Contents ix 8.4.3 Line of action of shear stress 227 8.4.4 Shear stress in any other direction on the plane 227 8.5 Principal stresses and strains in three dimensions-Mohr's circle representation 228 8.6 Graphical determination of the direction of the shear stress tn on an inclined plane in a three-dimensional principal stress system 229 8.7 The combined Mohr diagram for three-dimensional stress and strain systems 230 8.8 Application of the combined circle to two-dimensional stress systems 232 8.9 Graphical construction for the state of stress at a point 234 8.10 Construction for the state of strain on a general strain plane 235 8.11 State of stress-tensor notation 235 8.12 The stress equations of equilibrium 236 8.13 Principal stresses in a three-dimensional cartesian stress system 242 8.13.1 Solution of cubic equations 242 8.14 Stress invariants-Eigen values and Eigen vectors 243 8.15 Stress invariants 244 8.16 Reduced stresses 246 8.17 Strain invariants 247 8.18 Alternative procedure for determination of principal stresses 247 8.18.1 Evaluation of direction cosines for principal stresses 248 8.19 Octahedral planes and stresses 249 8.20 Deviatoric stresses 251 8.21 Deviatoric strains 253 8.22 Plane stress and plane strain 254 8.22.1 Plane stress 255 8.22.2 Plane strain 255 8.23 The stress-strain relations 256 8.24 The strain-displacement relationships 257 8.25 The strain equations of transformation 259 8.26 Compatibility 261 8.27 The stress function concept 263 8.27.1 Forms of Airy stress function in Cartesian coordinates 265 8.27.2 Case I-Bending of a simply supported beam by a uniformly distributed loading 267 8.27.3 The use of polar coordinates in two dimensions 271 8.27.4 Forms of stress function in polar coordinates 272 8.27.5 Case 2-Axi-symmetric case:solid shaft and thick cylinder radially loaded with uniform pressure 273 8.27.6 Case 3-The pure bending of a rectangular section curved beam 273 8.27.7 Case 4-Asymmetric case n =1.Shear loading of a circular arc cantilever beam 274 8.27.8 Case 5-The asymmetric cases n >2 -stress concentration at a circular hole in a tension field 276Contents ix 8.5 8.6 8.7 8.8 8.9 8.10 8.1 1 8.12 8.13 8.14 8.15 8.16 8.17 8.18 8.19 8.20 8.21 8.22 8.23 8.24 8.25 8.26 8.27 8.4.3 8.4.4 Principal stresses and strains in three dimensions - Mohr 's circle representation Graphical determination of the direction of the shear stress r,, on an inclined plane in a three-dimensional principal stress system The combined Mohr diagram for three-dimensional stress and strain systems Application of the combined circle to two-dimensional stress systems Graphical construction for the state of stress at a point Construction for the state of strain on a general strain plane State of stress-tensor notation The stress equations of equilibrium Principal stresses in a three-dimensional Cartesian stress system 8.13.1 Solution of cubic equations Stress invariants - Eigen values and Eigen vectors Stress invariants Reduced stresses Strain invariants Alternative procedure for determination of principal stresses 8.1 8.1 Evaluation of direction cosines for principal stresses Octahedral planes and stresses Deviatoric stresses Deviatoric strains Plane stress and plane strain 8.22.1 Plane stress 8.22.2 Plane strain The stress-strain relations The strain-displacement relationships The strain equations of transformation Compatibility The stress function concept 8.27.1 Forms of Airy stress function in Cartesian coordinates 8.27.2 Case 1 - Bending of a simply supported beam by a uniformly 8.27.3 The use of polar coordinates in two dimensions 8.27.4 Forms of stress function in polar coordinates 8.27.5 Case 2 - hi-symmetric case: solid shaft and thick cylinder radially loaded with uniform pressure 8.27.6 Case 3 - The pure bending of a rectangular section curved beam 8.27.7 Case 4 - Asymmetric case n = 1. Shear loading of a circular arc cantilever beam 8.27.8 Case 5 - The asymmetric cases n >, 2 -stress concentration at a circular hole in a tension$eld Line of action of shear stress Shear stress in any other direction on the plane distributed loading 227 227 228 229 230 232 234 235 235 236 242 242 243 244 246 247 247 248 249 25 1 25 3 254 255 255 256 257 259 26 1 263 265 267 27 1 272 273 273 274 276