Chapter 3 Stress and Equilibrium

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

2. F n A P3 P2 p P1 (Externally Loaded Body) (Sectioned Body) Traction Vector ElasticityTheory, Applications and NumericsM.H. Sadd , University of Rhode Island

3. y Tn n x z Stress Tensor Traction on an Oblique Plane ElasticityTheory, Applications and NumericsM.H. Sadd , University of Rhode Island

4. Stress Transformation ElasticityTheory, Applications and NumericsM.H. Sadd , University of Rhode Island

5. Two-Dimensional Stress Transformation ElasticityTheory, Applications and NumericsM.H. Sadd , University of Rhode Island

6. Principal Stresses & Directions (Principal Coordinate System) (General Coordinate System) ElasticityTheory, Applications and NumericsM.H. Sadd , University of Rhode Island

7. N A n Tn S Traction Vector Components Admissible N and Svalues lie in the shaded area Mohr’s Circles of Stress ElasticityTheory, Applications and NumericsM.H. Sadd , University of Rhode Island

8. Example 3-1 Stress Transformation ElasticityTheory, Applications and NumericsM.H. Sadd , University of Rhode Island

9. Spherical, Deviatoric, Octahedral and von Mises Stresses . . . Spherical Stress Tensor . . . Deviatoric Stress Tensor . . . Octahedral Normal and Shear Stresses . . . von Mises Stress ElasticityTheory, Applications and NumericsM.H. Sadd , University of Rhode Island

10. Stress Distribution Visualization Using2-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) ElasticityTheory, Applications and NumericsM.H. Sadd , University of Rhode Island

11. Example Stress Contour Distribution Plots Disk Under Diametrical Compression P P (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) ElasticityTheory, Applications and NumericsM.H. Sadd , University of Rhode Island

12. T n S V F Equilibrium Equations ElasticityTheory, Applications and NumericsM.H. Sadd , University of Rhode Island

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

14. Stress & Traction Components in Spherical Coordinates x3 R R R   R   x2  x1 Equilibrium Equations ElasticityTheory, Applications and NumericsM.H. Sadd , University of Rhode Island