CONTENTS ix 8.4 Buckling of Shells 384 8.4.1 Buckling of Cylinders 387 9 Finite Element Analysis 395 9.1 Three-Dimensional Element 396 9.2 Plate Element 397 9.3 Beam Element 397 9.4 Sublaminate 398 9.4.1 Step 1.Elements of [J]due to In-Plane Stresses 400 9.4.2 Step 2.Elements of [J]due to Out-of-Plane Normal Stresses 403 9.4.3 Step 3.Elements of [J]due to Out-of-Plane Shear Stresses 405 9.4.4 Step 4.The Stiffness Matrix 407 10 Failure Criteria 411 10.1 Quadratic Failure Criterion 413 10.1.1 Orthotropic Material 414 10.1.2 Transversely Isotropic Material 420 10.1.3 Isotropic Material 421 10.1.4 Plane-Strain and Plane-Stress Conditions 422 10.1.5 Proportional Loading-Stress Ratio 423 lO.2“Maximum Stress'”Failure Criterion 425 l0.3“Maximum Strain”Failure Criterion 426 10.4 Plate with a Hole or a Notch 430 10.4.1 Plate with a Circular Hole 431 10.4.2 Plate with a Notch 434 10.4.3 Characteristic Length 434 11 Micromechanics 436 11.1 Rule of Mixtures 436 11.1.1 Longitudinal Young Modulus E 438 11.1.2 Transverse Young Modulus E2 439 11.1.3 Longitudinal Shear Modulus Gi2 439 11.1.4 Transverse Shear Modulus G23 440 11.1.5 Longitudinal Poisson Ratio vi2 441 11.1.6 Transverse Poisson Ratio v23 442 11.1.7 Thermal Expansion Coefficients 443 11.1.8 Moisture Expansion Coefficients 445 11.1.9 Thermal Conductivity 446 11.1.10 Moisture Diffusivity 447 11.1.11 Specific Heat 448 11.2 Modified Rule of Mixtures 448 11.3 Note on the Micromechanics Models 449 Appendix A.Cross-Sectional Properties of Thin-Walled Composite Beams 453 Appendix B.Buckling Loads and Natural Frequencies of Orthotropic Beams with Shear Deformation 461 Appendix C.Typical Material Properties 464 Index 469CONTENTS ix 8.4 Buckling of Shells 384 8.4.1 Buckling of Cylinders 387 9 Finite Element Analysis 395 9.1 Three-Dimensional Element 396 9.2 Plate Element 397 9.3 Beam Element 397 9.4 Sublaminate 398 9.4.1 Step 1. Elements of [J ] due to In-Plane Stresses 400 9.4.2 Step 2. Elements of [J ] due to Out-of-Plane Normal Stresses 403 9.4.3 Step 3. Elements of [J ] due to Out-of-Plane Shear Stresses 405 9.4.4 Step 4. The Stiffness Matrix 407 10 Failure Criteria 411 10.1 Quadratic Failure Criterion 413 10.1.1 Orthotropic Material 414 10.1.2 Transversely Isotropic Material 420 10.1.3 Isotropic Material 421 10.1.4 Plane-Strain and Plane-Stress Conditions 422 10.1.5 Proportional Loading – Stress Ratio 423 10.2 “Maximum Stress” Failure Criterion 425 10.3 “Maximum Strain” Failure Criterion 426 10.4 Plate with a Hole or a Notch 430 10.4.1 Plate with a Circular Hole 431 10.4.2 Plate with a Notch 434 10.4.3 Characteristic Length 434 11 Micromechanics 436 11.1 Rule of Mixtures 436 11.1.1 Longitudinal Young Modulus E1 438 11.1.2 Transverse Young Modulus E2 439 11.1.3 Longitudinal Shear Modulus G12 439 11.1.4 Transverse Shear Modulus G23 440 11.1.5 Longitudinal Poisson Ratio ν12 441 11.1.6 Transverse Poisson Ratio ν23 442 11.1.7 Thermal Expansion Coefficients 443 11.1.8 Moisture Expansion Coefficients 445 11.1.9 Thermal Conductivity 446 11.1.10 Moisture Diffusivity 447 11.1.11 Specific Heat 448 11.2 Modified Rule of Mixtures 448 11.3 Note on the Micromechanics Models 449 Appendix A. Cross-Sectional Properties of Thin-Walled Composite Beams 453 Appendix B. Buckling Loads and Natural Frequencies of Orthotropic Beams with Shear Deformation 461 Appendix C. Typical Material Properties 464 Index 469