x)_ , is equal to the shear strain [CHE 08]:

[15.113]

Thus the constitutive equation of a beam subject to a shear force:

[15.114]

As here the shear force is constant, we find the deflection resulting from shear force _δ T_:

[15.115]

Where _G xz_ is the shear modulus of the honeycomb (here _G LT_), as we initially suppose that only the honeycomb supports the shear force. We observe that the deflection due to shear force remains lower than the one resulting from bending moment (but this is not always the case depending on the geometry of the problem and particularly if the thickness of the skins becomes small).

##### _Question 6_

The total deflection is therefore worth:

[15.116]

If we now suppose that there is no honeycomb, we can neglect the deflection resulting from shear force (it is then the case of a solid cross-section) and we have:

[15.117]

Where the _I y_ is that of a plate with a thickness of _2.e_ (equal to two times the thickness of the sandwich skins joined together), and _E_ is the equivalent modulus of the laminate formed of the two previous skins _E x = 51.4 GPa_. We then get a deflection:

[15.118]

Which is approximately 2 km!!! Of course this result makes no sense. In practice, we can show that a plate that is 2 mm thick, 100 mm wide and 2 m in length is of course unable to support a load of 2.5 kN, and that it will break far before reaching that load. To size it, we must study the most loaded cross-section (located at the clamp). This cross-section is subject to a bending resultant moment:

[15.119]

We must then determine the bending stiffness matrix composed of the two joined skins, i.e. _[0, 45, 90,_ − _45] 2S_:

[15.120]

We can then determine the curvatures:

[15.121]

Then the strains from:

[15.122]

as the membrane strains _ε 0_ are null. We find a strain at the surface of the plate:

[15.123]

This strain is huge! If you are not convinced, you can determine the stress along the _l_ -direction, _σ l_, of the ply at 0° at the surface of the plate, and you will find 166,000 MPa! In sum, this beam is indeed unable to support a load of 2.5 kN without the honeycomb.

##### _Question 7_

To determine whether there is fracture of the sandwich beam, we begin by calculating if there is fracture within the skins. These skins are subject to a resultant force of ±500 N/mm, thus under tension (and with a Yamada–Sun criterion):

  * – For plies at 0°: _RF = 1.13_
  * – For plies at 90°: _RF = 3.61_
  * – For plies at +45°: _RF = 1.27_
  * – For plies at −45°: _RF = 1.27_

And in compression:

  * – For plies at 0°: _RF = 1.05_
  * – For plies at 90°: _RF = 3.87_
  * – For plies at +45°: _RF = 1.25_
  * – For plies at −45°: _RF = 1.25_

Therefore, the skins withstand, but barely (the data of the exercise were designed for this result).

We must also verify that the honeycomb does not break under shearing by verifying that the shear stress remains lower than its shear stress limit:

[15.124]

Which is verified here.

We should also verify that there is no delamination between the skin and the honeycomb. We must therefore initially verify that the shear stress in the honeycomb remains lower than the shear stress limit of the glue. The real problem is more complicated and goes beyond the scope of the present lesson. Readers interested in learning more about this can refer to [BER 99].

Finally, if the lower skin is subject to compression, we would have to verify that it does not buckle. The calculation is complex because the skin is glued to the honeycomb, and the compressive resultant force is not constant (linear variation along _x_ -direction). This buckling would also tend to create a tensile stress along _z_ -direction at the location of the bond between the skin and the honeycomb that could cause the fracture of the glue. Once again, readers interested in learning more can refer to [BER 99, GAY 97].

## **15.8. Laminate plate under compression**

**Figure 15.17.** _Simply supported panel under compression. For a color version of this figure, seewww.iste.co.uk/bouvet/aeronautical2.zip_

##### _Question 1_

The only resultant force being the compression along _x_ -direction, we should take 100% of fibers at 0°. However, if we respect the 10% minimum fiber ratio in each direction rule, we will take 10% of plies at 90°, 10% at +45°, 10% at −45° and 70% at 0°. We must therefore determine the total thickness of the laminate to resist the compressive resultant force. Supposing that we know the thickness _h_ of the laminate, we can determine the thickness of the plies in each direction:

[15.125]

We then determine the membrane stiffness of the laminate, which is linear in _h_ :

[15.126]

Thus the strains, which in this case are homogeneous in the thickness of the laminate:

[15.127]

Thus:

  * – For plies at 0°:

[15.128]

  * – For plies at 90°:

[15.129]

  * – For plies at +45°:

[15.130]

  * – For plies at −45°:

[15.131]

Then, we apply a fracture criterion to each ply, for example a Yamada–Sun criterion:

  * – For plies at 0°: _0.10/h ≤ 1_
  * – For plies at 90°: _0.08/h ≤ 1_
  * – For plies at +45°: _0.08/h ≤ 1_
  * – For plies at −45°: _0.03/h ≤ 1_

We then use the maximum value of _h_ in all the four directions, i.e.:

[15.132]

This value is largely less than the thickness of a single ply! In other words, compression will not be the one to determine the size, but rather buckling (in sum, the calculations we just made serve no purpose).

In order to respect the 10% rule, we need at least two plies at ±45° (to respect mirror symmetry), at least one ply at 90° (if it is at the center), and at least two plies at 0°, so for example a stacking sequence: _[45,_ − _45, 0, 90 0.5]S_. This stacking sequence withstands the compressive loading without a problem:

  * – For plies at 0°: _RF = 9.96_
  * – For plies at 90°: _RF = 25.4_
  * – For plies at 45°: _RF = 11.2_
  * – For plies at −45°: _RF = 11.2_

However, we will see in the next question that it is far from supporting the buckling.

##### _Question 2_

With the stacking sequence _[45,_ − _45, 0, 90 0.5]S_, we obtain:

[15.133]

Thus:

[15.134]

and:

[15.135]

This value is well below the force of −100 N/mm and the plate will buckle. This result is classic; we observe that a panel under compression is generally more sensitive to buckling than compression, and it is its buckling resistance that governs the stacking sequence and in particular its thickness.

##### _Question 3_

We can make an initial estimation of the necessary thickness to withstand buckling along _x_ -direction noting that the buckling resistance increases with the cube of the thickness, thus the necessary thickness:

[15.136]

Upon the 1st approximation, we would thus have to multiply the thickness by 4.36 to resist buckling along _x_ -direction, resulting in a thickness of 7.6 mm, and thus the 8 mm proposed in the data of the exercise. Of course this approximation is only valid while using the same stacking sequence, i.e. one using this type _[45 n,_ − _45 n, 0n,_ _90 0.5n]S_. This type of stacking sequence should of course be avoided as it presents too many consecutive plies in the same direction, which has the effect of increasing interlaminar stresses.

