MCE 571 Theory of Elasticity

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

1Traction Vector

P1P2P3 p(Externally Loaded Body)FnA(Sectioned Body)Elasticity Theory, Applications and NumericsM.H. Sadd , University of Rhode Island

2Stress Tensor

Traction on an Oblique Plane

xzynTn Elasticity Theory, Applications and NumericsM.H. Sadd , University of Rhode Island

Stress Transformation

Elasticity Theory, Applications and NumericsM.H. Sadd , University of Rhode Island

4Two-Dimensional Stress Transformation

Elasticity Theory, Applications and NumericsM.H. Sadd , University of Rhode Island

5Principal Stresses & Directions

(General Coordinate System)(Principal Coordinate System)

Elasticity Theory, Applications and NumericsM.H. Sadd , University of Rhode Island

Traction Vector Components

Mohrs Circles of Stress

Admissible N and S values lie in the shaded area T nnASNElasticity Theory, Applications and NumericsM.H. Sadd , University of Rhode Island

Example 3-1 Stress Transformation

Elasticity Theory, Applications and NumericsM.H. Sadd , University of Rhode Island

Spherical, Deviatoric, Octahedral and von Mises Stresses

. . . Spherical Stress Tensor. . . Deviatoric Stress Tensor

Elasticity Theory, Applications and NumericsM.H. Sadd , University of Rhode Island

. . . Octahedral Normal and Shear Stresses

. . . von Mises StressStress Distribution Visualization Using2-D or 3-D Plots of Particular Contour LinesParticular Stress ComponentsPrincipal Stress ComponentsMaximum Shear Stressvon Mises StressIsochromatics (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 NumericsM.H. Sadd , University of Rhode Island

Example Stress Contour Distribution Plots Disk Under Diametrical CompressionElasticity Theory, Applications and NumericsM.H. Sadd , University of Rhode Island

PP(a) Disk Problem

(b) Max Shear Stress Contours (Isochromatic Lines)

(c) Max Principal Stress Contours

(d) Sum of Principal Stress Contours (Isopachic Lines)

(e) von Mises Stress Contours

(f) Stress Trajectories (Isostatic Lines)Equilibrium Equations

FT nVSElasticity Theory, Applications and NumericsM.H. Sadd , University of Rhode Island

Stress & Traction Components in Cylindrical Coordinates

Equilibrium Equations

x3x1x2rzdrzrrrzzdElasticity Theory, Applications and NumericsM.H. Sadd , University of Rhode Island

Stress & Traction Components in Spherical Coordinates

Equilibrium Equations

Rx3x1x2RRRElasticity Theory, Applications and NumericsM.H. Sadd , University of Rhode Island