Chapter 3 Stress and Equilibrium Elasticity Theory, Applications and Numerics M.H. Sadd , University of Rhode Island Chapter 3 Stress and Equilibrium Body and Surface Forces (a) Cantilever Beam Under Self-Weight Loading Body Forces: F(x) (b) Sectioned Axially Loaded Beam Surface Forces: T(x) S

Elasticity Theory, Applications and Numerics M. H Elasticity Theory, Applications and Numerics M.H. Sadd , University of Rhode Island Traction Vector P1 P2 P3 p (Externally Loaded Body) F n A (Sectioned Body)

Stress Tensor Traction on an Oblique Plane Elasticity Theory, Applications and Numerics M.H. Sadd , University of Rhode Island Stress Tensor x z y n Tn Traction on an Oblique Plane

Stress Transformation Elasticity Theory, Applications and Numerics M.H. Sadd , University of Rhode Island Stress Transformation

Two-Dimensional Stress Transformation Elasticity Theory, Applications and Numerics M.H. Sadd , University of Rhode Island Two-Dimensional Stress Transformation

Principal Stresses & Directions Elasticity Theory, Applications and Numerics M.H. Sadd , University of Rhode Island Principal Stresses & Directions (General Coordinate System) (Principal Coordinate System)

Traction Vector Components Elasticity Theory, Applications and Numerics M.H. Sadd , University of Rhode Island Traction Vector Components T n n A S N Admissible N and S values lie in the shaded area Mohr’s Circles of Stress

Example 3-1 Stress Transformation Elasticity Theory, Applications and Numerics M.H. Sadd , University of Rhode Island Example 3-1 Stress Transformation

Spherical, Deviatoric, Octahedral and von Mises Stresses . . . Spherical Stress Tensor . . . Deviatoric Stress Tensor . . . Octahedral Normal and Shear Stresses . . . von Mises Stress Elasticity Theory, Applications and Numerics M.H. Sadd , University of Rhode Island

Stress Distribution Visualization Using 2-D or 3-D Plots of Particular Contour Lines Particular Stress Components Principal Stress Components Maximum Shear Stress von Mises Stress Isochromatics (lines of principal stress difference = constant; same as max shear stress) Isoclinics (lines along which principal stresses have constant orientation) Isopachic lines (sum of principal stresses = constant) Isostatic lines (tangent oriented along a particular principal stress; sometimes called stress trajectories) Elasticity Theory, Applications and Numerics M.H. Sadd , University of Rhode Island

Example Stress Contour Distribution Plots Disk Under Diametrical Compression (b) Max Shear Stress Contours (Isochromatic Lines) (c) Max Principal Stress Contours (a) Disk Problem (d) Sum of Principal Stress Contours (Isopachic Lines) (e) von Mises Stress Contours (f) Stress Trajectories (Isostatic Lines) Elasticity Theory, Applications and Numerics M.H. Sadd , University of Rhode Island

Equilibrium Equations Elasticity Theory, Applications and Numerics M.H. Sadd , University of Rhode Island Equilibrium Equations F T n V S

Stress & Traction Components in Cylindrical Coordinates Elasticity Theory, Applications and Numerics M.H. Sadd , University of Rhode Island Stress & Traction Components in Cylindrical Coordinates  x3 x1 x2 r  z dr z r r rz z d Equilibrium Equations

Stress & Traction Components in Spherical Coordinates Elasticity Theory, Applications and Numerics M.H. Sadd , University of Rhode Island Stress & Traction Components in Spherical Coordinates R x3 x1 x2 R   R   R    Equilibrium Equations