The real problem is complicated because not only do we not know the thickness of the required laminate, but we do not know the ply ratio in each direction either. Unlike the case of a sizing with a given resultant force, where we can initially suppose that the ratio of fibers in each direction is proportional to the resultant force in that direction, here even those fiber ratios are unknown. The idea that fibers need to be placed at 0° to support the buckling from a compressive stress at 0° is false! This may seem odd, but it comes from the fact that a compressive buckling causes a 2D blister, and will therefore involve the bending stiffness of the laminate in all the directions. Add to that the fact that placing plies at ±45° disturbs the blister and thus tends to increase the buckling resistance; we reach the conclusion that there is nothing obvious! In practice, we often show that a ply at 90° is as effective as a ply at ±45°.

To simplify the problem, we will begin setting the total thickness of the laminate to 8 mm and review a number of stacking sequences. Using the method given by the ESDU 80023, we find the following critical buckling resultant forces:

**Table 15.2.** _Calculation of buckling resistance, bending stiffnesses and RF for eight stacking sequences (the two pure UD stacking sequences are entirely textbook cases only included for the purpose of the study, but are not real-world stacking sequences!)_

| _N x_ _cr_
(N/mm) | _D 11_
(N.mm) | _D 22_
(N.mm) | RF (0°) | RF (90°) | RF (±45°)
---|---|---|---|---|---|---
_[0 32] (textbook case)_ | 68 | 5.7 106 | 3.0 105 | 112 | – | –
_[90 32] (textbook case)_ | 35 | 3.0 105 | 5.7 106 | – | +∞ | –
_[0, 90] 8S_ | 68 | 4.0 106 | 2.0 106 | 59 | 5,364 | –
_[0, 45, 90,_ − _45] 4S_ | 84 | 3.9 106 | 1.3 106 | 42 | 155 | 50
_[45,_ − _45, 0, 90] 4S_ | 113 | 2.1 106 | 1.6 106 | 42 | 155 | 50
_[45,_ − _45] 8S_ | 119 | 1.7 106 | 1.7 106 | – | – | 12

You will note that for the stacking sequence composed only of plies at 90°, the RF is infinite, because in that case, only _σ t_ is non-zero, and the Yamada–Sun criterion does not account for matrix cracking. In sum, the Yamada–Sun criterion only makes sense for a real stacking sequence and not for pure UD (the pure UD stacking sequences are textbook cases that are only present for the comparison but are not real-world stacking sequences!).

Finally, we observe that there must be plies at ±45° to support buckling along _x_ -direction.

##### _Question 4_

The plies at 0° are better than plies at 90° because the blister caused by buckling may have two directions; the loading applied is along _x_ -direction and will cause more bending resultant moment _M x_ than _M y_. It should be kept in mind that even if applying a compressive resultant force will not cause bending resultant moment, once the plate is buckled, there will be many bending moments created (due to the deflection generated by buckling). Note, however, that this is not true in general cases! We can indeed show that in cases typically used in aviation (here the size of the plate is very large and square, whereas in aviation, they are generally smaller and rectangular, with the long side along the direction of the maximum compressive loading), adding a ply at 90° is better than adding a ply at 0°.

As for plies at ±45°, they tend to disturb the blister caused by buckling and therefore tend to increase the buckling resistance (see shape of the buckling blister in question 7). This result is always true and widely used in aviation!

##### _Question 5_

To make this panel lighter, we start by increasing its bending stiffness (the terms _D ij_ of the bending stiffness matrix). For this, we can either increase the stiffness of the material, but we must then design a stiffer material (which can be difficult!), or distance the material from the midplane. The bending stiffness varies with the cube of the thickness, and therefore this second solution is much more efficient. We can therefore either modify the shape of the panel, by, for example, adding stiffeners, or make a sandwich panel by adding honeycomb. For example, if we add 10-mm-thick honeycomb to the 6th stacking sequence from the previous table, we get a stacking sequence of:

[15.137]

We then find:

[15.138]

By taking 1,700 kg/m3 for the volumic mass of the composite and 64 kg/m3 for that of the honeycomb, the panel goes from 13.6 to 14.2 kg, whereas its buckling resistance has practically been multiplied by 10! We must nonetheless keep in mind that the sandwich panels present a few drawbacks:

  * – it is harder to introduce localized loading, and additional inserts are often required [GAY 97];
  * – they absorb humidity, and the honeycomb cells tend to fill up with water (which will then freeze at −50 °C when the aircraft is operating);
  * – they are more complicated to machine, and it is hard to make panels with complex shapes.

##### _Question 6_

We now focus on a stiffened panel with T-shaped stiffeners:

**Figure 15.18.** _Stiffened panel under buckling. For a color version of this figure, seewww.iste.co.uk/bouvet/aeronautical2.zip_

In practice, there exist two main shapes of stiffeners: T shape and omega shape:

**Figure 15.19.** _T-shaped stiffener and omega-shaped stiffener_

The T-shaped stiffeners are easy enough to manufacture because it is simply a matter of building them in the shape of a U, then joining them together to a flat plate. One final ply of composite is then generally added in order to avoid stiffener delamination. Here are two industrial fuselage panel realizations made up of San Diego Composites® and Muratec®; you will also be able to find more online.

**Figure 15.20.** _Fuselage with omega-shaped stiffeners (San Diego Composites ®) and a low-cost composite fuselage with T-shaped stiffeners (Muratec®)_

##### _Question 7_

To simplify the problem, we will limit ourselves to T-shaped stiffeners of 40 mm height, sitting at 200 mm from one another, and we will consider the same stacking sequence for the skin and the stiffeners:

**Figure 15.21.** _Stiffened panel with T-shaped stiffeners_

By keeping stacking sequences of type ±45°, we find that the following stacking sequence allows us to maintain a critical buckling load higher (in absolute value) than −100 N/mm:

[15.139]

Therefore, 22 plies are required instead of the 32 plies for the flat panel, meaning a stiffened panel of 10.8 kg rather than 13.6 kg for the flat panel. Of course this optimization is far from optimal, and we can test stiffeners with different geometries, stacking sequences, variable thicknesses in the panel and the stiffeners...

