viii CONTENTS 6.9 Buckling of Beams 290 6.9.1 Beams Subjected to Axial Load (Flexural-Torsional Buckling） 291 6.9.2 Lateral-Torsional Buckling of Orthotropic Beams with Symmetrical Cross Section 296 6.9.3 Local Buckling 300 6.10 Free Vibration of Beams(Flexural-Torsional Vibration) 306 6.10.1 Doubly Symmetrical Cross Sections 306 6.10.2 Beams with Symmetrical Cross Sections 309 6.10.3 Beams with Unsymmetrical Cross Sections 309 6.11 Summary 312 7 Beams with Shear Deformation 313 7.1 Governing Equations 314 7.1.1 Strain-Displacement Relationships 315 7.1.2 Force-Strain Relationships 315 7.1.3 Equilibrium Equations 320 7.1.4 Summary of Equations 320 7.1.5 Boundary Conditions 321 7.2 Stiffnesses and Compliances of Beams 321 7.2.1 Shear Stiffnesses and Compliances of Thin-Walled Open-Section Beams 322 7.2.2 Shear Stiffnesses and Compliances of Thin-Walled Closed-Section Beams 325 7.2.3 Stiffnesses of Sandwich Beams 326 7.3 Transversely Loaded Beams 329 7.4 Buckling of Beams 334 7.4.1 Axially Loaded Beams with Doubly Symmetrical Cross Sections(Flexural and Torsional Buckling) 335 7.4.2 Axially Loaded Beams with Symmetrical or Unsymmetrical Cross Sections (Flexural-Torsional Buckling) 341 7.4.3 Lateral-Torsional Buckling of Beams with Symmetrical Cross Section 345 7.4.4 Summary 346 7.5 Free Vibration of Beams 347 7.5.1 Beams with Doubly Symmetrical Cross Sections 347 7.5.2 Beams with Symmetrical or Unsymmetrical Cross Sections 356 7.5.3 Summary 359 7.6 Effect of Shear Deformation 359 8 Shells 365 8.1 Shells of Revolution with Axisymmetrical Loading 367 8.2 Cylindrical Shells 368 8.2.1 Membrane Theory 368 8.2.2 Built-In Ends 370 8.2.3 Temperature-Built-In Ends 379 8.3 Springback 380 8.3.1 Springback of Cylindrical Shells 380 8.3.2 Doubly Curved Shells 384viii CONTENTS 6.9 Buckling of Beams 290 6.9.1 Beams Subjected to Axial Load (Flexural–Torsional Buckling) 291 6.9.2 Lateral–Torsional Buckling of Orthotropic Beams with Symmetrical Cross Section 296 6.9.3 Local Buckling 300 6.10 Free Vibration of Beams (Flexural–Torsional Vibration) 306 6.10.1 Doubly Symmetrical Cross Sections 306 6.10.2 Beams with Symmetrical Cross Sections 309 6.10.3 Beams with Unsymmetrical Cross Sections 309 6.11 Summary 312 7 Beams with Shear Deformation 313 7.1 Governing Equations 314 7.1.1 Strain–Displacement Relationships 315 7.1.2 Force–Strain Relationships 315 7.1.3 Equilibrium Equations 320 7.1.4 Summary of Equations 320 7.1.5 Boundary Conditions 321 7.2 Stiffnesses and Compliances of Beams 321 7.2.1 Shear Stiffnesses and Compliances of Thin-Walled Open-Section Beams 322 7.2.2 Shear Stiffnesses and Compliances of Thin-Walled Closed-Section Beams 325 7.2.3 Stiffnesses of Sandwich Beams 326 7.3 Transversely Loaded Beams 329 7.4 Buckling of Beams 334 7.4.1 Axially Loaded Beams with Doubly Symmetrical Cross Sections (Flexural and Torsional Buckling) 335 7.4.2 Axially Loaded Beams with Symmetrical or Unsymmetrical Cross Sections (Flexural–Torsional Buckling) 341 7.4.3 Lateral–Torsional Buckling of Beams with Symmetrical Cross Section 345 7.4.4 Summary 346 7.5 Free Vibration of Beams 347 7.5.1 Beams with Doubly Symmetrical Cross Sections 347 7.5.2 Beams with Symmetrical or Unsymmetrical Cross Sections 356 7.5.3 Summary 359 7.6 Effect of Shear Deformation 359 8 Shells 365 8.1 Shells of Revolution with Axisymmetrical Loading 367 8.2 Cylindrical Shells 368 8.2.1 Membrane Theory 368 8.2.2 Built-In Ends 370 8.2.3 Temperature – Built-In Ends 379 8.3 Springback 380 8.3.1 Springback of Cylindrical Shells 380 8.3.2 Doubly Curved Shells 384