vi Contents 2.12 Struts with unsymmetrical cross-section 49 Examples 0 Problems 56 Strains Beyond the Elastic Limit 61 Summary 61 Introduction 62 3.1 Plastic bending of rectangular-sectioned beams 64 3.2 Shape factor-symmetrical sections 3.3 Application to I-section beams 61 3.4 Partially plastic bending of unsymmetrical sections 7 3.5 Shape factor-unsymmetrical sections 3.6 Deflections of partially plastic beams 69 3.7 Length of yielded area in beams 38 Collapse loads -plastic limit design 1 3.9 Residual stresses after yielding:elastic-perfectly plastic material 73 3.10 Torsion of shafts beyond the elastic limit-plastic torsion 3.11 Angles of twist of shafts strained beyond the elastic limit 3.12 Plastic torsion of hollow tubes 个 3.13 Plastic torsion of case-hardened shafts 3.14 Residual stresses after yield in torsion 7 3.15 Plastic bending and torsion of strain-hardening materials (a)Inelastic bending 6 (b)Inelastic torsion 83 3.16 Residual stresses-strain-hardening materials 84 3.17 Influence of residual stresses on bending and torsional strengths 84 3.18 Plastic yielding in the eccentric loading of rectangular sections 85 3.19 Plastic yielding and residual stresses under axial loading with stress concentrations 86 3.20 Plastic yielding of axially symmetric components 87 (a)Thick cylinders-collapse pressure 87 (b)Thick cylinders-"auto-frettage" 89 (c）Rotating discs 94 Examples 96 Problems 109 4 Rings,Discs and Cylinders Subjected to Rotation and Thermal Gradients 117 Summary 117 4.1 Thin rotating ring or cylinder 118 4.2 Rotating solid disc 119 4.3 Rotating disc with a central hole 122 4.4 Rotating thick cylinders or solid shafts 124 4.5 Rotating disc of uniform strength 125vi Contents 2.1 2 Struts with unsymmetrical cross-section Examples Problems 3 Strains Beyond the Elastic Limit Summary Introduction 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.1 1 3.12 3.13 3.14 3.15 3.16 3.17 3.18 3.19 3.20 Plastic bending of rectangular-sectioned beams Shape factor - symmetrical sections Application to I-section beams Partially plastic bending of unsymmetrical sections Shape factor - unsymmetrical sections Dejections of partially plastic beams Length of yielded area in beams Collapse loads - plastic limit design Residual stresses after yielding: elastic-perfectly plastic material Torsion of shafts beyond the elastic limit - plastic torsion Angles of twist of shafts strained beyond the elastic limit Plastic torsion of hollow tubes Plastic torsion of case-hardened shafts Residual stresses after yield in torsion Plastic bending and torsion of strain-hardening materials (a) Inelastic bending (b) Inelastic torsion Residual stresses - strain-hardening materials Influence of residual stresses on bending and torsional strengths Plastic yielding in the eccentric loadirzg of rectangular sections Plastic yielding and residual stresses under axial loading with stress concentrations Plastic yielding of axially symmetric components (a) Thick cylinders - collapse pressure (b) Thick cylinders - “auto-frettage ’’ (c) Rotating discs Examples Problems 4 Rings, Discs and Cylinders Subjected to Rotation and Thermal Gradients Summary 4.1 4.2 Rotating solid disc 4.3 4.4 4.5 Thin rotating ring or cylinder Rotating disc with a central hole Rotating thick cylinders or solid shafs Rotating disc of uniform strength 49 50 56 61 61 62 64 65 67 67 69 69 69 71 73 75 77 77 79 79 80 80 83 84 84 85 86 87 87 89 94 96 109 117 117 118 119 122 124 125