Furthermore, it becomes difficult to perform analytical calculations (even if methods exist, for example in [ESD 95]), and we often resort to calculations using the finite element method. For the studied case, the 1st buckling modes for flat and stiffened panels are as follows:

**Figure 15.22.** _First buckling modes of a flat panel 45, −45] 8S and a panel with T-shaped stiffeners [(45, −45)5, 45]S (the perspective of the stiffened panel can be deceiving; the panel is seen from below and the blister is facing away from the stiffeners). For a color version of this figure, see [www.iste.co.uk/bouvet/aeronautical2.zip_

We therefore typically find the 1st buckling mode in the form of a single buckling blister (which is not necessarily the case depending on the shape of the panel). The finite element calculation gave us _N xcr = 105 N/mm_ for the flat panel _[45, −45] 8S_, in comparison to _N xcr = 117 N/mm_ that we found analytically. The main difference comes from the chart of the ESDU 8023 which assumes that the components _D 16_ and _D 26_ of the bending matrix are null, which here is not the case. The finite element/analytical correlation nonetheless remains correct, but is even better for stacking sequences with null _D 16_ and _D 26_ components (or low). You will also note that in the previous figure, the maximum displacement is equal to 1. We can show that if the applied loading is equal to the critical buckling force, the solution to the problem is not unique, but rather determined to within a multiplicative coefficient (the buckling mode is an eigenvector of the stiffness matrix). In other words, if you take the solution from the previous figure and multiply the displacement field by a given coefficient, the new displacement field will always yield the solution to the problem. It is therefore common practice to arbitrarily select a maximum displacement of 1 (1 mm in this case), then apply a multiplicative coefficient to better identify the structure deformed shape (in this case, we used a coefficient 100). In reality, if we wished to study the development of the buckling mode in more detail, we would have to use a nonlinear calculation.

We can also optimize the panel by adding transverse stiffeners along _y_ -direction. In the case of aviation structures, we generally make sure we achieve the first buckling mode that buckles the skin-stiffened panel, i.e. the plate located between two consecutive stiffeners along _x-_ and _y_ -directions. In the case of a passenger plane type A320, the distance between the longitudinal stiffeners (these stiffeners are directed along the axis of the fuselage and are also called the stringers) is approximately 200 mm and approximately 600 mm between circumferential stiffeners (these stiffeners along the circumferential direction are therefore more or less circular and are called the frames). Using these distances approximately, we therefore choose to add a transverse T-shaped stiffener in _y_ -direction at the middle of the panel (this obviously avoids global buckling as was the case for the previous panel). In practice, the cross-section for these transverse stiffeners is higher than that of the longitudinal stiffeners; here, we select a height of 60 mm (40 mm for the longitudinal stiffeners). Evidently, the optimization problem then becomes far more complicated because the number of variable parameters becomes much higher: panel stacking sequence, stacking sequence and height of the longitudinal stiffeners, stacking sequence and height of the transverse stiffeners (and in reality, the shape of the stiffeners still needs optimizing). We choose to stay with a 22 ply stacking sequence _[(45, −45) 5, 45]S_ for the stiffeners and opt for a stacking sequence for the skins that will allow for buckling of the skin-stiffened panel (a 500 mm-long and 200 mm-wide plate). Maintaining a stacking sequence using plies at ±45° leads to a stacking sequence of 12 plies _[45, −45] 3S_ (you can also verify that the RF remain largely higher than 1). The optimization of the skin-stiffened panel can also be performed using the compression chart: you would find here a buckling for this panel that is 500 mm long and 200 mm wide for a resultant force of 158 N/mm. In this case, you can even demonstrate that 11 plies would have been enough, but a calculation using finite element method shows that the skin-stiffened panel would have buckled too early in the case of the full-sized panel (the stiffeners are obviously not infinitely stiff and therefore do not exactly impose null out-of-plane displacements).

Using a finite element calculation of the full-sized panel, we obtain buckling for Nxcr = 113 N/mm and a weight of the panel of 7.2 kg:

**Figure 15.23.** _Buckling of a stiffened panel with longitudinal and transverse stiffeners. For a color version of this figure, seewww.iste.co.uk/bouvet/aeronautical2.zip_

This design method using longitudinal and transverse stiffeners, making sure to buckle the skin-stiffened panel first, is very interesting for optimizing the mass of a panel, even if it does complicate the design process somewhat. It is obviously no coincidence that practically all planes are designed this way!

## **15.9. Tube under torsion/internal pressure**

**Figure 15.24.** _Composite tube under torsion/internal pressure. For a color version of this figure, seewww.iste.co.uk/bouvet/aeronautical2.zip_

##### _Question 1_

The torsion moment will cause a shear resultant force _T zθ_. If we cut the laminate along a face of normal vector _z_ , the resultant force is then directed along _θ_ -direction. To determine the shear resultant force, simply sum the moment caused by these arrows (Figure 15.25):

[15.140]

Where _(2.π.R)_ corresponds to the circle's perimeter, and the second _R_ to the lever arm for the calculation of the moment.

As for the internal pressure, it will cause a tensile resultant force _N θ_. This resultant force corresponds to a resultant force along _θ_ -direction for a cut by a face with normal vector _θ_. To determine this resultant force, simply state that the half-cylinder is balanced under the effect of _N θ_ and pressure _P_. To integrate pressure _P_ on a half-circle, it is important to account for the fact that the direction of _P_ varies depending on the considered point; therefore, either integrate considering the projection of _P_ along _x_ -direction (which will highlight an angle _θ_ that varies between – _π_ /2 and _π_ /2), or you can remember that the integral gives the pressure _P_ multiplied by the projected surface area. You will also note that the calculation does not highlight the length along _z_ -direction, in other words, it corresponds to a unit length (in N/mm). We then have:

[15.141]

**Figure 15.25.** _Resultant force under torsion a) and internal pressure b). For a color version of this figure, seewww.iste.co.uk/bouvet/aeronautical2.zip_

##### _Question 2_

We can then determine the ply ratio in each direction ( _α_ , _β_ , _γ_ ):

[15.142]

Omitting the 10% rule to simplify the problem. If we add that the sum of the ply ratios is obviously 100%, and considering a ratio of _γ_ at +45° and at −45° ( _α_ \+ _β_ \+ 2. _γ_ = 1) :

[15.143]

We can then reuse the proposed procedure in exercise 8, or simply test some numbers of plies. Here, you will find that two plies along ±45°-direction and four plies at 90° are enough (with the Yamada–Sun criterion):

  * – For 4 plies at 90°: _RF = 1.06_
  * – For 2 plies at 45°: _RF = 1.08_
  * – For 2 plies at −45°: _RF = 1.21_

Therefore, for example, the following stacking sequence:

[15.144]

This allows us to maintain a mirror symmetry and avoid too many consecutive plies in the same direction.

