1. 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

2. 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)

3. 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

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

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

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

7. 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

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

9. 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

10. 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

11. 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

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

13. 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

14. 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