##### _Question 3_

The tube being made from filament winding, we can easily create stacking sequences with any given directions. Nonetheless, in order to avoid tension/shear coupling, we will use as many plies at _+α_ as we do at _–α_ (you can verify that in this case, _A 16_ and _A 26_ are null). The calculation procedure in plies at _α_ is similar to classic plies at 0°, 90° or ±45°, outside of the shape of the stiffness matrices _ _which of course is more complicated (seeChapter 3). We can then vary angle _α_ and determine the number of plies that result in RF higher than 1. Since there is no loading along _z_ -direction, we will simply vary this angle between 45° and 90° (the solutions between 0° and 45° are also far less interesting). We denote as _n_ the number of plies at _α_ (and thus also at _–_ _α_ ), and get:

**Table 15.3.** _Calculation of a stacking sequence for a tube under torsion/internal pressure_

_α_ | 45° | 50° | 55° | 60° | 65° | 70° | 75° | 80° | 85° | 90°
---|---|---|---|---|---|---|---|---|---|---
_n_ | 14 | 12 | 9 | 7 | 6 | 5 | 6 | 8 | 12 | 14
RF | 1.03 | 1.07 | 1.01 | 1.04 | 1.13 | 1.05 | 1.12 | 1.07 | 1.07 | 1.02

The best solution is therefore 5 plies at 70° and 5 plies at −70°. We then find a stacking sequence of the following type:

[15.145]

And the optimal stacking sequence is therefore worse than the traditional stacking sequence with plies at 0°, ±45° and 90° (10 plies against 8). In conclusion, the obligation to place as many plies at _+α_ and _–α_ is more restrictive than placing plies at 0°, ±45° and 90°. Note, however, that this conclusion is not always true! Furthermore, in reality, filament winding allows us to stack in any direction. Lastly, keep in mind that using filament winding makes it simpler to stack plies at ± _α_ than at ±45° and 90°.

## **15.10. Optimization of a fabric with a strain fracture criterion**

**Figure 15.26.** _Ply of 8 harness satin fabric. For a color version of this figure, seewww.iste.co.uk/bouvet/aeronautical2.zip_

#### **_Part I: preamble_**

##### _Question 1_

Here, we are working with a balanced fabric, meaning that the fiber ratio is the same along _l-_ and _t_ -directions; thus, approximately 25% (of the total volume) is fibers in _l_ -direction, 25% are in _t_ -direction, and 50% fibers resin. In reality, even when a fabric is balanced, its characteristics are slightly different along _l-_ and _t_ -directions; we observe that the warp direction (the one that was under tension during weaving) is stiffer and more resistant than the weft direction (typically 10–20% more). This comes from the fact that the warp fibers oscillate less than the weft ones do, because the warp is stretched during weaving. In this exercise, we suppose that the characteristics are the same for both the directions _l_ and _t_ to simplify the calculations. Furthermore, the use of a strain criterion allows the calculations to be made even simpler.

The compliance matrix of the ply at 0° is simply (both in _(x, y)_ and _(l, t)_ ):

[15.146]

##### _Question 2_

The stiffness matrix of the ply at 0° is (both in _(x, y)_ and _(l, t)_ ):

[15.147]

##### _Question 3_

The compliance matrix of the ply at 45° is:

[15.148]

_S 16_ and _S 26_ (like _Q_ 16 and _Q 26_) are null because there are as many fibers at +45° as there are at −45° and because the behavior is assumed the same along _l_ -direction as it is along _t_ -direction.

##### _Question 4_

The stiffness matrix of the ply at 45° is:

[15.149]

The term _Q 66_ of a ply at 45° is of course much higher than that of the ply at 0°. We see that the one for the ply at 45° is linked to modulus _E_ , whereas the one for the ply at 0° is linked to _G_. This comes from the fact that a ply of fabric at 45° has fibers at +45° and at −45°, allowing it to support the stresses caused by shearing (a tension at +45° and a compression at −45°).

#### **_Part II: quasi-isotropic stacking sequence_**

##### _Question 5_

To determine the Young's modulus along _x_ -direction of a laminate composed of two plies of fabric at 0° and at 45°, we then begin by determining the stiffness matrix _A_ of the laminate:

[15.150]

We invert the matrix to determine the compliance matrix of the laminate:

[15.151]

We can then relate this matrix to the compliance matrix of a homogeneous material (without forgetting thickness _h_ ) (see exercise 3):

[15.152]

Thus:

[15.153]

##### _Question 6_

The material is quasi-isotropic because there are the same amount of fibers at 0°, at 90°, at +45° and at −45° (and the behavior of the fabric along the _l_ -direction is the same as along the _t_ -direction); thus, the modulus will be the same in all the directions (see Chapter 6). We then have:

[15.154]

##### _Question 7_

To determine the limit shear stress _τ xy0_, simply take a stress _τ xy0_, then determine the RF in each ply. The limit stress will be, by definition of the RF (the coefficient by which we must multiply the load to obtain fracture), the minimum of the RF multiplied by the selected stress _τ xy0_.

We restate that _τ xy0_ is the average shear stress, i.e.:

[15.155]

Where _e_ is the thickness of the laminate (here 2 mm). The drawback of this average stress is that no ply actually experiences it (therefore in particular, using this stress for a fracture criterion makes no sense).

Let us take an average stress of 100 MPa:

[15.156]

We can then determine the membrane strains:

[15.157]

Then the strains in each ply:

[15.158]

All that is left to do is determine the RF using the strain criterion and taking the minimum RF between directions _l_ and _t_ :

[15.159]

Thus:

  * – For plies at 0°: _RF = +∞_
  * – For plies at 45°: _RF = 1.89_

In the plies at 0°, the strains along _l-_ and _t_ -directions were null, and we find an infinite RF. This is due to the criterion we are using which does not account for shear strain, but rather only the strains in the fiber direction. In reality, this criterion only makes sense if we have at least one ply at 0° and one ply at 45°. In the present case, the infinite RF means that only plies at 45° support the loading (which is only true upon initial approximation).

The stress limit will therefore be the 100 MPa we selected, multiplied by the minimum RF; thus:

[15.160]

##### _Question 8_

To determine the RF for a given stress, we use the same procedure:

[15.161]

Thus, the strains in each ply:

[15.162]

Thus:

  * – For plies at 0°: _RF = 0.33_
  * – For plies at 45°: _RF = 0.38_

The proposed stacking method does not withstand the proposed load.

##### _Question 9_

To determine a stacking sequence that will withstand this loading, we simply note that the RF are linear depending on the thickness. Note, however, that this is only true under membrane loading, but not under bending loading. In other words, when we multiply the thickness of a laminate by two, we are dividing the RF by two as well (if we consider that the stacking sequence remains the same, and with plies twice as thick). Therefore, here it is just a matter of taking three times the stacking sequence plus a little extra something, an added ply at 0° for example, which would give us:

[15.163]

Furthermore, we avoid placing plies at 0° on the surface, because these are the ones that will support the most load ( _N x = −1,500 N/mm_). Redoing the calculations for this stacking sequence, we then get:

  * – For plies at 0°: _RF = 1.10_
  * – For plies at 45°: _RF = 1.18_

#### **_Part III: stacking sequence optimization_**

##### _Question 10_

We begin by supposing that the fiber ratio in each direction is proportional to the corresponding resultant force (note that it is impossible to place fibers at 0° without having any at 90° in this case):

[15.164]

If we add that the sum of the ratios is for obvious reasons 100% ( _α_ \+ _γ_ = 1) , then:

[15.165]

We name _h_ the thickness of the laminate, and get:

[15.166]

Thus the strains, which in this case are homogeneous throughout the thickness of the laminate:

[15.167]

Thus:

  * – For plies at 0°:

[15.168]

  * – For plies at +45°:

[15.169]

We then apply the strain criterion in each ply:

  * – For plies at 0°: _h ≥ 5.18 mm_
  * – For plies at +45°: _h ≥ 6.76 mm_

Of course we select the maximum value of _h_ for all four directions, i.e.:

[15.170]

Thus:

  * – For plies at 0°: _e_ 0° ≥ _α_. _h_ = 4.8 _mm_
  * – For plies at +45°: _e_ 45° ≥ _γ_ . _h_ = 2.0 _mm_

For example:

[15.171]

By redoing these calculations for this stacking sequence, we would find:

  * – For plies at 0°: _RF = 1.35_
  * – For plies at 45°: _RF = 1.03_

We find a solution that is heavier than the previous one (7 mm against 6.5 mm). This comes from the fact that the proposed method is adapted to a stacking sequence with plies at 0°, 90° and ±45°, but in this case, it is impossible to have fibers at 0° and none at 90°. In other words, when we place a ply at 0° to support the load along _x_ -direction, the fibers at 90° no longer serve a purpose.

#### **_Part IV: stacking sequence optimization under bending_**

##### _Question 11_

Consider a stacking sequence formed of two plies at 0° and six plies at 45°. In the general case, we would choose the following type of stacking sequence:

[15.172]

This type of stacking sequence avoids having too many consecutive plies in the same direction and avoids having the plies at 0°, the ones supporting the load, at the surface.

In the present case, if we want to support a resultant moment _M x_, we had best place the plies at 0° at the surface to support this moment, i.e.:

[15.173]

However, in practice, this stacking sequence is prohibited as it is forbidden to place the plies supporting the load (the ones at 0° in this case) at the surface, and we will always favor plies at ±45° on the surface to increase the buckling resistance. Furthermore, we cannot have six consecutive plies in the same direction to limit interlaminar stresses and thus delamination. Therefore, we will maintain the previous stacking sequence option:

[15.174]

To determine the fracture resultant moment _M xf_, it is simply a matter of supposing any given resultant moment _M x_, for example, _M x = 100 N_, then calculating the RF in each ply. Note, however, that in the case of a bending loading, the RF vary linearly throughout the thickness; they are therefore maximum/minimum at the top or bottom of each ply (or of each group of consecutive plies in the same direction). The fracture resultant moment will then be, by definition of the RF (the coefficient by which we must multiply the load to obtain the fracture), the minimum of the RF multiplied by the selected resultant moment _M x_.

We therefore begin by determining the stiffness matrix under bending:

[15.175]

We can then determine the curvatures:

[15.176]

And the strains from:

[15.177]

We then obtain a linear strain field along the thickness, null at the center and mainly composed of strains _ε x_:

**Figure 15.27.** _Strains of the laminate under bending M x. For a color version of this figure, see www.iste.co.uk/bouvet/aeronautical2.zip_

The minimum/maximum RF are achieved at the top of the upper ply at 0°, at the bottom of the lower ply at 0°, at the top of the upper surface ply at 45° and at the bottom of the lower surface ply at 45° (for plies at 45°, the strains in _(l, t)_ must of course be determined beforehand):

  * – At the top of the upper ply at 45°: _RF = 18.6_
  * – At the top of the upper ply at 0°: _RF = 6.93_
  * – At the bottom of the lower ply at 0°: _RF = 8.53_
  * – At the bottom of the lower ply at 45°: _RF = 22.8_

Thus:

[15.178]

## **15.11. Open hole tensile test**

##### _Question 1_

In the absence of a hole, it is just a matter of supposing a resultant force along _x_ -direction, for example _N x = 100 N/mm_, then calculating the RF within each ply. The fracture resultant force will then be, by definition of the RF (the coefficient by which we must multiply the loading to obtain the fracture), the minimum of the RF multiplied by the selected resultant force _N x_.

**Figure 15.28.** _Holed plate under tension_

We begin by determining the membrane stiffness matrix of the laminate:

[15.179]

We can then determine the membrane strains:

[15.180]

Then the strains in each ply:

[15.181]

All that is left to do is determine the RF using the strain criterion and using the minimum RF between directions _l_ and _t_ :

[15.182]

Thus:

  * – For plies at 0°: _RF = 20.4_
  * – For plies at 45°: _RF = 60.0_

Thus, the following fracture resultant force in the absence of a hole:

[15.183]

##### _Question 2_

To determine the equivalent moduli of the woven laminate, we begin by determining the stiffness matrix _ _of the laminate, inverting it in order to determine the compliance matrix of the laminate, then relating the matrix to the compliance matrix of a homogeneous material (without forgetting thickness _h_ ) (see exercise 3):

[15.184]

Thus:

[15.185]

##### _Question 3_

The resultant force concentration ratio at the hole edge (i.e. the ratio between the maximum resultant force obtained at the hole edge and the far-field resultant force) is here:

[15.186]

Supposing that the fracture takes place where the resultant force is highest, i.e. at the hole edge (which in this case is not so simple, as we will see in the coming questions), the fracture resultant force _N xhole_ in the presence of a hole will thus be:

[15.187]

Of course, this result does not depend on the size of the hole, just like the resultant force concentration ratio at the hole edge. In reality, we can show that this is not the case, because the smaller the hole, the more the composite will get damaged at the hole edge (in proportion to the size of the hole), and the less this hole will be a problem.

##### _Question 4_

The previous calculation that allows us to determine the stress at the hole edge only works if the material is linear elastic, and if the plate remains a plate. And at the hole edge, the material gets damaged. Therefore, not only is the behavior of the material no longer linear elastic, but also the laminated plate theory we presented previously is no longer valid (in particular, the damage at the hole edge will cause small delaminations and the plies of the laminate are no longer perfectly bonded, and the model of homogeneous strains through the thickness for a membrane loading is no longer true). And since the size of the damaged area is, upon initial approximation, constant depending on the size of the hole (the size of the damaged area will be proportionally larger if the hole size is small), a small hole is therefore less dangerous than a large hole. The "point stress" theory allows us to account for this effect, supposing that the calculation of the stresses is true only from a certain distance from the hole:

**Figure 15.29.** _Principle of the "point stress" theory. For a color version of this figure, seewww.iste.co.uk/bouvet/aeronautical2.zip_

We therefore begin by determining the resultant force concentration ratio at the hole edge for the three diameters:

[15.188]

**Figure 15.30.** _Hole size effect of holed plate under tension and accounting for this using the "point stress" theory. For a color version of this figure, seewww.iste.co.uk/bouvet/aeronautical2.zip_

We use this to evaluate from which value of _α_ we achieve this concentration ratio, and therefore the corresponding distance _d 0_:

[15.189]

We therefore find values of _d 0_ included between 1 and 2 mm, which are characteristic values of composite aircraft structures. Of course, if the point stress method were perfect, we would find the same value of d0 for all the radii of the holes, and if possible, for all the stacking sequences and thicknesses of the laminates. The reality is, as is often the case, more complicated than the small model you would like them to fit into. In industrial use, it is common to build charts of _d 0_ according to the radius and stacking sequence in use.

## **15.12. Multi-bolt composite joint**

##### _Question 1_

We will begin by evaluating the size of the current area (i.e. outside the joint area) of plate 1. The problem comes down to finding a stacking sequence that is resistant to resultant forces _N x = 1,500 N/mm_ and _N y = 1,000 N/mm_. I will let you verify that stacking sequence _[45, −45, 0, 90, 0 0.5]S_ (thus 9 plies and a total thickness of 2.25 mm) is suitable and gives using a Yamada–Sun criterion:

  * – For plies at 0°: _RF = 1.24_
  * – For plies at 90°: _RF = 1.78_
  * – For plies at 45°: _RF = 1.43_
  * – For plies at −45°: _RF = 1.43_

**Figure 15.31.** _Multi-bolt composite joint_

Now we size the current area of plate 2. The problem comes down to finding a stacking sequence that is resistant to resultant force _N x = 1,500 N/mm_. I will let you verify that a stacking sequence _[45, −45, 0, 90, 0 0.5]S_ (thus 9 plies and a total thickness of 2.25 mm) is still suitable and gives using a Yamada–Sun criterion:

  * – For plies at 0°: _RF = 1.00_
  * – For plies at 90°: _RF = 3.22_
  * – For plies at 45°: _RF = 1.13_
  * – For plies at −45°: _RF = 1.13_

And surprisingly, we obtain that the RF of the plies at 0° and ±45° decreases when the resultant force along _y_ -direction is null! This is fairly common and explains in part why we always have 10% of plies in each direction, to not have to verify all the intermediary loads, linear combination of maximum loads.

We will now focus on the joint area. We begin by choosing the pitch distance, i.e. the distance between two consecutive holes. In practice, the pitch distance must be at least equal to approximately four times the diameter of the hole in all the directions; we choose to use a pitch distance of _4.D_ (a typical value, often used in aviation):

[15.190]

Where _D_ is the diameter of the hole (and therefore the rivet) and _W_ the distance between two consecutive rivets. We can then determine the equivalent stress, _σ xeq_, defined by the stress leading to the same level of damage (or even the RF) as in the absence of the hole (see Chapter 11):

[15.191]

We must nonetheless note the thickness of the plate, because it can be different between the current area and the joint area (we often increase the thickness of the multi-bolt joint to withstand overloads due to the joint). Naming _e j_ the thickness of the joint area, we can determine the net stress at the joint:

[15.192]

_α_ is the hole ratio; therefore, for this case of tensile stress:

[15.193]

_K b_ is the bearing ratio (see Chapter 11):

[15.194]

and _σ b_ the bearing stress defined by:

[15.195]

Where _F rivet_ is the bearing force transmitted by each rivet (this force depends on the number of rivets). To determine this bearing force, it is necessary to know the number of rivets. We begin by choosing two rivets (and we will increase the number of rivets if they are not enough!); we therefore have 50% of the force passing through the 1st rivet and 50% through the second, or:

[15.196]

We then obtain:

[15.197]

Along _y_ -direction, the problem is simple; the force no longer passes in the rivets and only passes in plate 1. We then get:

[15.198]

**Figure 15.32.** _Diagram of the multi-bolt composite joint with varying thickness. For a color version of this figure, seewww.iste.co.uk/bouvet/aeronautical2.zip_

Plate 1 is therefore subject in the joint area to:

[15.199]

Where _e j_ is the thickness of the plate in the joint area, and it is therefore different to the thickness in the current area previously determined (it will likely be thicker). To determine the stacking sequence that is resistant to this load, it must also be noted that we will have to impose continuity between this stacking sequence and the current area (by performing ply drop-offs). It is therefore important to maintain the previous plies and add other to support this load. I will let you verify that the stacking sequence _[45, −45, 0 2, 90, 02, 90, 01.5]S_ (therefore, 19 plies and a total thickness of 4.75 mm) is suitable for the two plates and gives us, for plate 1:

  * – For plies at 0°: _RF = 1.16_
  * – For plies at 90°: _RF = 1.15_
  * – For plies at 45°: _RF = 1.16_
  * – For plies at −45°: _RF = 1.16_

And for plate 2:

  * – For plies at 0°: _RF = 1.06_
  * – For plies at 90°: _RF = 5.30_
  * – For plies at 45°: _RF = 1.24_
  * – For plies at −45°: _RF = 1.24_

We must then show that this stacking method resists bearing. This bearing load corresponds to the maximum load passing in a rivet; therefore, here:

[15.200]

Knowing that this stress must remain below the bearing limit stress, thus approximately 600 MPa (or −600 MPa to be specific, since it is a case of compression) for the considered T300/914. This condition is therefore not entirely verified, but if we add a ply, at 0° for example, for a stacking sequence such as _[45, 0, −45, 0 2, 90, 02, 90, 0]S_ (thus, 20 plies and a total thickness of 5 mm), the condition is respected:

[15.201]

And I will let you verify that the RF are still higher than 1.

To finish defining a multi-bolt joint, all that remains is to select the hole distance at the edge of the plate; for example, we can use a hole edge distance equal to 3.D (see previous figure).

##### _Question 2_

There is of course no point in taking the same stacking sequence in the joint as in the current area. We will therefore have to perform ply drop-offs to go from one stacking sequence to other, for example:

**Figure 15.33.** _Ply drop-offs at the edge of the multi-bolt joint. For a color version of this figure, seewww.iste.co.uk/bouvet/aeronautical2.zip_

In this diagram, you will note that we have opted to perform these ply drop-offs in a quasi-symmetrical manner, but this is not necessarily always the case. The rules for ply drop-offs are outside the scope of this book, and curious readers can refer to [BER 99, GAY 97] for more details. You can nonetheless retain the two main rules:

  * – You should distribute the ply drop-offs through the thickness and impose a slope of approximately 1/20, thus, 5 mm between two consecutive ply drop-offs for a ply thickness of 0.25 mm. In the present case, we would need a distance of approximately 55 mm to stop the 11 plies.
  * – Each stopped ply must, if possible, be covered by a continuous ply (this rule is often difficult to respect, and it is in fact not the case here).

In reality, the real geometry of a ply drop-off is of course more complex than the corresponding diagram. For example, here are the differences we can observe between a real and a theoretical ply drop-off [ABD 15]:

**Figure 15.34.** _Real and theoretical ply drop-offs ABD 15]. For a color version of this figure, see[www.iste.co.uk/bouvet/aeronautical2.zip_

We had also better perform these ply drop-offs as far as possible from the force transmission plane to avoid bending moments in the joint. You will note that in the previous figure (Figure 15.33), the sum of the applied moments at the joint is non-zero due to the gap of the midplane of the two plates. We had therefore also better align the two laminate midplanes to avoid this parasitic moment:

**Figure 15.35.** _Multi-bolt joint with alignment of the midplanes. For a color version of this figure, seewww.iste.co.uk/bouvet/aeronautical2.zip_

Even in reality, there is often a stiffener to support this bending moment. Since in aviation there are generally a lot of stiffeners, we practically always make sure that the joint coincides with a stiffener:

**Figure 15.36.** _Multi-bolt joint with stiffener. For a color version of this figure, seewww.iste.co.uk/bouvet/aeronautical2.zip_

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# Index

10% in each direction

## **A, B, C**

  * _A 16_
  * _A 26_
  * ahg
  * _A ij_
  * ageing
  * amorphous
    * state
  * anisotropic
  * area of damage tolerance
  * autoclave
  * band
    * kink
    * shear
  * barely visible impact damage (BVID)
  * bearing
    * ratio
  * behavior
    * bending
    * membrane
    * rubbery
  * bending
    * stiffness
  * _B ij_
  * blind fastener
  * Boeing
  * Bredt formula
  * buckling
    * Euler's
  * calculation of a bolt composite joint
  * carbon
    * /epoxy
    * fiber
    * fiber reinforced plastic (CFRP)
  * characterization of a UD ply
  * composite
    * unidirectional
  * compression
    * after impact (CAI)
    * longitudinal
    * transverse
  * coupling
    * bending/twisting
    * coefficient
    * tension/shear
  * criteria
    * fracture
      * criteria of a UD ply
    * Hill's
    * maximum
      * strain fracture
      * stress fracture
    * Tsai-Wu's
    * Von Mises
    * Yamada-Sun
  * critical buckling load
  * cross-linking
  * crystalline phase
  * crystallization
    * degree of
  * curvatures
  * cusps

## **D, E, F**

  * _d 0_
  * damage
    * detectable
    * undetectable
  * deflection
  * delamination
  * detectability of the impact
  * _D ij_
  * dispersed stacking sequence
  * elementary
    * stress states
    * stresses
  * ESDU 80023
  * ethylene
  * fabric
  * Federal Aviation Regulations (FAR)
    * FAR 25
  * filament winding
  * force
    * internal
    * shear
  * fracture
    * bearing
    * cleavage
    * fiber
    * interface
    * modes
    * of a UD ply
    * shear
    * structural
    * toughness
    * ultimate
  * Fybrflush

## **G, H, I**

  * glass
    * fiber reinforced plastic (GFRP)
    * transition temperature (Tg)
  * Hexcel
  * hole
    * ratio
    * size effect
  * honeycomb
  * hydrogen bonds
  * IM7/M21
  * impact
  * inorganic
  * internal pressure
  * isotropic
    * transversally

## **J, K, L**

  * Joint Aviation Requirement (JAR)
  * kink band
  * laminate
    * balanced
    * quasi-isotropic
    * unidirectional (UD)
  * load
    * given
    * highly improbable
    * limit (LL)
    * ultimate (UL)
  * longitudinal
  * Love-Kirschoff

## **M, N, O, P**

  * _M x_
  * _M xy_
  * _M y_
  * matter
    * inorganic
    * organic
  * matrix
    * bending stiffness
    * compliance
    * coupling
    * cracking
    * laminate stiffness
    * rotation
    * stiffness
  * membrane
  * mirror symmetry
  * modulus
    * equivalent
    * shear
    * Young's
  * Mohr-Coulomb
  * moment
    * bending
    * quadratic
      * bending
    * twisting
  * monomer
    * Nomex
  * multi-bolt composite joint
  * _N x_
  * _N y_
  * necking
  * off-axis loading
  * optimal stacking
  * optimization
    * of a laminate
    * stacking sequence
      * under bending
  * orthotropic
    * coordinate system
  * out-of-plane
  * panel
    * stiffened
    * under compression
  * pitch distance
  * plate
    * holed
    * model
    * under compression
  * ply
    * drop-off
    * fabric
    * off-axis UD
    * UD
  * point stress
  * Poisson's ratio
  * polycondensation
  * polyether ether ketone (PEEK)
  * polyethylene
  * polymer
  * polymerization
  * polyphenylene sulfide (PPS)
  * pre-preg
  * principle direction
  * pyrolysis

## **Q, R, S**

  * quasi-isotropic
  * reserve factor (RF)
  * residual strength after impact
  * resultant
    * force
    * moment
  * rivet
  * rotation
  * rules for stacking
  * sandwich
  * satin
  * shape factor
    * coefficient
  * shear
  * shearing
    * in-plane
    * off-axis
    * out-of-plane
  * sheet molding compound (SMC)
  * sizing of a multi-bolt composite joint
  * splitting
  * stiffener
    * omega-shaped
    * T-shaped
  * strain
    * membrane
    * principal
  * stress
    * bearing limit
    * concentration ratio
    * equivalent
    * hole edge
    * interlaminar
    * net
    * plane
  * stress/strain curve

## **T, U, V, W, Y**

  * T300/914
  * T700/M21
  * _T xy_
  * tabs
  * tensile specimen
  * tension
    * filled hole
    * hole
    * longitudinal
    * off-axis
    * open hole
    * transverse
  * test
    * compression
    * fracture
  * thermoforming
  * thermoplastic
  * thermoset
  * thermosetting
  * tolerance
    * to damage
    * to impact damage
  * Toray
  * torsion
  * transverse
  * Tsai-Wu
  * tube
    * composite
    * under torsion
  * twisting
  * unidirectional (UD)
  * Van der Waals
  * Visible Impact Damage (VID)
  * Von Mises
  * World Wide Failure Exercise
  * Yamada-Sun

**2017**

BOREL Michel, VÉNIZÉLOS Georges

_Movement Equations 2: Mathematical and Methodological Supplements
(Non-deformable Solid Mechanics Set – Volume 2)_

BOUVET Christophe

_Mechanics of Aeronautical Solids, Materials and Structures_

CHEBEL-MORELLO Brigitte _,_ NICOD Jean-Marc _,_ VARNIER Christophe

_From Prognostics and Health Systems Management to Predictive Maintenance 2: Knowledge, Traceability and Decision
(Reliability of Multiphysical Systems Set – Volume 7)_

EL HAMI Abdelkhalak, RADI Bouchaib

_Dynamics of Large Structures and Inverse Problems_
( _Mathematical and Mechanical Engineering Set – Volume 5_ )

_Fluid-Structure Interactions and Uncertainties: Ansys and Fluent Tools
(Reliability of Multiphysical Systems Set – Volume 6)_

KHARMANDA Ghias, EL HAMI Abdelkhalak

_Biomechanics: Optimization, Uncertainties and Reliability
(Reliability of Multiphysical Systems Set – Volume 5)_

LEDOUX Michel, EL HAMI Abdelkhalak

_Compressible Flow Propulsion and Digital Approaches in Fluid Mechanics
(Mathematical and Mechanical Engineering Set – Volume 4)_

_Fluid Mechanics: Analytical Methods
(Mathematical and Mechanical Engineering Set – Volume 3)_

MORI Yvon

_Mechanical Vibrations: Applications to Equipment_

**2016**

BOREL Michel, VÉNIZÉLOS Georges

_Movement Equations 1: Location, Kinematics and Kinetics
(Non-deformable Solid Mechanics Set – Volume 1)_

BOYARD Nicolas

_Heat Transfer in Polymer Composite Materials_

CARDON Alain, ITMI Mhamed

_New Autonomous Systems
(Reliability of Multiphysical Systems Set – Volume 1)_

DAHOO Pierre Richard, POUGNET Philippe, EL HAMI Abdelkhalak _Nanometer-scale Defect Detection Using Polarized Light
(Reliability of Multiphysical Systems Set – Volume 2)_

DE SAXCÉ Géry, VALLÉE Claude

_Galilean Mechanics and Thermodynamics of Continua_

DORMIEUX Luc, KONDO Djimédo

_Micromechanics of Fracture and Damage
(Micromechanics Set – Volume 1)_

EL HAMI Abdelkhalak, RADI Bouchaib

_Stochastic Dynamics of Structures
(Mathematical and Mechanical Engineering Set – Volume 2)_

GOURIVEAU Rafael, MEDJAHER Kamal, ZERHOUNI Noureddine

_From Prognostics and Health Systems Management to Predictive Maintenance 1: Monitoring and Prognostics
(Reliability of Multiphysical Systems Set – Volume 4)_

KHARMANDA Ghias, EL HAMI Abdelkhalak

_Reliability in Biomechanics
(Reliability of Multiphysical Systems Set –Volume 3)_

MOLIMARD Jérôme

_Experimental Mechanics of Solids and Structures_

RADI Bouchaib, EL HAMI Abdelkhalak

_Material Forming Processes: Simulation, Drawing, Hydroforming and Additive Manufacturing
(Mathematical and Mechanical Engineering Set – Volume 1)_

**2015**

KARLIČIĆ Danilo, MURMU Tony, ADHIKARI Sondipon, MCCARTHY Michael

_Non-local Structural Mechanics_

SAB Karam, LEBÉE Arthur

_Homogenization of Heterogeneous Thin and Thick Plates_

**2014**

ATANACKOVIC M. Teodor, PILIPOVIC Stevan, STANKOVIC Bogoljub, ZORICA Dusan

_Fractional Calculus with Applications in Mechanics: Vibrations and Diffusion Processes_

ATANACKOVIC M. Teodor, PILIPOVIC Stevan, STANKOVIC Bogoljub, ZORICA Dusan

_Fractional Calculus with Applications in Mechanics: Wave Propagation, Impact and Variational Principles_

CIBLAC Thierry, MOREL Jean-Claude

_Sustainable Masonry: Stability and Behavior of Structures_

ILANKO Sinniah, MONTERRUBIO Luis E., MOCHIDA Yusuke

_The Rayleigh−Ritz Method for Structural Analysis_

LALANNE Christian

_Mechanical Vibration and Shock Analysis – 5-volume series – 3 rd edition
Sinusoidal Vibration – Volume 1
Mechanical Shock – Volume 2
Random Vibration – Volume 3
Fatigue Damage – Volume 4
Specification Development – Volume 5_

LEMAIRE Maurice

_Uncertainty and Mechanics_

**2013**

ADHIKARI Sondipon

_Structural Dynamic Analysis with Generalized Damping Models: Analysis_

ADHIKARI Sondipon

_Structural Dynamic Analysis with Generalized Damping Models: Identification_

BAILLY Patrice

_Materials and Structures under Shock and Impact_

BASTIEN Jérôme, BERNARDIN Frédéric, LAMARQUE Claude-Henri

_Non-smooth Deterministic or Stochastic Discrete Dynamical Systems: Applications to Models with Friction or Impact_

EL HAMI Abdelkhalak, BOUCHAIB Radi

_Uncertainty and Optimization in Structural Mechanics_

KIRILLOV Oleg N., PELINOVSKY Dmitry E.

_Nonlinear Physical Systems: Spectral Analysis, Stability and Bifurcations_

LUONGO Angelo, ZULLI Daniele

_Mathematical Models of Beams and Cables_

SALENÇON Jean

_Yield Design_

**2012**

DAVIM J. Paulo

_Mechanical Engineering Education_

DUPEUX Michel, BRACCINI Muriel

_Mechanics of Solid Interfaces_

ELISHAKOFF Isaac _et al._

_Carbon Nanotubes and Nanosensors: Vibration, Buckling and Ballistic Impact_

GRÉDIAC Michel, HILD François

_Full-Field Measurements and Identification in Solid Mechanics_

GROUS Ammar

_Fracture Mechanics – 3-volume series
Analysis of Reliability and Quality Control – Volume 1
Applied Reliability – Volume 2
Applied Quality Control – Volume 3_

RECHO Naman

_Fracture Mechanics and Crack Growth_

**2011**

KRYSINSKI Tomasz, MALBURET François

_Mechanical Instability_

SOUSTELLE Michel

_An Introduction to Chemical Kinetics_

**2010**

BREITKOPF Piotr, FILOMENO COELHO Rajan

_Multidisciplinary Design Optimization in Computational Mechanics_

DAVIM J. Paulo

_Biotribolgy_

PAULTRE Patrick

_Dynamics of Structures_

SOUSTELLE Michel

_Handbook of Heterogenous Kinetics_

**2009**

BERLIOZ Alain, TROMPETTE Philippe

_Solid Mechanics using the Finite Element Method_

LEMAIRE Maurice

_Structural Reliability_

**2007**

GIRARD Alain, ROY Nicolas

_Structural Dynamics in Industry_

GUINEBRETIÈRE René

_X-ray Diffraction by Polycrystalline Materials_

KRYSINSKI Tomasz, MALBURET François

_Mechanical Vibrations_

KUNDU Tribikram

_Advanced Ultrasonic Methods for Material and Structure Inspection_

SIH George C. _et al._

_Particle and Continuum Aspects of Mesomechanics_

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